Average Error: 33.7 → 0.8
Time: 13.7s
Precision: 64
\[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}\]
↓
\[\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\frac{\sqrt[3]{t}}{\frac{z}{t}}}\right)\]
\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}
↓
\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\frac{\sqrt[3]{t}}{\frac{z}{t}}}\right)
double f(double x, double y, double z, double t) {
        double r632939 = x;
        double r632940 = r632939 * r632939;
        double r632941 = y;
        double r632942 = r632941 * r632941;
        double r632943 = r632940 / r632942;
        double r632944 = z;
        double r632945 = r632944 * r632944;
        double r632946 = t;
        double r632947 = r632946 * r632946;
        double r632948 = r632945 / r632947;
        double r632949 = r632943 + r632948;
        return r632949;
}
↓
double f(double x, double y, double z, double t) {
        double r632950 = x;
        double r632951 = y;
        double r632952 = r632950 / r632951;
        double r632953 = z;
        double r632954 = t;
        double r632955 = cbrt(r632954);
        double r632956 = r632955 * r632955;
        double r632957 = r632953 / r632956;
        double r632958 = r632953 / r632954;
        double r632959 = r632955 / r632958;
        double r632960 = r632957 / r632959;
        double r632961 = fma(r632952, r632952, r632960);
        return r632961;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original33.7
Target0.4
Herbie0.8
\[{\left(\frac{x}{y}\right)}^{2} + {\left(\frac{z}{t}\right)}^{2}\]

Derivation

  1. Initial program 33.7

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

  2. Simplified18.9

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{z \cdot z}{t \cdot t}\right)}\]
  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. Simplified4.0

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

  7. Applied *-un-lft-identity4.0

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

  8. Applied *-un-lft-identity4.0

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

  9. Applied times-frac4.0

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

  10. Applied add-cube-cbrt4.4

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

  11. Applied times-frac4.4

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

  12. Applied associate-/r*0.8

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

  13. Simplified0.8

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

  14. Final simplification0.8

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

Reproduce

herbie shell --seed 2020046 +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))))