Average Error: 0.4 → 0.3
Time: 1.4min
Precision: binary64
Cost: 39680
\[0 \leq u1 \land u1 \leq 1 \land 0 \leq u2 \land u2 \leq 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[0.5 + \cos \left(\sqrt{\left(2 \cdot \pi\right) \cdot u2} \cdot \sqrt{\left(2 \cdot \pi\right) \cdot u2}\right) \cdot \left(\sqrt{-\log u1} \cdot \left(0.16666666666666666 \cdot \sqrt{2}\right)\right)\]
\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 + \cos \left(\sqrt{\left(2 \cdot \pi\right) \cdot u2} \cdot \sqrt{\left(2 \cdot \pi\right) \cdot u2}\right) \cdot \left(\sqrt{-\log u1} \cdot \left(0.16666666666666666 \cdot \sqrt{2}\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
  (*
   (cos (* (sqrt (* (* 2.0 PI) u2)) (sqrt (* (* 2.0 PI) u2))))
   (* (sqrt (- (log u1))) (* 0.16666666666666666 (sqrt 2.0))))))
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 + (cos(sqrt((2.0 * ((double) M_PI)) * u2) * sqrt((2.0 * ((double) M_PI)) * u2)) * (sqrt(-log(u1)) * (0.16666666666666666 * sqrt(2.0))));
}

Error

Bits error versus u1

Bits error versus u2

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error0.3
Cost20032
\[0.5 + \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \frac{\sqrt{\log u1 \cdot -2}}{6}\]
Alternative 2
Error0.4
Cost20032
\[0.5 + \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \left(0.16666666666666666 \cdot \sqrt{\log u1 \cdot -2}\right)\]
Alternative 3
Error1.0
Cost13248
\[0.5 + \frac{\sqrt{\log u1 \cdot -2}}{6}\]
Alternative 4
Error53.0
Cost64
\[1\]

Error

Time

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-sqrt_binary640.4

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

  5. Applied *-un-lft-identity_binary640.4

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

  6. Taylor expanded around inf 0.3

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

  7. Simplified0.3

    \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot \left(\sqrt{2} \cdot \sqrt{-\log u1}\right)\right)} \cdot \cos \left(\sqrt{1 \cdot \left(\left(2 \cdot \pi\right) \cdot u2\right)} \cdot \sqrt{\left(2 \cdot \pi\right) \cdot u2}\right) + 0.5\]
  8. Using strategy rm

  9. Applied associate-*r*_binary640.3

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

  10. Simplified0.3

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

  11. Simplified0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2021045 
(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))