Average Error: 28.3 → 0.2
Time: 7.9s
Precision: binary64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
↓
\[-0.5 \cdot \mathsf{fma}\left(z + x, \frac{z - x}{y}, -y\right) \]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
-0.5 \cdot \mathsf{fma}\left(z + x, \frac{z - x}{y}, -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) (/ (- z x) y) (- 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), ((z - x) / y), -y);
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original28.3
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 28.3

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

  2. Simplified12.4

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

  3. Applied egg-rr0.2

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

  4. Taylor expanded in y around -inf 0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2022130 
(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)))