Average Error: 27.5 → 0.2
Time: 16.7s
Precision: binary64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[\left(y + \frac{z + x}{-\frac{y}{z - x}}\right) \cdot 0.5\]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
\left(y + \frac{z + x}{-\frac{y}{z - x}}\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
 (* (+ y (/ (+ z x) (- (/ y (- z x))))) 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 (y + ((z + x) / -(y / (z - x)))) * 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

Original27.5
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 27.5

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

  2. Simplified12.4

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

  4. Applied difference-of-squares_binary64_1846112.4

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

  5. Applied associate-/l*_binary64_184370.2

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

  7. Applied frac-2neg_binary64_185030.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2021076 
(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)))