normal distribution

Average Error: 0.4 → 0.2

Time: 6.5s

Precision: binary64

\[\left(0 \leq u1 \land u1 \leq 1\right) \land \left(0 \leq u2 \land u2 \leq 1\right)\]

Math

\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

↓

\[\mathsf{fma}\left(\sqrt{\log u1 \cdot -0.05555555555555555}, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right) \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (+
  (* (* (/ 1.0 6.0) (pow (* -2.0 (log u1)) 0.5)) (cos (* (* 2.0 PI) u2)))
  0.5))

↓

(FPCore (u1 u2)
 :precision binary64
 (fma (sqrt (* (log u1) -0.05555555555555555)) (cos (* (* 2.0 PI) u2)) 0.5))

C

double code(double u1, double u2) {
	return (((1.0 / 6.0) * pow((-2.0 * log(u1)), 0.5)) * cos(((2.0 * ((double) M_PI)) * u2))) + 0.5;
}

↓

double code(double u1, double u2) {
	return fma(sqrt((log(u1) * -0.05555555555555555)), cos(((2.0 * ((double) M_PI)) * u2)), 0.5);
}

Error

Bits error versus u1

Bits error versus u2

Derivation

1. Initial program 0.4

   \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

2. Applied egg-rr 0.2

   \[\leadsto \color{blue}{\sqrt{\left(-2 \cdot \log u1\right) \cdot 0.027777777777777776}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

3. Applied egg-rr 0.2

   \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(\sqrt{\log u1 \cdot -0.05555555555555555}, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)\right)}^{1}} \]

4. Final simplification 0.2

   \[\leadsto \mathsf{fma}\left(\sqrt{\log u1 \cdot -0.05555555555555555}, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right) \]

Reproduce

herbie shell --seed 2022129 
(FPCore (u1 u2)
  :name "normal distribution"
  :precision binary64
  :pre (and (and (<= 0.0 u1) (<= u1 1.0)) (and (<= 0.0 u2) (<= u2 1.0)))
  (+ (* (* (/ 1.0 6.0) (pow (* -2.0 (log u1)) 0.5)) (cos (* (* 2.0 PI) u2))) 0.5))