2cos (problem 3.3.5)

Percentage Accurate: 37.5% → 99.2%

Time: 20.4s

Alternatives: 20

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 37.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\sin x\\ \mathbf{if}\;\varepsilon \leq -0.0029:\\ \;\;\;\;\mathsf{fma}\left(\sin \varepsilon, t_0, \cos \varepsilon \cdot \cos x - \cos x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0018:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \varepsilon\right)\right), t_0, \cos x \cdot \left(-1 + \cos \varepsilon\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (sin x))))
   (if (<= eps -0.0029)
     (fma (sin eps) t_0 (- (* (cos eps) (cos x)) (cos x)))
     (if (<= eps 0.0018)
       (+
        (*
         (cos x)
         (+ (* -0.5 (pow eps 2.0)) (* 0.041666666666666664 (pow eps 4.0))))
        (* (sin x) (- (* 0.16666666666666666 (pow eps 3.0)) eps)))
       (fma (expm1 (log1p (sin eps))) t_0 (* (cos x) (+ -1.0 (cos eps))))))))

C

double code(double x, double eps) {
	double t_0 = -sin(x);
	double tmp;
	if (eps <= -0.0029) {
		tmp = fma(sin(eps), t_0, ((cos(eps) * cos(x)) - cos(x)));
	} else if (eps <= 0.0018) {
		tmp = (cos(x) * ((-0.5 * pow(eps, 2.0)) + (0.041666666666666664 * pow(eps, 4.0)))) + (sin(x) * ((0.16666666666666666 * pow(eps, 3.0)) - eps));
	} else {
		tmp = fma(expm1(log1p(sin(eps))), t_0, (cos(x) * (-1.0 + cos(eps))));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(-sin(x))
	tmp = 0.0
	if (eps <= -0.0029)
		tmp = fma(sin(eps), t_0, Float64(Float64(cos(eps) * cos(x)) - cos(x)));
	elseif (eps <= 0.0018)
		tmp = Float64(Float64(cos(x) * Float64(Float64(-0.5 * (eps ^ 2.0)) + Float64(0.041666666666666664 * (eps ^ 4.0)))) + Float64(sin(x) * Float64(Float64(0.16666666666666666 * (eps ^ 3.0)) - eps)));
	else
		tmp = fma(expm1(log1p(sin(eps))), t_0, Float64(cos(x) * Float64(-1.0 + cos(eps))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = (-N[Sin[x], $MachinePrecision])}, If[LessEqual[eps, -0.0029], N[(N[Sin[eps], $MachinePrecision] * t$95$0 + N[(N[(N[Cos[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.0018], N[(N[(N[Cos[x], $MachinePrecision] * N[(N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.041666666666666664 * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(N[(0.16666666666666666 * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision] - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Exp[N[Log[1 + N[Sin[eps], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision] * t$95$0 + N[(N[Cos[x], $MachinePrecision] * N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

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

Alternative 2: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sin \varepsilon, -\sin x, \frac{{\sin \varepsilon}^{2}}{\frac{-1 - \cos \varepsilon}{\cos x}}\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (sin eps)
  (- (sin x))
  (/ (pow (sin eps) 2.0) (/ (- -1.0 (cos eps)) (cos x)))))

C

double code(double x, double eps) {
	return fma(sin(eps), -sin(x), (pow(sin(eps), 2.0) / ((-1.0 - cos(eps)) / cos(x))));
}

Julia

function code(x, eps)
	return fma(sin(eps), Float64(-sin(x)), Float64((sin(eps) ^ 2.0) / Float64(Float64(-1.0 - cos(eps)) / cos(x))))
end

Wolfram

code[x_, eps_] := N[(N[Sin[eps], $MachinePrecision] * (-N[Sin[x], $MachinePrecision]) + N[(N[Power[N[Sin[eps], $MachinePrecision], 2.0], $MachinePrecision] / N[(N[(-1.0 - N[Cos[eps], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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

Alternative 3: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\sin x\\ \mathbf{if}\;\varepsilon \leq -0.00235:\\ \;\;\;\;\mathsf{fma}\left(\sin \varepsilon, t_0, \cos \varepsilon \cdot \cos x - \cos x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0018:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin \varepsilon, t_0, \cos x \cdot \left(-1 + \cos \varepsilon\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (sin x))))
   (if (<= eps -0.00235)
     (fma (sin eps) t_0 (- (* (cos eps) (cos x)) (cos x)))
     (if (<= eps 0.0018)
       (+
        (*
         (cos x)
         (+ (* -0.5 (pow eps 2.0)) (* 0.041666666666666664 (pow eps 4.0))))
        (* (sin x) (- (* 0.16666666666666666 (pow eps 3.0)) eps)))
       (fma (sin eps) t_0 (* (cos x) (+ -1.0 (cos eps))))))))

C

double code(double x, double eps) {
	double t_0 = -sin(x);
	double tmp;
	if (eps <= -0.00235) {
		tmp = fma(sin(eps), t_0, ((cos(eps) * cos(x)) - cos(x)));
	} else if (eps <= 0.0018) {
		tmp = (cos(x) * ((-0.5 * pow(eps, 2.0)) + (0.041666666666666664 * pow(eps, 4.0)))) + (sin(x) * ((0.16666666666666666 * pow(eps, 3.0)) - eps));
	} else {
		tmp = fma(sin(eps), t_0, (cos(x) * (-1.0 + cos(eps))));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(-sin(x))
	tmp = 0.0
	if (eps <= -0.00235)
		tmp = fma(sin(eps), t_0, Float64(Float64(cos(eps) * cos(x)) - cos(x)));
	elseif (eps <= 0.0018)
		tmp = Float64(Float64(cos(x) * Float64(Float64(-0.5 * (eps ^ 2.0)) + Float64(0.041666666666666664 * (eps ^ 4.0)))) + Float64(sin(x) * Float64(Float64(0.16666666666666666 * (eps ^ 3.0)) - eps)));
	else
		tmp = fma(sin(eps), t_0, Float64(cos(x) * Float64(-1.0 + cos(eps))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = (-N[Sin[x], $MachinePrecision])}, If[LessEqual[eps, -0.00235], N[(N[Sin[eps], $MachinePrecision] * t$95$0 + N[(N[(N[Cos[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.0018], N[(N[(N[Cos[x], $MachinePrecision] * N[(N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.041666666666666664 * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(N[(0.16666666666666666 * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision] - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[eps], $MachinePrecision] * t$95$0 + N[(N[Cos[x], $MachinePrecision] * N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

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

Alternative 4: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00215:\\ \;\;\;\;\cos \varepsilon \cdot \cos x - \left(\cos x + \sin \varepsilon \cdot \sin x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0018:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin \varepsilon, -\sin x, \cos x \cdot \left(-1 + \cos \varepsilon\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.00215)
   (- (* (cos eps) (cos x)) (+ (cos x) (* (sin eps) (sin x))))
   (if (<= eps 0.0018)
     (+
      (*
       (cos x)
       (+ (* -0.5 (pow eps 2.0)) (* 0.041666666666666664 (pow eps 4.0))))
      (* (sin x) (- (* 0.16666666666666666 (pow eps 3.0)) eps)))
     (fma (sin eps) (- (sin x)) (* (cos x) (+ -1.0 (cos eps)))))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.00215) {
		tmp = (cos(eps) * cos(x)) - (cos(x) + (sin(eps) * sin(x)));
	} else if (eps <= 0.0018) {
		tmp = (cos(x) * ((-0.5 * pow(eps, 2.0)) + (0.041666666666666664 * pow(eps, 4.0)))) + (sin(x) * ((0.16666666666666666 * pow(eps, 3.0)) - eps));
	} else {
		tmp = fma(sin(eps), -sin(x), (cos(x) * (-1.0 + cos(eps))));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -0.00215)
		tmp = Float64(Float64(cos(eps) * cos(x)) - Float64(cos(x) + Float64(sin(eps) * sin(x))));
	elseif (eps <= 0.0018)
		tmp = Float64(Float64(cos(x) * Float64(Float64(-0.5 * (eps ^ 2.0)) + Float64(0.041666666666666664 * (eps ^ 4.0)))) + Float64(sin(x) * Float64(Float64(0.16666666666666666 * (eps ^ 3.0)) - eps)));
	else
		tmp = fma(sin(eps), Float64(-sin(x)), Float64(cos(x) * Float64(-1.0 + cos(eps))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -0.00215], N[(N[(N[Cos[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[x], $MachinePrecision] + N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.0018], N[(N[(N[Cos[x], $MachinePrecision] * N[(N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.041666666666666664 * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(N[(0.16666666666666666 * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision] - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[eps], $MachinePrecision] * (-N[Sin[x], $MachinePrecision]) + N[(N[Cos[x], $MachinePrecision] * N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

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

Alternative 5: 74.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\varepsilon + x\right) - \cos x\\ \mathbf{if}\;t_0 \leq -0.0001:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - \varepsilon \cdot \sin x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (cos (+ eps x)) (cos x))))
   (if (<= t_0 -0.0001)
     t_0
     (- (* (cos x) (* -0.5 (pow eps 2.0))) (* eps (sin x))))))

C

double code(double x, double eps) {
	double t_0 = cos((eps + x)) - cos(x);
	double tmp;
	if (t_0 <= -0.0001) {
		tmp = t_0;
	} else {
		tmp = (cos(x) * (-0.5 * pow(eps, 2.0))) - (eps * sin(x));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((eps + x)) - cos(x)
    if (t_0 <= (-0.0001d0)) then
        tmp = t_0
    else
        tmp = (cos(x) * ((-0.5d0) * (eps ** 2.0d0))) - (eps * sin(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.cos((eps + x)) - Math.cos(x);
	double tmp;
	if (t_0 <= -0.0001) {
		tmp = t_0;
	} else {
		tmp = (Math.cos(x) * (-0.5 * Math.pow(eps, 2.0))) - (eps * Math.sin(x));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.cos((eps + x)) - math.cos(x)
	tmp = 0
	if t_0 <= -0.0001:
		tmp = t_0
	else:
		tmp = (math.cos(x) * (-0.5 * math.pow(eps, 2.0))) - (eps * math.sin(x))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(cos(Float64(eps + x)) - cos(x))
	tmp = 0.0
	if (t_0 <= -0.0001)
		tmp = t_0;
	else
		tmp = Float64(Float64(cos(x) * Float64(-0.5 * (eps ^ 2.0))) - Float64(eps * sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = cos((eps + x)) - cos(x);
	tmp = 0.0;
	if (t_0 <= -0.0001)
		tmp = t_0;
	else
		tmp = (cos(x) * (-0.5 * (eps ^ 2.0))) - (eps * sin(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Cos[N[(eps + x), $MachinePrecision]], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.0001], t$95$0, N[(N[(N[Cos[x], $MachinePrecision] * N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

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

Alternative 6: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00017 \lor \neg \left(\varepsilon \leq 0.00017\right):\\ \;\;\;\;\mathsf{fma}\left(\sin \varepsilon, -\sin x, \cos x \cdot \left(-1 + \cos \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right) + \cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.00017) (not (<= eps 0.00017)))
   (fma (sin eps) (- (sin x)) (* (cos x) (+ -1.0 (cos eps))))
   (+
    (* (sin x) (- (* 0.16666666666666666 (pow eps 3.0)) eps))
    (* (cos x) (* -0.5 (pow eps 2.0))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.00017) || !(eps <= 0.00017)) {
		tmp = fma(sin(eps), -sin(x), (cos(x) * (-1.0 + cos(eps))));
	} else {
		tmp = (sin(x) * ((0.16666666666666666 * pow(eps, 3.0)) - eps)) + (cos(x) * (-0.5 * pow(eps, 2.0)));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.00017) || !(eps <= 0.00017))
		tmp = fma(sin(eps), Float64(-sin(x)), Float64(cos(x) * Float64(-1.0 + cos(eps))));
	else
		tmp = Float64(Float64(sin(x) * Float64(Float64(0.16666666666666666 * (eps ^ 3.0)) - eps)) + Float64(cos(x) * Float64(-0.5 * (eps ^ 2.0))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.00017], N[Not[LessEqual[eps, 0.00017]], $MachinePrecision]], N[(N[Sin[eps], $MachinePrecision] * (-N[Sin[x], $MachinePrecision]) + N[(N[Cos[x], $MachinePrecision] * N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[x], $MachinePrecision] * N[(N[(0.16666666666666666 * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision] - eps), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

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

Alternative 7: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00017 \lor \neg \left(\varepsilon \leq 0.00017\right):\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right) + \cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.00017) (not (<= eps 0.00017)))
   (- (* (cos x) (+ -1.0 (cos eps))) (* (sin eps) (sin x)))
   (+
    (* (sin x) (- (* 0.16666666666666666 (pow eps 3.0)) eps))
    (* (cos x) (* -0.5 (pow eps 2.0))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.00017) || !(eps <= 0.00017)) {
		tmp = (cos(x) * (-1.0 + cos(eps))) - (sin(eps) * sin(x));
	} else {
		tmp = (sin(x) * ((0.16666666666666666 * pow(eps, 3.0)) - eps)) + (cos(x) * (-0.5 * pow(eps, 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 <= (-0.00017d0)) .or. (.not. (eps <= 0.00017d0))) then
        tmp = (cos(x) * ((-1.0d0) + cos(eps))) - (sin(eps) * sin(x))
    else
        tmp = (sin(x) * ((0.16666666666666666d0 * (eps ** 3.0d0)) - eps)) + (cos(x) * ((-0.5d0) * (eps ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.00017) || !(eps <= 0.00017)) {
		tmp = (Math.cos(x) * (-1.0 + Math.cos(eps))) - (Math.sin(eps) * Math.sin(x));
	} else {
		tmp = (Math.sin(x) * ((0.16666666666666666 * Math.pow(eps, 3.0)) - eps)) + (Math.cos(x) * (-0.5 * Math.pow(eps, 2.0)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.00017) or not (eps <= 0.00017):
		tmp = (math.cos(x) * (-1.0 + math.cos(eps))) - (math.sin(eps) * math.sin(x))
	else:
		tmp = (math.sin(x) * ((0.16666666666666666 * math.pow(eps, 3.0)) - eps)) + (math.cos(x) * (-0.5 * math.pow(eps, 2.0)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.00017) || !(eps <= 0.00017))
		tmp = Float64(Float64(cos(x) * Float64(-1.0 + cos(eps))) - Float64(sin(eps) * sin(x)));
	else
		tmp = Float64(Float64(sin(x) * Float64(Float64(0.16666666666666666 * (eps ^ 3.0)) - eps)) + Float64(cos(x) * Float64(-0.5 * (eps ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.00017) || ~((eps <= 0.00017)))
		tmp = (cos(x) * (-1.0 + cos(eps))) - (sin(eps) * sin(x));
	else
		tmp = (sin(x) * ((0.16666666666666666 * (eps ^ 3.0)) - eps)) + (cos(x) * (-0.5 * (eps ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.00017], N[Not[LessEqual[eps, 0.00017]], $MachinePrecision]], N[(N[(N[Cos[x], $MachinePrecision] * N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[x], $MachinePrecision] * N[(N[(0.16666666666666666 * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision] - eps), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

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

Alternative 8: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 2.9 \cdot 10^{-5}\right):\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - \varepsilon \cdot \sin x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -3e-5) (not (<= eps 2.9e-5)))
   (- (* (cos x) (+ -1.0 (cos eps))) (* (sin eps) (sin x)))
   (- (* (cos x) (* -0.5 (pow eps 2.0))) (* eps (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -3e-5) || !(eps <= 2.9e-5)) {
		tmp = (cos(x) * (-1.0 + cos(eps))) - (sin(eps) * sin(x));
	} else {
		tmp = (cos(x) * (-0.5 * pow(eps, 2.0))) - (eps * sin(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 <= (-3d-5)) .or. (.not. (eps <= 2.9d-5))) then
        tmp = (cos(x) * ((-1.0d0) + cos(eps))) - (sin(eps) * sin(x))
    else
        tmp = (cos(x) * ((-0.5d0) * (eps ** 2.0d0))) - (eps * sin(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -3e-5) || !(eps <= 2.9e-5)) {
		tmp = (Math.cos(x) * (-1.0 + Math.cos(eps))) - (Math.sin(eps) * Math.sin(x));
	} else {
		tmp = (Math.cos(x) * (-0.5 * Math.pow(eps, 2.0))) - (eps * Math.sin(x));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -3e-5) or not (eps <= 2.9e-5):
		tmp = (math.cos(x) * (-1.0 + math.cos(eps))) - (math.sin(eps) * math.sin(x))
	else:
		tmp = (math.cos(x) * (-0.5 * math.pow(eps, 2.0))) - (eps * math.sin(x))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -3e-5) || !(eps <= 2.9e-5))
		tmp = Float64(Float64(cos(x) * Float64(-1.0 + cos(eps))) - Float64(sin(eps) * sin(x)));
	else
		tmp = Float64(Float64(cos(x) * Float64(-0.5 * (eps ^ 2.0))) - Float64(eps * sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -3e-5) || ~((eps <= 2.9e-5)))
		tmp = (cos(x) * (-1.0 + cos(eps))) - (sin(eps) * sin(x));
	else
		tmp = (cos(x) * (-0.5 * (eps ^ 2.0))) - (eps * sin(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -3e-5], N[Not[LessEqual[eps, 2.9e-5]], $MachinePrecision]], N[(N[(N[Cos[x], $MachinePrecision] * N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[x], $MachinePrecision] * N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

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

Alternative 9: 64.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\varepsilon + x\right) - \cos x\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-14}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (cos (+ eps x)) (cos x))))
   (if (<= t_0 -1e-14) t_0 (* (sin x) (- eps)))))

C

double code(double x, double eps) {
	double t_0 = cos((eps + x)) - cos(x);
	double tmp;
	if (t_0 <= -1e-14) {
		tmp = t_0;
	} else {
		tmp = sin(x) * -eps;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((eps + x)) - cos(x)
    if (t_0 <= (-1d-14)) then
        tmp = t_0
    else
        tmp = sin(x) * -eps
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.cos((eps + x)) - Math.cos(x);
	double tmp;
	if (t_0 <= -1e-14) {
		tmp = t_0;
	} else {
		tmp = Math.sin(x) * -eps;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.cos((eps + x)) - math.cos(x)
	tmp = 0
	if t_0 <= -1e-14:
		tmp = t_0
	else:
		tmp = math.sin(x) * -eps
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(cos(Float64(eps + x)) - cos(x))
	tmp = 0.0
	if (t_0 <= -1e-14)
		tmp = t_0;
	else
		tmp = Float64(sin(x) * Float64(-eps));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = cos((eps + x)) - cos(x);
	tmp = 0.0;
	if (t_0 <= -1e-14)
		tmp = t_0;
	else
		tmp = sin(x) * -eps;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Cos[N[(eps + x), $MachinePrecision]], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-14], t$95$0, N[(N[Sin[x], $MachinePrecision] * (-eps)), $MachinePrecision]]]

Derivation

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

Alternative 10: 75.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* (sin (* 0.5 (fma 2.0 x eps))) (* -2.0 (sin (* eps 0.5)))))

C

double code(double x, double eps) {
	return sin((0.5 * fma(2.0, x, eps))) * (-2.0 * sin((eps * 0.5)));
}

Julia

function code(x, eps)
	return Float64(sin(Float64(0.5 * fma(2.0, x, eps))) * Float64(-2.0 * sin(Float64(eps * 0.5))))
end

Wolfram

code[x_, eps_] := N[(N[Sin[N[(0.5 * N[(2.0 * x + eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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

Alternative 11: 73.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-1 + \cos \varepsilon\right) - \sin \varepsilon \cdot x\\ t_1 := \sin x \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \mathbf{if}\;x \leq -0.00275:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2.7 \cdot 10^{-80}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-141}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\left(\varepsilon + x\right) - x}{2}\right) \cdot \sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)\\ \mathbf{elif}\;x \leq 0.000175:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (+ -1.0 (cos eps)) (* (sin eps) x)))
        (t_1 (* (sin x) (* -2.0 (sin (* eps 0.5))))))
   (if (<= x -0.00275)
     t_1
     (if (<= x -2.7e-80)
       t_0
       (if (<= x 2.5e-141)
         (*
          -2.0
          (* (sin (/ (- (+ eps x) x) 2.0)) (sin (/ (+ x (+ eps x)) 2.0))))
         (if (<= x 0.000175) t_0 t_1))))))

C

double code(double x, double eps) {
	double t_0 = (-1.0 + cos(eps)) - (sin(eps) * x);
	double t_1 = sin(x) * (-2.0 * sin((eps * 0.5)));
	double tmp;
	if (x <= -0.00275) {
		tmp = t_1;
	} else if (x <= -2.7e-80) {
		tmp = t_0;
	} else if (x <= 2.5e-141) {
		tmp = -2.0 * (sin((((eps + x) - x) / 2.0)) * sin(((x + (eps + x)) / 2.0)));
	} else if (x <= 0.000175) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((-1.0d0) + cos(eps)) - (sin(eps) * x)
    t_1 = sin(x) * ((-2.0d0) * sin((eps * 0.5d0)))
    if (x <= (-0.00275d0)) then
        tmp = t_1
    else if (x <= (-2.7d-80)) then
        tmp = t_0
    else if (x <= 2.5d-141) then
        tmp = (-2.0d0) * (sin((((eps + x) - x) / 2.0d0)) * sin(((x + (eps + x)) / 2.0d0)))
    else if (x <= 0.000175d0) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = (-1.0 + Math.cos(eps)) - (Math.sin(eps) * x);
	double t_1 = Math.sin(x) * (-2.0 * Math.sin((eps * 0.5)));
	double tmp;
	if (x <= -0.00275) {
		tmp = t_1;
	} else if (x <= -2.7e-80) {
		tmp = t_0;
	} else if (x <= 2.5e-141) {
		tmp = -2.0 * (Math.sin((((eps + x) - x) / 2.0)) * Math.sin(((x + (eps + x)) / 2.0)));
	} else if (x <= 0.000175) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = (-1.0 + math.cos(eps)) - (math.sin(eps) * x)
	t_1 = math.sin(x) * (-2.0 * math.sin((eps * 0.5)))
	tmp = 0
	if x <= -0.00275:
		tmp = t_1
	elif x <= -2.7e-80:
		tmp = t_0
	elif x <= 2.5e-141:
		tmp = -2.0 * (math.sin((((eps + x) - x) / 2.0)) * math.sin(((x + (eps + x)) / 2.0)))
	elif x <= 0.000175:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64(-1.0 + cos(eps)) - Float64(sin(eps) * x))
	t_1 = Float64(sin(x) * Float64(-2.0 * sin(Float64(eps * 0.5))))
	tmp = 0.0
	if (x <= -0.00275)
		tmp = t_1;
	elseif (x <= -2.7e-80)
		tmp = t_0;
	elseif (x <= 2.5e-141)
		tmp = Float64(-2.0 * Float64(sin(Float64(Float64(Float64(eps + x) - x) / 2.0)) * sin(Float64(Float64(x + Float64(eps + x)) / 2.0))));
	elseif (x <= 0.000175)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (-1.0 + cos(eps)) - (sin(eps) * x);
	t_1 = sin(x) * (-2.0 * sin((eps * 0.5)));
	tmp = 0.0;
	if (x <= -0.00275)
		tmp = t_1;
	elseif (x <= -2.7e-80)
		tmp = t_0;
	elseif (x <= 2.5e-141)
		tmp = -2.0 * (sin((((eps + x) - x) / 2.0)) * sin(((x + (eps + x)) / 2.0)));
	elseif (x <= 0.000175)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[eps], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[x], $MachinePrecision] * N[(-2.0 * N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.00275], t$95$1, If[LessEqual[x, -2.7e-80], t$95$0, If[LessEqual[x, 2.5e-141], N[(-2.0 * N[(N[Sin[N[(N[(N[(eps + x), $MachinePrecision] - x), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[(x + N[(eps + x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.000175], t$95$0, t$95$1]]]]]]

Derivation

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

Alternative 12: 72.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-1 + \cos \varepsilon\right) - \sin \varepsilon \cdot x\\ t_1 := \sin \left(\varepsilon \cdot 0.5\right)\\ t_2 := \sin x \cdot \left(-2 \cdot t_1\right)\\ \mathbf{if}\;x \leq -0.0062:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{-80}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-141}:\\ \;\;\;\;-2 \cdot {t_1}^{2}\\ \mathbf{elif}\;x \leq 0.0033:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (+ -1.0 (cos eps)) (* (sin eps) x)))
        (t_1 (sin (* eps 0.5)))
        (t_2 (* (sin x) (* -2.0 t_1))))
   (if (<= x -0.0062)
     t_2
     (if (<= x -6.2e-80)
       t_0
       (if (<= x 2.5e-141)
         (* -2.0 (pow t_1 2.0))
         (if (<= x 0.0033) t_0 t_2))))))

C

double code(double x, double eps) {
	double t_0 = (-1.0 + cos(eps)) - (sin(eps) * x);
	double t_1 = sin((eps * 0.5));
	double t_2 = sin(x) * (-2.0 * t_1);
	double tmp;
	if (x <= -0.0062) {
		tmp = t_2;
	} else if (x <= -6.2e-80) {
		tmp = t_0;
	} else if (x <= 2.5e-141) {
		tmp = -2.0 * pow(t_1, 2.0);
	} else if (x <= 0.0033) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = ((-1.0d0) + cos(eps)) - (sin(eps) * x)
    t_1 = sin((eps * 0.5d0))
    t_2 = sin(x) * ((-2.0d0) * t_1)
    if (x <= (-0.0062d0)) then
        tmp = t_2
    else if (x <= (-6.2d-80)) then
        tmp = t_0
    else if (x <= 2.5d-141) then
        tmp = (-2.0d0) * (t_1 ** 2.0d0)
    else if (x <= 0.0033d0) then
        tmp = t_0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = (-1.0 + Math.cos(eps)) - (Math.sin(eps) * x);
	double t_1 = Math.sin((eps * 0.5));
	double t_2 = Math.sin(x) * (-2.0 * t_1);
	double tmp;
	if (x <= -0.0062) {
		tmp = t_2;
	} else if (x <= -6.2e-80) {
		tmp = t_0;
	} else if (x <= 2.5e-141) {
		tmp = -2.0 * Math.pow(t_1, 2.0);
	} else if (x <= 0.0033) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = (-1.0 + math.cos(eps)) - (math.sin(eps) * x)
	t_1 = math.sin((eps * 0.5))
	t_2 = math.sin(x) * (-2.0 * t_1)
	tmp = 0
	if x <= -0.0062:
		tmp = t_2
	elif x <= -6.2e-80:
		tmp = t_0
	elif x <= 2.5e-141:
		tmp = -2.0 * math.pow(t_1, 2.0)
	elif x <= 0.0033:
		tmp = t_0
	else:
		tmp = t_2
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64(-1.0 + cos(eps)) - Float64(sin(eps) * x))
	t_1 = sin(Float64(eps * 0.5))
	t_2 = Float64(sin(x) * Float64(-2.0 * t_1))
	tmp = 0.0
	if (x <= -0.0062)
		tmp = t_2;
	elseif (x <= -6.2e-80)
		tmp = t_0;
	elseif (x <= 2.5e-141)
		tmp = Float64(-2.0 * (t_1 ^ 2.0));
	elseif (x <= 0.0033)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (-1.0 + cos(eps)) - (sin(eps) * x);
	t_1 = sin((eps * 0.5));
	t_2 = sin(x) * (-2.0 * t_1);
	tmp = 0.0;
	if (x <= -0.0062)
		tmp = t_2;
	elseif (x <= -6.2e-80)
		tmp = t_0;
	elseif (x <= 2.5e-141)
		tmp = -2.0 * (t_1 ^ 2.0);
	elseif (x <= 0.0033)
		tmp = t_0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[eps], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[x], $MachinePrecision] * N[(-2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0062], t$95$2, If[LessEqual[x, -6.2e-80], t$95$0, If[LessEqual[x, 2.5e-141], N[(-2.0 * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0033], t$95$0, t$95$2]]]]]]]

Derivation

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

Alternative 13: 69.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\varepsilon \cdot 0.5\right)\\ \mathbf{if}\;x \leq -3.4 \cdot 10^{-26} \lor \neg \left(x \leq 1.75 \cdot 10^{-5}\right):\\ \;\;\;\;\sin x \cdot \left(-2 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {t_0}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (* eps 0.5))))
   (if (or (<= x -3.4e-26) (not (<= x 1.75e-5)))
     (* (sin x) (* -2.0 t_0))
     (* -2.0 (pow t_0 2.0)))))

C

double code(double x, double eps) {
	double t_0 = sin((eps * 0.5));
	double tmp;
	if ((x <= -3.4e-26) || !(x <= 1.75e-5)) {
		tmp = sin(x) * (-2.0 * t_0);
	} else {
		tmp = -2.0 * pow(t_0, 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((eps * 0.5d0))
    if ((x <= (-3.4d-26)) .or. (.not. (x <= 1.75d-5))) then
        tmp = sin(x) * ((-2.0d0) * t_0)
    else
        tmp = (-2.0d0) * (t_0 ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin((eps * 0.5));
	double tmp;
	if ((x <= -3.4e-26) || !(x <= 1.75e-5)) {
		tmp = Math.sin(x) * (-2.0 * t_0);
	} else {
		tmp = -2.0 * Math.pow(t_0, 2.0);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.sin((eps * 0.5))
	tmp = 0
	if (x <= -3.4e-26) or not (x <= 1.75e-5):
		tmp = math.sin(x) * (-2.0 * t_0)
	else:
		tmp = -2.0 * math.pow(t_0, 2.0)
	return tmp

Julia

function code(x, eps)
	t_0 = sin(Float64(eps * 0.5))
	tmp = 0.0
	if ((x <= -3.4e-26) || !(x <= 1.75e-5))
		tmp = Float64(sin(x) * Float64(-2.0 * t_0));
	else
		tmp = Float64(-2.0 * (t_0 ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = sin((eps * 0.5));
	tmp = 0.0;
	if ((x <= -3.4e-26) || ~((x <= 1.75e-5)))
		tmp = sin(x) * (-2.0 * t_0);
	else
		tmp = -2.0 * (t_0 ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[x, -3.4e-26], N[Not[LessEqual[x, 1.75e-5]], $MachinePrecision]], N[(N[Sin[x], $MachinePrecision] * N[(-2.0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

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

Alternative 14: 66.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00045:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{elif}\;\varepsilon \leq 1.3 \cdot 10^{-27}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.00045)
   (- (cos eps) (cos x))
   (if (<= eps 1.3e-27)
     (* (sin x) (- eps))
     (* -2.0 (pow (sin (* eps 0.5)) 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.00045) {
		tmp = cos(eps) - cos(x);
	} else if (eps <= 1.3e-27) {
		tmp = sin(x) * -eps;
	} else {
		tmp = -2.0 * pow(sin((eps * 0.5)), 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 <= (-0.00045d0)) then
        tmp = cos(eps) - cos(x)
    else if (eps <= 1.3d-27) then
        tmp = sin(x) * -eps
    else
        tmp = (-2.0d0) * (sin((eps * 0.5d0)) ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -0.00045) {
		tmp = Math.cos(eps) - Math.cos(x);
	} else if (eps <= 1.3e-27) {
		tmp = Math.sin(x) * -eps;
	} else {
		tmp = -2.0 * Math.pow(Math.sin((eps * 0.5)), 2.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -0.00045:
		tmp = math.cos(eps) - math.cos(x)
	elif eps <= 1.3e-27:
		tmp = math.sin(x) * -eps
	else:
		tmp = -2.0 * math.pow(math.sin((eps * 0.5)), 2.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -0.00045)
		tmp = Float64(cos(eps) - cos(x));
	elseif (eps <= 1.3e-27)
		tmp = Float64(sin(x) * Float64(-eps));
	else
		tmp = Float64(-2.0 * (sin(Float64(eps * 0.5)) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -0.00045)
		tmp = cos(eps) - cos(x);
	elseif (eps <= 1.3e-27)
		tmp = sin(x) * -eps;
	else
		tmp = -2.0 * (sin((eps * 0.5)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -0.00045], N[(N[Cos[eps], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 1.3e-27], N[(N[Sin[x], $MachinePrecision] * (-eps)), $MachinePrecision], N[(-2.0 * N[Power[N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

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

Alternative 15: 66.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00045:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{elif}\;\varepsilon \leq 1.7 \cdot 10^{-7}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \cos \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.00045)
   (- (cos eps) (cos x))
   (if (<= eps 1.7e-7) (* (sin x) (- eps)) (+ -1.0 (cos eps)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.00045) {
		tmp = cos(eps) - cos(x);
	} else if (eps <= 1.7e-7) {
		tmp = sin(x) * -eps;
	} else {
		tmp = -1.0 + cos(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 <= (-0.00045d0)) then
        tmp = cos(eps) - cos(x)
    else if (eps <= 1.7d-7) then
        tmp = sin(x) * -eps
    else
        tmp = (-1.0d0) + cos(eps)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -0.00045) {
		tmp = Math.cos(eps) - Math.cos(x);
	} else if (eps <= 1.7e-7) {
		tmp = Math.sin(x) * -eps;
	} else {
		tmp = -1.0 + Math.cos(eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -0.00045:
		tmp = math.cos(eps) - math.cos(x)
	elif eps <= 1.7e-7:
		tmp = math.sin(x) * -eps
	else:
		tmp = -1.0 + math.cos(eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -0.00045)
		tmp = Float64(cos(eps) - cos(x));
	elseif (eps <= 1.7e-7)
		tmp = Float64(sin(x) * Float64(-eps));
	else
		tmp = Float64(-1.0 + cos(eps));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -0.00045)
		tmp = cos(eps) - cos(x);
	elseif (eps <= 1.7e-7)
		tmp = sin(x) * -eps;
	else
		tmp = -1.0 + cos(eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -0.00045], N[(N[Cos[eps], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 1.7e-7], N[(N[Sin[x], $MachinePrecision] * (-eps)), $MachinePrecision], N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision]]]

Derivation

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

Alternative 16: 48.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 + \cos \varepsilon\\ t_1 := -0.5 \cdot {\varepsilon}^{2}\\ \mathbf{if}\;\varepsilon \leq -4.6 \cdot 10^{-17}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -2.1 \cdot 10^{-121}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq 7.2 \cdot 10^{-155}:\\ \;\;\;\;\varepsilon \cdot \left(-x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00017:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ -1.0 (cos eps))) (t_1 (* -0.5 (pow eps 2.0))))
   (if (<= eps -4.6e-17)
     t_0
     (if (<= eps -2.1e-121)
       t_1
       (if (<= eps 7.2e-155) (* eps (- x)) (if (<= eps 0.00017) t_1 t_0))))))

C

double code(double x, double eps) {
	double t_0 = -1.0 + cos(eps);
	double t_1 = -0.5 * pow(eps, 2.0);
	double tmp;
	if (eps <= -4.6e-17) {
		tmp = t_0;
	} else if (eps <= -2.1e-121) {
		tmp = t_1;
	} else if (eps <= 7.2e-155) {
		tmp = eps * -x;
	} else if (eps <= 0.00017) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (-1.0d0) + cos(eps)
    t_1 = (-0.5d0) * (eps ** 2.0d0)
    if (eps <= (-4.6d-17)) then
        tmp = t_0
    else if (eps <= (-2.1d-121)) then
        tmp = t_1
    else if (eps <= 7.2d-155) then
        tmp = eps * -x
    else if (eps <= 0.00017d0) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = -1.0 + Math.cos(eps);
	double t_1 = -0.5 * Math.pow(eps, 2.0);
	double tmp;
	if (eps <= -4.6e-17) {
		tmp = t_0;
	} else if (eps <= -2.1e-121) {
		tmp = t_1;
	} else if (eps <= 7.2e-155) {
		tmp = eps * -x;
	} else if (eps <= 0.00017) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = -1.0 + math.cos(eps)
	t_1 = -0.5 * math.pow(eps, 2.0)
	tmp = 0
	if eps <= -4.6e-17:
		tmp = t_0
	elif eps <= -2.1e-121:
		tmp = t_1
	elif eps <= 7.2e-155:
		tmp = eps * -x
	elif eps <= 0.00017:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(-1.0 + cos(eps))
	t_1 = Float64(-0.5 * (eps ^ 2.0))
	tmp = 0.0
	if (eps <= -4.6e-17)
		tmp = t_0;
	elseif (eps <= -2.1e-121)
		tmp = t_1;
	elseif (eps <= 7.2e-155)
		tmp = Float64(eps * Float64(-x));
	elseif (eps <= 0.00017)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = -1.0 + cos(eps);
	t_1 = -0.5 * (eps ^ 2.0);
	tmp = 0.0;
	if (eps <= -4.6e-17)
		tmp = t_0;
	elseif (eps <= -2.1e-121)
		tmp = t_1;
	elseif (eps <= 7.2e-155)
		tmp = eps * -x;
	elseif (eps <= 0.00017)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -4.6e-17], t$95$0, If[LessEqual[eps, -2.1e-121], t$95$1, If[LessEqual[eps, 7.2e-155], N[(eps * (-x)), $MachinePrecision], If[LessEqual[eps, 0.00017], t$95$1, t$95$0]]]]]]

Derivation

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

Alternative 17: 65.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00045 \lor \neg \left(\varepsilon \leq 1.7 \cdot 10^{-7}\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.00045) (not (<= eps 1.7e-7)))
   (+ -1.0 (cos eps))
   (* (sin x) (- eps))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.00045) || !(eps <= 1.7e-7)) {
		tmp = -1.0 + cos(eps);
	} else {
		tmp = sin(x) * -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 <= (-0.00045d0)) .or. (.not. (eps <= 1.7d-7))) then
        tmp = (-1.0d0) + cos(eps)
    else
        tmp = sin(x) * -eps
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.00045) || !(eps <= 1.7e-7)) {
		tmp = -1.0 + Math.cos(eps);
	} else {
		tmp = Math.sin(x) * -eps;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.00045) or not (eps <= 1.7e-7):
		tmp = -1.0 + math.cos(eps)
	else:
		tmp = math.sin(x) * -eps
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.00045) || !(eps <= 1.7e-7))
		tmp = Float64(-1.0 + cos(eps));
	else
		tmp = Float64(sin(x) * Float64(-eps));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.00045) || ~((eps <= 1.7e-7)))
		tmp = -1.0 + cos(eps);
	else
		tmp = sin(x) * -eps;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.00045], N[Not[LessEqual[eps, 1.7e-7]], $MachinePrecision]], N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * (-eps)), $MachinePrecision]]

Derivation

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

Alternative 18: 43.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1 \cdot 10^{-8} \lor \neg \left(\varepsilon \leq 1.95 \cdot 10^{-87}\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -1e-8) (not (<= eps 1.95e-87)))
   (+ -1.0 (cos eps))
   (* eps (- x))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -1e-8) || !(eps <= 1.95e-87)) {
		tmp = -1.0 + cos(eps);
	} else {
		tmp = eps * -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 <= (-1d-8)) .or. (.not. (eps <= 1.95d-87))) then
        tmp = (-1.0d0) + cos(eps)
    else
        tmp = eps * -x
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -1e-8) || !(eps <= 1.95e-87)) {
		tmp = -1.0 + Math.cos(eps);
	} else {
		tmp = eps * -x;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -1e-8) or not (eps <= 1.95e-87):
		tmp = -1.0 + math.cos(eps)
	else:
		tmp = eps * -x
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -1e-8) || !(eps <= 1.95e-87))
		tmp = Float64(-1.0 + cos(eps));
	else
		tmp = Float64(eps * Float64(-x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -1e-8) || ~((eps <= 1.95e-87)))
		tmp = -1.0 + cos(eps);
	else
		tmp = eps * -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -1e-8], N[Not[LessEqual[eps, 1.95e-87]], $MachinePrecision]], N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision], N[(eps * (-x)), $MachinePrecision]]

Derivation

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

Alternative 19: 18.1% accurate, 51.3× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 (* eps (- x)))

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps * -x

Julia

function code(x, eps)
	return Float64(eps * Float64(-x))
end

MATLAB

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

Wolfram

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

Derivation

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

Alternative 20: 12.6% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

double code(double x, double eps) {
	return 0.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 0.0d0
end function

Java

public static double code(double x, double eps) {
	return 0.0;
}

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

function tmp = code(x, eps)
	tmp = 0.0;
end

Wolfram

code[x_, eps_] := 0.0

Derivation

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

Reproduce

herbie shell --seed 2023343 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))