Average Error: 10.4 → 0.0
Time: 2.4s
Precision: binary64
\[\frac{x + y \cdot \left(z - x\right)}{z}\]
↓
\[y + \frac{x}{z} \cdot \left(1 - y\right)\]
\frac{x + y \cdot \left(z - x\right)}{z}
↓
y + \frac{x}{z} \cdot \left(1 - y\right)
(FPCore (x y z) :precision binary64 (/ (+ x (* y (- z x))) z))
↓
(FPCore (x y z) :precision binary64 (+ y (* (/ x z) (- 1.0 y))))
double code(double x, double y, double z) {
	return (x + (y * (z - x))) / z;
}
↓
double code(double x, double y, double z) {
	return y + ((x / z) * (1.0 - y));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original10.4
Target0.0
Herbie0.0
\[\left(y + \frac{x}{z}\right) - \frac{y}{\frac{z}{x}}\]

Derivation

  1. Initial program 10.4

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

  3. Applied *-un-lft-identity_binary6410.4

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

  4. Applied add-cube-cbrt_binary6411.4

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

  5. Applied times-frac_binary6411.4

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

  6. Simplified11.4

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

  7. Taylor expanded around 0 3.5

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

  8. Simplified0.0

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

  10. Applied associate--l+_binary640.0

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

  11. Simplified0.0

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

  12. Final simplification0.0

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

Reproduce

herbie shell --seed 2020220 
(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))