Average Error: 28.4 → 0.2
Time: 7.3s
Precision: 64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[\left(\mathsf{fma}\left(\frac{{x}^{\left(\frac{2}{2}\right)}}{y}, x, y\right) - \frac{z}{\frac{y}{z}}\right) \cdot 0.5\]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
\left(\mathsf{fma}\left(\frac{{x}^{\left(\frac{2}{2}\right)}}{y}, x, y\right) - \frac{z}{\frac{y}{z}}\right) \cdot 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 ((fma((pow(x, (2.0 / 2.0)) / y), x, y) - (z / (y / z))) * 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

Original28.4
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 28.4

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

  2. Taylor expanded around 0 12.8

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

  3. Simplified12.8

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

  5. Applied unpow212.8

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

  6. Applied associate-/l*6.9

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

  8. Applied sqr-pow6.9

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

  9. Applied associate-/l*0.2

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

  10. Simplified0.2

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

  12. Applied *-un-lft-identity0.2

    \[\leadsto \color{blue}{\left(1 \cdot 0.5\right)} \cdot \left(\left(y + \frac{{x}^{\left(\frac{2}{2}\right)}}{\frac{y}{x}}\right) - \frac{z}{\frac{y}{z}}\right)\]

  13. Applied associate-*l*0.2

    \[\leadsto \color{blue}{1 \cdot \left(0.5 \cdot \left(\left(y + \frac{{x}^{\left(\frac{2}{2}\right)}}{\frac{y}{x}}\right) - \frac{z}{\frac{y}{z}}\right)\right)}\]

  14. Simplified0.2

    \[\leadsto 1 \cdot \color{blue}{\left(\left(\mathsf{fma}\left(\frac{{x}^{\left(\frac{2}{2}\right)}}{y}, x, y\right) - \frac{z}{\frac{y}{z}}\right) \cdot 0.5\right)}\]

  15. Final simplification0.2

    \[\leadsto \left(\mathsf{fma}\left(\frac{{x}^{\left(\frac{2}{2}\right)}}{y}, x, y\right) - \frac{z}{\frac{y}{z}}\right) \cdot 0.5\]

Reproduce

herbie shell --seed 2020057 +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)))