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

Average Error: 15.1 → 2.6

Time: 19.5s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r28593398 = x;
        double r28593399 = y;
        double r28593400 = z;
        double r28593401 = r28593399 / r28593400;
        double r28593402 = t;
        double r28593403 = r28593401 * r28593402;
        double r28593404 = r28593403 / r28593402;
        double r28593405 = r28593398 * r28593404;
        return r28593405;
}

↓

double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r28593406 = x;
        double r28593407 = cbrt(r28593406);
        double r28593408 = z;
        double r28593409 = cbrt(r28593408);
        double r28593410 = r28593407 / r28593409;
        double r28593411 = y;
        double r28593412 = r28593411 / r28593409;
        double r28593413 = r28593407 * r28593407;
        double r28593414 = r28593413 / r28593409;
        double r28593415 = r28593412 * r28593414;
        double r28593416 = r28593410 * r28593415;
        return r28593416;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	15.1
Target	1.4
Herbie	2.6

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

Derivation

1. Initial program 15.1

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

2. Simplified 5.9

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

4. Applied add-cube-cbrt 6.7

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

5. Applied *-un-lft-identity 6.7

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

6. Applied times-frac 6.7

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

7. Applied associate-*r* 5.5

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

8. Simplified 5.5

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

10. Applied *-un-lft-identity 5.5

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

11. Applied cbrt-prod 5.5

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

12. Applied add-cube-cbrt 5.7

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

13. Applied times-frac 5.7

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

14. Applied associate-*r* 4.6

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

15. Simplified 2.6

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

16. Final simplification 2.6

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

Reproduce

herbie shell --seed 2019170 
(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)))