Average Error: 27.7 → 0.2
Time: 5.2s
Precision: 64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[0.5 \cdot \left(\mathsf{fma}\left(\frac{x}{y}, x, y\right) - z \cdot \frac{z}{y}\right)\]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
0.5 \cdot \left(\mathsf{fma}\left(\frac{x}{y}, x, y\right) - z \cdot \frac{z}{y}\right)
double f(double x, double y, double z) {
        double r759910 = x;
        double r759911 = r759910 * r759910;
        double r759912 = y;
        double r759913 = r759912 * r759912;
        double r759914 = r759911 + r759913;
        double r759915 = z;
        double r759916 = r759915 * r759915;
        double r759917 = r759914 - r759916;
        double r759918 = 2.0;
        double r759919 = r759912 * r759918;
        double r759920 = r759917 / r759919;
        return r759920;
}
↓
double f(double x, double y, double z) {
        double r759921 = 0.5;
        double r759922 = x;
        double r759923 = y;
        double r759924 = r759922 / r759923;
        double r759925 = fma(r759924, r759922, r759923);
        double r759926 = z;
        double r759927 = r759926 / r759923;
        double r759928 = r759926 * r759927;
        double r759929 = r759925 - r759928;
        double r759930 = r759921 * r759929;
        return r759930;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original27.7
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 27.7

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

  2. Taylor expanded around 0 12.2

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

  3. Simplified12.2

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

  5. Applied unpow212.2

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

  6. Applied associate-/l*6.4

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

  8. Applied *-un-lft-identity6.4

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

  9. Applied add-sqr-sqrt35.1

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

  10. Applied unpow-prod-down35.1

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

  11. Applied times-frac31.9

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

  12. Simplified31.9

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

  13. Simplified0.2

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

  14. Taylor expanded around 0 12.2

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

  15. Simplified0.2

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

  16. Final simplification0.2

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

Reproduce

herbie shell --seed 2020100 +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)))