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

Average Error: 14.8 → 2.5

Time: 15.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -5.422773879432081316361824486038416445882 \cdot 10^{151}:\\ \;\;\;\;\frac{1}{\frac{z}{y \cdot x}}\\ \mathbf{elif}\;\frac{y}{z} \le -4.526201326686900451200991798578555798024 \cdot 10^{-297}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt[3]{z}} \cdot \left(\left(\frac{\sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{z}}} \cdot \frac{y}{\sqrt[3]{z}}\right) \cdot \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{z} \cdot \sqrt[3]{z}}}\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r26630653 = x;
        double r26630654 = y;
        double r26630655 = z;
        double r26630656 = r26630654 / r26630655;
        double r26630657 = t;
        double r26630658 = r26630656 * r26630657;
        double r26630659 = r26630658 / r26630657;
        double r26630660 = r26630653 * r26630659;
        return r26630660;
}

↓

double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r26630661 = y;
        double r26630662 = z;
        double r26630663 = r26630661 / r26630662;
        double r26630664 = -5.422773879432081e+151;
        bool r26630665 = r26630663 <= r26630664;
        double r26630666 = 1.0;
        double r26630667 = x;
        double r26630668 = r26630661 * r26630667;
        double r26630669 = r26630662 / r26630668;
        double r26630670 = r26630666 / r26630669;
        double r26630671 = -4.5262013266869005e-297;
        bool r26630672 = r26630663 <= r26630671;
        double r26630673 = r26630667 * r26630663;
        double r26630674 = cbrt(r26630662);
        double r26630675 = r26630666 / r26630674;
        double r26630676 = cbrt(r26630667);
        double r26630677 = cbrt(r26630674);
        double r26630678 = r26630676 / r26630677;
        double r26630679 = r26630661 / r26630674;
        double r26630680 = r26630678 * r26630679;
        double r26630681 = r26630676 * r26630676;
        double r26630682 = r26630674 * r26630674;
        double r26630683 = cbrt(r26630682);
        double r26630684 = r26630681 / r26630683;
        double r26630685 = r26630680 * r26630684;
        double r26630686 = r26630675 * r26630685;
        double r26630687 = r26630672 ? r26630673 : r26630686;
        double r26630688 = r26630665 ? r26630670 : r26630687;
        return r26630688;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	14.8
Target	1.6
Herbie	2.5

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

Derivation

1. Split input into 3 regimes

2. if (/ y z) < -5.422773879432081e+151

   1. Initial program 36.5

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

   2. Simplified 2.3

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

   4. Applied clear-num 2.4

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

   if -5.422773879432081e+151 < (/ y z) < -4.5262013266869005e-297

   1. Initial program 8.2

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

   2. Simplified 9.0

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

   4. Applied *-un-lft-identity 9.0

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

   5. Applied times-frac 0.2

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

   6. Simplified 0.2

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

   if -4.5262013266869005e-297 < (/ y z)

   1. Initial program 15.4

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

   2. Simplified 5.4

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

   4. Applied div-inv 5.4

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

   6. Applied add-cube-cbrt 6.1

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

   7. Applied *-un-lft-identity 6.1

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

   8. Applied times-frac 6.1

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

   9. Applied associate-*r* 6.1

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

   10. Simplified 4.6

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

   12. Applied add-cube-cbrt 4.6

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

   13. Applied cbrt-prod 4.7

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

   14. Applied add-cube-cbrt 4.8

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

   15. Applied times-frac 4.8

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

   16. Applied associate-*l* 3.8

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

4. Final simplification 2.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -5.422773879432081316361824486038416445882 \cdot 10^{151}:\\ \;\;\;\;\frac{1}{\frac{z}{y \cdot x}}\\ \mathbf{elif}\;\frac{y}{z} \le -4.526201326686900451200991798578555798024 \cdot 10^{-297}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt[3]{z}} \cdot \left(\left(\frac{\sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{z}}} \cdot \frac{y}{\sqrt[3]{z}}\right) \cdot \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{z} \cdot \sqrt[3]{z}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, B"

  :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)))