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

Average Error: 15.1 → 2.1

Time: 9.3s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r750827 = x;
        double r750828 = y;
        double r750829 = z;
        double r750830 = r750828 / r750829;
        double r750831 = t;
        double r750832 = r750830 * r750831;
        double r750833 = r750832 / r750831;
        double r750834 = r750827 * r750833;
        return r750834;
}

↓

double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r750835 = y;
        double r750836 = cbrt(r750835);
        double r750837 = x;
        double r750838 = cbrt(r750837);
        double r750839 = r750836 * r750838;
        double r750840 = z;
        double r750841 = r750839 / r750840;
        double r750842 = r750836 * r750836;
        double r750843 = r750838 * r750842;
        double r750844 = r750838 * r750843;
        double r750845 = r750841 * r750844;
        return r750845;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	15.1
Target	1.8
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 15.1

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

2. Simplified 6.2

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

4. Applied associate-/l* 6.1

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

6. Applied add-cube-cbrt 6.9

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

7. Applied *-un-lft-identity 6.9

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

8. Applied times-frac 6.9

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

9. Applied add-cube-cbrt 7.2

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

10. Applied times-frac 3.3

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

11. Simplified 3.3

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

12. Simplified 2.1

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

13. Final simplification 2.1

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

Reproduce

herbie shell --seed 2019174 
(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)))