normal_cdf

std.stats.normal_cdf · Level L5

The normal distribution's cumulative probability, Φ((x − μ)/σ), written with erfc so the lower tail keeps its relative accuracy where 1 + erf would cancel to 0.

Φ = ½·erfc(−(x − μ)/(σ√2))

Signature

normal_cdf(x: f64[n], mu: f64[], sigma: f64[]) → f64[n]

Structure

The function as NOVA stores it: one box per input, operation and output, and arrows that carry values. A double border marks another library function this one runs — called once, or by Scan once per element; select it to open that function.

xf64[n]muf64[]sigmaf64[]-1.41421SubtractdzMultiplyscaleDividetErfcc0.5Multiplyppf64[n]
  • input
  • operation
  • constant
  • call
  • output

Verification

  • Signature proven by NOVA’s shape solver, for every size.
  • Agrees with the reference (1 + erf((x − μ)/(σ√2)))/2 to 80 digits (100-digit arithmetic), on all 40 test cases.
  • All 170 float64 results inside the running error bound; the closest uses 38% of it.
  • Interpreter and NumPy backend return bit-identical results.
Accuracy in detail
correctly rounded (the float64 nearest the exact value)
69%
bit-equal to the NumPy formula in float64
80%
largest error, in units in the last place
30

Large ulp counts appear where a result is tiny next to the numbers it is computed from (after cancellation, for example), so one unit in the last place is tiny too; the absolute error is still inside the bound. Results within their own error of zero are not counted.

Identity

Calls
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Called by
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sha256:fc7306cf9f1f4a6b981b39963d34ef9b552f51433ba015936f7b64286410864b

The semantic hash of the graph. It changes when the program changes, and never when only its documentation does.