Average Error: 28.4 → 0.1
Time: 20.2s
Precision: 64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[\frac{y + \left(\frac{x + z}{\frac{y}{x - z}} + \frac{\mathsf{fma}\left(z, -1, z\right)}{\frac{y}{x + z}}\right)}{2}\]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
\frac{y + \left(\frac{x + z}{\frac{y}{x - z}} + \frac{\mathsf{fma}\left(z, -1, z\right)}{\frac{y}{x + z}}\right)}{2}
double f(double x, double y, double z) {
        double r496707 = x;
        double r496708 = r496707 * r496707;
        double r496709 = y;
        double r496710 = r496709 * r496709;
        double r496711 = r496708 + r496710;
        double r496712 = z;
        double r496713 = r496712 * r496712;
        double r496714 = r496711 - r496713;
        double r496715 = 2.0;
        double r496716 = r496709 * r496715;
        double r496717 = r496714 / r496716;
        return r496717;
}
↓
double f(double x, double y, double z) {
        double r496718 = y;
        double r496719 = x;
        double r496720 = z;
        double r496721 = r496719 + r496720;
        double r496722 = r496719 - r496720;
        double r496723 = r496718 / r496722;
        double r496724 = r496721 / r496723;
        double r496725 = -1.0;
        double r496726 = fma(r496720, r496725, r496720);
        double r496727 = r496718 / r496721;
        double r496728 = r496726 / r496727;
        double r496729 = r496724 + r496728;
        double r496730 = r496718 + r496729;
        double r496731 = 2.0;
        double r496732 = r496730 / r496731;
        return r496732;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original28.4
Target0.2
Herbie0.1
\[y \cdot 0.5 - \left(\frac{0.5}{y} \cdot \left(z + x\right)\right) \cdot \left(z - x\right)\]

Derivation

  1. Initial program 28.4

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

  2. Simplified0.2

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{x + z}{y}, x - z, y\right)}{2}}\]
  3. Using strategy rm

  4. Applied fma-udef0.2

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

  6. Applied add-cube-cbrt0.4

    \[\leadsto \frac{\frac{x + z}{y} \cdot \left(x - \color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}\right) + y}{2}\]

  7. Applied add-cube-cbrt0.6

    \[\leadsto \frac{\frac{x + z}{y} \cdot \left(\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}} - \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}\right) + y}{2}\]

  8. Applied prod-diff0.6

    \[\leadsto \frac{\frac{x + z}{y} \cdot \color{blue}{\left(\mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x}, -\sqrt[3]{z} \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{z}, \sqrt[3]{z} \cdot \sqrt[3]{z}, \sqrt[3]{z} \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right)\right)} + y}{2}\]

  9. Applied distribute-lft-in0.6

    \[\leadsto \frac{\color{blue}{\left(\frac{x + z}{y} \cdot \mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x}, -\sqrt[3]{z} \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) + \frac{x + z}{y} \cdot \mathsf{fma}\left(-\sqrt[3]{z}, \sqrt[3]{z} \cdot \sqrt[3]{z}, \sqrt[3]{z} \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right)\right)} + y}{2}\]

  10. Simplified0.4

    \[\leadsto \frac{\left(\color{blue}{\frac{x + z}{\frac{y}{{\left(\sqrt[3]{x}\right)}^{3} - z}}} + \frac{x + z}{y} \cdot \mathsf{fma}\left(-\sqrt[3]{z}, \sqrt[3]{z} \cdot \sqrt[3]{z}, \sqrt[3]{z} \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right)\right) + y}{2}\]

  11. Simplified0.4

    \[\leadsto \frac{\left(\frac{x + z}{\frac{y}{{\left(\sqrt[3]{x}\right)}^{3} - z}} + \color{blue}{\frac{\mathsf{fma}\left(z, -1, z\right)}{\frac{y}{x + z}}}\right) + y}{2}\]
  12. Using strategy rm

  13. Applied pow1/332.5

    \[\leadsto \frac{\left(\frac{x + z}{\frac{y}{{\color{blue}{\left({x}^{\frac{1}{3}}\right)}}^{3} - z}} + \frac{\mathsf{fma}\left(z, -1, z\right)}{\frac{y}{x + z}}\right) + y}{2}\]

  14. Applied pow-pow0.1

    \[\leadsto \frac{\left(\frac{x + z}{\frac{y}{\color{blue}{{x}^{\left(\frac{1}{3} \cdot 3\right)}} - z}} + \frac{\mathsf{fma}\left(z, -1, z\right)}{\frac{y}{x + z}}\right) + y}{2}\]

  15. Simplified0.1

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

  16. Final simplification0.1

    \[\leadsto \frac{y + \left(\frac{x + z}{\frac{y}{x - z}} + \frac{\mathsf{fma}\left(z, -1, z\right)}{\frac{y}{x + z}}\right)}{2}\]

Reproduce

herbie shell --seed 2019306 +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)))