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

Average Error: 14.5 → 2.1

Time: 23.0s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r342005 = x;
        double r342006 = y;
        double r342007 = z;
        double r342008 = r342006 / r342007;
        double r342009 = t;
        double r342010 = r342008 * r342009;
        double r342011 = r342010 / r342009;
        double r342012 = r342005 * r342011;
        return r342012;
}

↓

double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r342013 = y;
        double r342014 = z;
        double r342015 = cbrt(r342014);
        double r342016 = r342015 * r342015;
        double r342017 = x;
        double r342018 = cbrt(r342017);
        double r342019 = r342018 * r342018;
        double r342020 = r342016 / r342019;
        double r342021 = r342013 / r342020;
        double r342022 = r342015 / r342018;
        double r342023 = r342021 / r342022;
        return r342023;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	14.5
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. Initial program 14.5

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

2. Simplified 6.2

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

4. Applied div-inv 6.2

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

5. Applied associate-*l* 6.2

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

6. Simplified 6.1

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

8. Applied associate-*r/ 6.2

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

10. Applied associate-/l* 6.1

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

12. Applied add-cube-cbrt 6.9

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

13. Applied add-cube-cbrt 7.0

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

14. Applied times-frac 7.0

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

15. Applied associate-/r* 2.1

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

16. Final simplification 2.1

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

Reproduce

herbie shell --seed 2019304 +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) -1.20672205123045005e245) (/ y (/ z x)) (if (< (/ (* (/ y z) t) t) -5.90752223693390633e-275) (* x (/ y z)) (if (< (/ (* (/ y z) t) t) 5.65895442315341522e-65) (/ y (/ z x)) (if (< (/ (* (/ y z) t) t) 2.0087180502407133e217) (* x (/ y z)) (/ (* y x) z)))))

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