Average Error: 34.3 → 1.0
Time: 4.6s
Precision: 64
\[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}\]
↓
\[\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, {\left(\sqrt[3]{\frac{x}{y}}\right)}^{4} \cdot \left(\sqrt[3]{\frac{x}{y}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right)\]
\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}
↓
\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, {\left(\sqrt[3]{\frac{x}{y}}\right)}^{4} \cdot \left(\sqrt[3]{\frac{x}{y}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right)
double code(double x, double y, double z, double t) {
	return (((x * x) / (y * y)) + ((z * z) / (t * t)));
}
↓
double code(double x, double y, double z, double t) {
	return fma((z / t), (z / t), (pow(cbrt((x / y)), 4.0) * (cbrt((x / y)) * (cbrt(x) / cbrt(y)))));
}

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

Original34.3
Target0.4
Herbie1.0
\[{\left(\frac{x}{y}\right)}^{2} + {\left(\frac{z}{t}\right)}^{2}\]

Derivation

  1. Initial program 34.3

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

  2. Simplified19.5

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

  4. Applied times-frac0.4

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

  6. Applied add-cube-cbrt0.8

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

  7. Applied add-cube-cbrt1.1

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

  8. Applied swap-sqr1.1

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

  9. Simplified1.1

    \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \color{blue}{{\left(\sqrt[3]{\frac{x}{y}}\right)}^{4}} \cdot \left(\sqrt[3]{\frac{x}{y}} \cdot \sqrt[3]{\frac{x}{y}}\right)\right)\]
  10. Using strategy rm

  11. Applied cbrt-div1.0

    \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, {\left(\sqrt[3]{\frac{x}{y}}\right)}^{4} \cdot \left(\sqrt[3]{\frac{x}{y}} \cdot \color{blue}{\frac{\sqrt[3]{x}}{\sqrt[3]{y}}}\right)\right)\]

  12. Final simplification1.0

    \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, {\left(\sqrt[3]{\frac{x}{y}}\right)}^{4} \cdot \left(\sqrt[3]{\frac{x}{y}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right)\]

Reproduce

herbie shell --seed 2020091 +o rules:numerics
(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) (pow (/ z t) 2))

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