Average Error: 31.9 → 0.9
Time: 22.3s
Precision: 64
\[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}\]
↓
\[\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}} \cdot \frac{x}{y}\right)\right)\]
\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}
↓
\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}} \cdot \frac{x}{y}\right)\right)
double f(double x, double y, double z, double t) {
        double r30076478 = x;
        double r30076479 = r30076478 * r30076478;
        double r30076480 = y;
        double r30076481 = r30076480 * r30076480;
        double r30076482 = r30076479 / r30076481;
        double r30076483 = z;
        double r30076484 = r30076483 * r30076483;
        double r30076485 = t;
        double r30076486 = r30076485 * r30076485;
        double r30076487 = r30076484 / r30076486;
        double r30076488 = r30076482 + r30076487;
        return r30076488;
}
↓
double f(double x, double y, double z, double t) {
        double r30076489 = z;
        double r30076490 = t;
        double r30076491 = r30076489 / r30076490;
        double r30076492 = x;
        double r30076493 = cbrt(r30076492);
        double r30076494 = r30076493 * r30076493;
        double r30076495 = y;
        double r30076496 = cbrt(r30076495);
        double r30076497 = r30076496 * r30076496;
        double r30076498 = r30076494 / r30076497;
        double r30076499 = r30076493 / r30076496;
        double r30076500 = r30076492 / r30076495;
        double r30076501 = r30076499 * r30076500;
        double r30076502 = r30076498 * r30076501;
        double r30076503 = fma(r30076491, r30076491, r30076502);
        return r30076503;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original31.9
Target0.4
Herbie0.9
\[{\left(\frac{x}{y}\right)}^{2} + {\left(\frac{z}{t}\right)}^{2}\]

Derivation

  1. Initial program 31.9

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

  2. Simplified0.4

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

  4. Applied add-cube-cbrt0.8

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

  5. Applied add-cube-cbrt0.9

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

  6. Applied times-frac0.9

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

  7. Applied associate-*l*0.9

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

  8. Final simplification0.9

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

Reproduce

herbie shell --seed 2019158 +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))))