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

Average Error: 6.5 → 0.6

Time: 10.1s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -99560.267414621193893253803253173828125:\\ \;\;\;\;y \cdot \frac{t - z}{a} + x\\ \mathbf{elif}\;a \le 13720.53919203528494108468294143676757812:\\ \;\;\;\;\frac{y \cdot \left(t - z\right)}{a} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{\frac{a}{t - z}}}{\sqrt{1}} + x\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r373873 = x;
        double r373874 = y;
        double r373875 = z;
        double r373876 = t;
        double r373877 = r373875 - r373876;
        double r373878 = r373874 * r373877;
        double r373879 = a;
        double r373880 = r373878 / r373879;
        double r373881 = r373873 - r373880;
        return r373881;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r373882 = a;
        double r373883 = -99560.2674146212;
        bool r373884 = r373882 <= r373883;
        double r373885 = y;
        double r373886 = t;
        double r373887 = z;
        double r373888 = r373886 - r373887;
        double r373889 = r373888 / r373882;
        double r373890 = r373885 * r373889;
        double r373891 = x;
        double r373892 = r373890 + r373891;
        double r373893 = 13720.539192035285;
        bool r373894 = r373882 <= r373893;
        double r373895 = r373885 * r373888;
        double r373896 = r373895 / r373882;
        double r373897 = r373896 + r373891;
        double r373898 = r373882 / r373888;
        double r373899 = r373885 / r373898;
        double r373900 = 1.0;
        double r373901 = sqrt(r373900);
        double r373902 = r373899 / r373901;
        double r373903 = r373902 + r373891;
        double r373904 = r373894 ? r373897 : r373903;
        double r373905 = r373884 ? r373892 : r373904;
        return r373905;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	6.5
Target	0.7
Herbie	0.6

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

Derivation

1. Split input into 3 regimes

2. if a < -99560.2674146212

   1. Initial program 9.7

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

   2. Simplified 1.7

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

   4. Applied fma-udef 1.7

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

   6. Applied div-inv 1.7

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

   7. Applied associate-*l* 0.4

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

   8. Simplified 0.4

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

   if -99560.2674146212 < a < 13720.539192035285

   1. Initial program 0.8

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

   2. Simplified 3.6

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

   4. Applied fma-udef 3.6

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

   6. Applied *-un-lft-identity 3.6

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

   7. Applied add-cube-cbrt 4.3

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

   8. Applied times-frac 4.3

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

   9. Applied associate-*l* 10.7

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

   11. Applied add-sqr-sqrt 10.7

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

   12. Applied associate-*l* 10.7

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

   13. Simplified 10.2

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

   15. Applied associate-*r/ 4.9

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

   16. Applied sqrt-div 4.9

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

   17. Applied frac-times 1.5

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

   18. Simplified 0.8

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

   19. Simplified 0.8

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

   if 13720.539192035285 < a

   1. Initial program 10.4

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

   2. Simplified 1.8

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

   4. Applied fma-udef 1.8

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

   6. Applied *-un-lft-identity 1.8

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

   7. Applied add-cube-cbrt 2.2

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

   8. Applied times-frac 2.2

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

   9. Applied associate-*l* 0.9

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

   11. Applied add-sqr-sqrt 0.9

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

   12. Applied associate-*l* 0.9

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

   13. Simplified 3.0

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

   15. Applied sqrt-div 3.0

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

   16. Applied associate-*l/ 3.0

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

   17. Simplified 0.6

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -99560.267414621193893253803253173828125:\\ \;\;\;\;y \cdot \frac{t - z}{a} + x\\ \mathbf{elif}\;a \le 13720.53919203528494108468294143676757812:\\ \;\;\;\;\frac{y \cdot \left(t - z\right)}{a} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{\frac{a}{t - z}}}{\sqrt{1}} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 +o rules:numerics
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, F"
  :precision binary64

  :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)))