Average Error: 33.4 → 0.7
Time: 4.4s
Precision: 64
\[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}\]
↓
\[\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right) \cdot \left(\sqrt{\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)} \cdot \left(\sqrt{\sqrt{\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)}} \cdot \sqrt{\left|\sqrt[3]{\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)}\right| \cdot \sqrt{\sqrt[3]{\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)}}}\right)\right)\]
\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}
↓
\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right) \cdot \left(\sqrt{\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)} \cdot \left(\sqrt{\sqrt{\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)}} \cdot \sqrt{\left|\sqrt[3]{\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)}\right| \cdot \sqrt{\sqrt[3]{\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)}}}\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 (hypot((z / t), (x / y)) * (sqrt(hypot((z / t), (x / y))) * (sqrt(sqrt(hypot((z / t), (x / y)))) * sqrt((fabs(cbrt(hypot((z / t), (x / y)))) * sqrt(cbrt(hypot((z / t), (x / 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

Original33.4
Target0.4
Herbie0.7
\[{\left(\frac{x}{y}\right)}^{2} + {\left(\frac{z}{t}\right)}^{2}\]

Derivation

  1. Initial program 33.4

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

  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 add-sqr-sqrt19.6

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

  5. Simplified19.5

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

  6. Simplified0.4

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

  8. Applied add-sqr-sqrt0.6

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

  10. Applied add-sqr-sqrt0.6

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

  11. Applied sqrt-prod0.7

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

  13. Applied add-cube-cbrt0.7

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

  14. Applied sqrt-prod0.7

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

  15. Simplified0.7

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

  16. Final simplification0.7

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

Reproduce

herbie shell --seed 2020105 +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))))