Average Error: 27.9 → 0.1
Time: 5.9s
Precision: binary64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
↓
\[-0.5 \cdot \mathsf{fma}\left(\frac{z + x}{y}, z - x, -y\right) \]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
-0.5 \cdot \mathsf{fma}\left(\frac{z + x}{y}, z - x, -y\right)

(FPCore (x y z)
 :precision binary64
 (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)))
↓
(FPCore (x y z) :precision binary64 (* -0.5 (fma (/ (+ z x) y) (- z x) (- y))))
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 -0.5 * fma(((z + x) / y), (z - x), -y);
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original27.9
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 27.9

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

  2. Simplified12.7

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

  3. Applied difference-of-squares_binary6412.7

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

  4. Applied associate-/l*_binary640.1

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

  5. Applied associate-/r/_binary640.1

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

  6. Applied fma-neg_binary640.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2022125 
(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)))