Average Error: 0.4 → 0.7
Time: 9.1s
Precision: binary64
\[0 \leq u1 \land u1 \leq 1 \land 0 \leq u2 \land u2 \leq 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[e^{\log \left(0.5 + \left(0.16666666666666666 \cdot \sqrt{-2 \cdot \log u1}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\right)}\]
\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
↓
e^{\log \left(0.5 + \left(0.16666666666666666 \cdot \sqrt{-2 \cdot \log u1}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\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
 (exp
  (log
   (+
    0.5
    (*
     (* 0.16666666666666666 (sqrt (* -2.0 (log u1))))
     (cos (* (* 2.0 PI) u2)))))))
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 exp(log(0.5 + ((0.16666666666666666 * sqrt(-2.0 * log(u1))) * cos((2.0 * ((double) M_PI)) * u2))));
}

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. Simplified0.4

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

  4. Applied add-exp-log_binary640.7

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

  5. Final simplification0.7

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

Reproduce

herbie shell --seed 2020277 
(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))