Average Error: 0.4 → 0.4
Time: 36.1s
Precision: 64
\[0 \le u1 \le 1 \land 0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\frac{\left(\left(\frac{1}{6} \cdot {\left(\log u1 \cdot -2\right)}^{0.5}\right) \cdot \cos \left(\left(\pi \cdot 2\right) \cdot u2\right)\right) \cdot \left(\left(\frac{1}{6} \cdot {\left(\log u1 \cdot -2\right)}^{0.5}\right) \cdot \cos \left(\left(\pi \cdot 2\right) \cdot u2\right)\right) - 0.5 \cdot 0.5}{\left(\frac{1}{6} \cdot {\left(\log u1 \cdot -2\right)}^{0.5}\right) \cdot \cos \left(\left(\pi \cdot 2\right) \cdot u2\right) - 0.5}\]
\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
↓
\frac{\left(\left(\frac{1}{6} \cdot {\left(\log u1 \cdot -2\right)}^{0.5}\right) \cdot \cos \left(\left(\pi \cdot 2\right) \cdot u2\right)\right) \cdot \left(\left(\frac{1}{6} \cdot {\left(\log u1 \cdot -2\right)}^{0.5}\right) \cdot \cos \left(\left(\pi \cdot 2\right) \cdot u2\right)\right) - 0.5 \cdot 0.5}{\left(\frac{1}{6} \cdot {\left(\log u1 \cdot -2\right)}^{0.5}\right) \cdot \cos \left(\left(\pi \cdot 2\right) \cdot u2\right) - 0.5}
double f(double u1, double u2) {
        double r1409112 = 1.0;
        double r1409113 = 6.0;
        double r1409114 = r1409112 / r1409113;
        double r1409115 = -2.0;
        double r1409116 = u1;
        double r1409117 = log(r1409116);
        double r1409118 = r1409115 * r1409117;
        double r1409119 = 0.5;
        double r1409120 = pow(r1409118, r1409119);
        double r1409121 = r1409114 * r1409120;
        double r1409122 = 2.0;
        double r1409123 = atan2(1.0, 0.0);
        double r1409124 = r1409122 * r1409123;
        double r1409125 = u2;
        double r1409126 = r1409124 * r1409125;
        double r1409127 = cos(r1409126);
        double r1409128 = r1409121 * r1409127;
        double r1409129 = r1409128 + r1409119;
        return r1409129;
}
↓
double f(double u1, double u2) {
        double r1409130 = 0.16666666666666666;
        double r1409131 = u1;
        double r1409132 = log(r1409131);
        double r1409133 = -2.0;
        double r1409134 = r1409132 * r1409133;
        double r1409135 = 0.5;
        double r1409136 = pow(r1409134, r1409135);
        double r1409137 = r1409130 * r1409136;
        double r1409138 = atan2(1.0, 0.0);
        double r1409139 = 2.0;
        double r1409140 = r1409138 * r1409139;
        double r1409141 = u2;
        double r1409142 = r1409140 * r1409141;
        double r1409143 = cos(r1409142);
        double r1409144 = r1409137 * r1409143;
        double r1409145 = r1409144 * r1409144;
        double r1409146 = r1409135 * r1409135;
        double r1409147 = r1409145 - r1409146;
        double r1409148 = r1409144 - r1409135;
        double r1409149 = r1409147 / r1409148;
        return r1409149;
}

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. Using strategy rm

  6. Applied associate-*r*0.4

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

  7. Simplified0.4

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

  9. Applied flip-+0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019142 
(FPCore (u1 u2)
  :name "normal distribution"
  :pre (and (<= 0 u1 1) (<= 0 u2 1))
  (+ (* (* (/ 1 6) (pow (* -2 (log u1)) 0.5)) (cos (* (* 2 PI) u2))) 0.5))