Average Error: 28.1 → 0.2
Time: 4.5s
Precision: 64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[0.5 \cdot \left(\left(y + \left|x\right| \cdot \frac{\left|x\right|}{y}\right) - z \cdot \frac{z}{y}\right)\]
\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) - z \cdot \frac{z}{y}\right)
double f(double x, double y, double z) {
        double r684549 = x;
        double r684550 = r684549 * r684549;
        double r684551 = y;
        double r684552 = r684551 * r684551;
        double r684553 = r684550 + r684552;
        double r684554 = z;
        double r684555 = r684554 * r684554;
        double r684556 = r684553 - r684555;
        double r684557 = 2.0;
        double r684558 = r684551 * r684557;
        double r684559 = r684556 / r684558;
        return r684559;
}
↓
double f(double x, double y, double z) {
        double r684560 = 0.5;
        double r684561 = y;
        double r684562 = x;
        double r684563 = fabs(r684562);
        double r684564 = r684563 / r684561;
        double r684565 = r684563 * r684564;
        double r684566 = r684561 + r684565;
        double r684567 = z;
        double r684568 = r684567 / r684561;
        double r684569 = r684567 * r684568;
        double r684570 = r684566 - r684569;
        double r684571 = r684560 * r684570;
        return r684571;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

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Target

Original28.1
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 28.1

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

  2. Simplified28.1

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

  3. Taylor expanded around 0 12.6

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

  4. Simplified12.6

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

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

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

  7. Applied add-sqr-sqrt37.7

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

  8. Applied unpow-prod-down37.7

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

  9. Applied times-frac34.9

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

  10. Simplified34.8

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

  11. Simplified7.0

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

  13. Applied *-un-lft-identity7.0

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

  14. Applied add-sqr-sqrt7.0

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

  15. Applied times-frac7.0

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

  16. Simplified7.0

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

  17. Simplified0.2

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

  18. Final simplification0.2

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

Reproduce

herbie shell --seed 2020056 +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)))