Average Error: 33.6 → 3.0
Time: 5.0s
Precision: binary64
\[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.61883453027525452 \cdot 10^{240}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + \frac{z \cdot \frac{z}{t}}{t}\\ \mathbf{elif}\;t \le -2.1983701325920077 \cdot 10^{-41}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + z \cdot \left(\frac{z}{t} \cdot \frac{1}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{x}{y}}{y} + \frac{z}{t} \cdot \frac{z}{t}\\ \end{array}\]
\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}
↓
\begin{array}{l}
\mathbf{if}\;t \le -1.61883453027525452 \cdot 10^{240}:\\
\;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + \frac{z \cdot \frac{z}{t}}{t}\\

\mathbf{elif}\;t \le -2.1983701325920077 \cdot 10^{-41}:\\
\;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + z \cdot \left(\frac{z}{t} \cdot \frac{1}{t}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{x \cdot \frac{x}{y}}{y} + \frac{z}{t} \cdot \frac{z}{t}\\

\end{array}
double code(double x, double y, double z, double t) {
	return ((double) (((double) (((double) (x * x)) / ((double) (y * y)))) + ((double) (((double) (z * z)) / ((double) (t * t))))));
}
↓
double code(double x, double y, double z, double t) {
	double VAR;
	if ((t <= -1.6188345302752545e+240)) {
		VAR = ((double) (((double) (((double) (x / y)) * ((double) (x / y)))) + ((double) (((double) (z * ((double) (z / t)))) / t))));
	} else {
		double VAR_1;
		if ((t <= -2.1983701325920077e-41)) {
			VAR_1 = ((double) (((double) (((double) (x / y)) * ((double) (x / y)))) + ((double) (z * ((double) (((double) (z / t)) * ((double) (1.0 / t))))))));
		} else {
			VAR_1 = ((double) (((double) (((double) (x * ((double) (x / y)))) / y)) + ((double) (((double) (z / t)) * ((double) (z / t))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

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

Original33.6
Target0.4
Herbie3.0
\[{\left(\frac{x}{y}\right)}^{2} + {\left(\frac{z}{t}\right)}^{2}\]

Derivation

  1. Split input into 3 regimes

  2. if t < -1.61883453027525452e240

    1. Initial program 33.1

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

    2. Simplified24.5

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

    4. Applied add-sqr-sqrt43.4

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

    5. Applied times-frac39.7

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

    6. Applied add-sqr-sqrt39.8

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

    7. Applied unswap-sqr37.1

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

    8. Simplified37.0

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

    9. Simplified11.0

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

    11. Applied add-sqr-sqrt38.9

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

    12. Applied times-frac36.3

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

    13. Applied add-sqr-sqrt36.3

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

    14. Applied unswap-sqr33.1

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

    15. Simplified33.1

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

    16. Simplified0.4

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

    18. Applied associate-*r/1.5

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

    19. Simplified1.5

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

    if -1.61883453027525452e240 < t < -2.1983701325920077e-41

    1. Initial program 28.9

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

    2. Simplified19.1

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

    4. Applied add-sqr-sqrt40.7

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

    5. Applied times-frac36.2

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

    6. Applied add-sqr-sqrt36.2

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

    7. Applied unswap-sqr34.3

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

    8. Simplified34.2

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

    9. Simplified6.6

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

    11. Applied *-un-lft-identity6.6

      \[\leadsto \frac{x}{y} \cdot \frac{x}{y} + z \cdot \frac{\color{blue}{1 \cdot z}}{t \cdot t}\]

    12. Applied times-frac0.9

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

    if -2.1983701325920077e-41 < t

    1. Initial program 35.8

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

    2. Simplified27.3

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

    4. Applied add-sqr-sqrt45.1

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

    5. Applied times-frac41.9

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

    6. Applied add-sqr-sqrt41.9

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

    7. Applied unswap-sqr40.2

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

    8. Simplified40.1

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

    9. Simplified16.9

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

    11. Applied add-sqr-sqrt40.4

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

    12. Applied times-frac34.8

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

    13. Applied add-sqr-sqrt34.8

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

    14. Applied unswap-sqr32.5

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

    15. Simplified32.4

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

    16. Simplified0.4

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

    18. Applied associate-*r/4.2

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

    19. Simplified4.2

      \[\leadsto \frac{\color{blue}{x \cdot \frac{x}{y}}}{y} + \frac{z}{t} \cdot \frac{z}{t}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification3.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.61883453027525452 \cdot 10^{240}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + \frac{z \cdot \frac{z}{t}}{t}\\ \mathbf{elif}\;t \le -2.1983701325920077 \cdot 10^{-41}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + z \cdot \left(\frac{z}{t} \cdot \frac{1}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{x}{y}}{y} + \frac{z}{t} \cdot \frac{z}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2020179 
(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))))