Average Error: 33.9 → 0.7
Time: 20.8s
Precision: 64
\[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}\]
↓
\[\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \left(\sqrt[3]{z} \cdot \sqrt[3]{\frac{1}{t}}\right) \cdot \left(\left(\sqrt[3]{\frac{z}{t}} \cdot \sqrt[3]{\frac{z}{t}}\right) \cdot \frac{z}{t}\right)\right)\]
\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}
↓
\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \left(\sqrt[3]{z} \cdot \sqrt[3]{\frac{1}{t}}\right) \cdot \left(\left(\sqrt[3]{\frac{z}{t}} \cdot \sqrt[3]{\frac{z}{t}}\right) \cdot \frac{z}{t}\right)\right)
double f(double x, double y, double z, double t) {
        double r23489798 = x;
        double r23489799 = r23489798 * r23489798;
        double r23489800 = y;
        double r23489801 = r23489800 * r23489800;
        double r23489802 = r23489799 / r23489801;
        double r23489803 = z;
        double r23489804 = r23489803 * r23489803;
        double r23489805 = t;
        double r23489806 = r23489805 * r23489805;
        double r23489807 = r23489804 / r23489806;
        double r23489808 = r23489802 + r23489807;
        return r23489808;
}
↓
double f(double x, double y, double z, double t) {
        double r23489809 = x;
        double r23489810 = y;
        double r23489811 = r23489809 / r23489810;
        double r23489812 = z;
        double r23489813 = cbrt(r23489812);
        double r23489814 = 1.0;
        double r23489815 = t;
        double r23489816 = r23489814 / r23489815;
        double r23489817 = cbrt(r23489816);
        double r23489818 = r23489813 * r23489817;
        double r23489819 = r23489812 / r23489815;
        double r23489820 = cbrt(r23489819);
        double r23489821 = r23489820 * r23489820;
        double r23489822 = r23489821 * r23489819;
        double r23489823 = r23489818 * r23489822;
        double r23489824 = fma(r23489811, r23489811, r23489823);
        return r23489824;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original33.9
Target0.4
Herbie0.7
\[{\left(\frac{x}{y}\right)}^{2} + {\left(\frac{z}{t}\right)}^{2}\]

Derivation

  1. Initial program 33.9

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

  2. Simplified0.4

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

  4. Applied add-cube-cbrt0.8

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

  5. Applied associate-*r*0.8

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

  7. Applied div-inv0.8

    \[\leadsto \mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \left(\frac{z}{t} \cdot \left(\sqrt[3]{\frac{z}{t}} \cdot \sqrt[3]{\frac{z}{t}}\right)\right) \cdot \sqrt[3]{\color{blue}{z \cdot \frac{1}{t}}}\right)\]

  8. Applied cbrt-prod0.7

    \[\leadsto \mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \left(\frac{z}{t} \cdot \left(\sqrt[3]{\frac{z}{t}} \cdot \sqrt[3]{\frac{z}{t}}\right)\right) \cdot \color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{\frac{1}{t}}\right)}\right)\]

  9. Final simplification0.7

    \[\leadsto \mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \left(\sqrt[3]{z} \cdot \sqrt[3]{\frac{1}{t}}\right) \cdot \left(\left(\sqrt[3]{\frac{z}{t}} \cdot \sqrt[3]{\frac{z}{t}}\right) \cdot \frac{z}{t}\right)\right)\]

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rasterific.Svg.PathConverter:arcToSegments from rasterific-svg-0.2.3.1"

  :herbie-target
  (+ (pow (/ x y) 2.0) (pow (/ z t) 2.0))

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