Average Error: 33.7 → 1.4
Time: 5.9s
Precision: binary64
\[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}\]
↓
\[\left(x \cdot \left(\sqrt[3]{x} \cdot \frac{\sqrt[3]{x}}{y}\right)\right) \cdot \frac{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}}{y} + \frac{z}{t} \cdot \frac{z}{t}\]
\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}
↓
\left(x \cdot \left(\sqrt[3]{x} \cdot \frac{\sqrt[3]{x}}{y}\right)\right) \cdot \frac{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}}{y} + \frac{z}{t} \cdot \frac{z}{t}
double code(double x, double y, double z, double t) {
	return ((double) ((((double) (x * x)) / ((double) (y * y))) + (((double) (z * z)) / ((double) (t * t)))));
}
↓
double code(double x, double y, double z, double t) {
	return ((double) (((double) (((double) (x * ((double) (((double) cbrt(x)) * (((double) cbrt(x)) / y))))) * (((double) (((double) cbrt(((double) (((double) cbrt(x)) * ((double) cbrt(x)))))) * ((double) cbrt(((double) cbrt(x)))))) / y))) + ((double) ((z / t) * (z / t)))));
}

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

Original33.7
Target0.4
Herbie1.4
\[{\left(\frac{x}{y}\right)}^{2} + {\left(\frac{z}{t}\right)}^{2}\]

Derivation

  1. Initial program 33.7

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

  2. Simplified24.2

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

  4. Applied add-sqr-sqrt44.4

    \[\leadsto x \cdot \frac{x}{y \cdot y} + z \cdot \frac{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}{t \cdot t}\]

  5. Applied times-frac40.4

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

  6. Applied add-sqr-sqrt40.4

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

  7. Applied unswap-sqr38.7

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

  8. Simplified38.6

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

  9. Simplified13.0

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

  11. Applied add-cube-cbrt13.3

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

  12. Applied times-frac4.6

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

  13. Applied associate-*r*1.3

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

  14. Simplified1.3

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

  16. Applied add-cube-cbrt1.4

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

  17. Applied cbrt-prod1.4

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

  18. Final simplification1.4

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

Reproduce

herbie shell --seed 2020196 
(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) 2.0) (pow (/ z t) 2.0))

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