Average Error: 28.2 → 0.2
Time: 3.6s
Precision: binary64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[\left(\frac{\frac{z + x}{y}}{\frac{1}{z - x}} - y\right) \cdot -0.5\]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
\left(\frac{\frac{z + x}{y}}{\frac{1}{z - x}} - y\right) \cdot -0.5
(FPCore (x y z)
 :precision binary64
 (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)))
↓
(FPCore (x y z)
 :precision binary64
 (* (- (/ (/ (+ z x) y) (/ 1.0 (- z x))) y) -0.5))
double code(double x, double y, double z) {
	return (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
}
↓
double code(double x, double y, double z) {
	return ((((z + x) / y) / (1.0 / (z - x))) - y) * -0.5;
}

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

Original28.2
Target0.2
Herbie0.2
\[y \cdot 0.5 - \left(\frac{0.5}{y} \cdot \left(z + x\right)\right) \cdot \left(z - x\right)\]

Derivation

  1. Initial program 28.2

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

  2. Simplified12.4

    \[\leadsto \color{blue}{\left(\frac{z \cdot z - x \cdot x}{y} - y\right) \cdot -0.5}\]
  3. Using strategy rm

  4. Applied difference-of-squares_binary6412.4

    \[\leadsto \left(\frac{\color{blue}{\left(z + x\right) \cdot \left(z - x\right)}}{y} - y\right) \cdot -0.5\]

  5. Applied associate-/l*_binary640.2

    \[\leadsto \left(\color{blue}{\frac{z + x}{\frac{y}{z - x}}} - y\right) \cdot -0.5\]
  6. Using strategy rm

  7. Applied div-inv_binary640.2

    \[\leadsto \left(\frac{z + x}{\color{blue}{y \cdot \frac{1}{z - x}}} - y\right) \cdot -0.5\]

  8. Applied associate-/r*_binary640.2

    \[\leadsto \left(\color{blue}{\frac{\frac{z + x}{y}}{\frac{1}{z - x}}} - y\right) \cdot -0.5\]

  9. Final simplification0.2

    \[\leadsto \left(\frac{\frac{z + x}{y}}{\frac{1}{z - x}} - y\right) \cdot -0.5\]

Reproduce

herbie shell --seed 2020231 
(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)))