Average Error: 36.5 → 28.8
Time: 13.9s
Precision: binary64
Cost: 13568
\[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
↓
\[\frac{1}{2 \cdot e^{-0.0625 \cdot {\left(\frac{x}{y}\right)}^{2}} + -1} \]
(FPCore (x y)
 :precision binary64
 (/ (tan (/ x (* y 2.0))) (sin (/ x (* y 2.0)))))
↓
(FPCore (x y)
 :precision binary64
 (/ 1.0 (+ (* 2.0 (exp (* -0.0625 (pow (/ 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) {
	return 1.0 / ((2.0 * exp((-0.0625 * pow((x / y), 2.0)))) + -1.0);
}

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = tan((x / (y * 2.0d0))) / sin((x / (y * 2.0d0)))
end function
↓
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 / ((2.0d0 * exp(((-0.0625d0) * ((x / y) ** 2.0d0)))) + (-1.0d0))
end function

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 1.0 / ((2.0 * Math.exp((-0.0625 * Math.pow((x / y), 2.0)))) + -1.0);
}

def code(x, y):
	return math.tan((x / (y * 2.0))) / math.sin((x / (y * 2.0)))
↓
def code(x, y):
	return 1.0 / ((2.0 * math.exp((-0.0625 * math.pow((x / y), 2.0)))) + -1.0)

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)
	return Float64(1.0 / Float64(Float64(2.0 * exp(Float64(-0.0625 * (Float64(x / y) ^ 2.0)))) + -1.0))
end

function tmp = code(x, y)
	tmp = tan((x / (y * 2.0))) / sin((x / (y * 2.0)));
end
↓
function tmp = code(x, y)
	tmp = 1.0 / ((2.0 * exp((-0.0625 * ((x / y) ^ 2.0)))) + -1.0);
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_] := N[(1.0 / N[(N[(2.0 * N[Exp[N[(-0.0625 * N[Power[N[(x / y), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]

\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}
↓
\frac{1}{2 \cdot e^{-0.0625 \cdot {\left(\frac{x}{y}\right)}^{2}} + -1}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.5
Target29.7
Herbie28.8
\[\begin{array}{l} \mathbf{if}\;y < -1.2303690911306994 \cdot 10^{+114}:\\ \;\;\;\;1\\ \mathbf{elif}\;y < -9.102852406811914 \cdot 10^{-222}:\\ \;\;\;\;\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right) \cdot \log \left(e^{\cos \left(\frac{x}{y \cdot 2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Derivation

  1. Initial program 36.5

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

  2. Taylor expanded in x around inf 29.0

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

  3. Applied egg-rr29.0

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

  4. Taylor expanded in x around 0 32.8

    \[\leadsto \frac{1}{e^{\color{blue}{\log 2 + -0.0625 \cdot \frac{{x}^{2}}{{y}^{2}}}} - 1} \]

  5. Simplified28.8

    \[\leadsto \frac{1}{e^{\color{blue}{\mathsf{fma}\left(-0.0625, \frac{x}{y} \cdot \frac{x}{y}, \log 2\right)}} - 1} \]
    Proof

    [Start]32.8

    \[ \frac{1}{e^{\log 2 + -0.0625 \cdot \frac{{x}^{2}}{{y}^{2}}} - 1} \]

    +-commutative [=>]32.8

    \[ \frac{1}{e^{\color{blue}{-0.0625 \cdot \frac{{x}^{2}}{{y}^{2}} + \log 2}} - 1} \]

    fma-def [=>]32.8

    \[ \frac{1}{e^{\color{blue}{\mathsf{fma}\left(-0.0625, \frac{{x}^{2}}{{y}^{2}}, \log 2\right)}} - 1} \]

    unpow2 [=>]32.8

    \[ \frac{1}{e^{\mathsf{fma}\left(-0.0625, \frac{\color{blue}{x \cdot x}}{{y}^{2}}, \log 2\right)} - 1} \]

    unpow2 [=>]32.8

    \[ \frac{1}{e^{\mathsf{fma}\left(-0.0625, \frac{x \cdot x}{\color{blue}{y \cdot y}}, \log 2\right)} - 1} \]

    times-frac [=>]28.8

    \[ \frac{1}{e^{\mathsf{fma}\left(-0.0625, \color{blue}{\frac{x}{y} \cdot \frac{x}{y}}, \log 2\right)} - 1} \]

  6. Applied egg-rr28.8

    \[\leadsto \frac{1}{\color{blue}{e^{-0.0625 \cdot {\left(\frac{x}{y}\right)}^{2}} \cdot 2} - 1} \]

  7. Final simplification28.8

    \[\leadsto \frac{1}{2 \cdot e^{-0.0625 \cdot {\left(\frac{x}{y}\right)}^{2}} + -1} \]

Alternatives

Alternative 1
Error29.0
Cost64
\[1 \]

Error

Reproduce

herbie shell --seed 2022364 
(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)))))