Average Error: 21.2 → 0.2
Time: 12.0s
Precision: binary64
\[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
↓
\[{\left(\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)\right)}^{2} \]
\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}
↓
{\left(\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)\right)}^{2}

(FPCore (x y z t)
 :precision binary64
 (+ (/ (* x x) (* y y)) (/ (* z z) (* t t))))
↓
(FPCore (x y z t) :precision binary64 (pow (hypot (/ z t) (/ x y)) 2.0))
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 pow(hypot((z / t), (x / y)), 2.0);
}

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

Original21.2
Target0.2
Herbie0.2
\[{\left(\frac{x}{y}\right)}^{2} + {\left(\frac{z}{t}\right)}^{2} \]

Derivation

  1. Initial program 21.2

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

  2. Simplified16.9

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

  3. Applied add-sqr-sqrt_binary6417.0

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

  4. Simplified16.9

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

  5. Simplified0.2

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

  6. Applied add-sqr-sqrt_binary640.3

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

  7. Applied associate-*l*_binary640.3

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

  8. Applied pow1_binary640.3

    \[\leadsto \sqrt{\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)} \cdot \left(\sqrt{\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)} \cdot \color{blue}{{\left(\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)\right)}^{1}}\right) \]

  9. Applied pow1/2_binary640.3

    \[\leadsto \sqrt{\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)} \cdot \left(\color{blue}{{\left(\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)\right)}^{0.5}} \cdot {\left(\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)\right)}^{1}\right) \]

  10. Applied pow-prod-up_binary640.2

    \[\leadsto \sqrt{\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)} \cdot \color{blue}{{\left(\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)\right)}^{\left(0.5 + 1\right)}} \]

  11. Applied pow1/2_binary640.2

    \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)\right)}^{0.5}} \cdot {\left(\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)\right)}^{\left(0.5 + 1\right)} \]

  12. Applied pow-prod-up_binary640.2

    \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)\right)}^{\left(0.5 + \left(0.5 + 1\right)\right)}} \]

  13. Final simplification0.2

    \[\leadsto {\left(\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)\right)}^{2} \]

Reproduce

herbie shell --seed 2022068 
(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))))