Average Error: 0.4 → 0.3
Time: 16.9s
Precision: binary64
\[0 \leq u1 \land u1 \leq 1 \land 0 \leq u2 \land u2 \leq 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[0.5 + \left(0.16666666666666666 \cdot \left(\sqrt{-\log u1} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\sqrt[3]{2 \cdot \left(u2 \cdot \pi\right)} \cdot e^{2 \cdot \log \left(\sqrt[3]{2 \cdot \left(u2 \cdot \pi\right)}\right)}\right)\]
\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 + \left(0.16666666666666666 \cdot \left(\sqrt{-\log u1} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\sqrt[3]{2 \cdot \left(u2 \cdot \pi\right)} \cdot e^{2 \cdot \log \left(\sqrt[3]{2 \cdot \left(u2 \cdot \pi\right)}\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
  (*
   (* 0.16666666666666666 (* (sqrt (- (log u1))) (sqrt 2.0)))
   (cos
    (*
     (cbrt (* 2.0 (* u2 PI)))
     (exp (* 2.0 (log (cbrt (* 2.0 (* u2 PI)))))))))))
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 0.5 + ((0.16666666666666666 * (sqrt(-log(u1)) * sqrt(2.0))) * cos(cbrt(2.0 * (u2 * ((double) M_PI))) * exp(2.0 * log(cbrt(2.0 * (u2 * ((double) M_PI)))))));
}

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. Taylor expanded around inf 0.3

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

  4. Simplified0.3

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

  6. Applied add-cube-cbrt_binary640.3

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

  7. Simplified0.3

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

  8. Simplified0.3

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

  10. Applied add-exp-log_binary640.3

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

  11. Applied add-exp-log_binary640.3

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

  12. Applied prod-exp_binary640.3

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

  13. Simplified0.3

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

  14. Final simplification0.3

    \[\leadsto 0.5 + \left(0.16666666666666666 \cdot \left(\sqrt{-\log u1} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\sqrt[3]{2 \cdot \left(u2 \cdot \pi\right)} \cdot e^{2 \cdot \log \left(\sqrt[3]{2 \cdot \left(u2 \cdot \pi\right)}\right)}\right)\]

Reproduce

herbie shell --seed 2021075 
(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))