Average Error: 0.4 → 0.4
Time: 27.4s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
\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
↓
\mathsf{log1p}\left(\mathsf{expm1}\left(\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 r73028 = 1.0;
        double r73029 = 6.0;
        double r73030 = r73028 / r73029;
        double r73031 = -2.0;
        double r73032 = u1;
        double r73033 = log(r73032);
        double r73034 = r73031 * r73033;
        double r73035 = 0.5;
        double r73036 = pow(r73034, r73035);
        double r73037 = r73030 * r73036;
        double r73038 = 2.0;
        double r73039 = atan2(1.0, 0.0);
        double r73040 = r73038 * r73039;
        double r73041 = u2;
        double r73042 = r73040 * r73041;
        double r73043 = cos(r73042);
        double r73044 = r73037 * r73043;
        double r73045 = r73044 + r73035;
        return r73045;
}
↓
double f(double u1, double u2) {
        double r73046 = 1.0;
        double r73047 = 6.0;
        double r73048 = r73046 / r73047;
        double r73049 = -2.0;
        double r73050 = u1;
        double r73051 = log(r73050);
        double r73052 = r73049 * r73051;
        double r73053 = 0.5;
        double r73054 = pow(r73052, r73053);
        double r73055 = r73048 * r73054;
        double r73056 = expm1(r73055);
        double r73057 = log1p(r73056);
        double r73058 = 2.0;
        double r73059 = atan2(1.0, 0.0);
        double r73060 = r73058 * r73059;
        double r73061 = u2;
        double r73062 = r73060 * r73061;
        double r73063 = cos(r73062);
        double r73064 = r73057 * r73063;
        double r73065 = r73064 + r73053;
        return r73065;
}

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 log1p-expm1-u0.3

    \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt{\frac{1}{6}} \cdot \left(\sqrt{\frac{1}{6}} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right)\right)\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]

  7. Simplified0.4

    \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right)}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]

  8. Final simplification0.4

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]

Reproduce

herbie shell --seed 2019347 +o rules:numerics
(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))