Average Error: 33.7 → 0.6
Time: 3.8s
Precision: 64
\[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}\]
↓
\[\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \sqrt[3]{{\left(\left|\frac{x}{y}\right|\right)}^{\frac{3}{2}}} \cdot {\left(\left|\frac{x}{y}\right|\right)}^{\frac{3}{2}}\right)\]
\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}
↓
\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \sqrt[3]{{\left(\left|\frac{x}{y}\right|\right)}^{\frac{3}{2}}} \cdot {\left(\left|\frac{x}{y}\right|\right)}^{\frac{3}{2}}\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), (cbrt(pow(fabs((x / y)), 1.5)) * pow(fabs((x / y)), 1.5)));
}

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

Derivation

  1. Initial program 33.7

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

  2. Simplified19.3

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

  4. Applied add-sqr-sqrt19.3

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

  5. Simplified19.3

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

  6. Simplified0.4

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

  8. Applied add-sqr-sqrt0.5

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

  9. Applied associate-*l*0.5

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

  10. Simplified0.5

    \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \sqrt{\left|\frac{x}{y}\right|} \cdot \color{blue}{{\left(\left|\frac{x}{y}\right|\right)}^{\frac{3}{2}}}\right)\]
  11. Using strategy rm

  12. Applied add-cbrt-cube0.6

    \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \color{blue}{\sqrt[3]{\left(\sqrt{\left|\frac{x}{y}\right|} \cdot \sqrt{\left|\frac{x}{y}\right|}\right) \cdot \sqrt{\left|\frac{x}{y}\right|}}} \cdot {\left(\left|\frac{x}{y}\right|\right)}^{\frac{3}{2}}\right)\]

  13. Simplified0.6

    \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \sqrt[3]{\color{blue}{{\left(\left|\frac{x}{y}\right|\right)}^{\frac{3}{2}}}} \cdot {\left(\left|\frac{x}{y}\right|\right)}^{\frac{3}{2}}\right)\]

  14. Final simplification0.6

    \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \sqrt[3]{{\left(\left|\frac{x}{y}\right|\right)}^{\frac{3}{2}}} \cdot {\left(\left|\frac{x}{y}\right|\right)}^{\frac{3}{2}}\right)\]

Reproduce

herbie shell --seed 2020049 +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))))