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

Average Error: 10.0 → 0.0

Time: 2.7s

Precision: binary64

Math

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

↓

\[y + \frac{x}{z} \cdot \left(1 - y\right)\]

FPCore

(FPCore (x y z) :precision binary64 (/ (+ x (* y (- z x))) z))

↓

(FPCore (x y z) :precision binary64 (+ y (* (/ x z) (- 1.0 y))))

C

double code(double x, double y, double z) {
	return (((double) (x + ((double) (y * ((double) (z - x)))))) / z);
}

↓

double code(double x, double y, double z) {
	return ((double) (y + ((double) ((x / z) * ((double) (1.0 - y))))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	10.0
Target	0.0
Herbie	0.0

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

Derivation

1. Initial program 10.0

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

2. Simplified 3.7

   \[\leadsto \color{blue}{y + x \cdot \frac{1 - y}{z}}\]
3. Using strategy rm

4. Applied add-cube-cbrt 4.2

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

5. Applied *-un-lft-identity 4.2

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

6. Applied times-frac 4.2

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

7. Applied associate-*r* 1.6

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

8. Simplified 1.6

   \[\leadsto y + \color{blue}{\frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}}} \cdot \frac{1 - y}{\sqrt[3]{z}}\]
9. Using strategy rm

10. Applied pow1 1.6

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

11. Applied pow1 1.6

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

12. Applied pow-prod-down 1.6

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

13. Simplified 0.0

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

14. Final simplification 0.0

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

Reproduce

herbie shell --seed 2020198 
(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))