Average Error: 0.4 → 0.3
Time: 6.3s
Precision: binary64
\[\left(0 \leq u1 \land u1 \leq 1\right) \land \left(0 \leq u2 \land u2 \leq 1\right)\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
↓
\[\left(\sqrt{-\log u1} \cdot \left(\sqrt{2} \cdot 0.16666666666666666\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
\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
↓
\left(\sqrt{-\log u1} \cdot \left(\sqrt{2} \cdot 0.16666666666666666\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5

(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
 (+
  (*
   (* (sqrt (- (log u1))) (* (sqrt 2.0) 0.16666666666666666))
   (cos (* (* 2.0 PI) u2)))
  0.5))
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 ((sqrt(-log(u1)) * (sqrt(2.0) * 0.16666666666666666)) * cos((2.0 * ((double) M_PI)) * u2)) + 0.5;
}

Error

Bits error versus u1

Bits error versus u2

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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 add-sqr-sqrt_binary640.4

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

  3. Applied associate-*l*_binary640.4

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

  4. Simplified0.4

    \[\leadsto \left(\sqrt{\frac{1}{6}} \cdot \color{blue}{\left(\sqrt{-2 \cdot \log u1} \cdot \sqrt{0.16666666666666666}\right)}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

  5. Taylor expanded in u1 around inf 0.3

    \[\leadsto \color{blue}{\left(\sqrt{\log \left(\frac{1}{u1}\right)} \cdot \left(\sqrt{2} \cdot {\left(\sqrt{0.16666666666666666}\right)}^{2}\right)\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

  6. Simplified0.3

    \[\leadsto \color{blue}{\left(\sqrt{-\log u1} \cdot \left(\sqrt{2} \cdot 0.16666666666666666\right)\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

  7. Final simplification0.3

    \[\leadsto \left(\sqrt{-\log u1} \cdot \left(\sqrt{2} \cdot 0.16666666666666666\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

Reproduce

herbie shell --seed 2022081 
(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))