Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3

Average Error: 10.2 → 0.8

Time: 15.8s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

\[\frac{x + y \cdot \left(z - x\right)}{z}\]

↓

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

C

double f(double x, double y, double z) {
        double r896742 = x;
        double r896743 = y;
        double r896744 = z;
        double r896745 = r896744 - r896742;
        double r896746 = r896743 * r896745;
        double r896747 = r896742 + r896746;
        double r896748 = r896747 / r896744;
        return r896748;
}

↓

double f(double x, double y, double z) {
        double r896749 = x;
        double r896750 = z;
        double r896751 = r896749 / r896750;
        double r896752 = y;
        double r896753 = r896751 + r896752;
        double r896754 = r896752 * r896751;
        double r896755 = r896753 - r896754;
        double r896756 = cbrt(r896750);
        double r896757 = r896756 * r896756;
        double r896758 = r896749 / r896757;
        double r896759 = r896752 / r896756;
        double r896760 = -r896759;
        double r896761 = r896760 + r896759;
        double r896762 = r896758 * r896761;
        double r896763 = r896755 + r896762;
        return r896763;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	10.2
Target	0.0
Herbie	0.8

\[\left(y + \frac{x}{z}\right) - \frac{y}{\frac{z}{x}}\]

Derivation

1. Initial program 10.2

   \[\frac{x + y \cdot \left(z - x\right)}{z}\]

2. Simplified 10.2

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z - x, y, x\right)}{z}}\]

3. Taylor expanded around 0 3.4

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

5. Applied add-cube-cbrt 3.5

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

6. Applied times-frac 0.9

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

7. Applied add-sqr-sqrt 33.3

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

8. Applied prod-diff 33.3

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{x}{z} + y}, \sqrt{\frac{x}{z} + y}, -\frac{y}{\sqrt[3]{z}} \cdot \frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) + \mathsf{fma}\left(-\frac{y}{\sqrt[3]{z}}, \frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}}, \frac{y}{\sqrt[3]{z}} \cdot \frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right)}\]

9. Simplified 0.9

   \[\leadsto \color{blue}{\left(\left(\frac{x}{z} + y\right) - \frac{y}{\sqrt[3]{z}} \cdot \frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right)} + \mathsf{fma}\left(-\frac{y}{\sqrt[3]{z}}, \frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}}, \frac{y}{\sqrt[3]{z}} \cdot \frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right)\]

10. Simplified 0.9

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

12. Applied div-inv 1.0

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

13. Applied associate-*l* 0.9

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

14. Simplified 0.8

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

15. Final simplification 0.8

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

Reproduce

herbie shell --seed 2020043 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3"
  :precision binary64

  :herbie-target
  (- (+ y (/ x z)) (/ y (/ z x)))

  (/ (+ x (* y (- z x))) z))