2sin (example 3.3)

Percentage Accurate: 42.0% → 99.4%

Time: 13.0s

Alternatives: 10

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 42.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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Alternative 2: 99.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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Alternative 3: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot \sin \varepsilon - \frac{{\sin \varepsilon}^{2} \cdot \sin x}{1 + \cos \varepsilon} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (cos(x) * sin(eps)) - (((sin(eps) ** 2.0d0) * sin(x)) / (1.0d0 + cos(eps)))
end function

Java

public static double code(double x, double eps) {
	return (Math.cos(x) * Math.sin(eps)) - ((Math.pow(Math.sin(eps), 2.0) * Math.sin(x)) / (1.0 + Math.cos(eps)));
}

Python

def code(x, eps):
	return (math.cos(x) * math.sin(eps)) - ((math.pow(math.sin(eps), 2.0) * math.sin(x)) / (1.0 + math.cos(eps)))

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = (cos(x) * sin(eps)) - (((sin(eps) ^ 2.0) * sin(x)) / (1.0 + cos(eps)));
end

Wolfram

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

Derivation

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Alternative 4: 99.4% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (fma (cos x) (sin eps) (* (sin x) (+ -1.0 (cos eps)))))

C

double code(double x, double eps) {
	return fma(cos(x), sin(eps), (sin(x) * (-1.0 + cos(eps))));
}

Julia

function code(x, eps)
	return fma(cos(x), sin(eps), Float64(sin(x) * Float64(-1.0 + cos(eps))))
end

Wolfram

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

Derivation

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Alternative 5: 99.4% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (+ (* (cos x) (sin eps)) (* (sin x) (+ -1.0 (cos eps)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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Alternative 6: 75.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, eps_] := N[(N[Cos[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

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Alternative 7: 75.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0032 \lor \neg \left(x \leq 0.37\right):\\ \;\;\;\;\cos x \cdot \left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon + x \cdot \left(-1 + \cos \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -0.0032) (not (<= x 0.37)))
   (* (cos x) (* 2.0 (sin (* eps 0.5))))
   (+ (sin eps) (* x (+ -1.0 (cos eps))))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -0.0032) || !(x <= 0.37)) {
		tmp = cos(x) * (2.0 * sin((eps * 0.5)));
	} else {
		tmp = sin(eps) + (x * (-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 ((x <= (-0.0032d0)) .or. (.not. (x <= 0.37d0))) then
        tmp = cos(x) * (2.0d0 * sin((eps * 0.5d0)))
    else
        tmp = sin(eps) + (x * ((-1.0d0) + cos(eps)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -0.0032) || !(x <= 0.37)) {
		tmp = Math.cos(x) * (2.0 * Math.sin((eps * 0.5)));
	} else {
		tmp = Math.sin(eps) + (x * (-1.0 + Math.cos(eps)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -0.0032) or not (x <= 0.37):
		tmp = math.cos(x) * (2.0 * math.sin((eps * 0.5)))
	else:
		tmp = math.sin(eps) + (x * (-1.0 + math.cos(eps)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -0.0032) || !(x <= 0.37))
		tmp = Float64(cos(x) * Float64(2.0 * sin(Float64(eps * 0.5))));
	else
		tmp = Float64(sin(eps) + Float64(x * Float64(-1.0 + cos(eps))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -0.0032) || ~((x <= 0.37)))
		tmp = cos(x) * (2.0 * sin((eps * 0.5)));
	else
		tmp = sin(eps) + (x * (-1.0 + cos(eps)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -0.0032], N[Not[LessEqual[x, 0.37]], $MachinePrecision]], N[(N[Cos[x], $MachinePrecision] * N[(2.0 * N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[eps], $MachinePrecision] + N[(x * N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

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Alternative 8: 75.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0082 \lor \neg \left(\varepsilon \leq 0.00385\right):\\ \;\;\;\;\sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.0082) (not (<= eps 0.00385))) (sin eps) (* (cos x) eps)))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.0082) || !(eps <= 0.00385)) {
		tmp = sin(eps);
	} else {
		tmp = cos(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 <= (-0.0082d0)) .or. (.not. (eps <= 0.00385d0))) then
        tmp = sin(eps)
    else
        tmp = cos(x) * eps
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.0082) || !(eps <= 0.00385)) {
		tmp = Math.sin(eps);
	} else {
		tmp = Math.cos(x) * eps;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.0082) or not (eps <= 0.00385):
		tmp = math.sin(eps)
	else:
		tmp = math.cos(x) * eps
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.0082) || !(eps <= 0.00385))
		tmp = sin(eps);
	else
		tmp = Float64(cos(x) * eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.0082) || ~((eps <= 0.00385)))
		tmp = sin(eps);
	else
		tmp = cos(x) * eps;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.0082], N[Not[LessEqual[eps, 0.00385]], $MachinePrecision]], N[Sin[eps], $MachinePrecision], N[(N[Cos[x], $MachinePrecision] * eps), $MachinePrecision]]

Derivation

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Alternative 9: 54.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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Alternative 10: 29.3% 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

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Developer target: 99.4% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (fma (sin x) (- (cos eps) 1.0) (* (sin eps) (cos x))))

C

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

Julia

function code(x, eps)
	return fma(sin(x), Float64(cos(eps) - 1.0), Float64(sin(eps) * cos(x)))
end

Wolfram

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

Reproduce

herbie shell --seed 2024010 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64

  :herbie-target
  (fma (sin x) (- (cos eps) 1.0) (* (sin eps) (cos x)))

  (- (sin (+ x eps)) (sin x)))