Average Error: 29.1 → 0.2
Time: 9.0s
Precision: binary64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
↓
\[-0.5 \cdot \left(\left(x + z\right) \cdot \frac{z - x}{y} - y\right) \]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
-0.5 \cdot \left(\left(x + z\right) \cdot \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 (- (* (+ x z) (/ (- 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 * (((x + z) * ((z - x) / y)) - y);
}

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

Original29.1
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 29.1

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

  2. Simplified12.9

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

  3. Taylor expanded in z around 0 12.9

    \[\leadsto -0.5 \cdot \left(\color{blue}{\left(\frac{{z}^{2}}{y} - \frac{{x}^{2}}{y}\right)} - y\right) \]

  4. Simplified0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2022067 
(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)))