2tan (problem 3.3.2)

Percentage Accurate: 43.0% → 99.4%

Time: 20.8s

Alternatives: 15

Speedup: 2.0×

Specification

Math

\[\begin{array}{l} \\ \tan \left(x + \varepsilon\right) - \tan x \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- (tan (+ x eps)) (tan x)))

C

double code(double x, double eps) {
	return tan((x + eps)) - tan(x);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = tan((x + eps)) - tan(x)
end function

Java

public static double code(double x, double eps) {
	return Math.tan((x + eps)) - Math.tan(x);
}

Python

def code(x, eps):
	return math.tan((x + eps)) - math.tan(x)

Julia

function code(x, eps)
	return Float64(tan(Float64(x + eps)) - tan(x))
end

MATLAB

function tmp = code(x, eps)
	tmp = tan((x + eps)) - tan(x);
end

Wolfram

code[x_, eps_] := N[(N[Tan[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]

Initial Program: 43.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \tan \left(x + \varepsilon\right) - \tan x \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- (tan (+ x eps)) (tan x)))

C

double code(double x, double eps) {
	return tan((x + eps)) - tan(x);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = tan((x + eps)) - tan(x)
end function

Java

public static double code(double x, double eps) {
	return Math.tan((x + eps)) - Math.tan(x);
}

Python

def code(x, eps):
	return math.tan((x + eps)) - math.tan(x)

Julia

function code(x, eps)
	return Float64(tan(Float64(x + eps)) - tan(x))
end

MATLAB

function tmp = code(x, eps)
	tmp = tan((x + eps)) - tan(x);
end

Wolfram

code[x_, eps_] := N[(N[Tan[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \log \left({\left(e^{\tan \varepsilon}\right)}^{\tan x}\right)} + \frac{\tan \varepsilon \cdot \tan x}{\frac{1}{\tan x} - \tan \varepsilon} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (+
  (/ (/ (sin eps) (cos eps)) (- 1.0 (log (pow (exp (tan eps)) (tan x)))))
  (/ (* (tan eps) (tan x)) (- (/ 1.0 (tan x)) (tan eps)))))

C

double code(double x, double eps) {
	return ((sin(eps) / cos(eps)) / (1.0 - log(pow(exp(tan(eps)), tan(x))))) + ((tan(eps) * tan(x)) / ((1.0 / tan(x)) - tan(eps)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = ((sin(eps) / cos(eps)) / (1.0d0 - log((exp(tan(eps)) ** tan(x))))) + ((tan(eps) * tan(x)) / ((1.0d0 / tan(x)) - tan(eps)))
end function

Java

public static double code(double x, double eps) {
	return ((Math.sin(eps) / Math.cos(eps)) / (1.0 - Math.log(Math.pow(Math.exp(Math.tan(eps)), Math.tan(x))))) + ((Math.tan(eps) * Math.tan(x)) / ((1.0 / Math.tan(x)) - Math.tan(eps)));
}

Python

def code(x, eps):
	return ((math.sin(eps) / math.cos(eps)) / (1.0 - math.log(math.pow(math.exp(math.tan(eps)), math.tan(x))))) + ((math.tan(eps) * math.tan(x)) / ((1.0 / math.tan(x)) - math.tan(eps)))

Julia

function code(x, eps)
	return Float64(Float64(Float64(sin(eps) / cos(eps)) / Float64(1.0 - log((exp(tan(eps)) ^ tan(x))))) + Float64(Float64(tan(eps) * tan(x)) / Float64(Float64(1.0 / tan(x)) - tan(eps))))
end

MATLAB

function tmp = code(x, eps)
	tmp = ((sin(eps) / cos(eps)) / (1.0 - log((exp(tan(eps)) ^ tan(x))))) + ((tan(eps) * tan(x)) / ((1.0 / tan(x)) - tan(eps)));
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[Sin[eps], $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[Log[N[Power[N[Exp[N[Tan[eps], $MachinePrecision]], $MachinePrecision], N[Tan[x], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Tan[eps], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 / N[Tan[x], $MachinePrecision]), $MachinePrecision] - N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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1. Add Preprocessing

Alternative 2: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \varepsilon \cdot \tan x\\ \frac{t_0}{\frac{1}{\tan x} - \tan \varepsilon} + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - t_0} \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (tan eps) (tan x))))
   (+
    (/ t_0 (- (/ 1.0 (tan x)) (tan eps)))
    (/ (/ (sin eps) (cos eps)) (- 1.0 t_0)))))

C

double code(double x, double eps) {
	double t_0 = tan(eps) * tan(x);
	return (t_0 / ((1.0 / tan(x)) - tan(eps))) + ((sin(eps) / cos(eps)) / (1.0 - t_0));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    t_0 = tan(eps) * tan(x)
    code = (t_0 / ((1.0d0 / tan(x)) - tan(eps))) + ((sin(eps) / cos(eps)) / (1.0d0 - t_0))
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.tan(eps) * Math.tan(x);
	return (t_0 / ((1.0 / Math.tan(x)) - Math.tan(eps))) + ((Math.sin(eps) / Math.cos(eps)) / (1.0 - t_0));
}

Python

def code(x, eps):
	t_0 = math.tan(eps) * math.tan(x)
	return (t_0 / ((1.0 / math.tan(x)) - math.tan(eps))) + ((math.sin(eps) / math.cos(eps)) / (1.0 - t_0))

Julia

function code(x, eps)
	t_0 = Float64(tan(eps) * tan(x))
	return Float64(Float64(t_0 / Float64(Float64(1.0 / tan(x)) - tan(eps))) + Float64(Float64(sin(eps) / cos(eps)) / Float64(1.0 - t_0)))
end

MATLAB

function tmp = code(x, eps)
	t_0 = tan(eps) * tan(x);
	tmp = (t_0 / ((1.0 / tan(x)) - tan(eps))) + ((sin(eps) / cos(eps)) / (1.0 - t_0));
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[eps], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(t$95$0 / N[(N[(1.0 / N[Tan[x], $MachinePrecision]), $MachinePrecision] - N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sin[eps], $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision] / N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

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Alternative 3: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \varepsilon + \tan x\\ t_1 := 1 - \tan \varepsilon \cdot \tan x\\ \mathbf{if}\;\varepsilon \leq -4.7 \cdot 10^{-7}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan \varepsilon \cdot \sin x}{\cos x}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4.1 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \cos x + \tan x \cdot \left(\varepsilon \cdot \sin x\right)}{t_1}}{\cos x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{1}{t_1}, -\tan x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan eps) (tan x))) (t_1 (- 1.0 (* (tan eps) (tan x)))))
   (if (<= eps -4.7e-7)
     (- (/ t_0 (- 1.0 (/ (* (tan eps) (sin x)) (cos x)))) (tan x))
     (if (<= eps 4.1e-7)
       (/ (/ (+ (* eps (cos x)) (* (tan x) (* eps (sin x)))) t_1) (cos x))
       (fma t_0 (/ 1.0 t_1) (- (tan x)))))))

C

double code(double x, double eps) {
	double t_0 = tan(eps) + tan(x);
	double t_1 = 1.0 - (tan(eps) * tan(x));
	double tmp;
	if (eps <= -4.7e-7) {
		tmp = (t_0 / (1.0 - ((tan(eps) * sin(x)) / cos(x)))) - tan(x);
	} else if (eps <= 4.1e-7) {
		tmp = (((eps * cos(x)) + (tan(x) * (eps * sin(x)))) / t_1) / cos(x);
	} else {
		tmp = fma(t_0, (1.0 / t_1), -tan(x));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(tan(eps) + tan(x))
	t_1 = Float64(1.0 - Float64(tan(eps) * tan(x)))
	tmp = 0.0
	if (eps <= -4.7e-7)
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(Float64(tan(eps) * sin(x)) / cos(x)))) - tan(x));
	elseif (eps <= 4.1e-7)
		tmp = Float64(Float64(Float64(Float64(eps * cos(x)) + Float64(tan(x) * Float64(eps * sin(x)))) / t_1) / cos(x));
	else
		tmp = fma(t_0, Float64(1.0 / t_1), Float64(-tan(x)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[eps], $MachinePrecision] + N[Tan[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[Tan[eps], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -4.7e-7], N[(N[(t$95$0 / N[(1.0 - N[(N[(N[Tan[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 4.1e-7], N[(N[(N[(N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(N[Tan[x], $MachinePrecision] * N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(1.0 / t$95$1), $MachinePrecision] + (-N[Tan[x], $MachinePrecision])), $MachinePrecision]]]]]

Derivation

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1. Add Preprocessing

Alternative 4: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \varepsilon + \tan x\\ t_1 := 1 - \tan \varepsilon \cdot \tan x\\ \mathbf{if}\;\varepsilon \leq -4.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan \varepsilon \cdot \sin x}{\cos x}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 6.4 \cdot 10^{-7}:\\ \;\;\;\;\frac{\varepsilon}{\cos x} \cdot \frac{\cos x + \tan x \cdot \sin x}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{1}{t_1}, -\tan x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan eps) (tan x))) (t_1 (- 1.0 (* (tan eps) (tan x)))))
   (if (<= eps -4.2e-7)
     (- (/ t_0 (- 1.0 (/ (* (tan eps) (sin x)) (cos x)))) (tan x))
     (if (<= eps 6.4e-7)
       (* (/ eps (cos x)) (/ (+ (cos x) (* (tan x) (sin x))) t_1))
       (fma t_0 (/ 1.0 t_1) (- (tan x)))))))

C

double code(double x, double eps) {
	double t_0 = tan(eps) + tan(x);
	double t_1 = 1.0 - (tan(eps) * tan(x));
	double tmp;
	if (eps <= -4.2e-7) {
		tmp = (t_0 / (1.0 - ((tan(eps) * sin(x)) / cos(x)))) - tan(x);
	} else if (eps <= 6.4e-7) {
		tmp = (eps / cos(x)) * ((cos(x) + (tan(x) * sin(x))) / t_1);
	} else {
		tmp = fma(t_0, (1.0 / t_1), -tan(x));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(tan(eps) + tan(x))
	t_1 = Float64(1.0 - Float64(tan(eps) * tan(x)))
	tmp = 0.0
	if (eps <= -4.2e-7)
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(Float64(tan(eps) * sin(x)) / cos(x)))) - tan(x));
	elseif (eps <= 6.4e-7)
		tmp = Float64(Float64(eps / cos(x)) * Float64(Float64(cos(x) + Float64(tan(x) * sin(x))) / t_1));
	else
		tmp = fma(t_0, Float64(1.0 / t_1), Float64(-tan(x)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[eps], $MachinePrecision] + N[Tan[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[Tan[eps], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -4.2e-7], N[(N[(t$95$0 / N[(1.0 - N[(N[(N[Tan[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 6.4e-7], N[(N[(eps / N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] + N[(N[Tan[x], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(1.0 / t$95$1), $MachinePrecision] + (-N[Tan[x], $MachinePrecision])), $MachinePrecision]]]]]

Derivation

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1. Add Preprocessing

Alternative 5: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \varepsilon + \tan x\\ t_1 := 1 - \tan \varepsilon \cdot \tan x\\ \mathbf{if}\;\varepsilon \leq -4.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan \varepsilon \cdot \sin x}{\cos x}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 7.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\cos x + \tan x \cdot \sin x}{\cos x} \cdot \frac{\varepsilon}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{1}{t_1}, -\tan x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan eps) (tan x))) (t_1 (- 1.0 (* (tan eps) (tan x)))))
   (if (<= eps -4.8e-7)
     (- (/ t_0 (- 1.0 (/ (* (tan eps) (sin x)) (cos x)))) (tan x))
     (if (<= eps 7.8e-7)
       (* (/ (+ (cos x) (* (tan x) (sin x))) (cos x)) (/ eps t_1))
       (fma t_0 (/ 1.0 t_1) (- (tan x)))))))

C

double code(double x, double eps) {
	double t_0 = tan(eps) + tan(x);
	double t_1 = 1.0 - (tan(eps) * tan(x));
	double tmp;
	if (eps <= -4.8e-7) {
		tmp = (t_0 / (1.0 - ((tan(eps) * sin(x)) / cos(x)))) - tan(x);
	} else if (eps <= 7.8e-7) {
		tmp = ((cos(x) + (tan(x) * sin(x))) / cos(x)) * (eps / t_1);
	} else {
		tmp = fma(t_0, (1.0 / t_1), -tan(x));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(tan(eps) + tan(x))
	t_1 = Float64(1.0 - Float64(tan(eps) * tan(x)))
	tmp = 0.0
	if (eps <= -4.8e-7)
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(Float64(tan(eps) * sin(x)) / cos(x)))) - tan(x));
	elseif (eps <= 7.8e-7)
		tmp = Float64(Float64(Float64(cos(x) + Float64(tan(x) * sin(x))) / cos(x)) * Float64(eps / t_1));
	else
		tmp = fma(t_0, Float64(1.0 / t_1), Float64(-tan(x)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[eps], $MachinePrecision] + N[Tan[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[Tan[eps], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -4.8e-7], N[(N[(t$95$0 / N[(1.0 - N[(N[(N[Tan[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 7.8e-7], N[(N[(N[(N[Cos[x], $MachinePrecision] + N[(N[Tan[x], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(eps / t$95$1), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(1.0 / t$95$1), $MachinePrecision] + (-N[Tan[x], $MachinePrecision])), $MachinePrecision]]]]]

Derivation

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1. Add Preprocessing

Alternative 6: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \varepsilon + \tan x\\ t_1 := 1 - \tan \varepsilon \cdot \tan x\\ \mathbf{if}\;\varepsilon \leq -6.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan \varepsilon \cdot \sin x}{\cos x}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(\cos x + \tan x \cdot \sin x\right)}{\cos x}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{1}{t_1}, -\tan x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan eps) (tan x))) (t_1 (- 1.0 (* (tan eps) (tan x)))))
   (if (<= eps -6.2e-7)
     (- (/ t_0 (- 1.0 (/ (* (tan eps) (sin x)) (cos x)))) (tan x))
     (if (<= eps 4.5e-7)
       (/ (/ (* eps (+ (cos x) (* (tan x) (sin x)))) (cos x)) t_1)
       (fma t_0 (/ 1.0 t_1) (- (tan x)))))))

C

double code(double x, double eps) {
	double t_0 = tan(eps) + tan(x);
	double t_1 = 1.0 - (tan(eps) * tan(x));
	double tmp;
	if (eps <= -6.2e-7) {
		tmp = (t_0 / (1.0 - ((tan(eps) * sin(x)) / cos(x)))) - tan(x);
	} else if (eps <= 4.5e-7) {
		tmp = ((eps * (cos(x) + (tan(x) * sin(x)))) / cos(x)) / t_1;
	} else {
		tmp = fma(t_0, (1.0 / t_1), -tan(x));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(tan(eps) + tan(x))
	t_1 = Float64(1.0 - Float64(tan(eps) * tan(x)))
	tmp = 0.0
	if (eps <= -6.2e-7)
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(Float64(tan(eps) * sin(x)) / cos(x)))) - tan(x));
	elseif (eps <= 4.5e-7)
		tmp = Float64(Float64(Float64(eps * Float64(cos(x) + Float64(tan(x) * sin(x)))) / cos(x)) / t_1);
	else
		tmp = fma(t_0, Float64(1.0 / t_1), Float64(-tan(x)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[eps], $MachinePrecision] + N[Tan[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[Tan[eps], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -6.2e-7], N[(N[(t$95$0 / N[(1.0 - N[(N[(N[Tan[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 4.5e-7], N[(N[(N[(eps * N[(N[Cos[x], $MachinePrecision] + N[(N[Tan[x], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(t$95$0 * N[(1.0 / t$95$1), $MachinePrecision] + (-N[Tan[x], $MachinePrecision])), $MachinePrecision]]]]]

Derivation

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1. Add Preprocessing

Alternative 7: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \varepsilon + \tan x\\ t_1 := 1 - \tan \varepsilon \cdot \tan x\\ \mathbf{if}\;\varepsilon \leq -3.3 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{t_1} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 2.7 \cdot 10^{-9}:\\ \;\;\;\;\frac{\varepsilon}{\frac{\cos x}{\cos x + \frac{{\sin x}^{2}}{\cos x}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{1}{t_1}, -\tan x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan eps) (tan x))) (t_1 (- 1.0 (* (tan eps) (tan x)))))
   (if (<= eps -3.3e-9)
     (- (/ t_0 t_1) (tan x))
     (if (<= eps 2.7e-9)
       (/ eps (/ (cos x) (+ (cos x) (/ (pow (sin x) 2.0) (cos x)))))
       (fma t_0 (/ 1.0 t_1) (- (tan x)))))))

C

double code(double x, double eps) {
	double t_0 = tan(eps) + tan(x);
	double t_1 = 1.0 - (tan(eps) * tan(x));
	double tmp;
	if (eps <= -3.3e-9) {
		tmp = (t_0 / t_1) - tan(x);
	} else if (eps <= 2.7e-9) {
		tmp = eps / (cos(x) / (cos(x) + (pow(sin(x), 2.0) / cos(x))));
	} else {
		tmp = fma(t_0, (1.0 / t_1), -tan(x));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(tan(eps) + tan(x))
	t_1 = Float64(1.0 - Float64(tan(eps) * tan(x)))
	tmp = 0.0
	if (eps <= -3.3e-9)
		tmp = Float64(Float64(t_0 / t_1) - tan(x));
	elseif (eps <= 2.7e-9)
		tmp = Float64(eps / Float64(cos(x) / Float64(cos(x) + Float64((sin(x) ^ 2.0) / cos(x)))));
	else
		tmp = fma(t_0, Float64(1.0 / t_1), Float64(-tan(x)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[eps], $MachinePrecision] + N[Tan[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[Tan[eps], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -3.3e-9], N[(N[(t$95$0 / t$95$1), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 2.7e-9], N[(eps / N[(N[Cos[x], $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(1.0 / t$95$1), $MachinePrecision] + (-N[Tan[x], $MachinePrecision])), $MachinePrecision]]]]]

Derivation

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1. Add Preprocessing

Alternative 8: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \varepsilon + \tan x\\ \mathbf{if}\;\varepsilon \leq -1.35 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan \varepsilon \cdot \sin x}{\cos x}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{\varepsilon}{\frac{\cos x}{\cos x + \frac{{\sin x}^{2}}{\cos x}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{1}{1 - \tan \varepsilon \cdot \tan x}, -\tan x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan eps) (tan x))))
   (if (<= eps -1.35e-9)
     (- (/ t_0 (- 1.0 (/ (* (tan eps) (sin x)) (cos x)))) (tan x))
     (if (<= eps 3.2e-9)
       (/ eps (/ (cos x) (+ (cos x) (/ (pow (sin x) 2.0) (cos x)))))
       (fma t_0 (/ 1.0 (- 1.0 (* (tan eps) (tan x)))) (- (tan x)))))))

C

double code(double x, double eps) {
	double t_0 = tan(eps) + tan(x);
	double tmp;
	if (eps <= -1.35e-9) {
		tmp = (t_0 / (1.0 - ((tan(eps) * sin(x)) / cos(x)))) - tan(x);
	} else if (eps <= 3.2e-9) {
		tmp = eps / (cos(x) / (cos(x) + (pow(sin(x), 2.0) / cos(x))));
	} else {
		tmp = fma(t_0, (1.0 / (1.0 - (tan(eps) * tan(x)))), -tan(x));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(tan(eps) + tan(x))
	tmp = 0.0
	if (eps <= -1.35e-9)
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(Float64(tan(eps) * sin(x)) / cos(x)))) - tan(x));
	elseif (eps <= 3.2e-9)
		tmp = Float64(eps / Float64(cos(x) / Float64(cos(x) + Float64((sin(x) ^ 2.0) / cos(x)))));
	else
		tmp = fma(t_0, Float64(1.0 / Float64(1.0 - Float64(tan(eps) * tan(x)))), Float64(-tan(x)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[eps], $MachinePrecision] + N[Tan[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -1.35e-9], N[(N[(t$95$0 / N[(1.0 - N[(N[(N[Tan[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 3.2e-9], N[(eps / N[(N[Cos[x], $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(1.0 / N[(1.0 - N[(N[Tan[eps], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + (-N[Tan[x], $MachinePrecision])), $MachinePrecision]]]]

Derivation

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1. Add Preprocessing

Alternative 9: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.6 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 5.5 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \tan \varepsilon \cdot \tan x} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\frac{\cos x}{\cos x + \frac{{\sin x}^{2}}{\cos x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -3.6e-9) (not (<= eps 5.5e-9)))
   (- (/ (+ (tan eps) (tan x)) (- 1.0 (* (tan eps) (tan x)))) (tan x))
   (/ eps (/ (cos x) (+ (cos x) (/ (pow (sin x) 2.0) (cos x)))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.6e-9) || !(eps <= 5.5e-9)) {
		tmp = ((tan(eps) + tan(x)) / (1.0 - (tan(eps) * tan(x)))) - tan(x);
	} else {
		tmp = eps / (cos(x) / (cos(x) + (pow(sin(x), 2.0) / cos(x))));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-3.6d-9)) .or. (.not. (eps <= 5.5d-9))) then
        tmp = ((tan(eps) + tan(x)) / (1.0d0 - (tan(eps) * tan(x)))) - tan(x)
    else
        tmp = eps / (cos(x) / (cos(x) + ((sin(x) ** 2.0d0) / cos(x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.6e-9) || !(eps <= 5.5e-9)) {
		tmp = ((Math.tan(eps) + Math.tan(x)) / (1.0 - (Math.tan(eps) * Math.tan(x)))) - Math.tan(x);
	} else {
		tmp = eps / (Math.cos(x) / (Math.cos(x) + (Math.pow(Math.sin(x), 2.0) / Math.cos(x))));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -3.6e-9) or not (eps <= 5.5e-9):
		tmp = ((math.tan(eps) + math.tan(x)) / (1.0 - (math.tan(eps) * math.tan(x)))) - math.tan(x)
	else:
		tmp = eps / (math.cos(x) / (math.cos(x) + (math.pow(math.sin(x), 2.0) / math.cos(x))))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -3.6e-9) || !(eps <= 5.5e-9))
		tmp = Float64(Float64(Float64(tan(eps) + tan(x)) / Float64(1.0 - Float64(tan(eps) * tan(x)))) - tan(x));
	else
		tmp = Float64(eps / Float64(cos(x) / Float64(cos(x) + Float64((sin(x) ^ 2.0) / cos(x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -3.6e-9) || ~((eps <= 5.5e-9)))
		tmp = ((tan(eps) + tan(x)) / (1.0 - (tan(eps) * tan(x)))) - tan(x);
	else
		tmp = eps / (cos(x) / (cos(x) + ((sin(x) ^ 2.0) / cos(x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -3.6e-9], N[Not[LessEqual[eps, 5.5e-9]], $MachinePrecision]], N[(N[(N[(N[Tan[eps], $MachinePrecision] + N[Tan[x], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[eps], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], N[(eps / N[(N[Cos[x], $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 10: 78.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0082:\\ \;\;\;\;\log \left(e^{\frac{\sin \varepsilon}{\cos \varepsilon}}\right) - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.75 \cdot 10^{-6}:\\ \;\;\;\;\frac{\varepsilon}{\cos x} \cdot \left(\cos x + \frac{{\sin x}^{2}}{\cos x}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.0082)
   (- (log (exp (/ (sin eps) (cos eps)))) (tan x))
   (if (<= eps 3.75e-6)
     (* (/ eps (cos x)) (+ (cos x) (/ (pow (sin x) 2.0) (cos x))))
     (tan eps))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.0082) {
		tmp = log(exp((sin(eps) / cos(eps)))) - tan(x);
	} else if (eps <= 3.75e-6) {
		tmp = (eps / cos(x)) * (cos(x) + (pow(sin(x), 2.0) / cos(x)));
	} else {
		tmp = tan(eps);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= (-0.0082d0)) then
        tmp = log(exp((sin(eps) / cos(eps)))) - tan(x)
    else if (eps <= 3.75d-6) then
        tmp = (eps / cos(x)) * (cos(x) + ((sin(x) ** 2.0d0) / cos(x)))
    else
        tmp = tan(eps)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -0.0082) {
		tmp = Math.log(Math.exp((Math.sin(eps) / Math.cos(eps)))) - Math.tan(x);
	} else if (eps <= 3.75e-6) {
		tmp = (eps / Math.cos(x)) * (Math.cos(x) + (Math.pow(Math.sin(x), 2.0) / Math.cos(x)));
	} else {
		tmp = Math.tan(eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -0.0082:
		tmp = math.log(math.exp((math.sin(eps) / math.cos(eps)))) - math.tan(x)
	elif eps <= 3.75e-6:
		tmp = (eps / math.cos(x)) * (math.cos(x) + (math.pow(math.sin(x), 2.0) / math.cos(x)))
	else:
		tmp = math.tan(eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -0.0082)
		tmp = Float64(log(exp(Float64(sin(eps) / cos(eps)))) - tan(x));
	elseif (eps <= 3.75e-6)
		tmp = Float64(Float64(eps / cos(x)) * Float64(cos(x) + Float64((sin(x) ^ 2.0) / cos(x))));
	else
		tmp = tan(eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -0.0082)
		tmp = log(exp((sin(eps) / cos(eps)))) - tan(x);
	elseif (eps <= 3.75e-6)
		tmp = (eps / cos(x)) * (cos(x) + ((sin(x) ^ 2.0) / cos(x)));
	else
		tmp = tan(eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -0.0082], N[(N[Log[N[Exp[N[(N[Sin[eps], $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 3.75e-6], N[(N[(eps / N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Tan[eps], $MachinePrecision]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 11: 78.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.009:\\ \;\;\;\;\log \left(e^{\frac{\sin \varepsilon}{\cos \varepsilon}}\right) - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.3 \cdot 10^{-6}:\\ \;\;\;\;\frac{\varepsilon}{\frac{\cos x}{\cos x + \frac{{\sin x}^{2}}{\cos x}}}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.009)
   (- (log (exp (/ (sin eps) (cos eps)))) (tan x))
   (if (<= eps 3.3e-6)
     (/ eps (/ (cos x) (+ (cos x) (/ (pow (sin x) 2.0) (cos x)))))
     (tan eps))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.009) {
		tmp = log(exp((sin(eps) / cos(eps)))) - tan(x);
	} else if (eps <= 3.3e-6) {
		tmp = eps / (cos(x) / (cos(x) + (pow(sin(x), 2.0) / cos(x))));
	} else {
		tmp = tan(eps);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= (-0.009d0)) then
        tmp = log(exp((sin(eps) / cos(eps)))) - tan(x)
    else if (eps <= 3.3d-6) then
        tmp = eps / (cos(x) / (cos(x) + ((sin(x) ** 2.0d0) / cos(x))))
    else
        tmp = tan(eps)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -0.009) {
		tmp = Math.log(Math.exp((Math.sin(eps) / Math.cos(eps)))) - Math.tan(x);
	} else if (eps <= 3.3e-6) {
		tmp = eps / (Math.cos(x) / (Math.cos(x) + (Math.pow(Math.sin(x), 2.0) / Math.cos(x))));
	} else {
		tmp = Math.tan(eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -0.009:
		tmp = math.log(math.exp((math.sin(eps) / math.cos(eps)))) - math.tan(x)
	elif eps <= 3.3e-6:
		tmp = eps / (math.cos(x) / (math.cos(x) + (math.pow(math.sin(x), 2.0) / math.cos(x))))
	else:
		tmp = math.tan(eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -0.009)
		tmp = Float64(log(exp(Float64(sin(eps) / cos(eps)))) - tan(x));
	elseif (eps <= 3.3e-6)
		tmp = Float64(eps / Float64(cos(x) / Float64(cos(x) + Float64((sin(x) ^ 2.0) / cos(x)))));
	else
		tmp = tan(eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -0.009)
		tmp = log(exp((sin(eps) / cos(eps)))) - tan(x);
	elseif (eps <= 3.3e-6)
		tmp = eps / (cos(x) / (cos(x) + ((sin(x) ^ 2.0) / cos(x))));
	else
		tmp = tan(eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -0.009], N[(N[Log[N[Exp[N[(N[Sin[eps], $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 3.3e-6], N[(eps / N[(N[Cos[x], $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Tan[eps], $MachinePrecision]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 12: 78.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0082:\\ \;\;\;\;\log \left(e^{\frac{\sin \varepsilon}{\cos \varepsilon}}\right) - \tan x\\ \mathbf{elif}\;\varepsilon \leq 2.9 \cdot 10^{-6}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.0082)
   (- (log (exp (/ (sin eps) (cos eps)))) (tan x))
   (if (<= eps 2.9e-6)
     (* eps (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
     (tan eps))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.0082) {
		tmp = log(exp((sin(eps) / cos(eps)))) - tan(x);
	} else if (eps <= 2.9e-6) {
		tmp = eps * (1.0 + (pow(sin(x), 2.0) / pow(cos(x), 2.0)));
	} else {
		tmp = tan(eps);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= (-0.0082d0)) then
        tmp = log(exp((sin(eps) / cos(eps)))) - tan(x)
    else if (eps <= 2.9d-6) then
        tmp = eps * (1.0d0 + ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)))
    else
        tmp = tan(eps)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -0.0082) {
		tmp = Math.log(Math.exp((Math.sin(eps) / Math.cos(eps)))) - Math.tan(x);
	} else if (eps <= 2.9e-6) {
		tmp = eps * (1.0 + (Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)));
	} else {
		tmp = Math.tan(eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -0.0082:
		tmp = math.log(math.exp((math.sin(eps) / math.cos(eps)))) - math.tan(x)
	elif eps <= 2.9e-6:
		tmp = eps * (1.0 + (math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)))
	else:
		tmp = math.tan(eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -0.0082)
		tmp = Float64(log(exp(Float64(sin(eps) / cos(eps)))) - tan(x));
	elseif (eps <= 2.9e-6)
		tmp = Float64(eps * Float64(1.0 + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
	else
		tmp = tan(eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -0.0082)
		tmp = log(exp((sin(eps) / cos(eps)))) - tan(x);
	elseif (eps <= 2.9e-6)
		tmp = eps * (1.0 + ((sin(x) ^ 2.0) / (cos(x) ^ 2.0)));
	else
		tmp = tan(eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -0.0082], N[(N[Log[N[Exp[N[(N[Sin[eps], $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 2.9e-6], N[(eps * N[(1.0 + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Tan[eps], $MachinePrecision]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 13: 78.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0082 \lor \neg \left(\varepsilon \leq 4.8 \cdot 10^{-6}\right):\\ \;\;\;\;\tan \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.0082) (not (<= eps 4.8e-6)))
   (tan eps)
   (* eps (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.0082) || !(eps <= 4.8e-6)) {
		tmp = tan(eps);
	} else {
		tmp = eps * (1.0 + (pow(sin(x), 2.0) / pow(cos(x), 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-0.0082d0)) .or. (.not. (eps <= 4.8d-6))) then
        tmp = tan(eps)
    else
        tmp = eps * (1.0d0 + ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.0082) || !(eps <= 4.8e-6)) {
		tmp = Math.tan(eps);
	} else {
		tmp = eps * (1.0 + (Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.0082) or not (eps <= 4.8e-6):
		tmp = math.tan(eps)
	else:
		tmp = eps * (1.0 + (math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.0082) || !(eps <= 4.8e-6))
		tmp = tan(eps);
	else
		tmp = Float64(eps * Float64(1.0 + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.0082) || ~((eps <= 4.8e-6)))
		tmp = tan(eps);
	else
		tmp = eps * (1.0 + ((sin(x) ^ 2.0) / (cos(x) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.0082], N[Not[LessEqual[eps, 4.8e-6]], $MachinePrecision]], N[Tan[eps], $MachinePrecision], N[(eps * N[(1.0 + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 14: 59.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \tan \varepsilon \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (tan eps))

C

double code(double x, double eps) {
	return tan(eps);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = tan(eps)
end function

Java

public static double code(double x, double eps) {
	return Math.tan(eps);
}

Python

def code(x, eps):
	return math.tan(eps)

Julia

function code(x, eps)
	return tan(eps)
end

MATLAB

function tmp = code(x, eps)
	tmp = tan(eps);
end

Wolfram

code[x_, eps_] := N[Tan[eps], $MachinePrecision]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 15: 31.9% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ \varepsilon \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 eps)

C

double code(double x, double eps) {
	return eps;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps
end function

Java

public static double code(double x, double eps) {
	return eps;
}

Python

def code(x, eps):
	return eps

Julia

function code(x, eps)
	return eps
end

MATLAB

function tmp = code(x, eps)
	tmp = eps;
end

Wolfram

code[x_, eps_] := eps

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Developer target: 77.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (/ (sin eps) (* (cos x) (cos (+ x eps)))))

C

double code(double x, double eps) {
	return sin(eps) / (cos(x) * cos((x + eps)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = sin(eps) / (cos(x) * cos((x + eps)))
end function

Java

public static double code(double x, double eps) {
	return Math.sin(eps) / (Math.cos(x) * Math.cos((x + eps)));
}

Python

def code(x, eps):
	return math.sin(eps) / (math.cos(x) * math.cos((x + eps)))

Julia

function code(x, eps)
	return Float64(sin(eps) / Float64(cos(x) * cos(Float64(x + eps))))
end

MATLAB

function tmp = code(x, eps)
	tmp = sin(eps) / (cos(x) * cos((x + eps)));
end

Wolfram

code[x_, eps_] := N[(N[Sin[eps], $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * N[Cos[N[(x + eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024010 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :precision binary64

  :herbie-target
  (/ (sin eps) (* (cos x) (cos (+ x eps))))

  (- (tan (+ x eps)) (tan x)))