Average Error: 0.4 → 0.3
Time: 1.6m
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\frac{\sqrt{1} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}}{\frac{6}{\sqrt{1}}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
\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
↓
\frac{\sqrt{1} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}}{\frac{6}{\sqrt{1}}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 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(1.0) * pow((-2.0 * log(u1)), 0.5)) / (6.0 / sqrt(1.0))) * 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. Using strategy rm

  3. Applied add-sqr-sqrt0.4

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

  4. Applied associate-/l*0.4

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

  5. Applied associate-*l/0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2020066 
(FPCore (u1 u2)
  :name "normal distribution"
  :precision binary64
  :pre (and (<= 0.0 u1 1) (<= 0.0 u2 1))
  (+ (* (* (/ 1 6) (pow (* -2 (log u1)) 0.5)) (cos (* (* 2 PI) u2))) 0.5))