Average Error: 35.9 → 28.7
Time: 7.0s
Precision: binary64
\[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
↓
\[\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(0.5 \cdot \frac{x}{y}\right)}\right)\right) \]
(FPCore (x y)
 :precision binary64
 (/ (tan (/ x (* y 2.0))) (sin (/ x (* y 2.0)))))
↓
(FPCore (x y)
 :precision binary64
 (log1p (expm1 (/ 1.0 (cos (* 0.5 (/ x y)))))))
double code(double x, double y) {
	return tan((x / (y * 2.0))) / sin((x / (y * 2.0)));
}
↓
double code(double x, double y) {
	return log1p(expm1((1.0 / cos((0.5 * (x / y))))));
}

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) {
	return Math.log1p(Math.expm1((1.0 / Math.cos((0.5 * (x / y))))));
}

def code(x, y):
	return math.tan((x / (y * 2.0))) / math.sin((x / (y * 2.0)))
↓
def code(x, y):
	return math.log1p(math.expm1((1.0 / math.cos((0.5 * (x / y))))))

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)
	return log1p(expm1(Float64(1.0 / cos(Float64(0.5 * Float64(x / y))))))
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_] := N[Log[1 + N[(Exp[N[(1.0 / N[Cos[N[(0.5 * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]

\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}
↓
\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\cos \left(0.5 \cdot \frac{x}{y}\right)}\right)\right)

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original35.9
Target29.1
Herbie28.7
\[\begin{array}{l} \mathbf{if}\;y < -1.2303690911306994 \cdot 10^{+114}:\\ \;\;\;\;1\\ \mathbf{elif}\;y < -9.102852406811914 \cdot 10^{-222}:\\ \;\;\;\;\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right) \cdot \log \left(e^{\cos \left(\frac{x}{y \cdot 2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Derivation

  1. Initial program 35.9

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

  2. Applied egg-rr35.9

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

  3. Taylor expanded in x around inf 28.7

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

  4. Final simplification28.7

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

Reproduce

herbie shell --seed 2022153 
(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)))))