Average Error: 0.4 → 0.4
Time: 6.2s
Precision: binary64
\[0 \leq u1 \land u1 \leq 1 \land 0 \leq u2 \land u2 \leq 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[0.5 + 1 \cdot \left(\sqrt{\frac{\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)}{6}} \cdot \left({\left(-2 \cdot \log u1\right)}^{0.5} \cdot \sqrt{\frac{\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)}{6}}\right)\right)\]
\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
↓
0.5 + 1 \cdot \left(\sqrt{\frac{\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)}{6}} \cdot \left({\left(-2 \cdot \log u1\right)}^{0.5} \cdot \sqrt{\frac{\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)}{6}}\right)\right)
(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
   (*
    (sqrt (/ (cos (* 2.0 (* PI u2))) 6.0))
    (* (pow (* -2.0 (log u1)) 0.5) (sqrt (/ (cos (* 2.0 (* PI u2))) 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) sqrt((((double) cos(((double) (2.0 * ((double) (((double) M_PI) * u2)))))) / 6.0))) * ((double) (((double) pow(((double) (-2.0 * ((double) log(u1)))), 0.5)) * ((double) sqrt((((double) cos(((double) (2.0 * ((double) (((double) M_PI) * u2)))))) / 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 add-sqr-sqrtError: 0.5 bits

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

  5. Applied associate-*r*Error: 0.4 bits

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

  6. Final simplificationError: 0.4 bits

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

Reproduce

herbie shell --seed 2020204 
(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))