Average Error: 28.2 → 0.2
Time: 4.9s
Precision: 64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[0.5 \cdot \left(\left(y + \left|x\right| \cdot \frac{\left|x\right|}{y}\right) - \frac{{z}^{1}}{\frac{y}{z}}\right)\]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
0.5 \cdot \left(\left(y + \left|x\right| \cdot \frac{\left|x\right|}{y}\right) - \frac{{z}^{1}}{\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 + (fabs(x) * (fabs(x) / y))) - (pow(z, 1.0) / (y / z))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

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Target

Original28.2
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 28.2

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

  2. Simplified28.2

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

  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 *-un-lft-identity12.1

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

  7. Applied add-sqr-sqrt12.1

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

  8. Applied times-frac12.1

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

  9. Simplified12.1

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

  10. Simplified6.7

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

  12. Applied sqr-pow6.7

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

  13. Applied associate-/l*0.2

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

  14. Simplified0.2

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

  15. Final simplification0.2

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

Reproduce

herbie shell --seed 2020075 +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)))