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

Average Error: 14.6 → 2.1

Time: 17.6s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -2.726224425942599138418068187989578050083 \cdot 10^{-214} \lor \neg \left(\frac{y}{z} \le 7.182752616858212935481303668920257428991 \cdot 10^{-116}\right):\\ \;\;\;\;\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z}} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r667870 = x;
        double r667871 = y;
        double r667872 = z;
        double r667873 = r667871 / r667872;
        double r667874 = t;
        double r667875 = r667873 * r667874;
        double r667876 = r667875 / r667874;
        double r667877 = r667870 * r667876;
        return r667877;
}

↓

double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r667878 = y;
        double r667879 = z;
        double r667880 = r667878 / r667879;
        double r667881 = -2.726224425942599e-214;
        bool r667882 = r667880 <= r667881;
        double r667883 = 7.182752616858213e-116;
        bool r667884 = r667880 <= r667883;
        double r667885 = !r667884;
        bool r667886 = r667882 || r667885;
        double r667887 = cbrt(r667878);
        double r667888 = r667887 * r667887;
        double r667889 = cbrt(r667879);
        double r667890 = r667889 * r667889;
        double r667891 = r667888 / r667890;
        double r667892 = r667887 / r667889;
        double r667893 = x;
        double r667894 = r667892 * r667893;
        double r667895 = r667891 * r667894;
        double r667896 = r667893 * r667878;
        double r667897 = r667896 / r667879;
        double r667898 = r667886 ? r667895 : r667897;
        return r667898;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	14.6
Target	1.7
Herbie	2.1

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

Derivation

1. Split input into 2 regimes

2. if (/ y z) < -2.726224425942599e-214 or 7.182752616858213e-116 < (/ y z)

   1. Initial program 13.9

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

   2. Simplified 4.8

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

   4. Applied add-cube-cbrt 5.8

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

   5. Applied add-cube-cbrt 6.0

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

   6. Applied times-frac 6.0

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

   7. Applied associate-*l* 2.3

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

   if -2.726224425942599e-214 < (/ y z) < 7.182752616858213e-116

   1. Initial program 15.9

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

   2. Simplified 8.3

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

   4. Applied add-cube-cbrt 8.6

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

   5. Applied add-cube-cbrt 8.8

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

   6. Applied times-frac 8.8

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

   7. Applied associate-*l* 1.6

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

   9. Applied associate-*l/ 1.9

      \[\leadsto \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \color{blue}{\frac{\sqrt[3]{y} \cdot x}{\sqrt[3]{z}}}\]

   10. Applied frac-times 2.3

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

   11. Simplified 2.1

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

   12. Simplified 1.6

       \[\leadsto \frac{x \cdot y}{\color{blue}{z}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 2.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -2.726224425942599138418068187989578050083 \cdot 10^{-214} \lor \neg \left(\frac{y}{z} \le 7.182752616858212935481303668920257428991 \cdot 10^{-116}\right):\\ \;\;\;\;\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z}} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 +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)))