Average Error: 32.2 → 0.8
Time: 20.5s
Precision: 64
\[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}\]
↓
\[\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \left(\frac{z}{t} \cdot \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right) \cdot \sqrt[3]{\frac{z}{t}}\right)\]
\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}
↓
\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \left(\frac{z}{t} \cdot \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right) \cdot \sqrt[3]{\frac{z}{t}}\right)
double f(double x, double y, double z, double t) {
        double r26329169 = x;
        double r26329170 = r26329169 * r26329169;
        double r26329171 = y;
        double r26329172 = r26329171 * r26329171;
        double r26329173 = r26329170 / r26329172;
        double r26329174 = z;
        double r26329175 = r26329174 * r26329174;
        double r26329176 = t;
        double r26329177 = r26329176 * r26329176;
        double r26329178 = r26329175 / r26329177;
        double r26329179 = r26329173 + r26329178;
        return r26329179;
}
↓
double f(double x, double y, double z, double t) {
        double r26329180 = x;
        double r26329181 = y;
        double r26329182 = r26329180 / r26329181;
        double r26329183 = z;
        double r26329184 = t;
        double r26329185 = r26329183 / r26329184;
        double r26329186 = cbrt(r26329183);
        double r26329187 = r26329186 * r26329186;
        double r26329188 = cbrt(r26329184);
        double r26329189 = r26329188 * r26329188;
        double r26329190 = r26329187 / r26329189;
        double r26329191 = r26329185 * r26329190;
        double r26329192 = cbrt(r26329185);
        double r26329193 = r26329191 * r26329192;
        double r26329194 = fma(r26329182, r26329182, r26329193);
        return r26329194;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original32.2
Target0.4
Herbie0.8
\[{\left(\frac{x}{y}\right)}^{2} + {\left(\frac{z}{t}\right)}^{2}\]

Derivation

  1. Initial program 32.2

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

  2. Simplified0.4

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

  4. Applied add-cube-cbrt0.8

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

  5. Applied associate-*r*0.8

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

  7. Applied cbrt-div0.7

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

  8. Applied cbrt-div0.8

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

  9. Applied frac-times0.8

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

  10. Final simplification0.8

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

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rasterific.Svg.PathConverter:arcToSegments from rasterific-svg-0.2.3.1"

  :herbie-target
  (+ (pow (/ x y) 2) (pow (/ z t) 2))

  (+ (/ (* x x) (* y y)) (/ (* z z) (* t t))))