Average Error: 0.4 → 0.4
Time: 11.9s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\left({\left({\left(\log u1\right)}^{1} \cdot {-2}^{1}\right)}^{0.5} \cdot {\left(\sqrt{0.166666666666666657}\right)}^{2}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
\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
↓
\left({\left({\left(\log u1\right)}^{1} \cdot {-2}^{1}\right)}^{0.5} \cdot {\left(\sqrt{0.166666666666666657}\right)}^{2}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5
double f(double u1, double u2) {
        double r67576 = 1.0;
        double r67577 = 6.0;
        double r67578 = r67576 / r67577;
        double r67579 = -2.0;
        double r67580 = u1;
        double r67581 = log(r67580);
        double r67582 = r67579 * r67581;
        double r67583 = 0.5;
        double r67584 = pow(r67582, r67583);
        double r67585 = r67578 * r67584;
        double r67586 = 2.0;
        double r67587 = atan2(1.0, 0.0);
        double r67588 = r67586 * r67587;
        double r67589 = u2;
        double r67590 = r67588 * r67589;
        double r67591 = cos(r67590);
        double r67592 = r67585 * r67591;
        double r67593 = r67592 + r67583;
        return r67593;
}
↓
double f(double u1, double u2) {
        double r67594 = u1;
        double r67595 = log(r67594);
        double r67596 = 1.0;
        double r67597 = pow(r67595, r67596);
        double r67598 = -2.0;
        double r67599 = pow(r67598, r67596);
        double r67600 = r67597 * r67599;
        double r67601 = 0.5;
        double r67602 = pow(r67600, r67601);
        double r67603 = 0.16666666666666666;
        double r67604 = sqrt(r67603);
        double r67605 = 2.0;
        double r67606 = pow(r67604, r67605);
        double r67607 = r67602 * r67606;
        double r67608 = 2.0;
        double r67609 = atan2(1.0, 0.0);
        double r67610 = r67608 * r67609;
        double r67611 = u2;
        double r67612 = r67610 * r67611;
        double r67613 = cos(r67612);
        double r67614 = r67607 * r67613;
        double r67615 = r67614 + r67601;
        return r67615;
}

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. Using strategy rm

  3. Applied add-sqr-sqrt0.4

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

  4. Applied associate-*l*0.3

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

  5. Taylor expanded around 0 0.4

    \[\leadsto \color{blue}{\left({\left({\left(\log u1\right)}^{1} \cdot {-2}^{1}\right)}^{0.5} \cdot {\left(\sqrt{0.166666666666666657}\right)}^{2}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]

  6. Final simplification0.4

    \[\leadsto \left({\left({\left(\log u1\right)}^{1} \cdot {-2}^{1}\right)}^{0.5} \cdot {\left(\sqrt{0.166666666666666657}\right)}^{2}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]

Reproduce

herbie shell --seed 2020034 
(FPCore (u1 u2)
  :name "normal distribution"
  :precision binary64
  :pre (and (<= 0.0 u1 1) (<= 0.0 u2 1))
  (+ (* (* (/ 1 6) (pow (* -2 (log u1)) 0.5)) (cos (* (* 2 PI) u2))) 0.5))