Average Error: 33.9 → 0.8
Time: 16.8s
Precision: 64
\[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}\]
↓
\[\frac{z}{t} \cdot \frac{z}{t} + \sqrt[3]{\frac{x}{y}} \cdot \left(\left(\frac{x}{y} \cdot \sqrt[3]{\frac{x}{y}}\right) \cdot \left(\sqrt[3]{\frac{\sqrt[3]{x}}{y}} \cdot \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)\right)\]
\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}
↓
\frac{z}{t} \cdot \frac{z}{t} + \sqrt[3]{\frac{x}{y}} \cdot \left(\left(\frac{x}{y} \cdot \sqrt[3]{\frac{x}{y}}\right) \cdot \left(\sqrt[3]{\frac{\sqrt[3]{x}}{y}} \cdot \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)\right)
double f(double x, double y, double z, double t) {
        double r553261 = x;
        double r553262 = r553261 * r553261;
        double r553263 = y;
        double r553264 = r553263 * r553263;
        double r553265 = r553262 / r553264;
        double r553266 = z;
        double r553267 = r553266 * r553266;
        double r553268 = t;
        double r553269 = r553268 * r553268;
        double r553270 = r553267 / r553269;
        double r553271 = r553265 + r553270;
        return r553271;
}
↓
double f(double x, double y, double z, double t) {
        double r553272 = z;
        double r553273 = t;
        double r553274 = r553272 / r553273;
        double r553275 = r553274 * r553274;
        double r553276 = x;
        double r553277 = y;
        double r553278 = r553276 / r553277;
        double r553279 = cbrt(r553278);
        double r553280 = r553278 * r553279;
        double r553281 = cbrt(r553276);
        double r553282 = r553281 / r553277;
        double r553283 = cbrt(r553282);
        double r553284 = r553281 * r553281;
        double r553285 = cbrt(r553284);
        double r553286 = r553283 * r553285;
        double r553287 = r553280 * r553286;
        double r553288 = r553279 * r553287;
        double r553289 = r553275 + r553288;
        return r553289;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original33.9
Target0.4
Herbie0.8
\[{\left(\frac{x}{y}\right)}^{2} + {\left(\frac{z}{t}\right)}^{2}\]

Derivation

  1. Initial program 33.9

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

  2. Simplified13.1

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

  3. Taylor expanded around 0 33.9

    \[\leadsto \color{blue}{\frac{{x}^{2}}{{y}^{2}} + \frac{{z}^{2}}{{t}^{2}}}\]

  4. Simplified0.4

    \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{x}{y} + \frac{z}{t} \cdot \frac{z}{t}}\]
  5. Using strategy rm

  6. Applied add-cube-cbrt0.8

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

  7. Applied associate-*r*0.8

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

  8. Simplified0.8

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

  10. Applied *-un-lft-identity0.8

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

  11. Applied add-cube-cbrt0.8

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

  12. Applied times-frac0.8

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

  13. Applied cbrt-prod0.8

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

  14. Simplified0.8

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

  15. Final simplification0.8

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

Reproduce

herbie shell --seed 2019194 +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))))