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

Average Error: 5.9 → 1.4

Time: 13.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -401466375465460629504:\\ \;\;\;\;\left(\frac{t}{a} - \frac{z}{a}\right) \cdot y + x\\ \mathbf{elif}\;y \le 8.154301343153642018279670833547796995186 \cdot 10^{100}:\\ \;\;\;\;x + \left(\frac{t \cdot y}{a} - \frac{z}{\frac{a}{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt[3]{y}}{a} \cdot \left(t - z\right)\right) \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + x\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r288910 = x;
        double r288911 = y;
        double r288912 = z;
        double r288913 = t;
        double r288914 = r288912 - r288913;
        double r288915 = r288911 * r288914;
        double r288916 = a;
        double r288917 = r288915 / r288916;
        double r288918 = r288910 - r288917;
        return r288918;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r288919 = y;
        double r288920 = -4.014663754654606e+20;
        bool r288921 = r288919 <= r288920;
        double r288922 = t;
        double r288923 = a;
        double r288924 = r288922 / r288923;
        double r288925 = z;
        double r288926 = r288925 / r288923;
        double r288927 = r288924 - r288926;
        double r288928 = r288927 * r288919;
        double r288929 = x;
        double r288930 = r288928 + r288929;
        double r288931 = 8.154301343153642e+100;
        bool r288932 = r288919 <= r288931;
        double r288933 = r288922 * r288919;
        double r288934 = r288933 / r288923;
        double r288935 = r288923 / r288919;
        double r288936 = r288925 / r288935;
        double r288937 = r288934 - r288936;
        double r288938 = r288929 + r288937;
        double r288939 = cbrt(r288919);
        double r288940 = r288939 / r288923;
        double r288941 = r288922 - r288925;
        double r288942 = r288940 * r288941;
        double r288943 = r288939 * r288939;
        double r288944 = r288942 * r288943;
        double r288945 = r288944 + r288929;
        double r288946 = r288932 ? r288938 : r288945;
        double r288947 = r288921 ? r288930 : r288946;
        return r288947;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	5.9
Target	0.7
Herbie	1.4

\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if y < -4.014663754654606e+20

   1. Initial program 15.4

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

   2. Simplified 4.6

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

   4. Applied *-un-lft-identity 4.6

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

   5. Applied add-cube-cbrt 5.2

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

   6. Applied times-frac 5.2

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

   7. Applied associate-*l* 2.2

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

   8. Taylor expanded around 0 15.4

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

   9. Simplified 4.0

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

   11. Applied *-un-lft-identity 4.0

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

   12. Applied *-un-lft-identity 4.0

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

   13. Applied times-frac 4.0

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

   14. Applied *-un-lft-identity 4.0

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

   15. Applied times-frac 4.0

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

   16. Applied *-un-lft-identity 4.0

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

   17. Applied *-un-lft-identity 4.0

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

   18. Applied times-frac 4.0

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

   19. Applied *-un-lft-identity 4.0

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

   20. Applied times-frac 4.0

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

   21. Applied distribute-lft-out-- 4.0

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

   22. Simplified 0.7

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

   if -4.014663754654606e+20 < y < 8.154301343153642e+100

   1. Initial program 1.1

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

   2. Simplified 1.7

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

   4. Applied *-un-lft-identity 1.7

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

   5. Applied add-cube-cbrt 2.1

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

   6. Applied times-frac 2.1

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

   7. Applied associate-*l* 5.1

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

   8. Taylor expanded around 0 1.1

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

   9. Simplified 1.8

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

   10. Taylor expanded around 0 1.5

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

   11. Simplified 1.5

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

   if 8.154301343153642e+100 < y

   1. Initial program 23.0

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

   2. Simplified 5.8

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

   4. Applied *-un-lft-identity 5.8

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

   5. Applied add-cube-cbrt 6.4

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

   6. Applied times-frac 6.4

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

   7. Applied associate-*l* 1.6

      \[\leadsto x + \color{blue}{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{1} \cdot \left(\frac{\sqrt[3]{y}}{a} \cdot \left(t - z\right)\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -401466375465460629504:\\ \;\;\;\;\left(\frac{t}{a} - \frac{z}{a}\right) \cdot y + x\\ \mathbf{elif}\;y \le 8.154301343153642018279670833547796995186 \cdot 10^{100}:\\ \;\;\;\;x + \left(\frac{t \cdot y}{a} - \frac{z}{\frac{a}{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt[3]{y}}{a} \cdot \left(t - z\right)\right) \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(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 (/ 1.0 (/ (/ a (- z t)) y))) (if (< y 2.894426862792089e-49) (- x (/ (* y (- z t)) a)) (- x (/ y (/ a (- z t))))))

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