Average Error: 33.1 → 0.6
Time: 20.0s
Precision: 64
\[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}\]
↓
\[\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \left(\sqrt[3]{\frac{z}{t}} \cdot \frac{z}{t}\right) \cdot \sqrt[3]{\frac{z}{t} \cdot \frac{z}{t}}\right)\]
\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}
↓
\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \left(\sqrt[3]{\frac{z}{t}} \cdot \frac{z}{t}\right) \cdot \sqrt[3]{\frac{z}{t} \cdot \frac{z}{t}}\right)
double f(double x, double y, double z, double t) {
        double r15817185 = x;
        double r15817186 = r15817185 * r15817185;
        double r15817187 = y;
        double r15817188 = r15817187 * r15817187;
        double r15817189 = r15817186 / r15817188;
        double r15817190 = z;
        double r15817191 = r15817190 * r15817190;
        double r15817192 = t;
        double r15817193 = r15817192 * r15817192;
        double r15817194 = r15817191 / r15817193;
        double r15817195 = r15817189 + r15817194;
        return r15817195;
}
↓
double f(double x, double y, double z, double t) {
        double r15817196 = x;
        double r15817197 = y;
        double r15817198 = r15817196 / r15817197;
        double r15817199 = z;
        double r15817200 = t;
        double r15817201 = r15817199 / r15817200;
        double r15817202 = cbrt(r15817201);
        double r15817203 = r15817202 * r15817201;
        double r15817204 = r15817201 * r15817201;
        double r15817205 = cbrt(r15817204);
        double r15817206 = r15817203 * r15817205;
        double r15817207 = fma(r15817198, r15817198, r15817206);
        return r15817207;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original33.1
Target0.4
Herbie0.6
\[{\left(\frac{x}{y}\right)}^{2} + {\left(\frac{z}{t}\right)}^{2}\]

Derivation

  1. Initial program 33.1

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

  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

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

  5. Applied associate-*l*0.8

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

  7. Applied cbrt-unprod0.6

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

  8. Final simplification0.6

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

Reproduce

herbie shell --seed 2019192 +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) 2.0) (pow (/ z t) 2.0))

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