(/ (/ (exp (* (pow (/ (- x m) s) 2) (- 0.5))) (sqrt (* 2 PI))) s)

Average Error: 0.2 → 0.2

Time: 1.3s

Precision: binary64

Math

\[\frac{\frac{e^{{\left(\frac{x - m}{s}\right)}^{2} \cdot \left(-0.5\right)}}{\sqrt{2 \cdot \pi}}}{s}\]

↓

\[\frac{\frac{e^{{\left(\frac{x - m}{s}\right)}^{2} \cdot \left(-0.5\right)}}{\sqrt{2 \cdot \pi}}}{s}\]

Error

Bits error versus x

Bits error versus m

Bits error versus s

Derivation

1. Initial program 0.2

   \[\frac{\frac{e^{{\left(\frac{x - m}{s}\right)}^{2} \cdot \left(-0.5\right)}}{\sqrt{2 \cdot \pi}}}{s}\]

2. Final simplification 0.2

   \[\leadsto \frac{\frac{e^{{\left(\frac{x - m}{s}\right)}^{2} \cdot \left(-0.5\right)}}{\sqrt{2 \cdot \pi}}}{s}\]

Reproduce

herbie shell --seed 2020153 
(FPCore (x m s)
  :name "(/ (/ (exp (* (pow (/ (- x m) s) 2) (- 0.5))) (sqrt (* 2 PI))) s)"
  :precision binary64
  (/ (/ (exp (* (pow (/ (- x m) s) 2.0) (neg 0.5))) (sqrt (* 2.0 PI))) s))