# Example 975 — The car ten over was doing forty three where the other stopped `the_car_ten_over_was_doing_forty_three_where_the_other_stopped.eml` - A crate falls off a truck 34.6 metres ahead of two cars. The car at the 50 km/h limit stops just short of it; the driver of the car at 60 had reasoned that ten over is only a fifth faster and would need only about a fifth more road to stop. How fast the second car is still going when it reaches the crate is computed below. ## EML ```eml # Self-authored for the EML case corpus (no external origin). A crate falls off a # truck 34.6 metres ahead of two cars. The car at the 50 km/h limit stops just # short of it; the driver of the car at 60 had reasoned that ten over is only a # fifth faster and would need only about a fifth more road to stop. How fast the # second car is still going when it reaches the crate is computed below. # # The reasoning is careful. 60 is exactly a fifth more than 50; the driver's # reaction time is the same one and a half seconds at either speed; the brakes # and the road are the same; and the intent is exactly 'keep a safe margin'. # # The reaction time is spent at full speed, so the faster car covers 25 metres # before braking begins, where the slower one covers 20.8; and braking takes # away the square of the speed, not the speed. Where the car at 50 stops, the # car at 60 has braked for only 9.6 metres and is still doing 43 km/h, carrying # half of its energy into the crate. 50 => limit_kmh 60 => speed_kmh 15 => reaction_tenths_of_a_second 70 => braking_tenths_of_a_m_per_s2 int(limit_kmh * 1000 * reaction_tenths_of_a_second / 360) => reaction_distance_at_the_limit_cm int(speed_kmh * 1000 * reaction_tenths_of_a_second / 360) => reaction_distance_at_speed_cm int(limit_kmh * limit_kmh * 10000000 / (259200 * braking_tenths_of_a_m_per_s2)) => braking_distance_at_the_limit_cm int(speed_kmh * speed_kmh * 10000000 / (259200 * braking_tenths_of_a_m_per_s2)) => braking_distance_at_speed_cm reaction_distance_at_the_limit_cm + braking_distance_at_the_limit_cm => stop_at_the_limit_cm reaction_distance_at_speed_cm + braking_distance_at_speed_cm => stop_at_speed_cm stop_at_the_limit_cm - reaction_distance_at_speed_cm => braking_done_before_the_crate_cm int(braking_done_before_the_crate_cm * 259200 * braking_tenths_of_a_m_per_s2 / 10000000) => speed_squared_taken_away speed_kmh * speed_kmh - speed_squared_taken_away => speed_squared_at_the_crate 43 => speed_at_the_crate_kmh speed_at_the_crate_kmh * speed_at_the_crate_kmh => square_of_43 (speed_at_the_crate_kmh + 1) * (speed_at_the_crate_kmh + 1) => square_of_44 int(speed_squared_at_the_crate * 1000 / (speed_kmh * speed_kmh)) => energy_left_at_the_crate_per_mille int((speed_kmh - limit_kmh) * 100 / limit_kmh) => faster_by_percent int((stop_at_speed_cm - stop_at_the_limit_cm) * 100 / stop_at_the_limit_cm) => stopping_distance_longer_by_percent "limit : " + str(limit_kmh) + " km/h" ^0 "the faster car : " + str(speed_kmh) + " km/h, " + str(faster_by_percent) + " percent over" ^0 "reaction : " + str(reaction_tenths_of_a_second) + " tenths of a second; braking " + str(braking_tenths_of_a_m_per_s2) + " tenths of a m/s2" ^0 "" ^0 "at the limit : reaction " + str(reaction_distance_at_the_limit_cm) + " cm, braking " + str(braking_distance_at_the_limit_cm) + " cm, stops in " + str(stop_at_the_limit_cm) + " cm" ^0 "at " + str(speed_kmh) + " : reaction " + str(reaction_distance_at_speed_cm) + " cm, braking " + str(braking_distance_at_speed_cm) + " cm, stops in " + str(stop_at_speed_cm) + " cm" ^0 "stopping distance longer by : " + str(stopping_distance_longer_by_percent) + " percent" ^0 "" ^0 "the crate, " + str(stop_at_the_limit_cm) + " cm ahead : the car at " + str(speed_kmh) + " has braked " + str(braking_done_before_the_crate_cm) + " cm" ^0 "speed squared left : " + str(speed_squared_at_the_crate) + ", between " + str(square_of_43) + " and " + str(square_of_44) ^0 "speed at the crate : " + str(speed_at_the_crate_kmh) + " km/h" ^0 "energy carried into the crate : " + str(energy_left_at_the_crate_per_mille) + " per mille of what it had" ^0 "" ^0 # ---- what the driver verified ---- "the only-a-fifth reasoning" ^0 " speed : " + str(speed_kmh) + ", exactly a fifth over " + str(limit_kmh) ^0 " reaction : the same " + str(reaction_tenths_of_a_second) + " tenths of a second" ^0 " brakes and road : the same" ^0 " intent : keep a safe margin" ^0 " facts wrong : 0" ^0 " verdict : A FIFTH FASTER NEEDS ABOUT A FIFTH MORE ROAD" ^0 "" ^0 " keeping reaction, brakes and road the same is the part done" ^0 " right here, and it is why the two cars differ in exactly one" ^0 " thing, the speed" ^0 "" ^0 # ---- where the extra speed goes ---- "two parts of a stop" ^0 " the reaction : spent at full speed, so it grows with the speed," ^0 " " + str(reaction_distance_at_the_limit_cm) + " cm at " + str(limit_kmh) + " and " + str(reaction_distance_at_speed_cm) + " at " + str(speed_kmh) ^0 " the braking : takes away the square of the speed, so it grows" ^0 " with the square, " + str(braking_distance_at_the_limit_cm) + " cm against " + str(braking_distance_at_speed_cm) ^0 " at the crate : the car at " + str(speed_kmh) + " is only " + str(braking_done_before_the_crate_cm) + " cm into its braking," ^0 " which has taken " + str(speed_squared_taken_away) + " off its " + str(speed_kmh * speed_kmh) + " of speed squared" ^0 " what is left : " + str(speed_at_the_crate_kmh) + " km/h, most of the speed, because the" ^0 " first metres of braking take the least of it" ^0 "" ^0 # ---- what the driver got ---- "the crate" ^0 " believed : a stop a fifth further on, a small margin used" ^0 " actual : " + str(speed_at_the_crate_kmh) + " km/h at the point where the other car stood still" ^0 " was the arithmetic wrong : no; " + str(speed_kmh) + " is a fifth over " + str(limit_kmh) ^0 " does a fifth more speed use a fifth more road : no; it uses" ^0 " " + str(stopping_distance_longer_by_percent) + " percent more, and arrives with " + str(energy_left_at_the_crate_per_mille) + " per mille of its energy" ^0 "" ^0 # ---- null control ---- # The same stop computed from its two parts instead of scaled by the speed. 4152 => nc_stop_scaled_by_the_speed_cm 4484 => nc_stop_computed_from_reaction_and_braking_cm 332 => nc_cm_the_square_adds "null control - compute the stop, do not scale it" ^0 " stop, scaled by the speed : " + str(nc_stop_scaled_by_the_speed_cm) + " cm" ^0 " stop, from reaction and braking : " + str(nc_stop_computed_from_reaction_and_braking_cm) + " cm" ^0 " cm the square adds : " + str(nc_cm_the_square_adds) ^0 " no car and no road changed; the braking part was allowed to" ^0 " grow with the square it follows" ^0 "" ^0 # ---- the rule ---- "what driving a fifth over with the same brakes guarantees" ^0 " reaction, brakes and road as good as at the limit : exactly" ^0 " a stop about a fifth further on : not addressed; the reaction" ^0 " is spent at the higher speed and braking takes away speed" ^0 " squared, so where the car at " + str(limit_kmh) + " stops, the car at " + str(speed_kmh) + " is still" ^0 " doing " + str(speed_at_the_crate_kmh) + " km/h with " + str(energy_left_at_the_crate_per_mille) + " per mille of its energy" ^0 "" ^0 "a margin measured in speed is spent in distance, and distance is charged at" ^0 "the square; ten over is not a tenth of anything by the time the road runs out" ^0 "" ^0 "Sixty is a fifth over fifty - exactly. But the reaction is spent at sixty and" ^0 "braking removes the square of the speed, so the car at " + str(speed_kmh) + " needs " + str(stop_at_speed_cm) + " cm to" ^0 "stop, and at the " + str(stop_at_the_limit_cm) + " cm where the car at " + str(limit_kmh) + " stood still it is doing " + str(speed_at_the_crate_kmh) ^0 "km/h, until the stop is computed from its parts instead of scaled by the speed." ^0 ``` ## Python (deterministic transpilation) ```python limit_kmh = 50 speed_kmh = 60 reaction_tenths_of_a_second = 15 braking_tenths_of_a_m_per_s2 = 70 reaction_distance_at_the_limit_cm = int(limit_kmh * 1000 * reaction_tenths_of_a_second / 360) reaction_distance_at_speed_cm = int(speed_kmh * 1000 * reaction_tenths_of_a_second / 360) braking_distance_at_the_limit_cm = int(limit_kmh * limit_kmh * 10000000 / (259200 * braking_tenths_of_a_m_per_s2)) braking_distance_at_speed_cm = int(speed_kmh * speed_kmh * 10000000 / (259200 * braking_tenths_of_a_m_per_s2)) stop_at_the_limit_cm = reaction_distance_at_the_limit_cm + braking_distance_at_the_limit_cm stop_at_speed_cm = reaction_distance_at_speed_cm + braking_distance_at_speed_cm braking_done_before_the_crate_cm = stop_at_the_limit_cm - reaction_distance_at_speed_cm speed_squared_taken_away = int(braking_done_before_the_crate_cm * 259200 * braking_tenths_of_a_m_per_s2 / 10000000) speed_squared_at_the_crate = speed_kmh * speed_kmh - speed_squared_taken_away speed_at_the_crate_kmh = 43 square_of_43 = speed_at_the_crate_kmh * speed_at_the_crate_kmh square_of_44 = (speed_at_the_crate_kmh + 1) * (speed_at_the_crate_kmh + 1) energy_left_at_the_crate_per_mille = int(speed_squared_at_the_crate * 1000 / (speed_kmh * speed_kmh)) faster_by_percent = int((speed_kmh - limit_kmh) * 100 / limit_kmh) stopping_distance_longer_by_percent = int((stop_at_speed_cm - stop_at_the_limit_cm) * 100 / stop_at_the_limit_cm) print("limit : " + str(limit_kmh) + " km/h") print("the faster car : " + str(speed_kmh) + " km/h, " + str(faster_by_percent) + " percent over") print("reaction : " + str(reaction_tenths_of_a_second) + " tenths of a second; braking " + str(braking_tenths_of_a_m_per_s2) + " tenths of a m/s2") print("") print("at the limit : reaction " + str(reaction_distance_at_the_limit_cm) + " cm, braking " + str(braking_distance_at_the_limit_cm) + " cm, stops in " + str(stop_at_the_limit_cm) + " cm") print("at " + str(speed_kmh) + " : reaction " + str(reaction_distance_at_speed_cm) + " cm, braking " + str(braking_distance_at_speed_cm) + " cm, stops in " + str(stop_at_speed_cm) + " cm") print("stopping distance longer by : " + str(stopping_distance_longer_by_percent) + " percent") print("") print("the crate, " + str(stop_at_the_limit_cm) + " cm ahead : the car at " + str(speed_kmh) + " has braked " + str(braking_done_before_the_crate_cm) + " cm") print("speed squared left : " + str(speed_squared_at_the_crate) + ", between " + str(square_of_43) + " and " + str(square_of_44)) print("speed at the crate : " + str(speed_at_the_crate_kmh) + " km/h") print("energy carried into the crate : " + str(energy_left_at_the_crate_per_mille) + " per mille of what it had") print("") print("the only-a-fifth reasoning") print(" speed : " + str(speed_kmh) + ", exactly a fifth over " + str(limit_kmh)) print(" reaction : the same " + str(reaction_tenths_of_a_second) + " tenths of a second") print(" brakes and road : the same") print(" intent : keep a safe margin") print(" facts wrong : 0") print(" verdict : A FIFTH FASTER NEEDS ABOUT A FIFTH MORE ROAD") print("") print(" keeping reaction, brakes and road the same is the part done") print(" right here, and it is why the two cars differ in exactly one") print(" thing, the speed") print("") print("two parts of a stop") print(" the reaction : spent at full speed, so it grows with the speed,") print(" " + str(reaction_distance_at_the_limit_cm) + " cm at " + str(limit_kmh) + " and " + str(reaction_distance_at_speed_cm) + " at " + str(speed_kmh)) print(" the braking : takes away the square of the speed, so it grows") print(" with the square, " + str(braking_distance_at_the_limit_cm) + " cm against " + str(braking_distance_at_speed_cm)) print(" at the crate : the car at " + str(speed_kmh) + " is only " + str(braking_done_before_the_crate_cm) + " cm into its braking,") print(" which has taken " + str(speed_squared_taken_away) + " off its " + str(speed_kmh * speed_kmh) + " of speed squared") print(" what is left : " + str(speed_at_the_crate_kmh) + " km/h, most of the speed, because the") print(" first metres of braking take the least of it") print("") print("the crate") print(" believed : a stop a fifth further on, a small margin used") print(" actual : " + str(speed_at_the_crate_kmh) + " km/h at the point where the other car stood still") print(" was the arithmetic wrong : no; " + str(speed_kmh) + " is a fifth over " + str(limit_kmh)) print(" does a fifth more speed use a fifth more road : no; it uses") print(" " + str(stopping_distance_longer_by_percent) + " percent more, and arrives with " + str(energy_left_at_the_crate_per_mille) + " per mille of its energy") print("") nc_stop_scaled_by_the_speed_cm = 4152 nc_stop_computed_from_reaction_and_braking_cm = 4484 nc_cm_the_square_adds = 332 print("null control - compute the stop, do not scale it") print(" stop, scaled by the speed : " + str(nc_stop_scaled_by_the_speed_cm) + " cm") print(" stop, from reaction and braking : " + str(nc_stop_computed_from_reaction_and_braking_cm) + " cm") print(" cm the square adds : " + str(nc_cm_the_square_adds)) print(" no car and no road changed; the braking part was allowed to") print(" grow with the square it follows") print("") print("what driving a fifth over with the same brakes guarantees") print(" reaction, brakes and road as good as at the limit : exactly") print(" a stop about a fifth further on : not addressed; the reaction") print(" is spent at the higher speed and braking takes away speed") print(" squared, so where the car at " + str(limit_kmh) + " stops, the car at " + str(speed_kmh) + " is still") print(" doing " + str(speed_at_the_crate_kmh) + " km/h with " + str(energy_left_at_the_crate_per_mille) + " per mille of its energy") print("") print("a margin measured in speed is spent in distance, and distance is charged at") print("the square; ten over is not a tenth of anything by the time the road runs out") print("") print("Sixty is a fifth over fifty - exactly. But the reaction is spent at sixty and") print("braking removes the square of the speed, so the car at " + str(speed_kmh) + " needs " + str(stop_at_speed_cm) + " cm to") print("stop, and at the " + str(stop_at_the_limit_cm) + " cm where the car at " + str(limit_kmh) + " stood still it is doing " + str(speed_at_the_crate_kmh)) print("km/h, until the stop is computed from its parts instead of scaled by the speed.") ``` ## stdout (executed) ```text limit : 50 km/h the faster car : 60 km/h, 20 percent over reaction : 15 tenths of a second; braking 70 tenths of a m/s2 at the limit : reaction 2083 cm, braking 1377 cm, stops in 3460 cm at 60 : reaction 2500 cm, braking 1984 cm, stops in 4484 cm stopping distance longer by : 29 percent the crate, 3460 cm ahead : the car at 60 has braked 960 cm speed squared left : 1859, between 1849 and 1936 speed at the crate : 43 km/h energy carried into the crate : 516 per mille of what it had the only-a-fifth reasoning speed : 60, exactly a fifth over 50 reaction : the same 15 tenths of a second brakes and road : the same intent : keep a safe margin facts wrong : 0 verdict : A FIFTH FASTER NEEDS ABOUT A FIFTH MORE ROAD keeping reaction, brakes and road the same is the part done right here, and it is why the two cars differ in exactly one thing, the speed two parts of a stop the reaction : spent at full speed, so it grows with the speed, 2083 cm at 50 and 2500 at 60 the braking : takes away the square of the speed, so it grows with the square, 1377 cm against 1984 at the crate : the car at 60 is only 960 cm into its braking, which has taken 1741 off its 3600 of speed squared what is left : 43 km/h, most of the speed, because the first metres of braking take the least of it the crate believed : a stop a fifth further on, a small margin used actual : 43 km/h at the point where the other car stood still was the arithmetic wrong : no; 60 is a fifth over 50 does a fifth more speed use a fifth more road : no; it uses 29 percent more, and arrives with 516 per mille of its energy null control - compute the stop, do not scale it stop, scaled by the speed : 4152 cm stop, from reaction and braking : 4484 cm cm the square adds : 332 no car and no road changed; the braking part was allowed to grow with the square it follows what driving a fifth over with the same brakes guarantees reaction, brakes and road as good as at the limit : exactly a stop about a fifth further on : not addressed; the reaction is spent at the higher speed and braking takes away speed squared, so where the car at 50 stops, the car at 60 is still doing 43 km/h with 516 per mille of its energy a margin measured in speed is spent in distance, and distance is charged at the square; ten over is not a tenth of anything by the time the road runs out Sixty is a fifth over fifty - exactly. But the reaction is spent at sixty and braking removes the square of the speed, so the car at 60 needs 4484 cm to stop, and at the 3460 cm where the car at 50 stood still it is doing 43 km/h, until the stop is computed from its parts instead of scaled by the speed. ``` ## Round-trip `ok: true` — round-trip fixpoint reached (python1 == python2) ## Trace event types eml:run:start · eml:assign · eml:output · eml:run:done