Average Error: 33.7 → 1.2
Time: 21.6s
Precision: 64
\[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}\]
↓
\[\left(\sqrt[3]{\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{\frac{z}{t}}{\frac{t}{z}}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{\frac{z}{t}}{\frac{t}{z}}\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{\frac{z}{t}}{\frac{t}{z}}\right)}\]
\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}
↓
\left(\sqrt[3]{\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{\frac{z}{t}}{\frac{t}{z}}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{\frac{z}{t}}{\frac{t}{z}}\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{\frac{z}{t}}{\frac{t}{z}}\right)}
double f(double x, double y, double z, double t) {
        double r466899 = x;
        double r466900 = r466899 * r466899;
        double r466901 = y;
        double r466902 = r466901 * r466901;
        double r466903 = r466900 / r466902;
        double r466904 = z;
        double r466905 = r466904 * r466904;
        double r466906 = t;
        double r466907 = r466906 * r466906;
        double r466908 = r466905 / r466907;
        double r466909 = r466903 + r466908;
        return r466909;
}
↓
double f(double x, double y, double z, double t) {
        double r466910 = x;
        double r466911 = y;
        double r466912 = r466910 / r466911;
        double r466913 = z;
        double r466914 = t;
        double r466915 = r466913 / r466914;
        double r466916 = r466914 / r466913;
        double r466917 = r466915 / r466916;
        double r466918 = fma(r466912, r466912, r466917);
        double r466919 = cbrt(r466918);
        double r466920 = r466919 * r466919;
        double r466921 = r466920 * r466919;
        return r466921;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original33.7
Target0.4
Herbie1.2
\[{\left(\frac{x}{y}\right)}^{2} + {\left(\frac{z}{t}\right)}^{2}\]

Derivation

  1. Initial program 33.7

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

  2. Simplified18.8

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

  4. Applied associate-/l*13.0

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

  5. Simplified3.9

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

  7. Applied *-un-lft-identity3.9

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

  8. Applied times-frac4.1

    \[\leadsto \mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \color{blue}{\frac{1}{t} \cdot \frac{z}{\frac{t}{z}}}\right)\]
  9. Using strategy rm

  10. Applied add-cube-cbrt4.8

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

  11. Simplified4.8

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

  12. Simplified1.2

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

  13. Final simplification1.2

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

Reproduce

herbie shell --seed 2019326 +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))))