2cos (problem 3.3.5)

Percentage Accurate: 37.4% → 99.4%

Time: 18.8s

Alternatives: 13

Speedup: 1.8×

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: 37.4% 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: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\varepsilon \cdot 0.5\right)\\ \left(t_0 \cdot \cos x + \cos \left(\varepsilon \cdot 0.5\right) \cdot \sin x\right) \cdot \left(t_0 \cdot -2\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (* eps 0.5))))
   (* (+ (* t_0 (cos x)) (* (cos (* eps 0.5)) (sin x))) (* t_0 -2.0))))

C

double code(double x, double eps) {
	double t_0 = sin((eps * 0.5));
	return ((t_0 * cos(x)) + (cos((eps * 0.5)) * sin(x))) * (t_0 * -2.0);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    t_0 = sin((eps * 0.5d0))
    code = ((t_0 * cos(x)) + (cos((eps * 0.5d0)) * sin(x))) * (t_0 * (-2.0d0))
end function

Java

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

Python

def code(x, eps):
	t_0 = math.sin((eps * 0.5))
	return ((t_0 * math.cos(x)) + (math.cos((eps * 0.5)) * math.sin(x))) * (t_0 * -2.0)

Julia

function code(x, eps)
	t_0 = sin(Float64(eps * 0.5))
	return Float64(Float64(Float64(t_0 * cos(x)) + Float64(cos(Float64(eps * 0.5)) * sin(x))) * Float64(t_0 * -2.0))
end

MATLAB

function tmp = code(x, eps)
	t_0 = sin((eps * 0.5));
	tmp = ((t_0 * cos(x)) + (cos((eps * 0.5)) * sin(x))) * (t_0 * -2.0);
end

Wolfram

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

Derivation

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Alternative 2: 98.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.027:\\ \;\;\;\;\mathsf{fma}\left(\cos \varepsilon + -1, \cos x, \sin x \cdot \left(-\sin \varepsilon\right)\right)\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-52}:\\ \;\;\;\;\sin \left(\varepsilon \cdot 0.5\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5 + x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \cos x\right) - \sin x \cdot \sin \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -0.027)
   (fma (+ (cos eps) -1.0) (cos x) (* (sin x) (- (sin eps))))
   (if (<= x 6.8e-52)
     (* (sin (* eps 0.5)) (* -2.0 (sin (+ (* eps 0.5) x))))
     (- (- (* (cos x) (cos eps)) (cos x)) (* (sin x) (sin eps))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -0.027) {
		tmp = fma((cos(eps) + -1.0), cos(x), (sin(x) * -sin(eps)));
	} else if (x <= 6.8e-52) {
		tmp = sin((eps * 0.5)) * (-2.0 * sin(((eps * 0.5) + x)));
	} else {
		tmp = ((cos(x) * cos(eps)) - cos(x)) - (sin(x) * sin(eps));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -0.027)
		tmp = fma(Float64(cos(eps) + -1.0), cos(x), Float64(sin(x) * Float64(-sin(eps))));
	elseif (x <= 6.8e-52)
		tmp = Float64(sin(Float64(eps * 0.5)) * Float64(-2.0 * sin(Float64(Float64(eps * 0.5) + x))));
	else
		tmp = Float64(Float64(Float64(cos(x) * cos(eps)) - cos(x)) - Float64(sin(x) * sin(eps)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -0.027], N[(N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * (-N[Sin[eps], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.8e-52], N[(N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * N[Sin[N[(N[(eps * 0.5), $MachinePrecision] + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

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Alternative 3: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\varepsilon \cdot 0.5\right)\\ -2 \cdot \left(t_0 \cdot \left(t_0 \cdot \cos x + \cos \left(\varepsilon \cdot 0.5\right) \cdot \sin x\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (* eps 0.5))))
   (* -2.0 (* t_0 (+ (* t_0 (cos x)) (* (cos (* eps 0.5)) (sin x)))))))

C

double code(double x, double eps) {
	double t_0 = sin((eps * 0.5));
	return -2.0 * (t_0 * ((t_0 * cos(x)) + (cos((eps * 0.5)) * sin(x))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    t_0 = sin((eps * 0.5d0))
    code = (-2.0d0) * (t_0 * ((t_0 * cos(x)) + (cos((eps * 0.5d0)) * sin(x))))
end function

Java

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

Python

def code(x, eps):
	t_0 = math.sin((eps * 0.5))
	return -2.0 * (t_0 * ((t_0 * math.cos(x)) + (math.cos((eps * 0.5)) * math.sin(x))))

Julia

function code(x, eps)
	t_0 = sin(Float64(eps * 0.5))
	return Float64(-2.0 * Float64(t_0 * Float64(Float64(t_0 * cos(x)) + Float64(cos(Float64(eps * 0.5)) * sin(x)))))
end

MATLAB

function tmp = code(x, eps)
	t_0 = sin((eps * 0.5));
	tmp = -2.0 * (t_0 * ((t_0 * cos(x)) + (cos((eps * 0.5)) * sin(x))));
end

Wolfram

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

Derivation

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Alternative 4: 98.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \varepsilon + -1\\ \mathbf{if}\;x \leq -0.027:\\ \;\;\;\;\mathsf{fma}\left(t_0, \cos x, \sin x \cdot \left(-\sin \varepsilon\right)\right)\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-52}:\\ \;\;\;\;\sin \left(\varepsilon \cdot 0.5\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5 + x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot t_0 - \sin x \cdot \sin \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (cos eps) -1.0)))
   (if (<= x -0.027)
     (fma t_0 (cos x) (* (sin x) (- (sin eps))))
     (if (<= x 6.8e-52)
       (* (sin (* eps 0.5)) (* -2.0 (sin (+ (* eps 0.5) x))))
       (- (* (cos x) t_0) (* (sin x) (sin eps)))))))

C

double code(double x, double eps) {
	double t_0 = cos(eps) + -1.0;
	double tmp;
	if (x <= -0.027) {
		tmp = fma(t_0, cos(x), (sin(x) * -sin(eps)));
	} else if (x <= 6.8e-52) {
		tmp = sin((eps * 0.5)) * (-2.0 * sin(((eps * 0.5) + x)));
	} else {
		tmp = (cos(x) * t_0) - (sin(x) * sin(eps));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(cos(eps) + -1.0)
	tmp = 0.0
	if (x <= -0.027)
		tmp = fma(t_0, cos(x), Float64(sin(x) * Float64(-sin(eps))));
	elseif (x <= 6.8e-52)
		tmp = Float64(sin(Float64(eps * 0.5)) * Float64(-2.0 * sin(Float64(Float64(eps * 0.5) + x))));
	else
		tmp = Float64(Float64(cos(x) * t_0) - Float64(sin(x) * sin(eps)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[x, -0.027], N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * (-N[Sin[eps], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.8e-52], N[(N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * N[Sin[N[(N[(eps * 0.5), $MachinePrecision] + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

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Alternative 5: 98.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.027 \lor \neg \left(x \leq 5.6 \cdot 10^{-52}\right):\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right) - \sin x \cdot \sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\varepsilon \cdot 0.5\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5 + x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -0.027) (not (<= x 5.6e-52)))
   (- (* (cos x) (+ (cos eps) -1.0)) (* (sin x) (sin eps)))
   (* (sin (* eps 0.5)) (* -2.0 (sin (+ (* eps 0.5) x))))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -0.027) || !(x <= 5.6e-52)) {
		tmp = (cos(x) * (cos(eps) + -1.0)) - (sin(x) * sin(eps));
	} else {
		tmp = sin((eps * 0.5)) * (-2.0 * sin(((eps * 0.5) + x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= (-0.027d0)) .or. (.not. (x <= 5.6d-52))) then
        tmp = (cos(x) * (cos(eps) + (-1.0d0))) - (sin(x) * sin(eps))
    else
        tmp = sin((eps * 0.5d0)) * ((-2.0d0) * sin(((eps * 0.5d0) + x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -0.027) || !(x <= 5.6e-52)) {
		tmp = (Math.cos(x) * (Math.cos(eps) + -1.0)) - (Math.sin(x) * Math.sin(eps));
	} else {
		tmp = Math.sin((eps * 0.5)) * (-2.0 * Math.sin(((eps * 0.5) + x)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -0.027) or not (x <= 5.6e-52):
		tmp = (math.cos(x) * (math.cos(eps) + -1.0)) - (math.sin(x) * math.sin(eps))
	else:
		tmp = math.sin((eps * 0.5)) * (-2.0 * math.sin(((eps * 0.5) + x)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -0.027) || !(x <= 5.6e-52))
		tmp = Float64(Float64(cos(x) * Float64(cos(eps) + -1.0)) - Float64(sin(x) * sin(eps)));
	else
		tmp = Float64(sin(Float64(eps * 0.5)) * Float64(-2.0 * sin(Float64(Float64(eps * 0.5) + x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -0.027) || ~((x <= 5.6e-52)))
		tmp = (cos(x) * (cos(eps) + -1.0)) - (sin(x) * sin(eps));
	else
		tmp = sin((eps * 0.5)) * (-2.0 * sin(((eps * 0.5) + x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -0.027], N[Not[LessEqual[x, 5.6e-52]], $MachinePrecision]], N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * N[Sin[N[(N[(eps * 0.5), $MachinePrecision] + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

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1. Add Preprocessing

Alternative 6: 73.8% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (cos (+ eps x)) (cos x))))
   (if (<= t_0 -2e-5) t_0 (- (* -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 <= -2e-5) {
		tmp = t_0;
	} else {
		tmp = (-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 <= (-2d-5)) then
        tmp = t_0
    else
        tmp = ((-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 <= -2e-5) {
		tmp = t_0;
	} else {
		tmp = (-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 <= -2e-5:
		tmp = t_0
	else:
		tmp = (-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 <= -2e-5)
		tmp = t_0;
	else
		tmp = Float64(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 <= -2e-5)
		tmp = t_0;
	else
		tmp = (-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, -2e-5], t$95$0, N[(N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision] - N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

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1. Add Preprocessing

Alternative 7: 65.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\varepsilon + x\right) - \cos x\\ \mathbf{if}\;t_0 \leq -4 \cdot 10^{-14}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-\varepsilon \cdot \sin x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (cos (+ eps x)) (cos x))))
   (if (<= t_0 -4e-14) t_0 (- (* eps (sin x))))))

C

double code(double x, double eps) {
	double t_0 = cos((eps + x)) - cos(x);
	double tmp;
	if (t_0 <= -4e-14) {
		tmp = t_0;
	} else {
		tmp = -(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 <= (-4d-14)) then
        tmp = t_0
    else
        tmp = -(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 <= -4e-14) {
		tmp = t_0;
	} else {
		tmp = -(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 <= -4e-14:
		tmp = t_0
	else:
		tmp = -(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 <= -4e-14)
		tmp = t_0;
	else
		tmp = Float64(-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 <= -4e-14)
		tmp = t_0;
	else
		tmp = -(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, -4e-14], t$95$0, (-N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision])]]

Derivation

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1. Add Preprocessing

Alternative 8: 66.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -6.6 \cdot 10^{-7}:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{elif}\;\varepsilon \leq 2.4 \cdot 10^{-41}:\\ \;\;\;\;-\varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -6.6e-7)
   (- (cos eps) (cos x))
   (if (<= eps 2.4e-41)
     (- (* eps (sin x)))
     (* -2.0 (pow (sin (* eps 0.5)) 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -6.6e-7) {
		tmp = cos(eps) - cos(x);
	} else if (eps <= 2.4e-41) {
		tmp = -(eps * sin(x));
	} 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 <= (-6.6d-7)) then
        tmp = cos(eps) - cos(x)
    else if (eps <= 2.4d-41) then
        tmp = -(eps * sin(x))
    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 <= -6.6e-7) {
		tmp = Math.cos(eps) - Math.cos(x);
	} else if (eps <= 2.4e-41) {
		tmp = -(eps * Math.sin(x));
	} else {
		tmp = -2.0 * Math.pow(Math.sin((eps * 0.5)), 2.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -6.6e-7:
		tmp = math.cos(eps) - math.cos(x)
	elif eps <= 2.4e-41:
		tmp = -(eps * math.sin(x))
	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 <= -6.6e-7)
		tmp = Float64(cos(eps) - cos(x));
	elseif (eps <= 2.4e-41)
		tmp = Float64(-Float64(eps * sin(x)));
	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 <= -6.6e-7)
		tmp = cos(eps) - cos(x);
	elseif (eps <= 2.4e-41)
		tmp = -(eps * sin(x));
	else
		tmp = -2.0 * (sin((eps * 0.5)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -6.6e-7], N[(N[Cos[eps], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 2.4e-41], (-N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision]), N[(-2.0 * N[Power[N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

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1. Add Preprocessing

Alternative 9: 67.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000112 \lor \neg \left(\varepsilon \leq 2.2 \cdot 10^{-7}\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;-\varepsilon \cdot \sin x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.000112) (not (<= eps 2.2e-7)))
   (- (cos eps) (cos x))
   (- (* eps (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.000112) || !(eps <= 2.2e-7)) {
		tmp = cos(eps) - cos(x);
	} else {
		tmp = -(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 <= (-0.000112d0)) .or. (.not. (eps <= 2.2d-7))) then
        tmp = cos(eps) - cos(x)
    else
        tmp = -(eps * sin(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.000112) || !(eps <= 2.2e-7)) {
		tmp = Math.cos(eps) - Math.cos(x);
	} else {
		tmp = -(eps * Math.sin(x));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.000112) or not (eps <= 2.2e-7):
		tmp = math.cos(eps) - math.cos(x)
	else:
		tmp = -(eps * math.sin(x))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.000112) || !(eps <= 2.2e-7))
		tmp = Float64(cos(eps) - cos(x));
	else
		tmp = Float64(-Float64(eps * sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.000112) || ~((eps <= 2.2e-7)))
		tmp = cos(eps) - cos(x);
	else
		tmp = -(eps * sin(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.000112], N[Not[LessEqual[eps, 2.2e-7]], $MachinePrecision]], N[(N[Cos[eps], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], (-N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision])]

Derivation

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1. Add Preprocessing

Alternative 10: 75.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = sin((eps * 0.5d0)) * ((-2.0d0) * sin(((eps * 0.5d0) + x)))
end function

Java

public static double code(double x, double eps) {
	return Math.sin((eps * 0.5)) * (-2.0 * Math.sin(((eps * 0.5) + x)));
}

Python

def code(x, eps):
	return math.sin((eps * 0.5)) * (-2.0 * math.sin(((eps * 0.5) + x)))

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = sin((eps * 0.5)) * (-2.0 * sin(((eps * 0.5) + x)));
end

Wolfram

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

Derivation

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1. Add Preprocessing

Alternative 11: 66.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.7 \cdot 10^{-7} \lor \neg \left(\varepsilon \leq 2.2 \cdot 10^{-7}\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;-\varepsilon \cdot \sin x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -3.7e-7) (not (<= eps 2.2e-7)))
   (+ (cos eps) -1.0)
   (- (* eps (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.7e-7) || !(eps <= 2.2e-7)) {
		tmp = cos(eps) + -1.0;
	} else {
		tmp = -(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 <= (-3.7d-7)) .or. (.not. (eps <= 2.2d-7))) then
        tmp = cos(eps) + (-1.0d0)
    else
        tmp = -(eps * sin(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.7e-7) || !(eps <= 2.2e-7)) {
		tmp = Math.cos(eps) + -1.0;
	} else {
		tmp = -(eps * Math.sin(x));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -3.7e-7) or not (eps <= 2.2e-7):
		tmp = math.cos(eps) + -1.0
	else:
		tmp = -(eps * math.sin(x))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -3.7e-7) || !(eps <= 2.2e-7))
		tmp = Float64(cos(eps) + -1.0);
	else
		tmp = Float64(-Float64(eps * sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -3.7e-7) || ~((eps <= 2.2e-7)))
		tmp = cos(eps) + -1.0;
	else
		tmp = -(eps * sin(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -3.7e-7], N[Not[LessEqual[eps, 2.2e-7]], $MachinePrecision]], N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision], (-N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision])]

Derivation

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1. Add Preprocessing

Alternative 12: 43.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.75 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 2.5 \cdot 10^{-35}\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -1.75e-9) (not (<= eps 2.5e-35)))
   (+ (cos eps) -1.0)
   (* eps (- x))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.75e-9) || !(eps <= 2.5e-35)) {
		tmp = cos(eps) + -1.0;
	} 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 <= (-1.75d-9)) .or. (.not. (eps <= 2.5d-35))) then
        tmp = cos(eps) + (-1.0d0)
    else
        tmp = eps * -x
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.75e-9) || !(eps <= 2.5e-35)) {
		tmp = Math.cos(eps) + -1.0;
	} else {
		tmp = eps * -x;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -1.75e-9) or not (eps <= 2.5e-35):
		tmp = math.cos(eps) + -1.0
	else:
		tmp = eps * -x
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -1.75e-9) || !(eps <= 2.5e-35))
		tmp = Float64(cos(eps) + -1.0);
	else
		tmp = Float64(eps * Float64(-x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -1.75e-9) || ~((eps <= 2.5e-35)))
		tmp = cos(eps) + -1.0;
	else
		tmp = eps * -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -1.75e-9], N[Not[LessEqual[eps, 2.5e-35]], $MachinePrecision]], N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision], N[(eps * (-x)), $MachinePrecision]]

Derivation

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1. Add Preprocessing

Alternative 13: 17.8% 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

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1. Add Preprocessing

Reproduce

herbie shell --seed 2024006 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))