Average Error: 35.7 → 27.7
Time: 15.2s
Precision: binary64
\[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \leq 1.8859365892335298:\\ \;\;\;\;\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]
\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \leq 1.8859365892335298:\\
\;\;\;\;\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}\\

\mathbf{else}:\\
\;\;\;\;1\\

\end{array}
(FPCore (x y)
 :precision binary64
 (/ (tan (/ x (* y 2.0))) (sin (/ x (* y 2.0)))))
↓
(FPCore (x y)
 :precision binary64
 (if (<= (/ (tan (/ x (* y 2.0))) (sin (/ x (* y 2.0)))) 1.8859365892335298)
   (/ (tan (/ x (* y 2.0))) (sin (/ x (* y 2.0))))
   1.0))
double code(double x, double y) {
	return tan(x / (y * 2.0)) / sin(x / (y * 2.0));
}
↓
double code(double x, double y) {
	double tmp;
	if ((tan(x / (y * 2.0)) / sin(x / (y * 2.0))) <= 1.8859365892335298) {
		tmp = tan(x / (y * 2.0)) / sin(x / (y * 2.0));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original35.7
Target28.9
Herbie27.7
\[\begin{array}{l} \mathbf{if}\;y < -1.2303690911306994 \cdot 10^{+114}:\\ \;\;\;\;1\\ \mathbf{elif}\;y < -9.102852406811914 \cdot 10^{-222}:\\ \;\;\;\;\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right) \cdot \log \left(e^{\cos \left(\frac{x}{y \cdot 2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/.f64 (tan.f64 (/.f64 x (*.f64 y 2))) (sin.f64 (/.f64 x (*.f64 y 2)))) < 1.8859365892335298

    1. Initial program 24.2

      \[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}\]

    if 1.8859365892335298 < (/.f64 (tan.f64 (/.f64 x (*.f64 y 2))) (sin.f64 (/.f64 x (*.f64 y 2))))

    1. Initial program 62.0

      \[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}\]

    2. Taylor expanded around inf 38.3

      \[\leadsto \color{blue}{\frac{1}{\cos \left(0.5 \cdot \frac{x}{y}\right)}}\]

    3. Simplified38.2

      \[\leadsto \color{blue}{\frac{1}{\cos \left(\frac{0.5}{y} \cdot x\right)}}\]
    4. Using strategy rm

    5. Applied add-cube-cbrt_binary64_1307138.3

      \[\leadsto \frac{1}{\cos \left(\frac{0.5}{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}} \cdot x\right)}\]

    6. Applied *-un-lft-identity_binary64_1303638.3

      \[\leadsto \frac{1}{\cos \left(\frac{\color{blue}{1 \cdot 0.5}}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}} \cdot x\right)}\]

    7. Applied times-frac_binary64_1304238.2

      \[\leadsto \frac{1}{\cos \left(\color{blue}{\left(\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{0.5}{\sqrt[3]{y}}\right)} \cdot x\right)}\]

    8. Applied associate-*l*_binary64_1297738.2

      \[\leadsto \frac{1}{\cos \color{blue}{\left(\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \left(\frac{0.5}{\sqrt[3]{y}} \cdot x\right)\right)}}\]

    9. Simplified38.3

      \[\leadsto \frac{1}{\cos \left(\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \color{blue}{\frac{0.5 \cdot x}{\sqrt[3]{y}}}\right)}\]

    10. Taylor expanded around inf 35.7

      \[\leadsto \frac{1}{\color{blue}{1}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification27.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \leq 1.8859365892335298:\\ \;\;\;\;\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Reproduce

herbie shell --seed 2021176 
(FPCore (x y)
  :name "Diagrams.TwoD.Layout.CirclePacking:approxRadius from diagrams-contrib-1.3.0.5"
  :precision binary64

  :herbie-target
  (if (< y -1.2303690911306994e+114) 1.0 (if (< y -9.102852406811914e-222) (/ (sin (/ x (* y 2.0))) (* (sin (/ x (* y 2.0))) (log (exp (cos (/ x (* y 2.0))))))) 1.0))

  (/ (tan (/ x (* y 2.0))) (sin (/ x (* y 2.0)))))