Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, B

Average Error: 14.7 → 1.1

Time: 15.6s

Precision: 64

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

Math

\[x \cdot \frac{\frac{y}{z} \cdot t}{t}\]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -2.588366971627017444543194392656819407323 \cdot 10^{79}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;\frac{y}{z} \le -7.584778841906227496781487885537839730751 \cdot 10^{-279}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;\frac{y}{z} \le 0.0:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 1.406079879178553538420301463833789327206 \cdot 10^{177}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r703835 = x;
        double r703836 = y;
        double r703837 = z;
        double r703838 = r703836 / r703837;
        double r703839 = t;
        double r703840 = r703838 * r703839;
        double r703841 = r703840 / r703839;
        double r703842 = r703835 * r703841;
        return r703842;
}

↓

double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r703843 = y;
        double r703844 = z;
        double r703845 = r703843 / r703844;
        double r703846 = -2.5883669716270174e+79;
        bool r703847 = r703845 <= r703846;
        double r703848 = x;
        double r703849 = r703848 / r703844;
        double r703850 = r703849 * r703843;
        double r703851 = -7.584778841906227e-279;
        bool r703852 = r703845 <= r703851;
        double r703853 = r703845 * r703848;
        double r703854 = 0.0;
        bool r703855 = r703845 <= r703854;
        double r703856 = r703848 * r703843;
        double r703857 = r703856 / r703844;
        double r703858 = 1.4060798791785535e+177;
        bool r703859 = r703845 <= r703858;
        double r703860 = r703844 / r703843;
        double r703861 = r703848 / r703860;
        double r703862 = r703859 ? r703861 : r703850;
        double r703863 = r703855 ? r703857 : r703862;
        double r703864 = r703852 ? r703853 : r703863;
        double r703865 = r703847 ? r703850 : r703864;
        return r703865;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	14.7
Target	1.5
Herbie	1.1

\[\begin{array}{l} \mathbf{if}\;\frac{\frac{y}{z} \cdot t}{t} \lt -1.206722051230450047215521150762600712224 \cdot 10^{245}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \lt -5.90752223693390632993316700759382836344 \cdot 10^{-275}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \lt 5.658954423153415216825328199697215652986 \cdot 10^{-65}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \lt 2.008718050240713347941382056648619307142 \cdot 10^{217}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array}\]

Derivation

1. Split input into 4 regimes

2. if (/ y z) < -2.5883669716270174e+79 or 1.4060798791785535e+177 < (/ y z)

   1. Initial program 31.5

      \[x \cdot \frac{\frac{y}{z} \cdot t}{t}\]

   2. Simplified 16.2

      \[\leadsto \color{blue}{\frac{y}{z} \cdot x}\]
   3. Using strategy rm

   4. Applied pow1 16.2

      \[\leadsto \frac{y}{z} \cdot \color{blue}{{x}^{1}}\]

   5. Applied pow1 16.2

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

   6. Applied pow-prod-down 16.2

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

   7. Simplified 3.4

      \[\leadsto {\color{blue}{\left(\frac{x \cdot y}{z}\right)}}^{1}\]
   8. Using strategy rm

   9. Applied associate-/l* 14.7

      \[\leadsto {\color{blue}{\left(\frac{x}{\frac{z}{y}}\right)}}^{1}\]
   10. Using strategy rm

   11. Applied add-cube-cbrt 15.5

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

   12. Applied *-un-lft-identity 15.5

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

   13. Applied times-frac 15.5

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

   14. Applied associate-/r* 5.6

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

   15. Simplified 5.6

       \[\leadsto {\left(\frac{\color{blue}{x \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)}}{\frac{z}{\sqrt[3]{y}}}\right)}^{1}\]
   16. Using strategy rm

   17. Applied div-inv 5.6

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

   18. Applied times-frac 5.0

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

   19. Simplified 4.0

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

   if -2.5883669716270174e+79 < (/ y z) < -7.584778841906227e-279

   1. Initial program 8.4

      \[x \cdot \frac{\frac{y}{z} \cdot t}{t}\]

   2. Simplified 0.2

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

   if -7.584778841906227e-279 < (/ y z) < 0.0

   1. Initial program 18.3

      \[x \cdot \frac{\frac{y}{z} \cdot t}{t}\]

   2. Simplified 15.3

      \[\leadsto \color{blue}{\frac{y}{z} \cdot x}\]
   3. Using strategy rm

   4. Applied pow1 15.3

      \[\leadsto \frac{y}{z} \cdot \color{blue}{{x}^{1}}\]

   5. Applied pow1 15.3

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

   6. Applied pow-prod-down 15.3

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

   7. Simplified 0.1

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

   if 0.0 < (/ y z) < 1.4060798791785535e+177

   1. Initial program 10.0

      \[x \cdot \frac{\frac{y}{z} \cdot t}{t}\]

   2. Simplified 3.4

      \[\leadsto \color{blue}{\frac{y}{z} \cdot x}\]
   3. Using strategy rm

   4. Applied pow1 3.4

      \[\leadsto \frac{y}{z} \cdot \color{blue}{{x}^{1}}\]

   5. Applied pow1 3.4

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

   6. Applied pow-prod-down 3.4

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

   7. Simplified 7.3

      \[\leadsto {\color{blue}{\left(\frac{x \cdot y}{z}\right)}}^{1}\]
   8. Using strategy rm

   9. Applied associate-/l* 3.6

      \[\leadsto {\color{blue}{\left(\frac{x}{\frac{z}{y}}\right)}}^{1}\]
3. Recombined 4 regimes into one program.

4. Final simplification 1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -2.588366971627017444543194392656819407323 \cdot 10^{79}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;\frac{y}{z} \le -7.584778841906227496781487885537839730751 \cdot 10^{-279}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;\frac{y}{z} \le 0.0:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 1.406079879178553538420301463833789327206 \cdot 10^{177}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, B"
  :precision binary64

  :herbie-target
  (if (< (/ (* (/ y z) t) t) -1.20672205123045e+245) (/ y (/ z x)) (if (< (/ (* (/ y z) t) t) -5.907522236933906e-275) (* x (/ y z)) (if (< (/ (* (/ y z) t) t) 5.658954423153415e-65) (/ y (/ z x)) (if (< (/ (* (/ y z) t) t) 2.0087180502407133e+217) (* x (/ y z)) (/ (* y x) z)))))

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