2cos (problem 3.3.5)

Percentage Accurate: 38.6% → 99.3%

Time: 24.3s

Alternatives: 16

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: 38.6% 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: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\sin \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.0027:\\ \;\;\;\;\mathsf{fma}\left(\sin x, t_0, \cos x \cdot \left(-1 + \cos \varepsilon\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0019:\\ \;\;\;\;\mathsf{fma}\left(0.041666666666666664, \cos x \cdot {\varepsilon}^{4}, -0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin x \cdot t_0\right) - \cos x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (sin eps))))
   (if (<= eps -0.0027)
     (fma (sin x) t_0 (* (cos x) (+ -1.0 (cos eps))))
     (if (<= eps 0.0019)
       (fma
        0.041666666666666664
        (* (cos x) (pow eps 4.0))
        (+
         (* -0.5 (* eps (* eps (cos x))))
         (* (sin x) (- (* 0.16666666666666666 (pow eps 3.0)) eps))))
       (- (fma (cos x) (cos eps) (* (sin x) t_0)) (cos x))))))

C

double code(double x, double eps) {
	double t_0 = -sin(eps);
	double tmp;
	if (eps <= -0.0027) {
		tmp = fma(sin(x), t_0, (cos(x) * (-1.0 + cos(eps))));
	} else if (eps <= 0.0019) {
		tmp = fma(0.041666666666666664, (cos(x) * pow(eps, 4.0)), ((-0.5 * (eps * (eps * cos(x)))) + (sin(x) * ((0.16666666666666666 * pow(eps, 3.0)) - eps))));
	} else {
		tmp = fma(cos(x), cos(eps), (sin(x) * t_0)) - cos(x);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(-sin(eps))
	tmp = 0.0
	if (eps <= -0.0027)
		tmp = fma(sin(x), t_0, Float64(cos(x) * Float64(-1.0 + cos(eps))));
	elseif (eps <= 0.0019)
		tmp = fma(0.041666666666666664, Float64(cos(x) * (eps ^ 4.0)), Float64(Float64(-0.5 * Float64(eps * Float64(eps * cos(x)))) + Float64(sin(x) * Float64(Float64(0.16666666666666666 * (eps ^ 3.0)) - eps))));
	else
		tmp = Float64(fma(cos(x), cos(eps), Float64(sin(x) * t_0)) - cos(x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = (-N[Sin[eps], $MachinePrecision])}, If[LessEqual[eps, -0.0027], N[(N[Sin[x], $MachinePrecision] * t$95$0 + N[(N[Cos[x], $MachinePrecision] * N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.0019], N[(0.041666666666666664 * N[(N[Cos[x], $MachinePrecision] * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(eps * N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(N[(0.16666666666666666 * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision] - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.0027000000000000001

   1. Initial program 56.5%

      \[\cos \left(x + \varepsilon\right) - \cos x \]
   2. Step-by-step derivation

      1. cos-sum 99.1%

         \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x \]

      2. cancel-sign-sub-inv 99.1%

         \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon + \left(-\sin x\right) \cdot \sin \varepsilon\right)} - \cos x \]

      3. fma-def 99.1%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x, \cos \varepsilon, \left(-\sin x\right) \cdot \sin \varepsilon\right)} - \cos x \]

   3. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x, \cos \varepsilon, \left(-\sin x\right) \cdot \sin \varepsilon\right)} - \cos x \]

   4. Taylor expanded in x around inf 99.1%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\sin x \cdot \sin \varepsilon\right) + \cos \varepsilon \cdot \cos x\right) - \cos x} \]
   5. Step-by-step derivation

      1. neg-mul-1 99.1%

         \[\leadsto \left(\color{blue}{\left(-\sin x \cdot \sin \varepsilon\right)} + \cos \varepsilon \cdot \cos x\right) - \cos x \]

      2. distribute-lft-neg-in 99.1%

         \[\leadsto \left(\color{blue}{\left(-\sin x\right) \cdot \sin \varepsilon} + \cos \varepsilon \cdot \cos x\right) - \cos x \]

      3. associate--l+ 99.1%

         \[\leadsto \color{blue}{\left(-\sin x\right) \cdot \sin \varepsilon + \left(\cos \varepsilon \cdot \cos x - \cos x\right)} \]

      4. *-commutative 99.1%

         \[\leadsto \left(-\sin x\right) \cdot \sin \varepsilon + \left(\color{blue}{\cos x \cdot \cos \varepsilon} - \cos x\right) \]

      5. distribute-lft-neg-in 99.1%

         \[\leadsto \color{blue}{\left(-\sin x \cdot \sin \varepsilon\right)} + \left(\cos x \cdot \cos \varepsilon - \cos x\right) \]

      6. distribute-rgt-neg-in 99.1%

         \[\leadsto \color{blue}{\sin x \cdot \left(-\sin \varepsilon\right)} + \left(\cos x \cdot \cos \varepsilon - \cos x\right) \]

      7. fma-def 99.1%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \cos \varepsilon - \cos x\right)} \]

      8. *-rgt-identity 99.1%

         \[\leadsto \mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \cos \varepsilon - \color{blue}{\cos x \cdot 1}\right) \]

      9. distribute-lft-out-- 99.1%

         \[\leadsto \mathsf{fma}\left(\sin x, -\sin \varepsilon, \color{blue}{\cos x \cdot \left(\cos \varepsilon - 1\right)}\right) \]

      10. sub-neg 99.1%

          \[\leadsto \mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \color{blue}{\left(\cos \varepsilon + \left(-1\right)\right)}\right) \]

      11. metadata-eval 99.1%

          \[\leadsto \mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \left(\cos \varepsilon + \color{blue}{-1}\right)\right) \]

      12. +-commutative 99.1%

          \[\leadsto \mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \color{blue}{\left(-1 + \cos \varepsilon\right)}\right) \]

   6. Simplified 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \left(-1 + \cos \varepsilon\right)\right)} \]

   if -0.0027000000000000001 < eps < 0.0019

   1. Initial program 20.1%

      \[\cos \left(x + \varepsilon\right) - \cos x \]

   2. Taylor expanded in eps around 0 99.8%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + \left(0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + \left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right)\right)} \]
   3. Step-by-step derivation

      1. fma-def 99.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.041666666666666664, {\varepsilon}^{4} \cdot \cos x, 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + \left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right)\right)} \]

      2. *-commutative 99.8%

         \[\leadsto \mathsf{fma}\left(0.041666666666666664, \color{blue}{\cos x \cdot {\varepsilon}^{4}}, 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + \left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right)\right) \]

      3. +-commutative 99.8%

         \[\leadsto \mathsf{fma}\left(0.041666666666666664, \cos x \cdot {\varepsilon}^{4}, \color{blue}{\left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)}\right) \]

      4. associate-+l+ 99.8%

         \[\leadsto \mathsf{fma}\left(0.041666666666666664, \cos x \cdot {\varepsilon}^{4}, \color{blue}{-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + \left(-1 \cdot \left(\varepsilon \cdot \sin x\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right)}\right) \]

      5. unpow2 99.8%

         \[\leadsto \mathsf{fma}\left(0.041666666666666664, \cos x \cdot {\varepsilon}^{4}, -0.5 \cdot \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \cos x\right) + \left(-1 \cdot \left(\varepsilon \cdot \sin x\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right)\right) \]

      6. associate-*l* 99.8%

         \[\leadsto \mathsf{fma}\left(0.041666666666666664, \cos x \cdot {\varepsilon}^{4}, -0.5 \cdot \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right)} + \left(-1 \cdot \left(\varepsilon \cdot \sin x\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right)\right) \]

      7. associate-*r* 99.8%

         \[\leadsto \mathsf{fma}\left(0.041666666666666664, \cos x \cdot {\varepsilon}^{4}, -0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \left(\color{blue}{\left(-1 \cdot \varepsilon\right) \cdot \sin x} + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right)\right) \]

      8. associate-*r* 99.8%

         \[\leadsto \mathsf{fma}\left(0.041666666666666664, \cos x \cdot {\varepsilon}^{4}, -0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \left(\left(-1 \cdot \varepsilon\right) \cdot \sin x + \color{blue}{\left(0.16666666666666666 \cdot {\varepsilon}^{3}\right) \cdot \sin x}\right)\right) \]

      9. distribute-rgt-out 99.8%

         \[\leadsto \mathsf{fma}\left(0.041666666666666664, \cos x \cdot {\varepsilon}^{4}, -0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \color{blue}{\sin x \cdot \left(-1 \cdot \varepsilon + 0.16666666666666666 \cdot {\varepsilon}^{3}\right)}\right) \]

      10. mul-1-neg 99.8%

          \[\leadsto \mathsf{fma}\left(0.041666666666666664, \cos x \cdot {\varepsilon}^{4}, -0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \sin x \cdot \left(\color{blue}{\left(-\varepsilon\right)} + 0.16666666666666666 \cdot {\varepsilon}^{3}\right)\right) \]

   4. Simplified 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.041666666666666664, \cos x \cdot {\varepsilon}^{4}, -0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \sin x \cdot \left(\left(-\varepsilon\right) + 0.16666666666666666 \cdot {\varepsilon}^{3}\right)\right)} \]

   if 0.0019 < eps

   1. Initial program 46.8%

      \[\cos \left(x + \varepsilon\right) - \cos x \]
   2. Step-by-step derivation

      1. cos-sum 98.9%

         \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x \]

      2. cancel-sign-sub-inv 98.9%

         \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon + \left(-\sin x\right) \cdot \sin \varepsilon\right)} - \cos x \]

      3. fma-def 99.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x, \cos \varepsilon, \left(-\sin x\right) \cdot \sin \varepsilon\right)} - \cos x \]

   3. Applied egg-rr 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x, \cos \varepsilon, \left(-\sin x\right) \cdot \sin \varepsilon\right)} - \cos x \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0027:\\ \;\;\;\;\mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \left(-1 + \cos \varepsilon\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0019:\\ \;\;\;\;\mathsf{fma}\left(0.041666666666666664, \cos x \cdot {\varepsilon}^{4}, -0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin x \cdot \left(-\sin \varepsilon\right)\right) - \cos x\\ \end{array} \]

Alternative 2: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\sin \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.000165:\\ \;\;\;\;\mathsf{fma}\left(\sin x, t_0, \cos x \cdot \left(-1 + \cos \varepsilon\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00016:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin x \cdot t_0\right) - \cos x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (sin eps))))
   (if (<= eps -0.000165)
     (fma (sin x) t_0 (* (cos x) (+ -1.0 (cos eps))))
     (if (<= eps 0.00016)
       (+
        (* -0.5 (* eps (* eps (cos x))))
        (* (sin x) (- (* 0.16666666666666666 (pow eps 3.0)) eps)))
       (- (fma (cos x) (cos eps) (* (sin x) t_0)) (cos x))))))

C

double code(double x, double eps) {
	double t_0 = -sin(eps);
	double tmp;
	if (eps <= -0.000165) {
		tmp = fma(sin(x), t_0, (cos(x) * (-1.0 + cos(eps))));
	} else if (eps <= 0.00016) {
		tmp = (-0.5 * (eps * (eps * cos(x)))) + (sin(x) * ((0.16666666666666666 * pow(eps, 3.0)) - eps));
	} else {
		tmp = fma(cos(x), cos(eps), (sin(x) * t_0)) - cos(x);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(-sin(eps))
	tmp = 0.0
	if (eps <= -0.000165)
		tmp = fma(sin(x), t_0, Float64(cos(x) * Float64(-1.0 + cos(eps))));
	elseif (eps <= 0.00016)
		tmp = Float64(Float64(-0.5 * Float64(eps * Float64(eps * cos(x)))) + Float64(sin(x) * Float64(Float64(0.16666666666666666 * (eps ^ 3.0)) - eps)));
	else
		tmp = Float64(fma(cos(x), cos(eps), Float64(sin(x) * t_0)) - cos(x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = (-N[Sin[eps], $MachinePrecision])}, If[LessEqual[eps, -0.000165], N[(N[Sin[x], $MachinePrecision] * t$95$0 + N[(N[Cos[x], $MachinePrecision] * N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.00016], N[(N[(-0.5 * N[(eps * N[(eps * N[Cos[x], $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[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -1.65e-4

   1. Initial program 56.6%

      \[\cos \left(x + \varepsilon\right) - \cos x \]
   2. Step-by-step derivation

      1. cos-sum 98.5%

         \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x \]

      2. cancel-sign-sub-inv 98.5%

         \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon + \left(-\sin x\right) \cdot \sin \varepsilon\right)} - \cos x \]

      3. fma-def 98.5%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x, \cos \varepsilon, \left(-\sin x\right) \cdot \sin \varepsilon\right)} - \cos x \]

   3. Applied egg-rr 98.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x, \cos \varepsilon, \left(-\sin x\right) \cdot \sin \varepsilon\right)} - \cos x \]

   4. Taylor expanded in x around inf 98.5%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\sin x \cdot \sin \varepsilon\right) + \cos \varepsilon \cdot \cos x\right) - \cos x} \]
   5. Step-by-step derivation

      1. neg-mul-1 98.5%

         \[\leadsto \left(\color{blue}{\left(-\sin x \cdot \sin \varepsilon\right)} + \cos \varepsilon \cdot \cos x\right) - \cos x \]

      2. distribute-lft-neg-in 98.5%

         \[\leadsto \left(\color{blue}{\left(-\sin x\right) \cdot \sin \varepsilon} + \cos \varepsilon \cdot \cos x\right) - \cos x \]

      3. associate--l+ 98.5%

         \[\leadsto \color{blue}{\left(-\sin x\right) \cdot \sin \varepsilon + \left(\cos \varepsilon \cdot \cos x - \cos x\right)} \]

      4. *-commutative 98.5%

         \[\leadsto \left(-\sin x\right) \cdot \sin \varepsilon + \left(\color{blue}{\cos x \cdot \cos \varepsilon} - \cos x\right) \]

      5. distribute-lft-neg-in 98.5%

         \[\leadsto \color{blue}{\left(-\sin x \cdot \sin \varepsilon\right)} + \left(\cos x \cdot \cos \varepsilon - \cos x\right) \]

      6. distribute-rgt-neg-in 98.5%

         \[\leadsto \color{blue}{\sin x \cdot \left(-\sin \varepsilon\right)} + \left(\cos x \cdot \cos \varepsilon - \cos x\right) \]

      7. fma-def 98.5%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \cos \varepsilon - \cos x\right)} \]

      8. *-rgt-identity 98.5%

         \[\leadsto \mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \cos \varepsilon - \color{blue}{\cos x \cdot 1}\right) \]

      9. distribute-lft-out-- 98.5%

         \[\leadsto \mathsf{fma}\left(\sin x, -\sin \varepsilon, \color{blue}{\cos x \cdot \left(\cos \varepsilon - 1\right)}\right) \]

      10. sub-neg 98.5%

          \[\leadsto \mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \color{blue}{\left(\cos \varepsilon + \left(-1\right)\right)}\right) \]

      11. metadata-eval 98.5%

          \[\leadsto \mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \left(\cos \varepsilon + \color{blue}{-1}\right)\right) \]

      12. +-commutative 98.5%

          \[\leadsto \mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \color{blue}{\left(-1 + \cos \varepsilon\right)}\right) \]

   6. Simplified 98.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \left(-1 + \cos \varepsilon\right)\right)} \]

   if -1.65e-4 < eps < 1.60000000000000013e-4

   1. Initial program 19.7%

      \[\cos \left(x + \varepsilon\right) - \cos x \]

   2. Taylor expanded in eps around 0 99.7%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + \left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right)} \]
   3. Step-by-step derivation

      1. +-commutative 99.7%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)} \]

      2. associate-+l+ 99.7%

         \[\leadsto \color{blue}{-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + \left(-1 \cdot \left(\varepsilon \cdot \sin x\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right)} \]

      3. unpow2 99.7%

         \[\leadsto -0.5 \cdot \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \cos x\right) + \left(-1 \cdot \left(\varepsilon \cdot \sin x\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right) \]

      4. associate-*l* 99.7%

         \[\leadsto -0.5 \cdot \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right)} + \left(-1 \cdot \left(\varepsilon \cdot \sin x\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right) \]

      5. associate-*r* 99.7%

         \[\leadsto -0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \left(\color{blue}{\left(-1 \cdot \varepsilon\right) \cdot \sin x} + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right) \]

      6. associate-*r* 99.7%

         \[\leadsto -0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \left(\left(-1 \cdot \varepsilon\right) \cdot \sin x + \color{blue}{\left(0.16666666666666666 \cdot {\varepsilon}^{3}\right) \cdot \sin x}\right) \]

      7. distribute-rgt-out 99.7%

         \[\leadsto -0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \color{blue}{\sin x \cdot \left(-1 \cdot \varepsilon + 0.16666666666666666 \cdot {\varepsilon}^{3}\right)} \]

      8. mul-1-neg 99.7%

         \[\leadsto -0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \sin x \cdot \left(\color{blue}{\left(-\varepsilon\right)} + 0.16666666666666666 \cdot {\varepsilon}^{3}\right) \]

   4. Simplified 99.7%

      \[\leadsto \color{blue}{-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \sin x \cdot \left(\left(-\varepsilon\right) + 0.16666666666666666 \cdot {\varepsilon}^{3}\right)} \]

   if 1.60000000000000013e-4 < eps

   1. Initial program 46.8%

      \[\cos \left(x + \varepsilon\right) - \cos x \]
   2. Step-by-step derivation

      1. cos-sum 98.9%

         \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x \]

      2. cancel-sign-sub-inv 98.9%

         \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon + \left(-\sin x\right) \cdot \sin \varepsilon\right)} - \cos x \]

      3. fma-def 99.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x, \cos \varepsilon, \left(-\sin x\right) \cdot \sin \varepsilon\right)} - \cos x \]

   3. Applied egg-rr 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x, \cos \varepsilon, \left(-\sin x\right) \cdot \sin \varepsilon\right)} - \cos x \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000165:\\ \;\;\;\;\mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \left(-1 + \cos \varepsilon\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00016:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin x \cdot \left(-\sin \varepsilon\right)\right) - \cos x\\ \end{array} \]

Alternative 3: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000165:\\ \;\;\;\;\mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \left(-1 + \cos \varepsilon\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 0.000145:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.000165)
   (fma (sin x) (- (sin eps)) (* (cos x) (+ -1.0 (cos eps))))
   (if (<= eps 0.000145)
     (+
      (* -0.5 (* eps (* eps (cos x))))
      (* (sin x) (- (* 0.16666666666666666 (pow eps 3.0)) eps)))
     (- (* (cos x) (cos eps)) (+ (cos x) (* (sin x) (sin eps)))))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.000165) {
		tmp = fma(sin(x), -sin(eps), (cos(x) * (-1.0 + cos(eps))));
	} else if (eps <= 0.000145) {
		tmp = (-0.5 * (eps * (eps * cos(x)))) + (sin(x) * ((0.16666666666666666 * pow(eps, 3.0)) - eps));
	} else {
		tmp = (cos(x) * cos(eps)) - (cos(x) + (sin(x) * sin(eps)));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -0.000165)
		tmp = fma(sin(x), Float64(-sin(eps)), Float64(cos(x) * Float64(-1.0 + cos(eps))));
	elseif (eps <= 0.000145)
		tmp = Float64(Float64(-0.5 * Float64(eps * Float64(eps * cos(x)))) + Float64(sin(x) * Float64(Float64(0.16666666666666666 * (eps ^ 3.0)) - eps)));
	else
		tmp = Float64(Float64(cos(x) * cos(eps)) - Float64(cos(x) + Float64(sin(x) * sin(eps))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -0.000165], N[(N[Sin[x], $MachinePrecision] * (-N[Sin[eps], $MachinePrecision]) + N[(N[Cos[x], $MachinePrecision] * N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.000145], N[(N[(-0.5 * N[(eps * N[(eps * N[Cos[x], $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[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[x], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -1.65e-4

   1. Initial program 56.6%

      \[\cos \left(x + \varepsilon\right) - \cos x \]
   2. Step-by-step derivation

      1. cos-sum 98.5%

         \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x \]

      2. cancel-sign-sub-inv 98.5%

         \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon + \left(-\sin x\right) \cdot \sin \varepsilon\right)} - \cos x \]

      3. fma-def 98.5%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x, \cos \varepsilon, \left(-\sin x\right) \cdot \sin \varepsilon\right)} - \cos x \]

   3. Applied egg-rr 98.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x, \cos \varepsilon, \left(-\sin x\right) \cdot \sin \varepsilon\right)} - \cos x \]

   4. Taylor expanded in x around inf 98.5%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\sin x \cdot \sin \varepsilon\right) + \cos \varepsilon \cdot \cos x\right) - \cos x} \]
   5. Step-by-step derivation

      1. neg-mul-1 98.5%

         \[\leadsto \left(\color{blue}{\left(-\sin x \cdot \sin \varepsilon\right)} + \cos \varepsilon \cdot \cos x\right) - \cos x \]

      2. distribute-lft-neg-in 98.5%

         \[\leadsto \left(\color{blue}{\left(-\sin x\right) \cdot \sin \varepsilon} + \cos \varepsilon \cdot \cos x\right) - \cos x \]

      3. associate--l+ 98.5%

         \[\leadsto \color{blue}{\left(-\sin x\right) \cdot \sin \varepsilon + \left(\cos \varepsilon \cdot \cos x - \cos x\right)} \]

      4. *-commutative 98.5%

         \[\leadsto \left(-\sin x\right) \cdot \sin \varepsilon + \left(\color{blue}{\cos x \cdot \cos \varepsilon} - \cos x\right) \]

      5. distribute-lft-neg-in 98.5%

         \[\leadsto \color{blue}{\left(-\sin x \cdot \sin \varepsilon\right)} + \left(\cos x \cdot \cos \varepsilon - \cos x\right) \]

      6. distribute-rgt-neg-in 98.5%

         \[\leadsto \color{blue}{\sin x \cdot \left(-\sin \varepsilon\right)} + \left(\cos x \cdot \cos \varepsilon - \cos x\right) \]

      7. fma-def 98.5%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \cos \varepsilon - \cos x\right)} \]

      8. *-rgt-identity 98.5%

         \[\leadsto \mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \cos \varepsilon - \color{blue}{\cos x \cdot 1}\right) \]

      9. distribute-lft-out-- 98.5%

         \[\leadsto \mathsf{fma}\left(\sin x, -\sin \varepsilon, \color{blue}{\cos x \cdot \left(\cos \varepsilon - 1\right)}\right) \]

      10. sub-neg 98.5%

          \[\leadsto \mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \color{blue}{\left(\cos \varepsilon + \left(-1\right)\right)}\right) \]

      11. metadata-eval 98.5%

          \[\leadsto \mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \left(\cos \varepsilon + \color{blue}{-1}\right)\right) \]

      12. +-commutative 98.5%

          \[\leadsto \mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \color{blue}{\left(-1 + \cos \varepsilon\right)}\right) \]

   6. Simplified 98.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \left(-1 + \cos \varepsilon\right)\right)} \]

   if -1.65e-4 < eps < 1.45e-4

   1. Initial program 19.7%

      \[\cos \left(x + \varepsilon\right) - \cos x \]

   2. Taylor expanded in eps around 0 99.7%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + \left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right)} \]
   3. Step-by-step derivation

      1. +-commutative 99.7%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)} \]

      2. associate-+l+ 99.7%

         \[\leadsto \color{blue}{-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + \left(-1 \cdot \left(\varepsilon \cdot \sin x\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right)} \]

      3. unpow2 99.7%

         \[\leadsto -0.5 \cdot \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \cos x\right) + \left(-1 \cdot \left(\varepsilon \cdot \sin x\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right) \]

      4. associate-*l* 99.7%

         \[\leadsto -0.5 \cdot \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right)} + \left(-1 \cdot \left(\varepsilon \cdot \sin x\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right) \]

      5. associate-*r* 99.7%

         \[\leadsto -0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \left(\color{blue}{\left(-1 \cdot \varepsilon\right) \cdot \sin x} + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right) \]

      6. associate-*r* 99.7%

         \[\leadsto -0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \left(\left(-1 \cdot \varepsilon\right) \cdot \sin x + \color{blue}{\left(0.16666666666666666 \cdot {\varepsilon}^{3}\right) \cdot \sin x}\right) \]

      7. distribute-rgt-out 99.7%

         \[\leadsto -0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \color{blue}{\sin x \cdot \left(-1 \cdot \varepsilon + 0.16666666666666666 \cdot {\varepsilon}^{3}\right)} \]

      8. mul-1-neg 99.7%

         \[\leadsto -0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \sin x \cdot \left(\color{blue}{\left(-\varepsilon\right)} + 0.16666666666666666 \cdot {\varepsilon}^{3}\right) \]

   4. Simplified 99.7%

      \[\leadsto \color{blue}{-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \sin x \cdot \left(\left(-\varepsilon\right) + 0.16666666666666666 \cdot {\varepsilon}^{3}\right)} \]

   if 1.45e-4 < eps

   1. Initial program 46.8%

      \[\cos \left(x + \varepsilon\right) - \cos x \]
   2. Step-by-step derivation

      1. sub-neg 46.8%

         \[\leadsto \color{blue}{\cos \left(x + \varepsilon\right) + \left(-\cos x\right)} \]

      2. cos-sum 98.9%

         \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} + \left(-\cos x\right) \]

      3. associate-+l- 98.9%

         \[\leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon - \left(-\cos x\right)\right)} \]

      4. fma-neg 98.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x, \cos \varepsilon, -\left(\sin x \cdot \sin \varepsilon - \left(-\cos x\right)\right)\right)} \]

   3. Applied egg-rr 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x, \cos \varepsilon, -\left(\sin x \cdot \sin \varepsilon - \left(-\cos x\right)\right)\right)} \]

   4. Taylor expanded in x around -inf 98.9%

      \[\leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000165:\\ \;\;\;\;\mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \left(-1 + \cos \varepsilon\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 0.000145:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \end{array} \]

Alternative 4: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000165 \lor \neg \left(\varepsilon \leq 0.00013\right):\\ \;\;\;\;\mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \left(-1 + \cos \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.000165) (not (<= eps 0.00013)))
   (fma (sin x) (- (sin eps)) (* (cos x) (+ -1.0 (cos eps))))
   (+
    (* -0.5 (* eps (* eps (cos x))))
    (* (sin x) (- (* 0.16666666666666666 (pow eps 3.0)) eps)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.000165) || !(eps <= 0.00013)) {
		tmp = fma(sin(x), -sin(eps), (cos(x) * (-1.0 + cos(eps))));
	} else {
		tmp = (-0.5 * (eps * (eps * cos(x)))) + (sin(x) * ((0.16666666666666666 * pow(eps, 3.0)) - eps));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.000165) || !(eps <= 0.00013))
		tmp = fma(sin(x), Float64(-sin(eps)), Float64(cos(x) * Float64(-1.0 + cos(eps))));
	else
		tmp = Float64(Float64(-0.5 * Float64(eps * Float64(eps * cos(x)))) + Float64(sin(x) * Float64(Float64(0.16666666666666666 * (eps ^ 3.0)) - eps)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.000165], N[Not[LessEqual[eps, 0.00013]], $MachinePrecision]], N[(N[Sin[x], $MachinePrecision] * (-N[Sin[eps], $MachinePrecision]) + N[(N[Cos[x], $MachinePrecision] * N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(eps * N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(N[(0.16666666666666666 * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision] - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.65e-4 or 1.29999999999999989e-4 < eps

   1. Initial program 51.5%

      \[\cos \left(x + \varepsilon\right) - \cos x \]
   2. Step-by-step derivation

      1. cos-sum 98.7%

         \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x \]

      2. cancel-sign-sub-inv 98.7%

         \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon + \left(-\sin x\right) \cdot \sin \varepsilon\right)} - \cos x \]

      3. fma-def 98.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x, \cos \varepsilon, \left(-\sin x\right) \cdot \sin \varepsilon\right)} - \cos x \]

   3. Applied egg-rr 98.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x, \cos \varepsilon, \left(-\sin x\right) \cdot \sin \varepsilon\right)} - \cos x \]

   4. Taylor expanded in x around inf 98.7%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\sin x \cdot \sin \varepsilon\right) + \cos \varepsilon \cdot \cos x\right) - \cos x} \]
   5. Step-by-step derivation

      1. neg-mul-1 98.7%

         \[\leadsto \left(\color{blue}{\left(-\sin x \cdot \sin \varepsilon\right)} + \cos \varepsilon \cdot \cos x\right) - \cos x \]

      2. distribute-lft-neg-in 98.7%

         \[\leadsto \left(\color{blue}{\left(-\sin x\right) \cdot \sin \varepsilon} + \cos \varepsilon \cdot \cos x\right) - \cos x \]

      3. associate--l+ 98.7%

         \[\leadsto \color{blue}{\left(-\sin x\right) \cdot \sin \varepsilon + \left(\cos \varepsilon \cdot \cos x - \cos x\right)} \]

      4. *-commutative 98.7%

         \[\leadsto \left(-\sin x\right) \cdot \sin \varepsilon + \left(\color{blue}{\cos x \cdot \cos \varepsilon} - \cos x\right) \]

      5. distribute-lft-neg-in 98.7%

         \[\leadsto \color{blue}{\left(-\sin x \cdot \sin \varepsilon\right)} + \left(\cos x \cdot \cos \varepsilon - \cos x\right) \]

      6. distribute-rgt-neg-in 98.7%

         \[\leadsto \color{blue}{\sin x \cdot \left(-\sin \varepsilon\right)} + \left(\cos x \cdot \cos \varepsilon - \cos x\right) \]

      7. fma-def 98.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \cos \varepsilon - \cos x\right)} \]

      8. *-rgt-identity 98.7%

         \[\leadsto \mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \cos \varepsilon - \color{blue}{\cos x \cdot 1}\right) \]

      9. distribute-lft-out-- 98.7%

         \[\leadsto \mathsf{fma}\left(\sin x, -\sin \varepsilon, \color{blue}{\cos x \cdot \left(\cos \varepsilon - 1\right)}\right) \]

      10. sub-neg 98.7%

          \[\leadsto \mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \color{blue}{\left(\cos \varepsilon + \left(-1\right)\right)}\right) \]

      11. metadata-eval 98.7%

          \[\leadsto \mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \left(\cos \varepsilon + \color{blue}{-1}\right)\right) \]

      12. +-commutative 98.7%

          \[\leadsto \mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \color{blue}{\left(-1 + \cos \varepsilon\right)}\right) \]

   6. Simplified 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \left(-1 + \cos \varepsilon\right)\right)} \]

   if -1.65e-4 < eps < 1.29999999999999989e-4

   1. Initial program 19.7%

      \[\cos \left(x + \varepsilon\right) - \cos x \]

   2. Taylor expanded in eps around 0 99.7%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + \left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right)} \]
   3. Step-by-step derivation

      1. +-commutative 99.7%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)} \]

      2. associate-+l+ 99.7%

         \[\leadsto \color{blue}{-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + \left(-1 \cdot \left(\varepsilon \cdot \sin x\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right)} \]

      3. unpow2 99.7%

         \[\leadsto -0.5 \cdot \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \cos x\right) + \left(-1 \cdot \left(\varepsilon \cdot \sin x\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right) \]

      4. associate-*l* 99.7%

         \[\leadsto -0.5 \cdot \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right)} + \left(-1 \cdot \left(\varepsilon \cdot \sin x\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right) \]

      5. associate-*r* 99.7%

         \[\leadsto -0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \left(\color{blue}{\left(-1 \cdot \varepsilon\right) \cdot \sin x} + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right) \]

      6. associate-*r* 99.7%

         \[\leadsto -0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \left(\left(-1 \cdot \varepsilon\right) \cdot \sin x + \color{blue}{\left(0.16666666666666666 \cdot {\varepsilon}^{3}\right) \cdot \sin x}\right) \]

      7. distribute-rgt-out 99.7%

         \[\leadsto -0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \color{blue}{\sin x \cdot \left(-1 \cdot \varepsilon + 0.16666666666666666 \cdot {\varepsilon}^{3}\right)} \]

      8. mul-1-neg 99.7%

         \[\leadsto -0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \sin x \cdot \left(\color{blue}{\left(-\varepsilon\right)} + 0.16666666666666666 \cdot {\varepsilon}^{3}\right) \]

   4. Simplified 99.7%

      \[\leadsto \color{blue}{-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \sin x \cdot \left(\left(-\varepsilon\right) + 0.16666666666666666 \cdot {\varepsilon}^{3}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000165 \lor \neg \left(\varepsilon \leq 0.00013\right):\\ \;\;\;\;\mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \left(-1 + \cos \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\\ \end{array} \]

Alternative 5: 66.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\varepsilon + x\right) - \cos x\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (cos (+ eps x)) (cos x))))
   (if (<= t_0 -2e-8) 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 <= -2e-8) {
		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 <= (-2d-8)) 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 <= -2e-8) {
		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 <= -2e-8:
		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 <= -2e-8)
		tmp = t_0;
	else
		tmp = Float64(eps * Float64(-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-8)
		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, -2e-8], t$95$0, N[(eps * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cos.f64 (+.f64 x eps)) (cos.f64 x)) < -2e-8

   1. Initial program 80.4%

      \[\cos \left(x + \varepsilon\right) - \cos x \]

   if -2e-8 < (-.f64 (cos.f64 (+.f64 x eps)) (cos.f64 x))

   1. Initial program 15.9%

      \[\cos \left(x + \varepsilon\right) - \cos x \]

   2. Taylor expanded in eps around 0 57.6%

      \[\leadsto \color{blue}{-1 \cdot \left(\varepsilon \cdot \sin x\right)} \]
   3. Step-by-step derivation

      1. associate-*r* 57.6%

         \[\leadsto \color{blue}{\left(-1 \cdot \varepsilon\right) \cdot \sin x} \]

      2. mul-1-neg 57.6%

         \[\leadsto \color{blue}{\left(-\varepsilon\right)} \cdot \sin x \]

   4. Simplified 57.6%

      \[\leadsto \color{blue}{\left(-\varepsilon\right) \cdot \sin x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\varepsilon + x\right) - \cos x \leq -2 \cdot 10^{-8}:\\ \;\;\;\;\cos \left(\varepsilon + x\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \]

Alternative 6: 77.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.02 \lor \neg \left(\varepsilon \leq 0.66\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) - \varepsilon \cdot \sin x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.02) (not (<= eps 0.66)))
   (- (cos eps) (cos x))
   (- (* -0.5 (* eps (* eps (cos x)))) (* eps (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.02) || !(eps <= 0.66)) {
		tmp = cos(eps) - cos(x);
	} else {
		tmp = (-0.5 * (eps * (eps * cos(x)))) - (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 <= (-0.02d0)) .or. (.not. (eps <= 0.66d0))) then
        tmp = cos(eps) - cos(x)
    else
        tmp = ((-0.5d0) * (eps * (eps * cos(x)))) - (eps * sin(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.02) || !(eps <= 0.66)) {
		tmp = Math.cos(eps) - Math.cos(x);
	} else {
		tmp = (-0.5 * (eps * (eps * Math.cos(x)))) - (eps * Math.sin(x));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.02) or not (eps <= 0.66):
		tmp = math.cos(eps) - math.cos(x)
	else:
		tmp = (-0.5 * (eps * (eps * math.cos(x)))) - (eps * math.sin(x))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.02) || !(eps <= 0.66))
		tmp = Float64(cos(eps) - cos(x));
	else
		tmp = Float64(Float64(-0.5 * Float64(eps * Float64(eps * cos(x)))) - Float64(eps * sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.02) || ~((eps <= 0.66)))
		tmp = cos(eps) - cos(x);
	else
		tmp = (-0.5 * (eps * (eps * cos(x)))) - (eps * sin(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.02], N[Not[LessEqual[eps, 0.66]], $MachinePrecision]], N[(N[Cos[eps], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(eps * N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -0.0200000000000000004 or 0.660000000000000031 < eps

   1. Initial program 52.1%

      \[\cos \left(x + \varepsilon\right) - \cos x \]

   2. Taylor expanded in x around 0 54.1%

      \[\leadsto \color{blue}{\cos \varepsilon} - \cos x \]

   if -0.0200000000000000004 < eps < 0.660000000000000031

   1. Initial program 19.8%

      \[\cos \left(x + \varepsilon\right) - \cos x \]

   2. Taylor expanded in eps around 0 98.2%

      \[\leadsto \color{blue}{-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)} \]
   3. Step-by-step derivation

      1. mul-1-neg 98.2%

         \[\leadsto -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + \color{blue}{\left(-\varepsilon \cdot \sin x\right)} \]

      2. unsub-neg 98.2%

         \[\leadsto \color{blue}{-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) - \varepsilon \cdot \sin x} \]

      3. unpow2 98.2%

         \[\leadsto -0.5 \cdot \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \cos x\right) - \varepsilon \cdot \sin x \]

      4. associate-*l* 98.2%

         \[\leadsto -0.5 \cdot \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right)} - \varepsilon \cdot \sin x \]

   4. Simplified 98.2%

      \[\leadsto \color{blue}{-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) - \varepsilon \cdot \sin x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.02 \lor \neg \left(\varepsilon \leq 0.66\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) - \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 7: 77.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.1%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
2. Step-by-step derivation

   1. diff-cos 44.9%

      \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)} \]

   2. div-inv 44.9%

      \[\leadsto -2 \cdot \left(\sin \color{blue}{\left(\left(\left(x + \varepsilon\right) - x\right) \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]

   3. metadata-eval 44.9%

      \[\leadsto -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot \color{blue}{0.5}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]

   4. div-inv 44.9%

      \[\leadsto -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \color{blue}{\left(\left(\left(x + \varepsilon\right) + x\right) \cdot \frac{1}{2}\right)}\right) \]

   5. +-commutative 44.9%

      \[\leadsto -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \left(\color{blue}{\left(x + \left(x + \varepsilon\right)\right)} \cdot \frac{1}{2}\right)\right) \]

   6. metadata-eval 44.9%

      \[\leadsto -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot \color{blue}{0.5}\right)\right) \]

3. Applied egg-rr 44.9%

   \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative 44.9%

      \[\leadsto -2 \cdot \left(\sin \color{blue}{\left(0.5 \cdot \left(\left(x + \varepsilon\right) - x\right)\right)} \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]

   2. +-commutative 44.9%

      \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \left(\color{blue}{\left(\varepsilon + x\right)} - x\right)\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]

   3. associate--l+ 75.0%

      \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \color{blue}{\left(\varepsilon + \left(x - x\right)\right)}\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]

   4. +-inverses 75.0%

      \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon + \color{blue}{0}\right)\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]

   5. distribute-lft-in 75.0%

      \[\leadsto -2 \cdot \left(\sin \color{blue}{\left(0.5 \cdot \varepsilon + 0.5 \cdot 0\right)} \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]

   6. metadata-eval 75.0%

      \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + \color{blue}{0}\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]

   7. *-commutative 75.0%

      \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \sin \color{blue}{\left(0.5 \cdot \left(x + \left(x + \varepsilon\right)\right)\right)}\right) \]

   8. +-commutative 75.0%

      \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \sin \left(0.5 \cdot \left(x + \color{blue}{\left(\varepsilon + x\right)}\right)\right)\right) \]

5. Simplified 75.0%

   \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \sin \left(0.5 \cdot \left(x + \left(\varepsilon + x\right)\right)\right)\right)} \]
6. Step-by-step derivation

   1. log1p-expm1-u 75.0%

      \[\leadsto -2 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \sin \left(0.5 \cdot \left(x + \left(\varepsilon + x\right)\right)\right)\right)\right)} \]

   2. *-commutative 75.0%

      \[\leadsto -2 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\sin \left(0.5 \cdot \left(x + \left(\varepsilon + x\right)\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon + 0\right)}\right)\right) \]

   3. +-commutative 75.0%

      \[\leadsto -2 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(0.5 \cdot \left(x + \color{blue}{\left(x + \varepsilon\right)}\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon + 0\right)\right)\right) \]

   4. +-rgt-identity 75.0%

      \[\leadsto -2 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(0.5 \cdot \left(x + \left(x + \varepsilon\right)\right)\right) \cdot \sin \color{blue}{\left(0.5 \cdot \varepsilon\right)}\right)\right) \]

7. Applied egg-rr 75.0%

   \[\leadsto -2 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(0.5 \cdot \left(x + \left(x + \varepsilon\right)\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right)\right)} \]
8. Step-by-step derivation

   1. log1p-expm1-u 75.0%

      \[\leadsto -2 \cdot \color{blue}{\left(\sin \left(0.5 \cdot \left(x + \left(x + \varepsilon\right)\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right)} \]

   2. +-commutative 75.0%

      \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \left(x + \color{blue}{\left(\varepsilon + x\right)}\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]

9. Applied egg-rr 75.0%

   \[\leadsto -2 \cdot \color{blue}{\left(\sin \left(0.5 \cdot \left(x + \left(\varepsilon + x\right)\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right)} \]

10. Final simplification 75.0%

    \[\leadsto -2 \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \left(x + \left(\varepsilon + x\right)\right)\right)\right) \]

Alternative 8: 77.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.1%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
2. Step-by-step derivation

   1. diff-cos 44.9%

      \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)} \]

   2. div-inv 44.9%

      \[\leadsto -2 \cdot \left(\sin \color{blue}{\left(\left(\left(x + \varepsilon\right) - x\right) \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]

   3. metadata-eval 44.9%

      \[\leadsto -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot \color{blue}{0.5}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]

   4. div-inv 44.9%

      \[\leadsto -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \color{blue}{\left(\left(\left(x + \varepsilon\right) + x\right) \cdot \frac{1}{2}\right)}\right) \]

   5. +-commutative 44.9%

      \[\leadsto -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \left(\color{blue}{\left(x + \left(x + \varepsilon\right)\right)} \cdot \frac{1}{2}\right)\right) \]

   6. metadata-eval 44.9%

      \[\leadsto -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot \color{blue}{0.5}\right)\right) \]

3. Applied egg-rr 44.9%

   \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative 44.9%

      \[\leadsto -2 \cdot \left(\sin \color{blue}{\left(0.5 \cdot \left(\left(x + \varepsilon\right) - x\right)\right)} \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]

   2. +-commutative 44.9%

      \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \left(\color{blue}{\left(\varepsilon + x\right)} - x\right)\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]

   3. associate--l+ 75.0%

      \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \color{blue}{\left(\varepsilon + \left(x - x\right)\right)}\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]

   4. +-inverses 75.0%

      \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon + \color{blue}{0}\right)\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]

   5. distribute-lft-in 75.0%

      \[\leadsto -2 \cdot \left(\sin \color{blue}{\left(0.5 \cdot \varepsilon + 0.5 \cdot 0\right)} \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]

   6. metadata-eval 75.0%

      \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + \color{blue}{0}\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]

   7. *-commutative 75.0%

      \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \sin \color{blue}{\left(0.5 \cdot \left(x + \left(x + \varepsilon\right)\right)\right)}\right) \]

   8. +-commutative 75.0%

      \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \sin \left(0.5 \cdot \left(x + \color{blue}{\left(\varepsilon + x\right)}\right)\right)\right) \]

5. Simplified 75.0%

   \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \sin \left(0.5 \cdot \left(x + \left(\varepsilon + x\right)\right)\right)\right)} \]

6. Taylor expanded in x around -inf 75.0%

   \[\leadsto -2 \cdot \color{blue}{\left(\sin \left(0.5 \cdot \left(\varepsilon - -2 \cdot x\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right)} \]

7. Final simplification 75.0%

   \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon - x \cdot -2\right)\right) \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]

Alternative 9: 67.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \varepsilon - \cos x\\ t_1 := \varepsilon \cdot \left(-\sin x\right)\\ \mathbf{if}\;\varepsilon \leq -2.4 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 2.6 \cdot 10^{-55}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq 7 \cdot 10^{-13}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \mathbf{elif}\;\varepsilon \leq 0.66:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (cos eps) (cos x))) (t_1 (* eps (- (sin x)))))
   (if (<= eps -2.4e-5)
     t_0
     (if (<= eps 2.6e-55)
       t_1
       (if (<= eps 7e-13) (* eps (* eps -0.5)) (if (<= eps 0.66) t_1 t_0))))))

C

double code(double x, double eps) {
	double t_0 = cos(eps) - cos(x);
	double t_1 = eps * -sin(x);
	double tmp;
	if (eps <= -2.4e-5) {
		tmp = t_0;
	} else if (eps <= 2.6e-55) {
		tmp = t_1;
	} else if (eps <= 7e-13) {
		tmp = eps * (eps * -0.5);
	} else if (eps <= 0.66) {
		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 = cos(eps) - cos(x)
    t_1 = eps * -sin(x)
    if (eps <= (-2.4d-5)) then
        tmp = t_0
    else if (eps <= 2.6d-55) then
        tmp = t_1
    else if (eps <= 7d-13) then
        tmp = eps * (eps * (-0.5d0))
    else if (eps <= 0.66d0) 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 = Math.cos(eps) - Math.cos(x);
	double t_1 = eps * -Math.sin(x);
	double tmp;
	if (eps <= -2.4e-5) {
		tmp = t_0;
	} else if (eps <= 2.6e-55) {
		tmp = t_1;
	} else if (eps <= 7e-13) {
		tmp = eps * (eps * -0.5);
	} else if (eps <= 0.66) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.cos(eps) - math.cos(x)
	t_1 = eps * -math.sin(x)
	tmp = 0
	if eps <= -2.4e-5:
		tmp = t_0
	elif eps <= 2.6e-55:
		tmp = t_1
	elif eps <= 7e-13:
		tmp = eps * (eps * -0.5)
	elif eps <= 0.66:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(cos(eps) - cos(x))
	t_1 = Float64(eps * Float64(-sin(x)))
	tmp = 0.0
	if (eps <= -2.4e-5)
		tmp = t_0;
	elseif (eps <= 2.6e-55)
		tmp = t_1;
	elseif (eps <= 7e-13)
		tmp = Float64(eps * Float64(eps * -0.5));
	elseif (eps <= 0.66)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = cos(eps) - cos(x);
	t_1 = eps * -sin(x);
	tmp = 0.0;
	if (eps <= -2.4e-5)
		tmp = t_0;
	elseif (eps <= 2.6e-55)
		tmp = t_1;
	elseif (eps <= 7e-13)
		tmp = eps * (eps * -0.5);
	elseif (eps <= 0.66)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Cos[eps], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(eps * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]}, If[LessEqual[eps, -2.4e-5], t$95$0, If[LessEqual[eps, 2.6e-55], t$95$1, If[LessEqual[eps, 7e-13], N[(eps * N[(eps * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.66], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -2.4000000000000001e-5 or 0.660000000000000031 < eps

   1. Initial program 51.8%

      \[\cos \left(x + \varepsilon\right) - \cos x \]

   2. Taylor expanded in x around 0 53.8%

      \[\leadsto \color{blue}{\cos \varepsilon} - \cos x \]

   if -2.4000000000000001e-5 < eps < 2.5999999999999999e-55 or 7.0000000000000005e-13 < eps < 0.660000000000000031

   1. Initial program 20.7%

      \[\cos \left(x + \varepsilon\right) - \cos x \]

   2. Taylor expanded in eps around 0 84.0%

      \[\leadsto \color{blue}{-1 \cdot \left(\varepsilon \cdot \sin x\right)} \]
   3. Step-by-step derivation

      1. associate-*r* 84.0%

         \[\leadsto \color{blue}{\left(-1 \cdot \varepsilon\right) \cdot \sin x} \]

      2. mul-1-neg 84.0%

         \[\leadsto \color{blue}{\left(-\varepsilon\right)} \cdot \sin x \]

   4. Simplified 84.0%

      \[\leadsto \color{blue}{\left(-\varepsilon\right) \cdot \sin x} \]

   if 2.5999999999999999e-55 < eps < 7.0000000000000005e-13

   1. Initial program 3.7%

      \[\cos \left(x + \varepsilon\right) - \cos x \]

   2. Taylor expanded in x around 0 3.7%

      \[\leadsto \color{blue}{\cos \varepsilon - 1} \]

   3. Taylor expanded in eps around 0 87.7%

      \[\leadsto \color{blue}{-0.5 \cdot {\varepsilon}^{2}} \]
   4. Step-by-step derivation

      1. *-commutative 87.7%

         \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot -0.5} \]

      2. unpow2 87.7%

         \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot -0.5 \]

      3. associate-*l* 87.7%

         \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)} \]

   5. Simplified 87.7%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.4 \cdot 10^{-5}:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{elif}\;\varepsilon \leq 2.6 \cdot 10^{-55}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{elif}\;\varepsilon \leq 7 \cdot 10^{-13}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \mathbf{elif}\;\varepsilon \leq 0.66:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \end{array} \]

Alternative 10: 71.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\varepsilon \cdot 0.5\right)\\ \mathbf{if}\;x \leq -2.95 \cdot 10^{-18} \lor \neg \left(x \leq 1.5 \cdot 10^{-27}\right):\\ \;\;\;\;-2 \cdot \left(\sin x \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {t_0}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (* eps 0.5))))
   (if (or (<= x -2.95e-18) (not (<= x 1.5e-27)))
     (* -2.0 (* (sin x) 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 <= -2.95e-18) || !(x <= 1.5e-27)) {
		tmp = -2.0 * (sin(x) * 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 <= (-2.95d-18)) .or. (.not. (x <= 1.5d-27))) then
        tmp = (-2.0d0) * (sin(x) * 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 <= -2.95e-18) || !(x <= 1.5e-27)) {
		tmp = -2.0 * (Math.sin(x) * 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 <= -2.95e-18) or not (x <= 1.5e-27):
		tmp = -2.0 * (math.sin(x) * 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 <= -2.95e-18) || !(x <= 1.5e-27))
		tmp = Float64(-2.0 * Float64(sin(x) * 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 <= -2.95e-18) || ~((x <= 1.5e-27)))
		tmp = -2.0 * (sin(x) * 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, -2.95e-18], N[Not[LessEqual[x, 1.5e-27]], $MachinePrecision]], N[(-2.0 * N[(N[Sin[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.9500000000000001e-18 or 1.5000000000000001e-27 < x

   1. Initial program 10.2%

      \[\cos \left(x + \varepsilon\right) - \cos x \]
   2. Step-by-step derivation

      1. diff-cos 9.3%

         \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)} \]

      2. div-inv 9.3%

         \[\leadsto -2 \cdot \left(\sin \color{blue}{\left(\left(\left(x + \varepsilon\right) - x\right) \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]

      3. metadata-eval 9.3%

         \[\leadsto -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot \color{blue}{0.5}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]

      4. div-inv 9.3%

         \[\leadsto -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \color{blue}{\left(\left(\left(x + \varepsilon\right) + x\right) \cdot \frac{1}{2}\right)}\right) \]

      5. +-commutative 9.3%

         \[\leadsto -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \left(\color{blue}{\left(x + \left(x + \varepsilon\right)\right)} \cdot \frac{1}{2}\right)\right) \]

      6. metadata-eval 9.3%

         \[\leadsto -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot \color{blue}{0.5}\right)\right) \]

   3. Applied egg-rr 9.3%

      \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative 9.3%

         \[\leadsto -2 \cdot \left(\sin \color{blue}{\left(0.5 \cdot \left(\left(x + \varepsilon\right) - x\right)\right)} \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]

      2. +-commutative 9.3%

         \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \left(\color{blue}{\left(\varepsilon + x\right)} - x\right)\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]

      3. associate--l+ 56.4%

         \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \color{blue}{\left(\varepsilon + \left(x - x\right)\right)}\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]

      4. +-inverses 56.4%

         \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon + \color{blue}{0}\right)\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]

      5. distribute-lft-in 56.4%

         \[\leadsto -2 \cdot \left(\sin \color{blue}{\left(0.5 \cdot \varepsilon + 0.5 \cdot 0\right)} \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]

      6. metadata-eval 56.4%

         \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + \color{blue}{0}\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]

      7. *-commutative 56.4%

         \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \sin \color{blue}{\left(0.5 \cdot \left(x + \left(x + \varepsilon\right)\right)\right)}\right) \]

      8. +-commutative 56.4%

         \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \sin \left(0.5 \cdot \left(x + \color{blue}{\left(\varepsilon + x\right)}\right)\right)\right) \]

   5. Simplified 56.4%

      \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \sin \left(0.5 \cdot \left(x + \left(\varepsilon + x\right)\right)\right)\right)} \]

   6. Taylor expanded in x around -inf 56.4%

      \[\leadsto -2 \cdot \color{blue}{\left(\sin \left(0.5 \cdot \left(\varepsilon - -2 \cdot x\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right)} \]

   7. Taylor expanded in eps around 0 52.9%

      \[\leadsto -2 \cdot \left(\color{blue}{\sin x} \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]

   if -2.9500000000000001e-18 < x < 1.5000000000000001e-27

   1. Initial program 70.5%

      \[\cos \left(x + \varepsilon\right) - \cos x \]
   2. Step-by-step derivation

      1. diff-cos 92.2%

         \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)} \]

      2. div-inv 92.2%

         \[\leadsto -2 \cdot \left(\sin \color{blue}{\left(\left(\left(x + \varepsilon\right) - x\right) \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]

      3. metadata-eval 92.2%

         \[\leadsto -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot \color{blue}{0.5}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]

      4. div-inv 92.2%

         \[\leadsto -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \color{blue}{\left(\left(\left(x + \varepsilon\right) + x\right) \cdot \frac{1}{2}\right)}\right) \]

      5. +-commutative 92.2%

         \[\leadsto -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \left(\color{blue}{\left(x + \left(x + \varepsilon\right)\right)} \cdot \frac{1}{2}\right)\right) \]

      6. metadata-eval 92.2%

         \[\leadsto -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot \color{blue}{0.5}\right)\right) \]

   3. Applied egg-rr 92.2%

      \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative 92.2%

         \[\leadsto -2 \cdot \left(\sin \color{blue}{\left(0.5 \cdot \left(\left(x + \varepsilon\right) - x\right)\right)} \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]

      2. +-commutative 92.2%

         \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \left(\color{blue}{\left(\varepsilon + x\right)} - x\right)\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]

      3. associate--l+ 99.7%

         \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \color{blue}{\left(\varepsilon + \left(x - x\right)\right)}\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]

      4. +-inverses 99.7%

         \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon + \color{blue}{0}\right)\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]

      5. distribute-lft-in 99.7%

         \[\leadsto -2 \cdot \left(\sin \color{blue}{\left(0.5 \cdot \varepsilon + 0.5 \cdot 0\right)} \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]

      6. metadata-eval 99.7%

         \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + \color{blue}{0}\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]

      7. *-commutative 99.7%

         \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \sin \color{blue}{\left(0.5 \cdot \left(x + \left(x + \varepsilon\right)\right)\right)}\right) \]

      8. +-commutative 99.7%

         \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \sin \left(0.5 \cdot \left(x + \color{blue}{\left(\varepsilon + x\right)}\right)\right)\right) \]

   5. Simplified 99.7%

      \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \sin \left(0.5 \cdot \left(x + \left(\varepsilon + x\right)\right)\right)\right)} \]

   6. Taylor expanded in x around 0 91.2%

      \[\leadsto -2 \cdot \color{blue}{{\sin \left(0.5 \cdot \varepsilon\right)}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.95 \cdot 10^{-18} \lor \neg \left(x \leq 1.5 \cdot 10^{-27}\right):\\ \;\;\;\;-2 \cdot \left(\sin x \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array} \]

Alternative 11: 69.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{-18} \lor \neg \left(x \leq 2.15 \cdot 10^{-26}\right):\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -4.7e-18) (not (<= x 2.15e-26)))
   (* eps (- (sin x)))
   (* -2.0 (pow (sin (* eps 0.5)) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -4.7e-18) || !(x <= 2.15e-26)) {
		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 ((x <= (-4.7d-18)) .or. (.not. (x <= 2.15d-26))) 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 ((x <= -4.7e-18) || !(x <= 2.15e-26)) {
		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 (x <= -4.7e-18) or not (x <= 2.15e-26):
		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 ((x <= -4.7e-18) || !(x <= 2.15e-26))
		tmp = Float64(eps * Float64(-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 ((x <= -4.7e-18) || ~((x <= 2.15e-26)))
		tmp = eps * -sin(x);
	else
		tmp = -2.0 * (sin((eps * 0.5)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -4.7e-18], N[Not[LessEqual[x, 2.15e-26]], $MachinePrecision]], 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

1. Split input into 2 regimes

2. if x < -4.6999999999999996e-18 or 2.14999999999999994e-26 < x

   1. Initial program 10.2%

      \[\cos \left(x + \varepsilon\right) - \cos x \]

   2. Taylor expanded in eps around 0 50.0%

      \[\leadsto \color{blue}{-1 \cdot \left(\varepsilon \cdot \sin x\right)} \]
   3. Step-by-step derivation

      1. associate-*r* 50.0%

         \[\leadsto \color{blue}{\left(-1 \cdot \varepsilon\right) \cdot \sin x} \]

      2. mul-1-neg 50.0%

         \[\leadsto \color{blue}{\left(-\varepsilon\right)} \cdot \sin x \]

   4. Simplified 50.0%

      \[\leadsto \color{blue}{\left(-\varepsilon\right) \cdot \sin x} \]

   if -4.6999999999999996e-18 < x < 2.14999999999999994e-26

   1. Initial program 70.5%

      \[\cos \left(x + \varepsilon\right) - \cos x \]
   2. Step-by-step derivation

      1. diff-cos 92.2%

         \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)} \]

      2. div-inv 92.2%

         \[\leadsto -2 \cdot \left(\sin \color{blue}{\left(\left(\left(x + \varepsilon\right) - x\right) \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]

      3. metadata-eval 92.2%

         \[\leadsto -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot \color{blue}{0.5}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]

      4. div-inv 92.2%

         \[\leadsto -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \color{blue}{\left(\left(\left(x + \varepsilon\right) + x\right) \cdot \frac{1}{2}\right)}\right) \]

      5. +-commutative 92.2%

         \[\leadsto -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \left(\color{blue}{\left(x + \left(x + \varepsilon\right)\right)} \cdot \frac{1}{2}\right)\right) \]

      6. metadata-eval 92.2%

         \[\leadsto -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot \color{blue}{0.5}\right)\right) \]

   3. Applied egg-rr 92.2%

      \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative 92.2%

         \[\leadsto -2 \cdot \left(\sin \color{blue}{\left(0.5 \cdot \left(\left(x + \varepsilon\right) - x\right)\right)} \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]

      2. +-commutative 92.2%

         \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \left(\color{blue}{\left(\varepsilon + x\right)} - x\right)\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]

      3. associate--l+ 99.7%

         \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \color{blue}{\left(\varepsilon + \left(x - x\right)\right)}\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]

      4. +-inverses 99.7%

         \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon + \color{blue}{0}\right)\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]

      5. distribute-lft-in 99.7%

         \[\leadsto -2 \cdot \left(\sin \color{blue}{\left(0.5 \cdot \varepsilon + 0.5 \cdot 0\right)} \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]

      6. metadata-eval 99.7%

         \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + \color{blue}{0}\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]

      7. *-commutative 99.7%

         \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \sin \color{blue}{\left(0.5 \cdot \left(x + \left(x + \varepsilon\right)\right)\right)}\right) \]

      8. +-commutative 99.7%

         \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \sin \left(0.5 \cdot \left(x + \color{blue}{\left(\varepsilon + x\right)}\right)\right)\right) \]

   5. Simplified 99.7%

      \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \sin \left(0.5 \cdot \left(x + \left(\varepsilon + x\right)\right)\right)\right)} \]

   6. Taylor expanded in x around 0 91.2%

      \[\leadsto -2 \cdot \color{blue}{{\sin \left(0.5 \cdot \varepsilon\right)}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{-18} \lor \neg \left(x \leq 2.15 \cdot 10^{-26}\right):\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array} \]

Alternative 12: 66.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 + \cos \varepsilon\\ t_1 := \varepsilon \cdot \left(-\sin x\right)\\ \mathbf{if}\;\varepsilon \leq -8.5 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 3.6 \cdot 10^{-53}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq 8.2 \cdot 10^{-13}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \mathbf{elif}\;\varepsilon \leq 0.66:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ -1.0 (cos eps))) (t_1 (* eps (- (sin x)))))
   (if (<= eps -8.5e-5)
     t_0
     (if (<= eps 3.6e-53)
       t_1
       (if (<= eps 8.2e-13)
         (* eps (* eps -0.5))
         (if (<= eps 0.66) t_1 t_0))))))

C

double code(double x, double eps) {
	double t_0 = -1.0 + cos(eps);
	double t_1 = eps * -sin(x);
	double tmp;
	if (eps <= -8.5e-5) {
		tmp = t_0;
	} else if (eps <= 3.6e-53) {
		tmp = t_1;
	} else if (eps <= 8.2e-13) {
		tmp = eps * (eps * -0.5);
	} else if (eps <= 0.66) {
		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 = eps * -sin(x)
    if (eps <= (-8.5d-5)) then
        tmp = t_0
    else if (eps <= 3.6d-53) then
        tmp = t_1
    else if (eps <= 8.2d-13) then
        tmp = eps * (eps * (-0.5d0))
    else if (eps <= 0.66d0) 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 = eps * -Math.sin(x);
	double tmp;
	if (eps <= -8.5e-5) {
		tmp = t_0;
	} else if (eps <= 3.6e-53) {
		tmp = t_1;
	} else if (eps <= 8.2e-13) {
		tmp = eps * (eps * -0.5);
	} else if (eps <= 0.66) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = -1.0 + math.cos(eps)
	t_1 = eps * -math.sin(x)
	tmp = 0
	if eps <= -8.5e-5:
		tmp = t_0
	elif eps <= 3.6e-53:
		tmp = t_1
	elif eps <= 8.2e-13:
		tmp = eps * (eps * -0.5)
	elif eps <= 0.66:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(-1.0 + cos(eps))
	t_1 = Float64(eps * Float64(-sin(x)))
	tmp = 0.0
	if (eps <= -8.5e-5)
		tmp = t_0;
	elseif (eps <= 3.6e-53)
		tmp = t_1;
	elseif (eps <= 8.2e-13)
		tmp = Float64(eps * Float64(eps * -0.5));
	elseif (eps <= 0.66)
		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 = eps * -sin(x);
	tmp = 0.0;
	if (eps <= -8.5e-5)
		tmp = t_0;
	elseif (eps <= 3.6e-53)
		tmp = t_1;
	elseif (eps <= 8.2e-13)
		tmp = eps * (eps * -0.5);
	elseif (eps <= 0.66)
		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[(eps * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]}, If[LessEqual[eps, -8.5e-5], t$95$0, If[LessEqual[eps, 3.6e-53], t$95$1, If[LessEqual[eps, 8.2e-13], N[(eps * N[(eps * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.66], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -8.500000000000001e-5 or 0.660000000000000031 < eps

   1. Initial program 51.8%

      \[\cos \left(x + \varepsilon\right) - \cos x \]

   2. Taylor expanded in x around 0 50.8%

      \[\leadsto \color{blue}{\cos \varepsilon - 1} \]

   if -8.500000000000001e-5 < eps < 3.5999999999999999e-53 or 8.2000000000000004e-13 < eps < 0.660000000000000031

   1. Initial program 20.7%

      \[\cos \left(x + \varepsilon\right) - \cos x \]

   2. Taylor expanded in eps around 0 84.0%

      \[\leadsto \color{blue}{-1 \cdot \left(\varepsilon \cdot \sin x\right)} \]
   3. Step-by-step derivation

      1. associate-*r* 84.0%

         \[\leadsto \color{blue}{\left(-1 \cdot \varepsilon\right) \cdot \sin x} \]

      2. mul-1-neg 84.0%

         \[\leadsto \color{blue}{\left(-\varepsilon\right)} \cdot \sin x \]

   4. Simplified 84.0%

      \[\leadsto \color{blue}{\left(-\varepsilon\right) \cdot \sin x} \]

   if 3.5999999999999999e-53 < eps < 8.2000000000000004e-13

   1. Initial program 3.7%

      \[\cos \left(x + \varepsilon\right) - \cos x \]

   2. Taylor expanded in x around 0 3.7%

      \[\leadsto \color{blue}{\cos \varepsilon - 1} \]

   3. Taylor expanded in eps around 0 87.7%

      \[\leadsto \color{blue}{-0.5 \cdot {\varepsilon}^{2}} \]
   4. Step-by-step derivation

      1. *-commutative 87.7%

         \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot -0.5} \]

      2. unpow2 87.7%

         \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot -0.5 \]

      3. associate-*l* 87.7%

         \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)} \]

   5. Simplified 87.7%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -8.5 \cdot 10^{-5}:\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 3.6 \cdot 10^{-53}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{elif}\;\varepsilon \leq 8.2 \cdot 10^{-13}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \mathbf{elif}\;\varepsilon \leq 0.66:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \cos \varepsilon\\ \end{array} \]

Alternative 13: 47.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.76 \cdot 10^{-6} \lor \neg \left(\varepsilon \leq 0.00013\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -1.76e-6) (not (<= eps 0.00013)))
   (+ -1.0 (cos eps))
   (* eps (* eps -0.5))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.76e-6) || !(eps <= 0.00013)) {
		tmp = -1.0 + cos(eps);
	} else {
		tmp = eps * (eps * -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-1.76d-6)) .or. (.not. (eps <= 0.00013d0))) then
        tmp = (-1.0d0) + cos(eps)
    else
        tmp = eps * (eps * (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.76e-6) || !(eps <= 0.00013)) {
		tmp = -1.0 + Math.cos(eps);
	} else {
		tmp = eps * (eps * -0.5);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -1.76e-6) or not (eps <= 0.00013):
		tmp = -1.0 + math.cos(eps)
	else:
		tmp = eps * (eps * -0.5)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -1.76e-6) || !(eps <= 0.00013))
		tmp = Float64(-1.0 + cos(eps));
	else
		tmp = Float64(eps * Float64(eps * -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -1.76e-6) || ~((eps <= 0.00013)))
		tmp = -1.0 + cos(eps);
	else
		tmp = eps * (eps * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -1.76e-6], N[Not[LessEqual[eps, 0.00013]], $MachinePrecision]], N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision], N[(eps * N[(eps * -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.7600000000000001e-6 or 1.29999999999999989e-4 < eps

   1. Initial program 51.1%

      \[\cos \left(x + \varepsilon\right) - \cos x \]

   2. Taylor expanded in x around 0 50.1%

      \[\leadsto \color{blue}{\cos \varepsilon - 1} \]

   if -1.7600000000000001e-6 < eps < 1.29999999999999989e-4

   1. Initial program 19.9%

      \[\cos \left(x + \varepsilon\right) - \cos x \]

   2. Taylor expanded in x around 0 19.7%

      \[\leadsto \color{blue}{\cos \varepsilon - 1} \]

   3. Taylor expanded in eps around 0 38.1%

      \[\leadsto \color{blue}{-0.5 \cdot {\varepsilon}^{2}} \]
   4. Step-by-step derivation

      1. *-commutative 38.1%

         \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot -0.5} \]

      2. unpow2 38.1%

         \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot -0.5 \]

      3. associate-*l* 38.1%

         \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)} \]

   5. Simplified 38.1%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.76 \cdot 10^{-6} \lor \neg \left(\varepsilon \leq 0.00013\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \end{array} \]

Alternative 14: 23.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 6.2 \cdot 10^{+31}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \cos x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 6.2e+31) (* eps (* eps -0.5)) (- 1.0 (cos x))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 6.2e+31) {
		tmp = eps * (eps * -0.5);
	} else {
		tmp = 1.0 - cos(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= 6.2d+31) then
        tmp = eps * (eps * (-0.5d0))
    else
        tmp = 1.0d0 - cos(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= 6.2e+31) {
		tmp = eps * (eps * -0.5);
	} else {
		tmp = 1.0 - Math.cos(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= 6.2e+31:
		tmp = eps * (eps * -0.5)
	else:
		tmp = 1.0 - math.cos(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 6.2e+31)
		tmp = Float64(eps * Float64(eps * -0.5));
	else
		tmp = Float64(1.0 - cos(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= 6.2e+31)
		tmp = eps * (eps * -0.5);
	else
		tmp = 1.0 - cos(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 6.2e+31], N[(eps * N[(eps * -0.5), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 6.2000000000000004e31

   1. Initial program 32.7%

      \[\cos \left(x + \varepsilon\right) - \cos x \]

   2. Taylor expanded in x around 0 32.6%

      \[\leadsto \color{blue}{\cos \varepsilon - 1} \]

   3. Taylor expanded in eps around 0 25.9%

      \[\leadsto \color{blue}{-0.5 \cdot {\varepsilon}^{2}} \]
   4. Step-by-step derivation

      1. *-commutative 25.9%

         \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot -0.5} \]

      2. unpow2 25.9%

         \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot -0.5 \]

      3. associate-*l* 25.9%

         \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)} \]

   5. Simplified 25.9%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)} \]

   if 6.2000000000000004e31 < eps

   1. Initial program 46.7%

      \[\cos \left(x + \varepsilon\right) - \cos x \]

   2. Taylor expanded in x around 0 42.2%

      \[\leadsto \color{blue}{\left(\cos \varepsilon + -1 \cdot \left(x \cdot \sin \varepsilon\right)\right)} - \cos x \]
   3. Step-by-step derivation

      1. mul-1-neg 42.2%

         \[\leadsto \left(\cos \varepsilon + \color{blue}{\left(-x \cdot \sin \varepsilon\right)}\right) - \cos x \]

      2. unsub-neg 42.2%

         \[\leadsto \color{blue}{\left(\cos \varepsilon - x \cdot \sin \varepsilon\right)} - \cos x \]

   4. Simplified 42.2%

      \[\leadsto \color{blue}{\left(\cos \varepsilon - x \cdot \sin \varepsilon\right)} - \cos x \]

   5. Taylor expanded in eps around 0 9.8%

      \[\leadsto \color{blue}{1 - \cos x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 22.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 6.2 \cdot 10^{+31}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \cos x\\ \end{array} \]

Alternative 15: 9.4% accurate, 41.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(x \cdot x\right) \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* 0.5 (* x x)))

C

double code(double x, double eps) {
	return 0.5 * (x * x);
}

Fortran

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

Java

public static double code(double x, double eps) {
	return 0.5 * (x * x);
}

Python

def code(x, eps):
	return 0.5 * (x * x)

Julia

function code(x, eps)
	return Float64(0.5 * Float64(x * x))
end

MATLAB

function tmp = code(x, eps)
	tmp = 0.5 * (x * x);
end

Wolfram

code[x_, eps_] := N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.1%

   \[\cos \left(x + \varepsilon\right) - \cos x \]

2. Taylor expanded in x around 0 33.5%

   \[\leadsto \color{blue}{\left(\cos \varepsilon + -1 \cdot \left(x \cdot \sin \varepsilon\right)\right)} - \cos x \]
3. Step-by-step derivation

   1. mul-1-neg 33.5%

      \[\leadsto \left(\cos \varepsilon + \color{blue}{\left(-x \cdot \sin \varepsilon\right)}\right) - \cos x \]

   2. unsub-neg 33.5%

      \[\leadsto \color{blue}{\left(\cos \varepsilon - x \cdot \sin \varepsilon\right)} - \cos x \]

4. Simplified 33.5%

   \[\leadsto \color{blue}{\left(\cos \varepsilon - x \cdot \sin \varepsilon\right)} - \cos x \]

5. Taylor expanded in eps around 0 13.0%

   \[\leadsto \color{blue}{1 - \cos x} \]

6. Taylor expanded in x around 0 8.0%

   \[\leadsto \color{blue}{0.5 \cdot {x}^{2}} \]
7. Step-by-step derivation

   1. unpow2 8.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(x \cdot x\right)} \]

8. Simplified 8.0%

   \[\leadsto \color{blue}{0.5 \cdot \left(x \cdot x\right)} \]

9. Final simplification 8.0%

   \[\leadsto 0.5 \cdot \left(x \cdot x\right) \]

Alternative 16: 22.5% accurate, 41.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 (* eps (* eps -0.5)))

C

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

Fortran

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

Java

public static double code(double x, double eps) {
	return eps * (eps * -0.5);
}

Python

def code(x, eps):
	return eps * (eps * -0.5)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.1%

   \[\cos \left(x + \varepsilon\right) - \cos x \]

2. Taylor expanded in x around 0 35.5%

   \[\leadsto \color{blue}{\cos \varepsilon - 1} \]

3. Taylor expanded in eps around 0 20.3%

   \[\leadsto \color{blue}{-0.5 \cdot {\varepsilon}^{2}} \]
4. Step-by-step derivation

   1. *-commutative 20.3%

      \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot -0.5} \]

   2. unpow2 20.3%

      \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot -0.5 \]

   3. associate-*l* 20.3%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)} \]

5. Simplified 20.3%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)} \]

6. Final simplification 20.3%

   \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot -0.5\right) \]

Reproduce

herbie shell --seed 2023274 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))