Average Error: 0.4 → 0.3
Time: 25.5s
Precision: 64
\[0 \le u1 \le 1 \land 0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\mathsf{fma}\left(\cos \left(\left(\pi \cdot 2\right) \cdot u2\right), \left({\left(-2 \cdot \log u1\right)}^{0.5} \cdot \sqrt{\frac{1}{6}}\right) \cdot \sqrt{\frac{1}{6}}, 0.5\right)\]
\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({\left(-2 \cdot \log u1\right)}^{0.5} \cdot \sqrt{\frac{1}{6}}\right) \cdot \sqrt{\frac{1}{6}}, 0.5\right)
double f(double u1, double u2) {
        double r917112 = 1.0;
        double r917113 = 6.0;
        double r917114 = r917112 / r917113;
        double r917115 = -2.0;
        double r917116 = u1;
        double r917117 = log(r917116);
        double r917118 = r917115 * r917117;
        double r917119 = 0.5;
        double r917120 = pow(r917118, r917119);
        double r917121 = r917114 * r917120;
        double r917122 = 2.0;
        double r917123 = atan2(1.0, 0.0);
        double r917124 = r917122 * r917123;
        double r917125 = u2;
        double r917126 = r917124 * r917125;
        double r917127 = cos(r917126);
        double r917128 = r917121 * r917127;
        double r917129 = r917128 + r917119;
        return r917129;
}
↓
double f(double u1, double u2) {
        double r917130 = atan2(1.0, 0.0);
        double r917131 = 2.0;
        double r917132 = r917130 * r917131;
        double r917133 = u2;
        double r917134 = r917132 * r917133;
        double r917135 = cos(r917134);
        double r917136 = -2.0;
        double r917137 = u1;
        double r917138 = log(r917137);
        double r917139 = r917136 * r917138;
        double r917140 = 0.5;
        double r917141 = pow(r917139, r917140);
        double r917142 = 0.16666666666666666;
        double r917143 = sqrt(r917142);
        double r917144 = r917141 * r917143;
        double r917145 = r917144 * r917143;
        double r917146 = fma(r917135, r917145, r917140);
        return r917146;
}

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

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

  4. Applied add-sqr-sqrt0.4

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

  5. Applied associate-*r*0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019156 +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))