Average Error: 33.5 → 0.5
Time: 3.6s
Precision: 64
\[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}\]
↓
\[\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \sqrt[3]{{\left(\left|\frac{x}{y}\right|\right)}^{\frac{3}{2}}} \cdot {\left(\left|\frac{x}{y}\right|\right)}^{\frac{3}{2}}\right)\]
\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}
↓
\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \sqrt[3]{{\left(\left|\frac{x}{y}\right|\right)}^{\frac{3}{2}}} \cdot {\left(\left|\frac{x}{y}\right|\right)}^{\frac{3}{2}}\right)
double f(double x, double y, double z, double t) {
        double r618881 = x;
        double r618882 = r618881 * r618881;
        double r618883 = y;
        double r618884 = r618883 * r618883;
        double r618885 = r618882 / r618884;
        double r618886 = z;
        double r618887 = r618886 * r618886;
        double r618888 = t;
        double r618889 = r618888 * r618888;
        double r618890 = r618887 / r618889;
        double r618891 = r618885 + r618890;
        return r618891;
}
↓
double f(double x, double y, double z, double t) {
        double r618892 = z;
        double r618893 = t;
        double r618894 = r618892 / r618893;
        double r618895 = x;
        double r618896 = y;
        double r618897 = r618895 / r618896;
        double r618898 = fabs(r618897);
        double r618899 = 1.5;
        double r618900 = pow(r618898, r618899);
        double r618901 = cbrt(r618900);
        double r618902 = r618901 * r618900;
        double r618903 = fma(r618894, r618894, r618902);
        return r618903;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original33.5
Target0.4
Herbie0.5
\[{\left(\frac{x}{y}\right)}^{2} + {\left(\frac{z}{t}\right)}^{2}\]

Derivation

  1. Initial program 33.5

    \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}\]

  2. Simplified18.5

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

  4. Applied add-sqr-sqrt18.5

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

  5. Simplified18.5

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

  6. Simplified0.4

    \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \left|\frac{x}{y}\right| \cdot \color{blue}{\left|\frac{x}{y}\right|}\right)\]
  7. Using strategy rm

  8. Applied add-sqr-sqrt0.5

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

  9. Applied associate-*l*0.5

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

  10. Simplified0.5

    \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \sqrt{\left|\frac{x}{y}\right|} \cdot \color{blue}{{\left(\left|\frac{x}{y}\right|\right)}^{\frac{3}{2}}}\right)\]
  11. Using strategy rm

  12. Applied add-cbrt-cube0.5

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

  13. Simplified0.5

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

  14. Final simplification0.5

    \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \sqrt[3]{{\left(\left|\frac{x}{y}\right|\right)}^{\frac{3}{2}}} \cdot {\left(\left|\frac{x}{y}\right|\right)}^{\frac{3}{2}}\right)\]

Reproduce

herbie shell --seed 2019353 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rasterific.Svg.PathConverter:arcToSegments from rasterific-svg-0.2.3.1"
  :precision binary64

  :herbie-target
  (+ (pow (/ x y) 2) (pow (/ z t) 2))

  (+ (/ (* x x) (* y y)) (/ (* z z) (* t t))))