Average Error: 34.2 → 1.2
Time: 22.8s
Precision: 64
\[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}\]
↓
\[\left(\left(\sqrt{\sqrt[3]{\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{\frac{z}{t}}{\frac{t}{z}}\right)}} \cdot \sqrt{\sqrt[3]{\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{\frac{z}{t}}{\frac{t}{z}}\right)}}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{\frac{z}{t}}{\frac{t}{z}}\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{\frac{z}{t}}{\frac{t}{z}}\right)}\]
\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}
↓
\left(\left(\sqrt{\sqrt[3]{\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{\frac{z}{t}}{\frac{t}{z}}\right)}} \cdot \sqrt{\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)}\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 r463852 = x;
        double r463853 = r463852 * r463852;
        double r463854 = y;
        double r463855 = r463854 * r463854;
        double r463856 = r463853 / r463855;
        double r463857 = z;
        double r463858 = r463857 * r463857;
        double r463859 = t;
        double r463860 = r463859 * r463859;
        double r463861 = r463858 / r463860;
        double r463862 = r463856 + r463861;
        return r463862;
}
↓
double f(double x, double y, double z, double t) {
        double r463863 = x;
        double r463864 = y;
        double r463865 = r463863 / r463864;
        double r463866 = z;
        double r463867 = t;
        double r463868 = r463866 / r463867;
        double r463869 = r463867 / r463866;
        double r463870 = r463868 / r463869;
        double r463871 = fma(r463865, r463865, r463870);
        double r463872 = cbrt(r463871);
        double r463873 = sqrt(r463872);
        double r463874 = r463873 * r463873;
        double r463875 = r463874 * r463872;
        double r463876 = r463875 * r463872;
        return r463876;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original34.2
Target0.4
Herbie1.2
\[{\left(\frac{x}{y}\right)}^{2} + {\left(\frac{z}{t}\right)}^{2}\]

Derivation

  1. Initial program 34.2

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

  2. Simplified19.4

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

  4. Applied associate-/l*13.2

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

  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 associate-/r*0.4

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

  9. Applied add-cube-cbrt1.2

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

  11. Applied add-sqr-sqrt1.2

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

  12. Final simplification1.2

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

Reproduce

herbie shell --seed 2019303 +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))))