Average Error: 28.6 → 0.2
Time: 3.5s
Precision: 64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[0.5 \cdot \left(\left(y + \frac{{x}^{1}}{\frac{y}{x}}\right) - z \cdot \frac{z}{y}\right)\]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
0.5 \cdot \left(\left(y + \frac{{x}^{1}}{\frac{y}{x}}\right) - z \cdot \frac{z}{y}\right)
double f(double x, double y, double z) {
        double r571858 = x;
        double r571859 = r571858 * r571858;
        double r571860 = y;
        double r571861 = r571860 * r571860;
        double r571862 = r571859 + r571861;
        double r571863 = z;
        double r571864 = r571863 * r571863;
        double r571865 = r571862 - r571864;
        double r571866 = 2.0;
        double r571867 = r571860 * r571866;
        double r571868 = r571865 / r571867;
        return r571868;
}
↓
double f(double x, double y, double z) {
        double r571869 = 0.5;
        double r571870 = y;
        double r571871 = x;
        double r571872 = 1.0;
        double r571873 = pow(r571871, r571872);
        double r571874 = r571870 / r571871;
        double r571875 = r571873 / r571874;
        double r571876 = r571870 + r571875;
        double r571877 = z;
        double r571878 = r571877 / r571870;
        double r571879 = r571877 * r571878;
        double r571880 = r571876 - r571879;
        double r571881 = r571869 * r571880;
        return r571881;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

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Target

Original28.6
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 28.6

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

  2. Taylor expanded around 0 12.5

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

  3. Simplified12.5

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

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

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

  6. Applied add-sqr-sqrt38.7

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

  7. Applied unpow-prod-down38.7

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

  8. Applied times-frac36.0

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

  9. Simplified36.0

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

  10. Simplified6.7

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

  12. Applied sqr-pow6.7

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

  13. Applied associate-/l*0.2

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

  14. Simplified0.2

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

  15. Final simplification0.2

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

Reproduce

herbie shell --seed 2020001 +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)))