Average Error: 10.8 → 0.1
Time: 3.1s
Precision: binary64
\[\frac{x + y \cdot \left(z - x\right)}{z}\]
↓
\[\left(\frac{x}{z} + y\right) - \frac{\frac{x}{z}}{\frac{1}{y}}\]
\frac{x + y \cdot \left(z - x\right)}{z}
↓
\left(\frac{x}{z} + y\right) - \frac{\frac{x}{z}}{\frac{1}{y}}
double code(double x, double y, double z) {
	return ((double) (((double) (x + ((double) (y * ((double) (z - x)))))) / z));
}
↓
double code(double x, double y, double z) {
	return ((double) (((double) (((double) (x / z)) + y)) - ((double) (((double) (x / z)) / ((double) (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

Original10.8
Target0.0
Herbie0.1
\[\left(y + \frac{x}{z}\right) - \frac{y}{\frac{z}{x}}\]

Derivation

  1. Initial program 10.8

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

  2. Taylor expanded around 0 3.7

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

  4. Applied associate-/l*3.1

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

  6. Applied div-inv3.1

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

  7. Applied associate-/r*0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2020171 
(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))