Average Error: 0.4 → 0.3
Time: 26.8s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\left(\sqrt{\frac{1}{6}} \cdot \left(\sqrt{\frac{1}{6}} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
\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(\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
double f(double u1, double u2) {
        double r72017 = 1.0;
        double r72018 = 6.0;
        double r72019 = r72017 / r72018;
        double r72020 = -2.0;
        double r72021 = u1;
        double r72022 = log(r72021);
        double r72023 = r72020 * r72022;
        double r72024 = 0.5;
        double r72025 = pow(r72023, r72024);
        double r72026 = r72019 * r72025;
        double r72027 = 2.0;
        double r72028 = atan2(1.0, 0.0);
        double r72029 = r72027 * r72028;
        double r72030 = u2;
        double r72031 = r72029 * r72030;
        double r72032 = cos(r72031);
        double r72033 = r72026 * r72032;
        double r72034 = r72033 + r72024;
        return r72034;
}
↓
double f(double u1, double u2) {
        double r72035 = 1.0;
        double r72036 = 6.0;
        double r72037 = r72035 / r72036;
        double r72038 = sqrt(r72037);
        double r72039 = -2.0;
        double r72040 = u1;
        double r72041 = log(r72040);
        double r72042 = r72039 * r72041;
        double r72043 = 0.5;
        double r72044 = pow(r72042, r72043);
        double r72045 = r72038 * r72044;
        double r72046 = r72038 * r72045;
        double r72047 = 2.0;
        double r72048 = atan2(1.0, 0.0);
        double r72049 = r72047 * r72048;
        double r72050 = u2;
        double r72051 = r72049 * r72050;
        double r72052 = cos(r72051);
        double r72053 = r72046 * r72052;
        double r72054 = r72053 + r72043;
        return r72054;
}

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. Final simplification0.3

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

Reproduce

herbie shell --seed 2019304 
(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))