Average Error: 0.4 → 0.3
Time: 5.8s
Precision: binary64
\[0 \leq u1 \land u1 \leq 1 \land 0 \leq u2 \land u2 \leq 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[0.5 + 1 \cdot \frac{\cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot {\left(-2 \cdot \log u1\right)}^{0.5}}{6}\]
\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
↓
0.5 + 1 \cdot \frac{\cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot {\left(-2 \cdot \log u1\right)}^{0.5}}{6}
(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
 (+
  0.5
  (* 1.0 (/ (* (cos (* 2.0 (* PI u2))) (pow (* -2.0 (log u1)) 0.5)) 6.0))))
double code(double u1, double u2) {
	return ((double) (((double) (((double) ((1.0 / 6.0) * ((double) pow(((double) (-2.0 * ((double) log(u1)))), 0.5)))) * ((double) cos(((double) (((double) (2.0 * ((double) M_PI))) * u2)))))) + 0.5));
}
↓
double code(double u1, double u2) {
	return ((double) (0.5 + ((double) (1.0 * (((double) (((double) cos(((double) (2.0 * ((double) (((double) M_PI) * u2)))))) * ((double) pow(((double) (-2.0 * ((double) log(u1)))), 0.5)))) / 6.0)))));
}

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 Error: 0.4 bits

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

  2. SimplifiedError: 0.4 bits

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

  4. Applied associate-*r/Error: 0.3 bits

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

  5. SimplifiedError: 0.3 bits

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

  6. Final simplificationError: 0.3 bits

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

Reproduce

herbie shell --seed 2020203 
(FPCore (u1 u2)
  :name "normal distribution"
  :precision binary64
  :pre (and (<= 0.0 u1 1.0) (<= 0.0 u2 1.0))
  (+ (* (* (/ 1.0 6.0) (pow (* -2.0 (log u1)) 0.5)) (cos (* (* 2.0 PI) u2))) 0.5))