Average Error: 28.2 → 0.2
Time: 13.2s
Precision: binary64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[\frac{\frac{x}{\frac{y}{x}} + \left(y - \frac{\frac{z}{y}}{\frac{1}{z}}\right)}{2}\]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
\frac{\frac{x}{\frac{y}{x}} + \left(y - \frac{\frac{z}{y}}{\frac{1}{z}}\right)}{2}
(FPCore (x y z)
 :precision binary64
 (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)))
↓
(FPCore (x y z)
 :precision binary64
 (/ (+ (/ x (/ y x)) (- y (/ (/ z y) (/ 1.0 z)))) 2.0))
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 ((x / (y / x)) + (y - ((z / y) / (1.0 / z)))) / 2.0;
}

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. Simplified12.1

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

  3. Taylor expanded around 0 12.1

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

  4. Simplified0.1

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

  6. Applied div-inv_binary64_133740.2

    \[\leadsto \frac{\frac{x}{\frac{y}{x}} + \left(y - \frac{z}{\color{blue}{y \cdot \frac{1}{z}}}\right)}{2}\]

  7. Applied associate-/r*_binary64_133210.2

    \[\leadsto \frac{\frac{x}{\frac{y}{x}} + \left(y - \color{blue}{\frac{\frac{z}{y}}{\frac{1}{z}}}\right)}{2}\]

  8. Final simplification0.2

    \[\leadsto \frac{\frac{x}{\frac{y}{x}} + \left(y - \frac{\frac{z}{y}}{\frac{1}{z}}\right)}{2}\]

Reproduce

herbie shell --seed 2021176 
(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)))