Average Error: 33.5 → 1.8
Time: 6.4s
Precision: binary64
\[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot y \le 7.60861 \cdot 10^{-322}:\\ \;\;\;\;\frac{x \cdot \frac{x}{y}}{y} + \frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{elif}\;y \cdot y \le 4.5636386635572052 \cdot 10^{291}:\\ \;\;\;\;\frac{x}{\frac{y \cdot y}{x}} + \frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \left(\frac{x}{\sqrt[3]{y}} \cdot \frac{x}{y}\right) + \frac{z}{t} \cdot \frac{z}{t}\\ \end{array}\]
\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}
↓
\begin{array}{l}
\mathbf{if}\;y \cdot y \le 7.60861 \cdot 10^{-322}:\\
\;\;\;\;\frac{x \cdot \frac{x}{y}}{y} + \frac{z}{t} \cdot \frac{z}{t}\\

\mathbf{elif}\;y \cdot y \le 4.5636386635572052 \cdot 10^{291}:\\
\;\;\;\;\frac{x}{\frac{y \cdot y}{x}} + \frac{z}{t} \cdot \frac{z}{t}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \left(\frac{x}{\sqrt[3]{y}} \cdot \frac{x}{y}\right) + \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 ((((double) (y * y)) <= 7.6086109459552e-322)) {
		VAR = ((double) (((double) (((double) (x * ((double) (x / y)))) / y)) + ((double) (((double) (z / t)) * ((double) (z / t))))));
	} else {
		double VAR_1;
		if ((((double) (y * y)) <= 4.563638663557205e+291)) {
			VAR_1 = ((double) (((double) (x / ((double) (((double) (y * y)) / x)))) + ((double) (((double) (z / t)) * ((double) (z / t))))));
		} else {
			VAR_1 = ((double) (((double) (((double) (1.0 / ((double) (((double) cbrt(y)) * ((double) cbrt(y)))))) * ((double) (((double) (x / ((double) cbrt(y)))) * ((double) (x / 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

Original33.5
Target0.4
Herbie1.8
\[{\left(\frac{x}{y}\right)}^{2} + {\left(\frac{z}{t}\right)}^{2}\]

Derivation

  1. Split input into 3 regimes

  2. if (* y y) < 7.60861e-322

    1. Initial program 63.8

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}\]
    2. Using strategy rm

    3. Applied times-frac63.7

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

    5. Applied times-frac0.5

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

    7. Applied associate-*r/5.4

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

    8. Simplified5.4

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

    if 7.60861e-322 < (* y y) < 4.5636386635572052e291

    1. Initial program 24.4

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}\]
    2. Using strategy rm

    3. Applied times-frac7.6

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

    5. Applied associate-/l*0.6

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

    if 4.5636386635572052e291 < (* y y)

    1. Initial program 35.4

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}\]
    2. Using strategy rm

    3. Applied times-frac19.6

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

    5. Applied times-frac0.4

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

    7. Applied add-cube-cbrt0.6

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

    8. Applied *-un-lft-identity0.6

      \[\leadsto \frac{\color{blue}{1 \cdot x}}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}} \cdot \frac{x}{y} + \frac{z}{t} \cdot \frac{z}{t}\]

    9. Applied times-frac0.6

      \[\leadsto \color{blue}{\left(\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{x}{\sqrt[3]{y}}\right)} \cdot \frac{x}{y} + \frac{z}{t} \cdot \frac{z}{t}\]

    10. Applied associate-*l*2.1

      \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \left(\frac{x}{\sqrt[3]{y}} \cdot \frac{x}{y}\right)} + \frac{z}{t} \cdot \frac{z}{t}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification1.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \le 7.60861 \cdot 10^{-322}:\\ \;\;\;\;\frac{x \cdot \frac{x}{y}}{y} + \frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{elif}\;y \cdot y \le 4.5636386635572052 \cdot 10^{291}:\\ \;\;\;\;\frac{x}{\frac{y \cdot y}{x}} + \frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \left(\frac{x}{\sqrt[3]{y}} \cdot \frac{x}{y}\right) + \frac{z}{t} \cdot \frac{z}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2020173 
(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))))