Average Error: 10.1 → 0.0
Time: 2.7s
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 (((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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program Error: 10.1 bits

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

  2. SimplifiedError: 3.4 bits

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

  4. Applied add-cube-cbrtError: 3.9 bits

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

  5. Applied *-un-lft-identityError: 3.9 bits

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

  6. Applied times-fracError: 3.9 bits

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

  7. Applied associate-*r*Error: 1.4 bits

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

  8. SimplifiedError: 1.4 bits

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

  10. Applied pow1Error: 1.4 bits

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

  11. Applied pow1Error: 1.4 bits

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

  12. Applied pow-prod-downError: 1.4 bits

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

  13. SimplifiedError: 0.0 bits

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

  14. Final simplificationError: 0.0 bits

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

Reproduce

herbie shell --seed 2020203 
(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))