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 r429683 = x;
        double r429684 = r429683 * r429683;
        double r429685 = y;
        double r429686 = r429685 * r429685;
        double r429687 = r429684 / r429686;
        double r429688 = z;
        double r429689 = r429688 * r429688;
        double r429690 = t;
        double r429691 = r429690 * r429690;
        double r429692 = r429689 / r429691;
        double r429693 = r429687 + r429692;
        return r429693;
}
↓
double f(double x, double y, double z, double t) {
        double r429694 = x;
        double r429695 = y;
        double r429696 = r429694 / r429695;
        double r429697 = z;
        double r429698 = t;
        double r429699 = r429697 / r429698;
        double r429700 = r429698 / r429697;
        double r429701 = r429699 / r429700;
        double r429702 = fma(r429696, r429696, r429701);
        double r429703 = cbrt(r429702);
        double r429704 = r429703 * r429703;
        double r429705 = r429704 * r429703;
        return r429705;
}

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))))