Average Error: 33.8 → 1.7
Time: 6.1s
Precision: binary64
\[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \cdot t \leq 9.7500396747249 \cdot 10^{-319} \lor \neg \left(t \cdot t \leq 2.995238407764539 \cdot 10^{+307}\right):\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + \frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \left(\frac{z}{t} \cdot \frac{z}{\sqrt[3]{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{x}}{y} \cdot \left(x \cdot \left(\sqrt[3]{x} \cdot \frac{\sqrt[3]{x}}{y}\right)\right) + z \cdot \frac{z}{t \cdot t}\\ \end{array}\]
\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}
↓
\begin{array}{l}
\mathbf{if}\;t \cdot t \leq 9.7500396747249 \cdot 10^{-319} \lor \neg \left(t \cdot t \leq 2.995238407764539 \cdot 10^{+307}\right):\\
\;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + \frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \left(\frac{z}{t} \cdot \frac{z}{\sqrt[3]{t}}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt[3]{x}}{y} \cdot \left(x \cdot \left(\sqrt[3]{x} \cdot \frac{\sqrt[3]{x}}{y}\right)\right) + z \cdot \frac{z}{t \cdot t}\\

\end{array}
(FPCore (x y z t)
 :precision binary64
 (+ (/ (* x x) (* y y)) (/ (* z z) (* t t))))
↓
(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* t t) 9.7500396747249e-319)
         (not (<= (* t t) 2.995238407764539e+307)))
   (+
    (* (/ x y) (/ x y))
    (* (/ 1.0 (* (cbrt t) (cbrt t))) (* (/ z t) (/ z (cbrt t)))))
   (+
    (* (/ (cbrt x) y) (* x (* (cbrt x) (/ (cbrt x) y))))
    (* z (/ z (* t t))))))
double code(double x, double y, double z, double t) {
	return ((double) ((((double) (x * x)) / ((double) (y * y))) + (((double) (z * z)) / ((double) (t * t)))));
}
↓
double code(double x, double y, double z, double t) {
	double tmp;
	if (((((double) (t * t)) <= 9.7500396747249e-319) || !(((double) (t * t)) <= 2.995238407764539e+307))) {
		tmp = ((double) (((double) ((x / y) * (x / y))) + ((double) ((1.0 / ((double) (((double) cbrt(t)) * ((double) cbrt(t))))) * ((double) ((z / t) * (z / ((double) cbrt(t)))))))));
	} else {
		tmp = ((double) (((double) ((((double) cbrt(x)) / y) * ((double) (x * ((double) (((double) cbrt(x)) * (((double) cbrt(x)) / y))))))) + ((double) (z * (z / ((double) (t * t)))))));
	}
	return tmp;
}

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

Original33.8
Target0.4
Herbie1.7
\[{\left(\frac{x}{y}\right)}^{2} + {\left(\frac{z}{t}\right)}^{2}\]

Derivation

  1. Split input into 2 regimes

  2. if (* t t) < 9.7500397e-319 or 2.995238407764539e307 < (* t t)

    1. Initial program Error: 43.4 bits

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

    2. SimplifiedError: 37.5 bits

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

    4. Applied add-sqr-sqrtError: 51.0 bits

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

    5. Applied times-fracError: 42.8 bits

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

    6. Applied add-sqr-sqrtError: 42.8 bits

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

    7. Applied unswap-sqrError: 38.9 bits

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

    8. SimplifiedError: 38.8 bits

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

    9. SimplifiedError: 13.3 bits

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

    11. Applied add-sqr-sqrtError: 38.7 bits

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

    12. Applied times-fracError: 34.2 bits

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

    13. Applied add-sqr-sqrtError: 34.3 bits

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

    14. Applied unswap-sqrError: 32.5 bits

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

    15. SimplifiedError: 32.4 bits

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

    16. SimplifiedError: 0.4 bits

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

    18. Applied add-cube-cbrtError: 0.7 bits

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

    19. Applied *-un-lft-identityError: 0.7 bits

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

    20. Applied times-fracError: 0.7 bits

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

    21. Applied associate-*l*Error: 2.0 bits

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

    22. SimplifiedError: 2.0 bits

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

    if 9.7500397e-319 < (* t t) < 2.995238407764539e307

    1. Initial program Error: 25.1 bits

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

    2. SimplifiedError: 13.1 bits

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

    4. Applied add-cube-cbrtError: 13.4 bits

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

    5. Applied times-fracError: 4.6 bits

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

    6. Applied associate-*r*Error: 1.4 bits

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

    7. SimplifiedError: 1.4 bits

      \[\leadsto \color{blue}{\left(x \cdot \left(\sqrt[3]{x} \cdot \frac{\sqrt[3]{x}}{y}\right)\right)} \cdot \frac{\sqrt[3]{x}}{y} + z \cdot \frac{z}{t \cdot t}\]
  3. Recombined 2 regimes into one program.

  4. Final simplificationError: 1.7 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 9.7500396747249 \cdot 10^{-319} \lor \neg \left(t \cdot t \leq 2.995238407764539 \cdot 10^{+307}\right):\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + \frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \left(\frac{z}{t} \cdot \frac{z}{\sqrt[3]{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{x}}{y} \cdot \left(x \cdot \left(\sqrt[3]{x} \cdot \frac{\sqrt[3]{x}}{y}\right)\right) + z \cdot \frac{z}{t \cdot t}\\ \end{array}\]

Reproduce

herbie shell --seed 2020204 
(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))))