Average Error: 35.3 → 27.1
Time: 7.6s
Precision: binary64
\[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
↓
\[\begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ t_1 := \frac{\tan t_0}{\sin t_0}\\ \mathbf{if}\;t_1 \leq 6.559048086219479:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{expm1}\left(t_1\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}
↓
\begin{array}{l}
t_0 := \frac{x}{y \cdot 2}\\
t_1 := \frac{\tan t_0}{\sin t_0}\\
\mathbf{if}\;t_1 \leq 6.559048086219479:\\
\;\;\;\;\mathsf{log1p}\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{expm1}\left(t_1\right)\right)\right)\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
 (let* ((t_0 (/ x (* y 2.0))) (t_1 (/ (tan t_0) (sin t_0))))
   (if (<= t_1 6.559048086219479) (log1p (log1p (expm1 (expm1 t_1)))) 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 t_0 = x / (y * 2.0);
	double t_1 = tan(t_0) / sin(t_0);
	double tmp;
	if (t_1 <= 6.559048086219479) {
		tmp = log1p(log1p(expm1(expm1(t_1))));
	} 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

Original35.3
Target28.5
Herbie27.1
\[\begin{array}{l} \mathbf{if}\;y < -1.2303690911306994 \cdot 10^{+114}:\\ \;\;\;\;1\\ \mathbf{elif}\;y < -9.102852406811914 \cdot 10^{-222}:\\ \;\;\;\;\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right) \cdot \log \left(e^{\cos \left(\frac{x}{y \cdot 2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if (/.f64 (tan.f64 (/.f64 x (*.f64 y 2))) (sin.f64 (/.f64 x (*.f64 y 2)))) < 6.55904808621947932

    1. Initial program 25.8

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

    2. Applied log1p-expm1-u_binary6425.8

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}\right)\right)} \]

    3. Applied log1p-expm1-u_binary6425.8

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{expm1}\left(\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}\right)\right)\right)}\right) \]

    if 6.55904808621947932 < (/.f64 (tan.f64 (/.f64 x (*.f64 y 2))) (sin.f64 (/.f64 x (*.f64 y 2))))

    1. Initial program 63.4

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

    2. Taylor expanded in x around 0 31.2

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

  4. Final simplification27.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \leq 6.559048086219479:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{expm1}\left(\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Reproduce

herbie shell --seed 2022081 
(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)))))