Average Error: 0.4 → 0.3
Time: 44.0s
Precision: 64
\[0 \le u1 \le 1 \land 0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\mathsf{fma}\left(\cos \left(\left(\pi \cdot 2\right) \cdot u2\right), \left(\sqrt{\frac{1}{6}} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \sqrt{\frac{1}{6}}, 0.5\right)\]
\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
↓
\mathsf{fma}\left(\cos \left(\left(\pi \cdot 2\right) \cdot u2\right), \left(\sqrt{\frac{1}{6}} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \sqrt{\frac{1}{6}}, 0.5\right)
double f(double u1, double u2) {
        double r1778020 = 1.0;
        double r1778021 = 6.0;
        double r1778022 = r1778020 / r1778021;
        double r1778023 = -2.0;
        double r1778024 = u1;
        double r1778025 = log(r1778024);
        double r1778026 = r1778023 * r1778025;
        double r1778027 = 0.5;
        double r1778028 = pow(r1778026, r1778027);
        double r1778029 = r1778022 * r1778028;
        double r1778030 = 2.0;
        double r1778031 = atan2(1.0, 0.0);
        double r1778032 = r1778030 * r1778031;
        double r1778033 = u2;
        double r1778034 = r1778032 * r1778033;
        double r1778035 = cos(r1778034);
        double r1778036 = r1778029 * r1778035;
        double r1778037 = r1778036 + r1778027;
        return r1778037;
}
↓
double f(double u1, double u2) {
        double r1778038 = atan2(1.0, 0.0);
        double r1778039 = 2.0;
        double r1778040 = r1778038 * r1778039;
        double r1778041 = u2;
        double r1778042 = r1778040 * r1778041;
        double r1778043 = cos(r1778042);
        double r1778044 = 0.16666666666666666;
        double r1778045 = sqrt(r1778044);
        double r1778046 = -2.0;
        double r1778047 = u1;
        double r1778048 = log(r1778047);
        double r1778049 = r1778046 * r1778048;
        double r1778050 = 0.5;
        double r1778051 = pow(r1778049, r1778050);
        double r1778052 = r1778045 * r1778051;
        double r1778053 = r1778052 * r1778045;
        double r1778054 = fma(r1778043, r1778053, r1778050);
        return r1778054;
}

Error

Bits error versus u1

Bits error versus u2

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. Simplified0.4

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

  4. Applied add-sqr-sqrt0.4

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

  5. Applied associate-*l*0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019142 +o rules:numerics
(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))