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

Average Error: 5.5 → 0.5

Time: 19.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(z - t\right) \cdot y = -\infty:\\ \;\;\;\;x - \frac{z - t}{a} \cdot y\\ \mathbf{elif}\;\left(z - t\right) \cdot y \le 4.602631262379337 \cdot 10^{+178}:\\ \;\;\;\;x - \frac{\left(z - t\right) \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \left(z - t\right) \cdot \frac{y}{a}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r17660795 = x;
        double r17660796 = y;
        double r17660797 = z;
        double r17660798 = t;
        double r17660799 = r17660797 - r17660798;
        double r17660800 = r17660796 * r17660799;
        double r17660801 = a;
        double r17660802 = r17660800 / r17660801;
        double r17660803 = r17660795 - r17660802;
        return r17660803;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r17660804 = z;
        double r17660805 = t;
        double r17660806 = r17660804 - r17660805;
        double r17660807 = y;
        double r17660808 = r17660806 * r17660807;
        double r17660809 = -inf.0;
        bool r17660810 = r17660808 <= r17660809;
        double r17660811 = x;
        double r17660812 = a;
        double r17660813 = r17660806 / r17660812;
        double r17660814 = r17660813 * r17660807;
        double r17660815 = r17660811 - r17660814;
        double r17660816 = 4.602631262379337e+178;
        bool r17660817 = r17660808 <= r17660816;
        double r17660818 = r17660808 / r17660812;
        double r17660819 = r17660811 - r17660818;
        double r17660820 = r17660807 / r17660812;
        double r17660821 = r17660806 * r17660820;
        double r17660822 = r17660811 - r17660821;
        double r17660823 = r17660817 ? r17660819 : r17660822;
        double r17660824 = r17660810 ? r17660815 : r17660823;
        return r17660824;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	5.5
Target	0.7
Herbie	0.5

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

Derivation

1. Split input into 3 regimes

2. if (* y (- z t)) < -inf.0

   1. Initial program 60.1

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

   2. Taylor expanded around 0 60.3

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

   3. Simplified 0.2

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

   5. Applied associate-/r/ 0.2

      \[\leadsto x - \color{blue}{\frac{z - t}{a} \cdot y}\]

   if -inf.0 < (* y (- z t)) < 4.602631262379337e+178

   1. Initial program 0.4

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

   2. Taylor expanded around 0 0.4

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

   3. Simplified 2.6

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

   5. Applied div-inv 2.8

      \[\leadsto x - \color{blue}{\left(z - t\right) \cdot \frac{1}{\frac{a}{y}}}\]

   6. Simplified 2.7

      \[\leadsto x - \left(z - t\right) \cdot \color{blue}{\frac{y}{a}}\]
   7. Using strategy rm

   8. Applied *-un-lft-identity 2.7

      \[\leadsto x - \left(z - t\right) \cdot \frac{y}{\color{blue}{1 \cdot a}}\]

   9. Applied add-cube-cbrt 3.1

      \[\leadsto x - \left(z - t\right) \cdot \frac{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}{1 \cdot a}\]

   10. Applied times-frac 3.1

       \[\leadsto x - \left(z - t\right) \cdot \color{blue}{\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{1} \cdot \frac{\sqrt[3]{y}}{a}\right)}\]

   11. Applied associate-*r* 1.2

       \[\leadsto x - \color{blue}{\left(\left(z - t\right) \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{1}\right) \cdot \frac{\sqrt[3]{y}}{a}}\]

   12. Simplified 1.2

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

   14. Applied associate-*r/ 0.9

       \[\leadsto x - \color{blue}{\frac{\left(\left(\left(z - t\right) \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}{a}}\]

   15. Simplified 0.4

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

   if 4.602631262379337e+178 < (* y (- z t))

   1. Initial program 23.3

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

   2. Taylor expanded around 0 23.3

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

   3. Simplified 0.9

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

   5. Applied div-inv 1.0

      \[\leadsto x - \color{blue}{\left(z - t\right) \cdot \frac{1}{\frac{a}{y}}}\]

   6. Simplified 0.7

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z - t\right) \cdot y = -\infty:\\ \;\;\;\;x - \frac{z - t}{a} \cdot y\\ \mathbf{elif}\;\left(z - t\right) \cdot y \le 4.602631262379337 \cdot 10^{+178}:\\ \;\;\;\;x - \frac{\left(z - t\right) \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \left(z - t\right) \cdot \frac{y}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 
(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)))