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

Average Error: 15.2 → 3.1

Time: 5.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \le 6.572772375852078491850975400123842729186 \cdot 10^{50}:\\ \;\;\;\;\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{z}} \cdot \left(\frac{\sqrt[3]{x}}{\sqrt[3]{z}} \cdot \frac{y}{\sqrt[3]{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \sqrt[3]{\sqrt[3]{z}}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{x \cdot y}{z}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r677925 = x;
        double r677926 = y;
        double r677927 = z;
        double r677928 = r677926 / r677927;
        double r677929 = t;
        double r677930 = r677928 * r677929;
        double r677931 = r677930 / r677929;
        double r677932 = r677925 * r677931;
        return r677932;
}

↓

double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r677933 = y;
        double r677934 = z;
        double r677935 = r677933 / r677934;
        double r677936 = 6.5727723758520785e+50;
        bool r677937 = r677935 <= r677936;
        double r677938 = x;
        double r677939 = cbrt(r677938);
        double r677940 = r677939 * r677939;
        double r677941 = cbrt(r677934);
        double r677942 = r677940 / r677941;
        double r677943 = r677939 / r677941;
        double r677944 = r677941 * r677941;
        double r677945 = cbrt(r677944);
        double r677946 = cbrt(r677941);
        double r677947 = r677945 * r677946;
        double r677948 = r677933 / r677947;
        double r677949 = r677943 * r677948;
        double r677950 = r677942 * r677949;
        double r677951 = 1.0;
        double r677952 = r677938 * r677933;
        double r677953 = r677952 / r677934;
        double r677954 = r677951 * r677953;
        double r677955 = r677937 ? r677950 : r677954;
        return r677955;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	15.2
Target	1.7
Herbie	3.1

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

Derivation

1. Split input into 2 regimes

2. if (/ y z) < 6.5727723758520785e+50

   1. Initial program 13.5

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

   2. Simplified 5.7

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

   4. Applied add-cube-cbrt 6.5

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

   5. Applied *-un-lft-identity 6.5

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

   6. Applied times-frac 6.5

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

   7. Applied associate-*r* 5.8

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

   8. Simplified 5.8

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

   10. Applied add-cube-cbrt 5.8

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

   11. Applied cbrt-prod 5.9

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

   13. Applied add-cube-cbrt 6.0

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

   14. Applied times-frac 6.0

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

   15. Applied associate-*l* 2.6

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

   if 6.5727723758520785e+50 < (/ y z)

   1. Initial program 24.7

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

   2. Simplified 11.0

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

   4. Applied add-cube-cbrt 12.0

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

   5. Applied *-un-lft-identity 12.0

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

   6. Applied times-frac 12.0

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

   7. Applied associate-*r* 6.4

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

   8. Simplified 6.4

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

   10. Applied *-un-lft-identity 6.4

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

   11. Applied associate-*l* 6.4

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

   12. Simplified 5.9

       \[\leadsto 1 \cdot \color{blue}{\frac{x \cdot y}{z}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 3.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \le 6.572772375852078491850975400123842729186 \cdot 10^{50}:\\ \;\;\;\;\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{z}} \cdot \left(\frac{\sqrt[3]{x}}{\sqrt[3]{z}} \cdot \frac{y}{\sqrt[3]{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \sqrt[3]{\sqrt[3]{z}}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{x \cdot y}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 
(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)))