Average Error: 35.8 → 28.6
Time: 6.6s
Precision: binary64
\[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
↓
\[\begin{array}{l} t_0 := \log \left(\sqrt{e^{\cos \left(x \cdot \frac{0.5}{y}\right)}}\right)\\ \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{t_0 + t_0}\right)\right) \end{array} \]
(FPCore (x y)
 :precision binary64
 (/ (tan (/ x (* y 2.0))) (sin (/ x (* y 2.0)))))
↓
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (log (sqrt (exp (cos (* x (/ 0.5 y))))))))
   (log1p (expm1 (/ 1.0 (+ t_0 t_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 = log(sqrt(exp(cos((x * (0.5 / y))))));
	return log1p(expm1((1.0 / (t_0 + t_0))));
}

public static double code(double x, double y) {
	return Math.tan((x / (y * 2.0))) / Math.sin((x / (y * 2.0)));
}
↓
public static double code(double x, double y) {
	double t_0 = Math.log(Math.sqrt(Math.exp(Math.cos((x * (0.5 / y))))));
	return Math.log1p(Math.expm1((1.0 / (t_0 + t_0))));
}

def code(x, y):
	return math.tan((x / (y * 2.0))) / math.sin((x / (y * 2.0)))
↓
def code(x, y):
	t_0 = math.log(math.sqrt(math.exp(math.cos((x * (0.5 / y))))))
	return math.log1p(math.expm1((1.0 / (t_0 + t_0))))

function code(x, y)
	return Float64(tan(Float64(x / Float64(y * 2.0))) / sin(Float64(x / Float64(y * 2.0))))
end
↓
function code(x, y)
	t_0 = log(sqrt(exp(cos(Float64(x * Float64(0.5 / y))))))
	return log1p(expm1(Float64(1.0 / Float64(t_0 + t_0))))
end

code[x_, y_] := N[(N[Tan[N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sin[N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_] := Block[{t$95$0 = N[Log[N[Sqrt[N[Exp[N[Cos[N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, N[Log[1 + N[(Exp[N[(1.0 / N[(t$95$0 + t$95$0), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]]

\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}
↓
\begin{array}{l}
t_0 := \log \left(\sqrt{e^{\cos \left(x \cdot \frac{0.5}{y}\right)}}\right)\\
\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{t_0 + t_0}\right)\right)
\end{array}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original35.8
Target28.9
Herbie28.6
\[\begin{array}{l} \mathbf{if}\;y < -1.2303690911306994 \cdot 10^{+114}:\\ \;\;\;\;1\\ \mathbf{elif}\;y < -9.102852406811914 \cdot 10^{-222}:\\ \;\;\;\;\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right) \cdot \log \left(e^{\cos \left(\frac{x}{y \cdot 2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Derivation

  1. Initial program 35.8

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

  2. Applied egg-rr35.8

    \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\tan \left(\frac{x}{y} \cdot 0.5\right)}{\sin \left(\frac{x}{y} \cdot 0.5\right)}\right)\right)} \]

  3. Taylor expanded in x around inf 28.6

    \[\leadsto \mathsf{log1p}\left(\color{blue}{e^{\frac{1}{\cos \left(0.5 \cdot \frac{x}{y}\right)}} - 1}\right) \]

  4. Simplified28.6

    \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\frac{1}{\cos \left(\frac{0.5}{\frac{y}{x}}\right)}\right)}\right) \]

  5. Applied egg-rr28.6

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\color{blue}{\log \left(\sqrt{e^{\cos \left(x \cdot \frac{0.5}{y}\right)}}\right) + \log \left(\sqrt{e^{\cos \left(x \cdot \frac{0.5}{y}\right)}}\right)}}\right)\right) \]

  6. Final simplification28.6

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\log \left(\sqrt{e^{\cos \left(x \cdot \frac{0.5}{y}\right)}}\right) + \log \left(\sqrt{e^{\cos \left(x \cdot \frac{0.5}{y}\right)}}\right)}\right)\right) \]

Reproduce

herbie shell --seed 2022210 
(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)))))