Average Error: 28.0 → 0.2
Time: 2.5s
Precision: 64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[0.5 \cdot \left(\left(y + \frac{{x}^{1}}{-y} \cdot \left(-x\right)\right) - \frac{z}{\frac{y}{z}}\right)\]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
0.5 \cdot \left(\left(y + \frac{{x}^{1}}{-y} \cdot \left(-x\right)\right) - \frac{z}{\frac{y}{z}}\right)
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 * ((y + ((pow(x, 1.0) / -y) * -x)) - (z / (y / z))));
}

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

Original28.0
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 28.0

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

  2. Simplified28.0

    \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{2}}{y}}\]

  3. Taylor expanded around 0 12.1

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

  4. Simplified12.1

    \[\leadsto \color{blue}{0.5 \cdot \left(\left(y + \frac{{x}^{2}}{y}\right) - \frac{{z}^{2}}{y}\right)}\]
  5. Using strategy rm

  6. Applied unpow212.1

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

  7. Applied associate-/l*6.6

    \[\leadsto 0.5 \cdot \left(\left(y + \frac{{x}^{2}}{y}\right) - \color{blue}{\frac{z}{\frac{y}{z}}}\right)\]
  8. Using strategy rm

  9. Applied sqr-pow6.6

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

  10. Applied associate-/l*0.2

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

  11. Simplified0.2

    \[\leadsto 0.5 \cdot \left(\left(y + \frac{{x}^{\left(\frac{2}{2}\right)}}{\color{blue}{\frac{y}{x}}}\right) - \frac{z}{\frac{y}{z}}\right)\]
  12. Using strategy rm

  13. Applied frac-2neg0.2

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

  14. Applied associate-/r/0.2

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

  15. Simplified0.2

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

  16. Final simplification0.2

    \[\leadsto 0.5 \cdot \left(\left(y + \frac{{x}^{1}}{-y} \cdot \left(-x\right)\right) - \frac{z}{\frac{y}{z}}\right)\]

Reproduce

herbie shell --seed 2020071 +o rules:numerics
(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)))