Average Error: 28.4 → 0.2
Time: 13.1s
Precision: 64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[\frac{1}{2} \cdot \left(\left(y + x \cdot \frac{x}{y}\right) - \left|z\right| \cdot \frac{\left|z\right|}{y}\right)\]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
\frac{1}{2} \cdot \left(\left(y + x \cdot \frac{x}{y}\right) - \left|z\right| \cdot \frac{\left|z\right|}{y}\right)
double f(double x, double y, double z) {
        double r469627 = x;
        double r469628 = r469627 * r469627;
        double r469629 = y;
        double r469630 = r469629 * r469629;
        double r469631 = r469628 + r469630;
        double r469632 = z;
        double r469633 = r469632 * r469632;
        double r469634 = r469631 - r469633;
        double r469635 = 2.0;
        double r469636 = r469629 * r469635;
        double r469637 = r469634 / r469636;
        return r469637;
}
↓
double f(double x, double y, double z) {
        double r469638 = 1.0;
        double r469639 = 2.0;
        double r469640 = r469638 / r469639;
        double r469641 = y;
        double r469642 = x;
        double r469643 = r469642 / r469641;
        double r469644 = r469642 * r469643;
        double r469645 = r469641 + r469644;
        double r469646 = z;
        double r469647 = fabs(r469646);
        double r469648 = r469647 / r469641;
        double r469649 = r469647 * r469648;
        double r469650 = r469645 - r469649;
        double r469651 = r469640 * r469650;
        return r469651;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

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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 12.6

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

  3. Simplified12.6

    \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(y + \frac{{x}^{2}}{y}\right) - \frac{{z}^{2}}{y}\right)}\]
  4. Using strategy rm

  5. Applied *-un-lft-identity12.6

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

  6. Applied add-sqr-sqrt12.6

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

  7. Applied times-frac12.6

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

  8. Simplified12.6

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

  9. Simplified7.3

    \[\leadsto \frac{1}{2} \cdot \left(\left(y + \frac{{x}^{2}}{y}\right) - \left|z\right| \cdot \color{blue}{\frac{\left|z\right|}{y}}\right)\]
  10. Using strategy rm

  11. Applied *-un-lft-identity7.3

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

  12. Applied add-sqr-sqrt36.0

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

  13. Applied unpow-prod-down36.0

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

  14. Applied times-frac32.6

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

  15. Simplified32.6

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

  16. Simplified0.2

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

  17. Final simplification0.2

    \[\leadsto \frac{1}{2} \cdot \left(\left(y + x \cdot \frac{x}{y}\right) - \left|z\right| \cdot \frac{\left|z\right|}{y}\right)\]

Reproduce

herbie shell --seed 2019303 
(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)))