Average Error: 33.6 → 1.2
Time: 5.1s
Precision: 64
\[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}\]
↓
\[\left(\sqrt[3]{\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x}{y} \cdot \frac{x}{y}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x}{y} \cdot \frac{x}{y}\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x}{y} \cdot \frac{x}{y}\right)}\]
\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}
↓
\left(\sqrt[3]{\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x}{y} \cdot \frac{x}{y}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x}{y} \cdot \frac{x}{y}\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x}{y} \cdot \frac{x}{y}\right)}
double f(double x, double y, double z, double t) {
        double r2115086 = x;
        double r2115087 = r2115086 * r2115086;
        double r2115088 = y;
        double r2115089 = r2115088 * r2115088;
        double r2115090 = r2115087 / r2115089;
        double r2115091 = z;
        double r2115092 = r2115091 * r2115091;
        double r2115093 = t;
        double r2115094 = r2115093 * r2115093;
        double r2115095 = r2115092 / r2115094;
        double r2115096 = r2115090 + r2115095;
        return r2115096;
}
↓
double f(double x, double y, double z, double t) {
        double r2115097 = z;
        double r2115098 = t;
        double r2115099 = r2115097 / r2115098;
        double r2115100 = x;
        double r2115101 = y;
        double r2115102 = r2115100 / r2115101;
        double r2115103 = r2115102 * r2115102;
        double r2115104 = fma(r2115099, r2115099, r2115103);
        double r2115105 = cbrt(r2115104);
        double r2115106 = r2115105 * r2115105;
        double r2115107 = r2115106 * r2115105;
        return r2115107;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original33.6
Target0.4
Herbie1.2
\[{\left(\frac{x}{y}\right)}^{2} + {\left(\frac{z}{t}\right)}^{2}\]

Derivation

  1. Initial program 33.6

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

  2. Simplified18.4

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

  4. Applied times-frac0.4

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

  6. Applied add-cube-cbrt1.2

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

  7. Final simplification1.2

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

Reproduce

herbie shell --seed 2020018 +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))))