Optimisation.CirclePacking:place from circle-packing-0.1.0.4, F

Average Error: 6.1 → 0.6

Time: 37.2s

Precision: 64

Math

\[x - \frac{y \cdot \left(z - t\right)}{a}\]

↓

\[\begin{array}{l} \mathbf{if}\;a \le -748199.857216111:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{a}, y, x\right)\\ \mathbf{elif}\;a \le 2.3164280398826682 \cdot 10^{+52}:\\ \;\;\;\;\frac{y \cdot \left(t - z\right)}{a} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{a}, y, x\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r13432877 = x;
        double r13432878 = y;
        double r13432879 = z;
        double r13432880 = t;
        double r13432881 = r13432879 - r13432880;
        double r13432882 = r13432878 * r13432881;
        double r13432883 = a;
        double r13432884 = r13432882 / r13432883;
        double r13432885 = r13432877 - r13432884;
        return r13432885;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r13432886 = a;
        double r13432887 = -748199.857216111;
        bool r13432888 = r13432886 <= r13432887;
        double r13432889 = t;
        double r13432890 = z;
        double r13432891 = r13432889 - r13432890;
        double r13432892 = r13432891 / r13432886;
        double r13432893 = y;
        double r13432894 = x;
        double r13432895 = fma(r13432892, r13432893, r13432894);
        double r13432896 = 2.3164280398826682e+52;
        bool r13432897 = r13432886 <= r13432896;
        double r13432898 = r13432893 * r13432891;
        double r13432899 = r13432898 / r13432886;
        double r13432900 = r13432899 + r13432894;
        double r13432901 = r13432897 ? r13432900 : r13432895;
        double r13432902 = r13432888 ? r13432895 : r13432901;
        return r13432902;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	6.1
Target	0.6
Herbie	0.6

\[\begin{array}{l} \mathbf{if}\;y \lt -1.0761266216389975 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if a < -748199.857216111 or 2.3164280398826682e+52 < a

   1. Initial program 10.2

      \[x - \frac{y \cdot \left(z - t\right)}{a}\]

   2. Simplified 0.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - z}{a}, y, x\right)}\]
   3. Using strategy rm

   4. Applied fma-udef 0.5

      \[\leadsto \color{blue}{\frac{t - z}{a} \cdot y + x}\]
   5. Using strategy rm

   6. Applied fma-def 0.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - z}{a}, y, x\right)}\]

   if -748199.857216111 < a < 2.3164280398826682e+52

   1. Initial program 0.8

      \[x - \frac{y \cdot \left(z - t\right)}{a}\]

   2. Simplified 12.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - z}{a}, y, x\right)}\]
   3. Using strategy rm

   4. Applied fma-udef 12.5

      \[\leadsto \color{blue}{\frac{t - z}{a} \cdot y + x}\]
   5. Using strategy rm

   6. Applied associate-*l/ 0.8

      \[\leadsto \color{blue}{\frac{\left(t - z\right) \cdot y}{a}} + x\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -748199.857216111:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{a}, y, x\right)\\ \mathbf{elif}\;a \le 2.3164280398826682 \cdot 10^{+52}:\\ \;\;\;\;\frac{y \cdot \left(t - z\right)}{a} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{a}, y, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, F"

  :herbie-target
  (if (< y -1.0761266216389975e-10) (- x (/ 1 (/ (/ a (- z t)) y))) (if (< y 2.894426862792089e-49) (- x (/ (* y (- z t)) a)) (- x (/ y (/ a (- z t))))))

  (- x (/ (* y (- z t)) a)))