2cos (problem 3.3.5)

Percentage Accurate: 37.8% → 99.2%

Time: 21.5s

Alternatives: 17

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 37.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0028:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, -\mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0024:\\ \;\;\;\;\left(0.16666666666666666 \cdot \left(\sin x \cdot {\varepsilon}^{3}\right) + \cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right)\right) - \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin \varepsilon \cdot \left(-\sin x\right)\right) - \cos x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.0028)
   (fma (cos x) (cos eps) (- (fma (sin eps) (sin x) (cos x))))
   (if (<= eps 0.0024)
     (-
      (+
       (* 0.16666666666666666 (* (sin x) (pow eps 3.0)))
       (*
        (cos x)
        (+ (* -0.5 (pow eps 2.0)) (* 0.041666666666666664 (pow eps 4.0)))))
      (* eps (sin x)))
     (- (fma (cos x) (cos eps) (* (sin eps) (- (sin x)))) (cos x)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.0028) {
		tmp = fma(cos(x), cos(eps), -fma(sin(eps), sin(x), cos(x)));
	} else if (eps <= 0.0024) {
		tmp = ((0.16666666666666666 * (sin(x) * pow(eps, 3.0))) + (cos(x) * ((-0.5 * pow(eps, 2.0)) + (0.041666666666666664 * pow(eps, 4.0))))) - (eps * sin(x));
	} else {
		tmp = fma(cos(x), cos(eps), (sin(eps) * -sin(x))) - cos(x);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -0.0028)
		tmp = fma(cos(x), cos(eps), Float64(-fma(sin(eps), sin(x), cos(x))));
	elseif (eps <= 0.0024)
		tmp = Float64(Float64(Float64(0.16666666666666666 * Float64(sin(x) * (eps ^ 3.0))) + Float64(cos(x) * Float64(Float64(-0.5 * (eps ^ 2.0)) + Float64(0.041666666666666664 * (eps ^ 4.0))))) - Float64(eps * sin(x)));
	else
		tmp = Float64(fma(cos(x), cos(eps), Float64(sin(eps) * Float64(-sin(x)))) - cos(x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -0.0028], N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + (-N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision] + N[Cos[x], $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[eps, 0.0024], N[(N[(N[(0.16666666666666666 * N[(N[Sin[x], $MachinePrecision] * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.041666666666666664 * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + N[(N[Sin[eps], $MachinePrecision] * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.00279999999999999997

   1. Initial program 42.1%

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

      1. sub-neg 42.1%

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

      2. +-commutative 42.1%

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

      3. cos-sum 98.3%

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

      4. cancel-sign-sub-inv 98.3%

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

      5. associate-+r+ 98.5%

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

      6. *-commutative 98.5%

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

   3. Applied egg-rr 98.5%

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

      1. +-commutative 98.5%

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

      2. unsub-neg 98.5%

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

   5. Applied egg-rr 98.5%

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

      1. associate-+l- 98.4%

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

      2. fma-neg 98.5%

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

      3. *-commutative 98.5%

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

      4. cancel-sign-sub 98.5%

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

      5. +-commutative 98.5%

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

      6. *-commutative 98.5%

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

      7. fma-def 98.5%

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

   7. Applied egg-rr 98.5%

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

   if -0.00279999999999999997 < eps < 0.00239999999999999979

   1. Initial program 20.6%

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

   2. Taylor expanded in eps around 0 99.8%

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

      1. +-commutative 99.8%

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

      2. mul-1-neg 99.8%

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

      3. unsub-neg 99.8%

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

      4. associate-+r+ 99.8%

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

      5. +-commutative 99.8%

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

      6. *-commutative 99.8%

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

      7. associate-*r* 99.8%

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

      8. associate-*r* 99.8%

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

      9. distribute-rgt-out 99.8%

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

   4. Simplified 99.8%

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

   if 0.00239999999999999979 < eps

   1. Initial program 57.1%

      \[\cos \left(x + \varepsilon\right) - \cos x \]
   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. fma-neg 98.8%

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

      3. distribute-lft-neg-in 98.8%

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

      4. *-commutative 98.8%

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

   3. Applied egg-rr 98.8%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0028:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, -\mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0024:\\ \;\;\;\;\left(0.16666666666666666 \cdot \left(\sin x \cdot {\varepsilon}^{3}\right) + \cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right)\right) - \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin \varepsilon \cdot \left(-\sin x\right)\right) - \cos x\\ \end{array} \]

Alternative 2: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0025:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \cos x\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{elif}\;\varepsilon \leq 0.0024:\\ \;\;\;\;\left(0.16666666666666666 \cdot \left(\sin x \cdot {\varepsilon}^{3}\right) + \cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right)\right) - \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin \varepsilon \cdot \left(-\sin x\right)\right) - \cos x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.0025)
   (- (- (* (cos x) (cos eps)) (cos x)) (* (sin eps) (sin x)))
   (if (<= eps 0.0024)
     (-
      (+
       (* 0.16666666666666666 (* (sin x) (pow eps 3.0)))
       (*
        (cos x)
        (+ (* -0.5 (pow eps 2.0)) (* 0.041666666666666664 (pow eps 4.0)))))
      (* eps (sin x)))
     (- (fma (cos x) (cos eps) (* (sin eps) (- (sin x)))) (cos x)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.0025) {
		tmp = ((cos(x) * cos(eps)) - cos(x)) - (sin(eps) * sin(x));
	} else if (eps <= 0.0024) {
		tmp = ((0.16666666666666666 * (sin(x) * pow(eps, 3.0))) + (cos(x) * ((-0.5 * pow(eps, 2.0)) + (0.041666666666666664 * pow(eps, 4.0))))) - (eps * sin(x));
	} else {
		tmp = fma(cos(x), cos(eps), (sin(eps) * -sin(x))) - cos(x);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -0.0025)
		tmp = Float64(Float64(Float64(cos(x) * cos(eps)) - cos(x)) - Float64(sin(eps) * sin(x)));
	elseif (eps <= 0.0024)
		tmp = Float64(Float64(Float64(0.16666666666666666 * Float64(sin(x) * (eps ^ 3.0))) + Float64(cos(x) * Float64(Float64(-0.5 * (eps ^ 2.0)) + Float64(0.041666666666666664 * (eps ^ 4.0))))) - Float64(eps * sin(x)));
	else
		tmp = Float64(fma(cos(x), cos(eps), Float64(sin(eps) * Float64(-sin(x)))) - cos(x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -0.0025], N[(N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.0024], N[(N[(N[(0.16666666666666666 * N[(N[Sin[x], $MachinePrecision] * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.041666666666666664 * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + N[(N[Sin[eps], $MachinePrecision] * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.00250000000000000005

   1. Initial program 42.1%

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

      1. sub-neg 42.1%

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

      2. +-commutative 42.1%

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

      3. cos-sum 98.3%

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

      4. cancel-sign-sub-inv 98.3%

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

      5. associate-+r+ 98.5%

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

      6. *-commutative 98.5%

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

   3. Applied egg-rr 98.5%

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

      1. +-commutative 98.5%

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

      2. unsub-neg 98.5%

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

   5. Applied egg-rr 98.5%

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

   if -0.00250000000000000005 < eps < 0.00239999999999999979

   1. Initial program 20.6%

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

   2. Taylor expanded in eps around 0 99.8%

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

      1. +-commutative 99.8%

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

      2. mul-1-neg 99.8%

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

      3. unsub-neg 99.8%

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

      4. associate-+r+ 99.8%

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

      5. +-commutative 99.8%

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

      6. *-commutative 99.8%

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

      7. associate-*r* 99.8%

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

      8. associate-*r* 99.8%

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

      9. distribute-rgt-out 99.8%

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

   4. Simplified 99.8%

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

   if 0.00239999999999999979 < eps

   1. Initial program 57.1%

      \[\cos \left(x + \varepsilon\right) - \cos x \]
   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. fma-neg 98.8%

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

      3. distribute-lft-neg-in 98.8%

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

      4. *-commutative 98.8%

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

   3. Applied egg-rr 98.8%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0025:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \cos x\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{elif}\;\varepsilon \leq 0.0024:\\ \;\;\;\;\left(0.16666666666666666 \cdot \left(\sin x \cdot {\varepsilon}^{3}\right) + \cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right)\right) - \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin \varepsilon \cdot \left(-\sin x\right)\right) - \cos x\\ \end{array} \]

Alternative 3: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.1 \cdot 10^{-5}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \cos x\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{elif}\;\varepsilon \leq 3 \cdot 10^{-5}:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin \varepsilon \cdot \left(-\sin x\right)\right) - \cos x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -4.1e-5)
   (- (- (* (cos x) (cos eps)) (cos x)) (* (sin eps) (sin x)))
   (if (<= eps 3e-5)
     (- (* (cos x) (* -0.5 (pow eps 2.0))) (* eps (sin x)))
     (- (fma (cos x) (cos eps) (* (sin eps) (- (sin x)))) (cos x)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -4.1e-5) {
		tmp = ((cos(x) * cos(eps)) - cos(x)) - (sin(eps) * sin(x));
	} else if (eps <= 3e-5) {
		tmp = (cos(x) * (-0.5 * pow(eps, 2.0))) - (eps * sin(x));
	} else {
		tmp = fma(cos(x), cos(eps), (sin(eps) * -sin(x))) - cos(x);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -4.1e-5)
		tmp = Float64(Float64(Float64(cos(x) * cos(eps)) - cos(x)) - Float64(sin(eps) * sin(x)));
	elseif (eps <= 3e-5)
		tmp = Float64(Float64(cos(x) * Float64(-0.5 * (eps ^ 2.0))) - Float64(eps * sin(x)));
	else
		tmp = Float64(fma(cos(x), cos(eps), Float64(sin(eps) * Float64(-sin(x)))) - cos(x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -4.1e-5], N[(N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 3e-5], N[(N[(N[Cos[x], $MachinePrecision] * N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + N[(N[Sin[eps], $MachinePrecision] * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -4.10000000000000005e-5

   1. Initial program 42.1%

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

      1. sub-neg 42.1%

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

      2. +-commutative 42.1%

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

      3. cos-sum 98.3%

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

      4. cancel-sign-sub-inv 98.3%

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

      5. associate-+r+ 98.5%

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

      6. *-commutative 98.5%

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

   3. Applied egg-rr 98.5%

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

      1. +-commutative 98.5%

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

      2. unsub-neg 98.5%

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

   5. Applied egg-rr 98.5%

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

   if -4.10000000000000005e-5 < eps < 3.00000000000000008e-5

   1. Initial program 20.6%

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

   2. Taylor expanded in eps around 0 99.7%

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

      1. +-commutative 99.7%

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

      2. mul-1-neg 99.7%

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

      3. unsub-neg 99.7%

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

      4. associate-*r* 99.7%

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

      5. *-commutative 99.7%

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

   4. Simplified 99.7%

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

   if 3.00000000000000008e-5 < eps

   1. Initial program 57.1%

      \[\cos \left(x + \varepsilon\right) - \cos x \]
   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. fma-neg 98.8%

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

      3. distribute-lft-neg-in 98.8%

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

      4. *-commutative 98.8%

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

   3. Applied egg-rr 98.8%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.1 \cdot 10^{-5}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \cos x\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{elif}\;\varepsilon \leq 3 \cdot 10^{-5}:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin \varepsilon \cdot \left(-\sin x\right)\right) - \cos x\\ \end{array} \]

Alternative 4: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.1 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 3.6 \cdot 10^{-5}\right):\\ \;\;\;\;\mathsf{fma}\left(\cos \varepsilon + -1, \cos x, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - \varepsilon \cdot \sin x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -4.1e-5) (not (<= eps 3.6e-5)))
   (fma (+ (cos eps) -1.0) (cos x) (* (sin eps) (- (sin x))))
   (- (* (cos x) (* -0.5 (pow eps 2.0))) (* eps (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -4.1e-5) || !(eps <= 3.6e-5)) {
		tmp = fma((cos(eps) + -1.0), cos(x), (sin(eps) * -sin(x)));
	} else {
		tmp = (cos(x) * (-0.5 * pow(eps, 2.0))) - (eps * sin(x));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -4.1e-5) || !(eps <= 3.6e-5))
		tmp = fma(Float64(cos(eps) + -1.0), cos(x), Float64(sin(eps) * Float64(-sin(x))));
	else
		tmp = Float64(Float64(cos(x) * Float64(-0.5 * (eps ^ 2.0))) - Float64(eps * sin(x)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -4.1e-5], N[Not[LessEqual[eps, 3.6e-5]], $MachinePrecision]], N[(N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[Sin[eps], $MachinePrecision] * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[x], $MachinePrecision] * N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -4.10000000000000005e-5 or 3.60000000000000009e-5 < eps

   1. Initial program 50.7%

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

      1. sub-neg 50.7%

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

      2. cos-sum 98.5%

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

      3. cancel-sign-sub-inv 98.5%

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

      4. associate-+l+ 98.6%

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

      5. *-commutative 98.6%

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

   3. Applied egg-rr 98.6%

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

      1. +-commutative 98.6%

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

      2. distribute-rgt-neg-out 98.6%

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

      3. *-commutative 98.6%

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

      4. unsub-neg 98.6%

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

      5. associate-+r- 98.7%

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

      6. *-commutative 98.7%

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

      7. neg-mul-1 98.7%

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

      8. distribute-rgt-out 98.6%

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

      9. *-commutative 98.6%

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

   5. Simplified 98.6%

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

      1. *-commutative 98.6%

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

      2. fma-neg 98.6%

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

      3. distribute-rgt-neg-in 98.6%

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

   7. Applied egg-rr 98.6%

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

   if -4.10000000000000005e-5 < eps < 3.60000000000000009e-5

   1. Initial program 20.6%

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

   2. Taylor expanded in eps around 0 99.7%

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

      1. +-commutative 99.7%

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

      2. mul-1-neg 99.7%

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

      3. unsub-neg 99.7%

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

      4. associate-*r* 99.7%

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

      5. *-commutative 99.7%

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

   4. Simplified 99.7%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.1 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 3.6 \cdot 10^{-5}\right):\\ \;\;\;\;\mathsf{fma}\left(\cos \varepsilon + -1, \cos x, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 5: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.3 \cdot 10^{-5}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin \varepsilon \cdot \sin x\right)\\ \mathbf{elif}\;\varepsilon \leq 4.2 \cdot 10^{-5}:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos \varepsilon + -1, \cos x, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -3.3e-5)
   (- (* (cos x) (cos eps)) (+ (cos x) (* (sin eps) (sin x))))
   (if (<= eps 4.2e-5)
     (- (* (cos x) (* -0.5 (pow eps 2.0))) (* eps (sin x)))
     (fma (+ (cos eps) -1.0) (cos x) (* (sin eps) (- (sin x)))))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -3.3e-5) {
		tmp = (cos(x) * cos(eps)) - (cos(x) + (sin(eps) * sin(x)));
	} else if (eps <= 4.2e-5) {
		tmp = (cos(x) * (-0.5 * pow(eps, 2.0))) - (eps * sin(x));
	} else {
		tmp = fma((cos(eps) + -1.0), cos(x), (sin(eps) * -sin(x)));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -3.3e-5)
		tmp = Float64(Float64(cos(x) * cos(eps)) - Float64(cos(x) + Float64(sin(eps) * sin(x))));
	elseif (eps <= 4.2e-5)
		tmp = Float64(Float64(cos(x) * Float64(-0.5 * (eps ^ 2.0))) - Float64(eps * sin(x)));
	else
		tmp = fma(Float64(cos(eps) + -1.0), cos(x), Float64(sin(eps) * Float64(-sin(x))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -3.3e-5], N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[x], $MachinePrecision] + N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 4.2e-5], N[(N[(N[Cos[x], $MachinePrecision] * N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[Sin[eps], $MachinePrecision] * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -3.3000000000000003e-5

   1. Initial program 42.1%

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

      1. sub-neg 42.1%

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

      2. +-commutative 42.1%

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

      3. cos-sum 98.3%

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

      4. cancel-sign-sub-inv 98.3%

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

      5. associate-+r+ 98.5%

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

      6. *-commutative 98.5%

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

   3. Applied egg-rr 98.5%

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

      1. +-commutative 98.5%

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

      2. +-commutative 98.5%

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

      3. associate-+r+ 98.3%

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

      4. +-commutative 98.3%

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

      5. *-commutative 98.3%

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

      6. cancel-sign-sub-inv 98.3%

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

      7. sub-neg 98.3%

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

      8. associate--l- 98.4%

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

      9. +-commutative 98.4%

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

      10. *-commutative 98.4%

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

   5. Applied egg-rr 98.4%

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

   if -3.3000000000000003e-5 < eps < 4.19999999999999977e-5

   1. Initial program 20.6%

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

   2. Taylor expanded in eps around 0 99.7%

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

      1. +-commutative 99.7%

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

      2. mul-1-neg 99.7%

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

      3. unsub-neg 99.7%

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

      4. associate-*r* 99.7%

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

      5. *-commutative 99.7%

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

   4. Simplified 99.7%

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

   if 4.19999999999999977e-5 < eps

   1. Initial program 57.1%

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

      1. sub-neg 57.1%

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

      2. cos-sum 98.7%

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

      3. cancel-sign-sub-inv 98.7%

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

      4. associate-+l+ 98.7%

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

      5. *-commutative 98.7%

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

   3. Applied egg-rr 98.7%

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

      1. +-commutative 98.7%

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

      2. distribute-rgt-neg-out 98.7%

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

      3. *-commutative 98.7%

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

      4. unsub-neg 98.7%

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

      5. associate-+r- 98.8%

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

      6. *-commutative 98.8%

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

      7. neg-mul-1 98.8%

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

      8. distribute-rgt-out 98.8%

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

      9. *-commutative 98.8%

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

   5. Simplified 98.8%

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

      1. *-commutative 98.8%

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

      2. fma-neg 98.8%

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

      3. distribute-rgt-neg-in 98.8%

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

   7. Applied egg-rr 98.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \varepsilon + -1, \cos x, \sin \varepsilon \cdot \left(-\sin x\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.3 \cdot 10^{-5}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin \varepsilon \cdot \sin x\right)\\ \mathbf{elif}\;\varepsilon \leq 4.2 \cdot 10^{-5}:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos \varepsilon + -1, \cos x, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \end{array} \]

Alternative 6: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.8 \cdot 10^{-5}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \cos x\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{elif}\;\varepsilon \leq 2.8 \cdot 10^{-5}:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos \varepsilon + -1, \cos x, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -3.8e-5)
   (- (- (* (cos x) (cos eps)) (cos x)) (* (sin eps) (sin x)))
   (if (<= eps 2.8e-5)
     (- (* (cos x) (* -0.5 (pow eps 2.0))) (* eps (sin x)))
     (fma (+ (cos eps) -1.0) (cos x) (* (sin eps) (- (sin x)))))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -3.8e-5) {
		tmp = ((cos(x) * cos(eps)) - cos(x)) - (sin(eps) * sin(x));
	} else if (eps <= 2.8e-5) {
		tmp = (cos(x) * (-0.5 * pow(eps, 2.0))) - (eps * sin(x));
	} else {
		tmp = fma((cos(eps) + -1.0), cos(x), (sin(eps) * -sin(x)));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -3.8e-5)
		tmp = Float64(Float64(Float64(cos(x) * cos(eps)) - cos(x)) - Float64(sin(eps) * sin(x)));
	elseif (eps <= 2.8e-5)
		tmp = Float64(Float64(cos(x) * Float64(-0.5 * (eps ^ 2.0))) - Float64(eps * sin(x)));
	else
		tmp = fma(Float64(cos(eps) + -1.0), cos(x), Float64(sin(eps) * Float64(-sin(x))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -3.8e-5], N[(N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 2.8e-5], N[(N[(N[Cos[x], $MachinePrecision] * N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[Sin[eps], $MachinePrecision] * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -3.8000000000000002e-5

   1. Initial program 42.1%

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

      1. sub-neg 42.1%

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

      2. +-commutative 42.1%

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

      3. cos-sum 98.3%

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

      4. cancel-sign-sub-inv 98.3%

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

      5. associate-+r+ 98.5%

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

      6. *-commutative 98.5%

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

   3. Applied egg-rr 98.5%

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

      1. +-commutative 98.5%

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

      2. unsub-neg 98.5%

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

   5. Applied egg-rr 98.5%

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

   if -3.8000000000000002e-5 < eps < 2.79999999999999996e-5

   1. Initial program 20.6%

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

   2. Taylor expanded in eps around 0 99.7%

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

      1. +-commutative 99.7%

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

      2. mul-1-neg 99.7%

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

      3. unsub-neg 99.7%

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

      4. associate-*r* 99.7%

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

      5. *-commutative 99.7%

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

   4. Simplified 99.7%

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

   if 2.79999999999999996e-5 < eps

   1. Initial program 57.1%

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

      1. sub-neg 57.1%

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

      2. cos-sum 98.7%

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

      3. cancel-sign-sub-inv 98.7%

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

      4. associate-+l+ 98.7%

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

      5. *-commutative 98.7%

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

   3. Applied egg-rr 98.7%

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

      1. +-commutative 98.7%

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

      2. distribute-rgt-neg-out 98.7%

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

      3. *-commutative 98.7%

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

      4. unsub-neg 98.7%

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

      5. associate-+r- 98.8%

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

      6. *-commutative 98.8%

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

      7. neg-mul-1 98.8%

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

      8. distribute-rgt-out 98.8%

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

      9. *-commutative 98.8%

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

   5. Simplified 98.8%

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

      1. *-commutative 98.8%

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

      2. fma-neg 98.8%

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

      3. distribute-rgt-neg-in 98.8%

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

   7. Applied egg-rr 98.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \varepsilon + -1, \cos x, \sin \varepsilon \cdot \left(-\sin x\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.8 \cdot 10^{-5}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \cos x\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{elif}\;\varepsilon \leq 2.8 \cdot 10^{-5}:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos \varepsilon + -1, \cos x, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \end{array} \]

Alternative 7: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.4 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 3 \cdot 10^{-5}\right):\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - \varepsilon \cdot \sin x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -4.4e-5) (not (<= eps 3e-5)))
   (- (* (cos x) (+ (cos eps) -1.0)) (* (sin eps) (sin x)))
   (- (* (cos x) (* -0.5 (pow eps 2.0))) (* eps (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -4.4e-5) || !(eps <= 3e-5)) {
		tmp = (cos(x) * (cos(eps) + -1.0)) - (sin(eps) * sin(x));
	} else {
		tmp = (cos(x) * (-0.5 * pow(eps, 2.0))) - (eps * sin(x));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-4.4d-5)) .or. (.not. (eps <= 3d-5))) then
        tmp = (cos(x) * (cos(eps) + (-1.0d0))) - (sin(eps) * sin(x))
    else
        tmp = (cos(x) * ((-0.5d0) * (eps ** 2.0d0))) - (eps * sin(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -4.4e-5) || !(eps <= 3e-5)) {
		tmp = (Math.cos(x) * (Math.cos(eps) + -1.0)) - (Math.sin(eps) * Math.sin(x));
	} else {
		tmp = (Math.cos(x) * (-0.5 * Math.pow(eps, 2.0))) - (eps * Math.sin(x));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -4.4e-5) or not (eps <= 3e-5):
		tmp = (math.cos(x) * (math.cos(eps) + -1.0)) - (math.sin(eps) * math.sin(x))
	else:
		tmp = (math.cos(x) * (-0.5 * math.pow(eps, 2.0))) - (eps * math.sin(x))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -4.4e-5) || !(eps <= 3e-5))
		tmp = Float64(Float64(cos(x) * Float64(cos(eps) + -1.0)) - Float64(sin(eps) * sin(x)));
	else
		tmp = Float64(Float64(cos(x) * Float64(-0.5 * (eps ^ 2.0))) - Float64(eps * sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -4.4e-5) || ~((eps <= 3e-5)))
		tmp = (cos(x) * (cos(eps) + -1.0)) - (sin(eps) * sin(x));
	else
		tmp = (cos(x) * (-0.5 * (eps ^ 2.0))) - (eps * sin(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -4.4e-5], N[Not[LessEqual[eps, 3e-5]], $MachinePrecision]], N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[x], $MachinePrecision] * N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -4.3999999999999999e-5 or 3.00000000000000008e-5 < eps

   1. Initial program 50.7%

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

      1. sub-neg 50.7%

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

      2. cos-sum 98.5%

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

      3. cancel-sign-sub-inv 98.5%

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

      4. associate-+l+ 98.6%

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

      5. *-commutative 98.6%

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

   3. Applied egg-rr 98.6%

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

      1. +-commutative 98.6%

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

      2. distribute-rgt-neg-out 98.6%

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

      3. *-commutative 98.6%

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

      4. unsub-neg 98.6%

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

      5. associate-+r- 98.7%

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

      6. *-commutative 98.7%

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

      7. neg-mul-1 98.7%

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

      8. distribute-rgt-out 98.6%

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

      9. *-commutative 98.6%

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

   5. Simplified 98.6%

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

   if -4.3999999999999999e-5 < eps < 3.00000000000000008e-5

   1. Initial program 20.6%

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

   2. Taylor expanded in eps around 0 99.7%

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

      1. +-commutative 99.7%

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

      2. mul-1-neg 99.7%

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

      3. unsub-neg 99.7%

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

      4. associate-*r* 99.7%

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

      5. *-commutative 99.7%

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

   4. Simplified 99.7%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.4 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 3 \cdot 10^{-5}\right):\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 8: 65.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\varepsilon + x\right) - \cos x\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-12}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-\varepsilon \cdot \sin x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (cos (+ eps x)) (cos x))))
   (if (<= t_0 -1e-12) 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 <= -1e-12) {
		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 <= (-1d-12)) 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 <= -1e-12) {
		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 <= -1e-12:
		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 <= -1e-12)
		tmp = t_0;
	else
		tmp = Float64(-Float64(eps * sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = cos((eps + x)) - cos(x);
	tmp = 0.0;
	if (t_0 <= -1e-12)
		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, -1e-12], 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)) < -9.9999999999999998e-13

   1. Initial program 75.0%

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

   if -9.9999999999999998e-13 < (-.f64 (cos.f64 (+.f64 x eps)) (cos.f64 x))

   1. Initial program 16.8%

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

   2. Taylor expanded in eps around 0 62.4%

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

      1. associate-*r* 62.4%

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

      2. mul-1-neg 62.4%

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

   4. Simplified 62.4%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\varepsilon + x\right) - \cos x \leq -1 \cdot 10^{-12}:\\ \;\;\;\;\cos \left(\varepsilon + x\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;-\varepsilon \cdot \sin x\\ \end{array} \]

Alternative 9: 68.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\varepsilon + x\right) - \cos x\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-12}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon \cdot \left(-\sin x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (cos (+ eps x)) (cos x))))
   (if (<= t_0 -1e-12) t_0 (* (sin eps) (- (sin x))))))

C

double code(double x, double eps) {
	double t_0 = cos((eps + x)) - cos(x);
	double tmp;
	if (t_0 <= -1e-12) {
		tmp = t_0;
	} else {
		tmp = sin(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 <= (-1d-12)) then
        tmp = t_0
    else
        tmp = sin(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 <= -1e-12) {
		tmp = t_0;
	} else {
		tmp = Math.sin(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 <= -1e-12:
		tmp = t_0
	else:
		tmp = math.sin(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 <= -1e-12)
		tmp = t_0;
	else
		tmp = Float64(sin(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 <= -1e-12)
		tmp = t_0;
	else
		tmp = sin(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, -1e-12], t$95$0, N[(N[Sin[eps], $MachinePrecision] * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cos.f64 (+.f64 x eps)) (cos.f64 x)) < -9.9999999999999998e-13

   1. Initial program 75.0%

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

   if -9.9999999999999998e-13 < (-.f64 (cos.f64 (+.f64 x eps)) (cos.f64 x))

   1. Initial program 16.8%

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

      1. sub-neg 16.8%

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

      2. +-commutative 16.8%

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

      3. cos-sum 41.4%

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

      4. cancel-sign-sub-inv 41.4%

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

      5. associate-+r+ 87.6%

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

      6. *-commutative 87.6%

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

   3. Applied egg-rr 87.6%

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

   4. Taylor expanded in x around 0 64.6%

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

   5. Taylor expanded in eps around 0 65.7%

      \[\leadsto \left(\color{blue}{1} - 1\right) + \sin \varepsilon \cdot \left(-\sin x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\varepsilon + x\right) - \cos x \leq -1 \cdot 10^{-12}:\\ \;\;\;\;\cos \left(\varepsilon + x\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon \cdot \left(-\sin x\right)\\ \end{array} \]

Alternative 10: 76.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.027 \lor \neg \left(\varepsilon \leq 0.0038\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot {\varepsilon}^{2} - \varepsilon \cdot \sin x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.027) (not (<= eps 0.0038)))
   (- (cos eps) (cos x))
   (- (* -0.5 (pow eps 2.0)) (* eps (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.027) || !(eps <= 0.0038)) {
		tmp = cos(eps) - cos(x);
	} else {
		tmp = (-0.5 * pow(eps, 2.0)) - (eps * sin(x));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-0.027d0)) .or. (.not. (eps <= 0.0038d0))) then
        tmp = cos(eps) - cos(x)
    else
        tmp = ((-0.5d0) * (eps ** 2.0d0)) - (eps * sin(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.027) || !(eps <= 0.0038)) {
		tmp = Math.cos(eps) - Math.cos(x);
	} else {
		tmp = (-0.5 * Math.pow(eps, 2.0)) - (eps * Math.sin(x));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.027) or not (eps <= 0.0038):
		tmp = math.cos(eps) - math.cos(x)
	else:
		tmp = (-0.5 * math.pow(eps, 2.0)) - (eps * math.sin(x))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.027) || !(eps <= 0.0038))
		tmp = Float64(cos(eps) - cos(x));
	else
		tmp = Float64(Float64(-0.5 * (eps ^ 2.0)) - Float64(eps * sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.027) || ~((eps <= 0.0038)))
		tmp = cos(eps) - cos(x);
	else
		tmp = (-0.5 * (eps ^ 2.0)) - (eps * sin(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.027], N[Not[LessEqual[eps, 0.0038]], $MachinePrecision]], N[(N[Cos[eps], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision] - N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -0.0269999999999999997 or 0.00379999999999999999 < eps

   1. Initial program 50.7%

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

   2. Taylor expanded in x around 0 52.5%

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

   if -0.0269999999999999997 < eps < 0.00379999999999999999

   1. Initial program 20.6%

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

      1. sub-neg 20.6%

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

      2. +-commutative 20.6%

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

      3. cos-sum 21.0%

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

      4. cancel-sign-sub-inv 21.0%

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

      5. associate-+r+ 83.3%

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

      6. *-commutative 83.3%

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

   3. Applied egg-rr 83.3%

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

   4. Taylor expanded in x around 0 83.3%

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

   5. Taylor expanded in eps around 0 99.5%

      \[\leadsto \color{blue}{-1 \cdot \left(\varepsilon \cdot \sin x\right) + -0.5 \cdot {\varepsilon}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.027 \lor \neg \left(\varepsilon \leq 0.0038\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot {\varepsilon}^{2} - \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 11: 67.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.4 \cdot 10^{-6} \lor \neg \left(\varepsilon \leq 5.8 \cdot 10^{-6}\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;-\varepsilon \cdot \sin x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -5.4e-6) (not (<= eps 5.8e-6)))
   (- (cos eps) (cos x))
   (- (* eps (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -5.4e-6) || !(eps <= 5.8e-6)) {
		tmp = cos(eps) - cos(x);
	} else {
		tmp = -(eps * sin(x));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-5.4d-6)) .or. (.not. (eps <= 5.8d-6))) then
        tmp = cos(eps) - cos(x)
    else
        tmp = -(eps * sin(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -5.4e-6) || !(eps <= 5.8e-6)) {
		tmp = Math.cos(eps) - Math.cos(x);
	} else {
		tmp = -(eps * Math.sin(x));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -5.4e-6) or not (eps <= 5.8e-6):
		tmp = math.cos(eps) - math.cos(x)
	else:
		tmp = -(eps * math.sin(x))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -5.4e-6) || !(eps <= 5.8e-6))
		tmp = Float64(cos(eps) - cos(x));
	else
		tmp = Float64(-Float64(eps * sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -5.4e-6) || ~((eps <= 5.8e-6)))
		tmp = cos(eps) - cos(x);
	else
		tmp = -(eps * sin(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -5.4e-6], N[Not[LessEqual[eps, 5.8e-6]], $MachinePrecision]], N[(N[Cos[eps], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], (-N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if eps < -5.39999999999999997e-6 or 5.8000000000000004e-6 < eps

   1. Initial program 50.7%

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

   2. Taylor expanded in x around 0 52.5%

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

   if -5.39999999999999997e-6 < eps < 5.8000000000000004e-6

   1. Initial program 20.6%

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

   2. Taylor expanded in eps around 0 83.2%

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

      1. associate-*r* 83.2%

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

      2. mul-1-neg 83.2%

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

   4. Simplified 83.2%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.4 \cdot 10^{-6} \lor \neg \left(\varepsilon \leq 5.8 \cdot 10^{-6}\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;-\varepsilon \cdot \sin x\\ \end{array} \]

Alternative 12: 76.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.7%

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

   1. diff-cos 43.7%

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

   2. div-inv 43.7%

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

   3. +-commutative 43.7%

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

   4. associate--l+ 75.8%

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

   5. *-un-lft-identity 75.8%

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

   6. *-un-lft-identity 75.8%

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

   7. distribute-rgt-out-- 75.8%

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

   8. metadata-eval 75.8%

      \[\leadsto -2 \cdot \left(\sin \left(\left(\varepsilon + x \cdot \color{blue}{0}\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]

   9. metadata-eval 75.8%

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

   10. div-inv 75.8%

       \[\leadsto -2 \cdot \left(\sin \left(\left(\varepsilon + x \cdot 0\right) \cdot 0.5\right) \cdot \sin \color{blue}{\left(\left(\left(x + \varepsilon\right) + x\right) \cdot \frac{1}{2}\right)}\right) \]

   11. +-commutative 75.8%

       \[\leadsto -2 \cdot \left(\sin \left(\left(\varepsilon + x \cdot 0\right) \cdot 0.5\right) \cdot \sin \left(\left(\color{blue}{\left(\varepsilon + x\right)} + x\right) \cdot \frac{1}{2}\right)\right) \]

   12. associate-+l+ 75.8%

       \[\leadsto -2 \cdot \left(\sin \left(\left(\varepsilon + x \cdot 0\right) \cdot 0.5\right) \cdot \sin \left(\color{blue}{\left(\varepsilon + \left(x + x\right)\right)} \cdot \frac{1}{2}\right)\right) \]

   13. count-2 75.8%

       \[\leadsto -2 \cdot \left(\sin \left(\left(\varepsilon + x \cdot 0\right) \cdot 0.5\right) \cdot \sin \left(\left(\varepsilon + \color{blue}{2 \cdot x}\right) \cdot \frac{1}{2}\right)\right) \]

   14. *-commutative 75.8%

       \[\leadsto -2 \cdot \left(\sin \left(\left(\varepsilon + x \cdot 0\right) \cdot 0.5\right) \cdot \sin \left(\left(\varepsilon + \color{blue}{x \cdot 2}\right) \cdot \frac{1}{2}\right)\right) \]

   15. metadata-eval 75.8%

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

3. Applied egg-rr 75.8%

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

   1. associate-*r* 75.8%

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

   2. *-commutative 75.8%

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

   3. *-commutative 75.8%

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

   4. *-commutative 75.8%

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

   5. mul0-rgt 75.8%

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

5. Simplified 75.8%

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

6. Final simplification 75.8%

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

Alternative 13: 46.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000125 \lor \neg \left(\varepsilon \leq 0.00014\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot {\varepsilon}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.000125) (not (<= eps 0.00014)))
   (+ (cos eps) -1.0)
   (* -0.5 (pow eps 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.000125) || !(eps <= 0.00014)) {
		tmp = cos(eps) + -1.0;
	} else {
		tmp = -0.5 * pow(eps, 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-0.000125d0)) .or. (.not. (eps <= 0.00014d0))) then
        tmp = cos(eps) + (-1.0d0)
    else
        tmp = (-0.5d0) * (eps ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.000125) || !(eps <= 0.00014)) {
		tmp = Math.cos(eps) + -1.0;
	} else {
		tmp = -0.5 * Math.pow(eps, 2.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.000125) or not (eps <= 0.00014):
		tmp = math.cos(eps) + -1.0
	else:
		tmp = -0.5 * math.pow(eps, 2.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.000125) || !(eps <= 0.00014))
		tmp = Float64(cos(eps) + -1.0);
	else
		tmp = Float64(-0.5 * (eps ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.000125) || ~((eps <= 0.00014)))
		tmp = cos(eps) + -1.0;
	else
		tmp = -0.5 * (eps ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.000125], N[Not[LessEqual[eps, 0.00014]], $MachinePrecision]], N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision], N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.25e-4 or 1.3999999999999999e-4 < eps

   1. Initial program 50.7%

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

   2. Taylor expanded in x around 0 51.0%

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

   if -1.25e-4 < eps < 1.3999999999999999e-4

   1. Initial program 20.6%

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

   2. Taylor expanded in x around 0 20.5%

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

   3. Taylor expanded in eps around 0 34.4%

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

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000125 \lor \neg \left(\varepsilon \leq 0.00014\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot {\varepsilon}^{2}\\ \end{array} \]

Alternative 14: 66.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.1 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 2.7 \cdot 10^{-6}\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;-\varepsilon \cdot \sin x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -1.1e-5) (not (<= eps 2.7e-6)))
   (+ (cos eps) -1.0)
   (- (* eps (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.1e-5) || !(eps <= 2.7e-6)) {
		tmp = cos(eps) + -1.0;
	} else {
		tmp = -(eps * sin(x));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-1.1d-5)) .or. (.not. (eps <= 2.7d-6))) then
        tmp = cos(eps) + (-1.0d0)
    else
        tmp = -(eps * sin(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.1e-5) || !(eps <= 2.7e-6)) {
		tmp = Math.cos(eps) + -1.0;
	} else {
		tmp = -(eps * Math.sin(x));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -1.1e-5) or not (eps <= 2.7e-6):
		tmp = math.cos(eps) + -1.0
	else:
		tmp = -(eps * math.sin(x))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -1.1e-5) || !(eps <= 2.7e-6))
		tmp = Float64(cos(eps) + -1.0);
	else
		tmp = Float64(-Float64(eps * sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -1.1e-5) || ~((eps <= 2.7e-6)))
		tmp = cos(eps) + -1.0;
	else
		tmp = -(eps * sin(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -1.1e-5], N[Not[LessEqual[eps, 2.7e-6]], $MachinePrecision]], N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision], (-N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if eps < -1.1e-5 or 2.69999999999999998e-6 < eps

   1. Initial program 50.7%

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

   2. Taylor expanded in x around 0 51.0%

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

   if -1.1e-5 < eps < 2.69999999999999998e-6

   1. Initial program 20.6%

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

   2. Taylor expanded in eps around 0 83.2%

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

      1. associate-*r* 83.2%

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

      2. mul-1-neg 83.2%

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

   4. Simplified 83.2%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.1 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 2.7 \cdot 10^{-6}\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;-\varepsilon \cdot \sin x\\ \end{array} \]

Alternative 15: 14.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 1 - \cos \varepsilon \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- 1.0 (cos eps)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.7%

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

2. Taylor expanded in x around 0 35.7%

   \[\leadsto \color{blue}{\cos \varepsilon - 1} \]
3. Step-by-step derivation

   1. add-log-exp 35.7%

      \[\leadsto \color{blue}{\log \left(e^{\cos \varepsilon}\right)} - 1 \]

4. Applied egg-rr 35.7%

   \[\leadsto \color{blue}{\log \left(e^{\cos \varepsilon}\right)} - 1 \]
5. Step-by-step derivation

   1. add-log-exp 35.7%

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

   2. flip-- 35.7%

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

   3. metadata-eval 35.7%

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

   4. sub-1-cos 42.8%

      \[\leadsto \frac{\color{blue}{-\sin \varepsilon \cdot \sin \varepsilon}}{\cos \varepsilon + 1} \]

   5. unpow2 42.7%

      \[\leadsto \frac{-\color{blue}{{\sin \varepsilon}^{2}}}{\cos \varepsilon + 1} \]

   6. neg-mul-1 42.7%

      \[\leadsto \frac{\color{blue}{-1 \cdot {\sin \varepsilon}^{2}}}{\cos \varepsilon + 1} \]

   7. *-un-lft-identity 42.7%

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

   8. times-frac 42.7%

      \[\leadsto \color{blue}{\frac{-1}{1} \cdot \frac{{\sin \varepsilon}^{2}}{\cos \varepsilon + 1}} \]

   9. metadata-eval 42.7%

      \[\leadsto \color{blue}{-1} \cdot \frac{{\sin \varepsilon}^{2}}{\cos \varepsilon + 1} \]

   10. add-sqr-sqrt 42.8%

       \[\leadsto -1 \cdot \frac{\color{blue}{\sqrt{{\sin \varepsilon}^{2}} \cdot \sqrt{{\sin \varepsilon}^{2}}}}{\cos \varepsilon + 1} \]

   11. sqrt-unprod 39.4%

       \[\leadsto -1 \cdot \frac{\color{blue}{\sqrt{{\sin \varepsilon}^{2} \cdot {\sin \varepsilon}^{2}}}}{\cos \varepsilon + 1} \]

   12. sqr-neg 39.4%

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

   13. sqrt-unprod 8.6%

       \[\leadsto -1 \cdot \frac{\color{blue}{\sqrt{-{\sin \varepsilon}^{2}} \cdot \sqrt{-{\sin \varepsilon}^{2}}}}{\cos \varepsilon + 1} \]

   14. add-sqr-sqrt 13.3%

       \[\leadsto -1 \cdot \frac{\color{blue}{-{\sin \varepsilon}^{2}}}{\cos \varepsilon + 1} \]

   15. unpow2 13.3%

       \[\leadsto -1 \cdot \frac{-\color{blue}{\sin \varepsilon \cdot \sin \varepsilon}}{\cos \varepsilon + 1} \]

   16. sub-1-cos 13.4%

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

   17. metadata-eval 13.4%

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

   18. flip-- 13.4%

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

   19. sub-neg 13.4%

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

   20. metadata-eval 13.4%

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

6. Applied egg-rr 13.4%

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

   1. +-commutative 13.4%

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

   2. distribute-lft-in 13.4%

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

   3. metadata-eval 13.4%

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

   4. neg-mul-1 13.4%

      \[\leadsto 1 + \color{blue}{\left(-\cos \varepsilon\right)} \]

   5. sub-neg 13.4%

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

8. Simplified 13.4%

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

9. Final simplification 13.4%

   \[\leadsto 1 - \cos \varepsilon \]

Alternative 16: 38.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \cos \varepsilon + -1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.7%

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

2. Taylor expanded in x around 0 35.7%

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

3. Final simplification 35.7%

   \[\leadsto \cos \varepsilon + -1 \]

Alternative 17: 12.9% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

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

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 35.7%

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

   1. diff-cos 43.7%

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

   2. sin-mult 35.5%

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

   3. associate-*r/ 35.5%

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

3. Applied egg-rr 35.7%

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

   1. *-commutative 35.7%

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

   2. associate-/l* 35.7%

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

   3. associate--l+ 35.7%

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

   4. mul0-rgt 35.7%

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

   5. mul0-rgt 35.7%

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

   6. metadata-eval 35.7%

      \[\leadsto \frac{\cos \left(0.5 \cdot \left(\varepsilon + \left(0 - \left(\varepsilon + x \cdot 2\right)\right)\right)\right) - \cos \left(0.5 \cdot \left(\left(\varepsilon + 0\right) + \left(\varepsilon + x \cdot 2\right)\right)\right)}{\color{blue}{-1}} \]

5. Simplified 35.7%

   \[\leadsto \color{blue}{\frac{\cos \left(0.5 \cdot \left(\varepsilon + \left(0 - \left(\varepsilon + x \cdot 2\right)\right)\right)\right) - \cos \left(0.5 \cdot \left(\left(\varepsilon + 0\right) + \left(\varepsilon + x \cdot 2\right)\right)\right)}{-1}} \]
6. Step-by-step derivation

   1. sub-neg 35.7%

      \[\leadsto \frac{\color{blue}{\cos \left(0.5 \cdot \left(\varepsilon + \left(0 - \left(\varepsilon + x \cdot 2\right)\right)\right)\right) + \left(-\cos \left(0.5 \cdot \left(\left(\varepsilon + 0\right) + \left(\varepsilon + x \cdot 2\right)\right)\right)\right)}}{-1} \]

7. Applied egg-rr 11.7%

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

   1. sub-neg 11.7%

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

   2. +-inverses 11.7%

      \[\leadsto \frac{\color{blue}{0}}{-1} \]

9. Simplified 11.7%

   \[\leadsto \frac{\color{blue}{0}}{-1} \]

10. Final simplification 11.7%

    \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023336 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))