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

Average Error: 14.4 → 2.6

Time: 46.2s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r24383651 = x;
        double r24383652 = y;
        double r24383653 = z;
        double r24383654 = r24383652 / r24383653;
        double r24383655 = t;
        double r24383656 = r24383654 * r24383655;
        double r24383657 = r24383656 / r24383655;
        double r24383658 = r24383651 * r24383657;
        return r24383658;
}

↓

double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r24383659 = x;
        double r24383660 = cbrt(r24383659);
        double r24383661 = cbrt(r24383660);
        double r24383662 = z;
        double r24383663 = cbrt(r24383662);
        double r24383664 = r24383661 / r24383663;
        double r24383665 = r24383660 * r24383660;
        double r24383666 = r24383665 / r24383663;
        double r24383667 = cbrt(r24383665);
        double r24383668 = y;
        double r24383669 = r24383667 * r24383668;
        double r24383670 = r24383669 / r24383663;
        double r24383671 = r24383666 * r24383670;
        double r24383672 = r24383664 * r24383671;
        return r24383672;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	14.4
Target	1.4
Herbie	2.6

\[\begin{array}{l} \mathbf{if}\;\frac{\frac{y}{z} \cdot t}{t} \lt -1.20672205123045 \cdot 10^{+245}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \lt -5.907522236933906 \cdot 10^{-275}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \lt 5.658954423153415 \cdot 10^{-65}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \lt 2.0087180502407133 \cdot 10^{+217}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array}\]

Derivation

1. Initial program 14.4

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

2. Simplified 6.2

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

4. Applied *-un-lft-identity 6.2

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

5. Applied add-cube-cbrt 7.0

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

6. Applied times-frac 7.0

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

7. Applied associate-*r* 5.6

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

8. Simplified 5.6

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

10. Applied add-cube-cbrt 5.8

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

11. Applied add-cube-cbrt 5.8

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

12. Applied cbrt-prod 5.9

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

13. Applied times-frac 5.9

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

14. Applied associate-*r* 4.7

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

15. Simplified 2.6

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

16. Final simplification 2.6

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

Reproduce

herbie shell --seed 2019165 +o rules:numerics
(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)))