Average Error: 27.6 → 0.1
Time: 4.4s
Precision: binary64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[\frac{y}{2} - \frac{z - x}{y} \cdot \frac{z + x}{2}\]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
\frac{y}{2} - \frac{z - x}{y} \cdot \frac{z + x}{2}
double code(double x, double y, double z) {
	return ((double) (((double) (((double) (((double) (x * x)) + ((double) (y * y)))) - ((double) (z * z)))) / ((double) (y * 2.0))));
}
↓
double code(double x, double y, double z) {
	return ((double) (((double) (y / 2.0)) - ((double) (((double) (((double) (z - x)) / y)) * ((double) (((double) (z + x)) / 2.0))))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

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Target

Original27.6
Target0.2
Herbie0.1
\[y \cdot 0.5 - \left(\frac{0.5}{y} \cdot \left(z + x\right)\right) \cdot \left(z - x\right)\]

Derivation

  1. Initial program 27.6

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

  2. Simplified0.1

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

  4. Applied div-sub0.1

    \[\leadsto \color{blue}{\frac{y}{2} - \frac{\left(x + z\right) \cdot \frac{z - x}{y}}{2}}\]

  5. Simplified0.1

    \[\leadsto \frac{y}{2} - \color{blue}{\frac{z - x}{y} \cdot \frac{x + z}{2}}\]

  6. Final simplification0.1

    \[\leadsto \frac{y}{2} - \frac{z - x}{y} \cdot \frac{z + x}{2}\]

Reproduce

herbie shell --seed 2020181 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A"
  :precision binary64

  :herbie-target
  (- (* y 0.5) (* (* (/ 0.5 y) (+ z x)) (- z x)))

  (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)))