2cos (problem 3.3.5)

Percentage Accurate: 38.1% → 99.2%

Time: 24.2s

Alternatives: 19

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 38.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin x \cdot \sin \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.0062:\\ \;\;\;\;\log \left(e^{\cos x \cdot \left(\cos \varepsilon + -1\right) - t_0}\right)\\ \mathbf{elif}\;\varepsilon \leq 0.005:\\ \;\;\;\;\cos x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4} + -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, -\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (sin x) (sin eps))))
   (if (<= eps -0.0062)
     (log (exp (- (* (cos x) (+ (cos eps) -1.0)) t_0)))
     (if (<= eps 0.005)
       (-
        (*
         (cos x)
         (+ (* 0.041666666666666664 (pow eps 4.0)) (* -0.5 (* eps eps))))
        t_0)
       (fma (cos x) (cos eps) (- (fma (sin x) (sin eps) (cos x))))))))

C

double code(double x, double eps) {
	double t_0 = sin(x) * sin(eps);
	double tmp;
	if (eps <= -0.0062) {
		tmp = log(exp(((cos(x) * (cos(eps) + -1.0)) - t_0)));
	} else if (eps <= 0.005) {
		tmp = (cos(x) * ((0.041666666666666664 * pow(eps, 4.0)) + (-0.5 * (eps * eps)))) - t_0;
	} else {
		tmp = fma(cos(x), cos(eps), -fma(sin(x), sin(eps), cos(x)));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(sin(x) * sin(eps))
	tmp = 0.0
	if (eps <= -0.0062)
		tmp = log(exp(Float64(Float64(cos(x) * Float64(cos(eps) + -1.0)) - t_0)));
	elseif (eps <= 0.005)
		tmp = Float64(Float64(cos(x) * Float64(Float64(0.041666666666666664 * (eps ^ 4.0)) + Float64(-0.5 * Float64(eps * eps)))) - t_0);
	else
		tmp = fma(cos(x), cos(eps), Float64(-fma(sin(x), sin(eps), cos(x))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.0062], N[Log[N[Exp[N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[eps, 0.005], N[(N[(N[Cos[x], $MachinePrecision] * N[(N[(0.041666666666666664 * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + (-N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision] + N[Cos[x], $MachinePrecision]), $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.00619999999999999978

   1. Initial program 59.5%

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

      1. cos-sum 98.3%

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

   3. Applied egg-rr 98.3%

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

   4. Taylor expanded in x around inf 98.4%

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

      1. associate--r+ 98.2%

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

      2. *-commutative 98.2%

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

      3. *-commutative 98.2%

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

   6. Simplified 98.2%

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

   7. Taylor expanded in x around inf 98.2%

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

      1. sub-neg 98.2%

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

      2. neg-mul-1 98.2%

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

      3. distribute-rgt-out 98.3%

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

   9. Simplified 98.3%

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

       1. add-log-exp 98.5%

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

   11. Applied egg-rr 98.5%

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

   if -0.00619999999999999978 < eps < 0.0050000000000000001

   1. Initial program 24.7%

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

      1. cos-sum 27.1%

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

   3. Applied egg-rr 27.1%

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

   4. Taylor expanded in x around inf 27.1%

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

      1. associate--r+ 84.4%

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

      2. *-commutative 84.4%

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

      3. *-commutative 84.4%

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

   6. Simplified 84.4%

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

   7. Taylor expanded in x around inf 84.4%

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

      1. sub-neg 84.4%

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

      2. neg-mul-1 84.4%

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

      3. distribute-rgt-out 84.4%

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

   9. Simplified 84.4%

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

   10. Taylor expanded in eps around 0 99.8%

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

       1. associate-*r* 99.8%

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

       2. associate-*r* 99.8%

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

       3. distribute-rgt-out 99.8%

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

       4. unpow2 99.8%

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

   12. Simplified 99.8%

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

   if 0.0050000000000000001 < eps

   1. Initial program 53.9%

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

      1. cos-sum 98.5%

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

      2. associate--l- 98.5%

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

      3. fma-neg 98.6%

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

      4. fma-def 98.7%

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

   3. Applied egg-rr 98.7%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0062:\\ \;\;\;\;\log \left(e^{\cos x \cdot \left(\cos \varepsilon + -1\right) - \sin x \cdot \sin \varepsilon}\right)\\ \mathbf{elif}\;\varepsilon \leq 0.005:\\ \;\;\;\;\cos x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4} + -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \sin x \cdot \sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, -\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right)\\ \end{array} \]

Alternative 2: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin x \cdot \sin \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.0062:\\ \;\;\;\;\log \left(e^{\cos x \cdot \left(\cos \varepsilon + -1\right) - t_0}\right)\\ \mathbf{elif}\;\varepsilon \leq 0.005:\\ \;\;\;\;\cos x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4} + -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \left(-\cos x\right) - t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (sin x) (sin eps))))
   (if (<= eps -0.0062)
     (log (exp (- (* (cos x) (+ (cos eps) -1.0)) t_0)))
     (if (<= eps 0.005)
       (-
        (*
         (cos x)
         (+ (* 0.041666666666666664 (pow eps 4.0)) (* -0.5 (* eps eps))))
        t_0)
       (fma (cos x) (cos eps) (- (- (cos x)) t_0))))))

C

double code(double x, double eps) {
	double t_0 = sin(x) * sin(eps);
	double tmp;
	if (eps <= -0.0062) {
		tmp = log(exp(((cos(x) * (cos(eps) + -1.0)) - t_0)));
	} else if (eps <= 0.005) {
		tmp = (cos(x) * ((0.041666666666666664 * pow(eps, 4.0)) + (-0.5 * (eps * eps)))) - t_0;
	} else {
		tmp = fma(cos(x), cos(eps), (-cos(x) - t_0));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(sin(x) * sin(eps))
	tmp = 0.0
	if (eps <= -0.0062)
		tmp = log(exp(Float64(Float64(cos(x) * Float64(cos(eps) + -1.0)) - t_0)));
	elseif (eps <= 0.005)
		tmp = Float64(Float64(cos(x) * Float64(Float64(0.041666666666666664 * (eps ^ 4.0)) + Float64(-0.5 * Float64(eps * eps)))) - t_0);
	else
		tmp = fma(cos(x), cos(eps), Float64(Float64(-cos(x)) - t_0));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.0062], N[Log[N[Exp[N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[eps, 0.005], N[(N[(N[Cos[x], $MachinePrecision] * N[(N[(0.041666666666666664 * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + N[((-N[Cos[x], $MachinePrecision]) - t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.00619999999999999978

   1. Initial program 59.5%

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

      1. cos-sum 98.3%

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

   3. Applied egg-rr 98.3%

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

   4. Taylor expanded in x around inf 98.4%

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

      1. associate--r+ 98.2%

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

      2. *-commutative 98.2%

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

      3. *-commutative 98.2%

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

   6. Simplified 98.2%

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

   7. Taylor expanded in x around inf 98.2%

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

      1. sub-neg 98.2%

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

      2. neg-mul-1 98.2%

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

      3. distribute-rgt-out 98.3%

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

   9. Simplified 98.3%

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

       1. add-log-exp 98.5%

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

   11. Applied egg-rr 98.5%

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

   if -0.00619999999999999978 < eps < 0.0050000000000000001

   1. Initial program 24.7%

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

      1. cos-sum 27.1%

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

   3. Applied egg-rr 27.1%

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

   4. Taylor expanded in x around inf 27.1%

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

      1. associate--r+ 84.4%

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

      2. *-commutative 84.4%

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

      3. *-commutative 84.4%

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

   6. Simplified 84.4%

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

   7. Taylor expanded in x around inf 84.4%

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

      1. sub-neg 84.4%

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

      2. neg-mul-1 84.4%

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

      3. distribute-rgt-out 84.4%

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

   9. Simplified 84.4%

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

   10. Taylor expanded in eps around 0 99.8%

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

       1. associate-*r* 99.8%

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

       2. associate-*r* 99.8%

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

       3. distribute-rgt-out 99.8%

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

       4. unpow2 99.8%

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

   12. Simplified 99.8%

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

   if 0.0050000000000000001 < eps

   1. Initial program 53.9%

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

      1. sub-neg 53.9%

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

      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. associate-+l- 98.5%

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

      4. fma-neg 98.6%

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

   3. Applied egg-rr 98.6%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0062:\\ \;\;\;\;\log \left(e^{\cos x \cdot \left(\cos \varepsilon + -1\right) - \sin x \cdot \sin \varepsilon}\right)\\ \mathbf{elif}\;\varepsilon \leq 0.005:\\ \;\;\;\;\cos x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4} + -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \sin x \cdot \sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \left(-\cos x\right) - \sin x \cdot \sin \varepsilon\right)\\ \end{array} \]

Alternative 3: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \varepsilon + -1\\ t_1 := \sin x \cdot \sin \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.0062:\\ \;\;\;\;\log \left(e^{\cos x \cdot t_0 - t_1}\right)\\ \mathbf{elif}\;\varepsilon \leq 0.005:\\ \;\;\;\;\cos x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4} + -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \cos x, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (cos eps) -1.0)) (t_1 (* (sin x) (sin eps))))
   (if (<= eps -0.0062)
     (log (exp (- (* (cos x) t_0) t_1)))
     (if (<= eps 0.005)
       (-
        (*
         (cos x)
         (+ (* 0.041666666666666664 (pow eps 4.0)) (* -0.5 (* eps eps))))
        t_1)
       (fma t_0 (cos x) (* (sin eps) (- (sin x))))))))

C

double code(double x, double eps) {
	double t_0 = cos(eps) + -1.0;
	double t_1 = sin(x) * sin(eps);
	double tmp;
	if (eps <= -0.0062) {
		tmp = log(exp(((cos(x) * t_0) - t_1)));
	} else if (eps <= 0.005) {
		tmp = (cos(x) * ((0.041666666666666664 * pow(eps, 4.0)) + (-0.5 * (eps * eps)))) - t_1;
	} else {
		tmp = fma(t_0, cos(x), (sin(eps) * -sin(x)));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(cos(eps) + -1.0)
	t_1 = Float64(sin(x) * sin(eps))
	tmp = 0.0
	if (eps <= -0.0062)
		tmp = log(exp(Float64(Float64(cos(x) * t_0) - t_1)));
	elseif (eps <= 0.005)
		tmp = Float64(Float64(cos(x) * Float64(Float64(0.041666666666666664 * (eps ^ 4.0)) + Float64(-0.5 * Float64(eps * eps)))) - t_1);
	else
		tmp = fma(t_0, cos(x), Float64(sin(eps) * Float64(-sin(x))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.0062], N[Log[N[Exp[N[(N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[eps, 0.005], N[(N[(N[Cos[x], $MachinePrecision] * N[(N[(0.041666666666666664 * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(t$95$0 * 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 < -0.00619999999999999978

   1. Initial program 59.5%

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

      1. cos-sum 98.3%

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

   3. Applied egg-rr 98.3%

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

   4. Taylor expanded in x around inf 98.4%

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

      1. associate--r+ 98.2%

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

      2. *-commutative 98.2%

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

      3. *-commutative 98.2%

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

   6. Simplified 98.2%

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

   7. Taylor expanded in x around inf 98.2%

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

      1. sub-neg 98.2%

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

      2. neg-mul-1 98.2%

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

      3. distribute-rgt-out 98.3%

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

   9. Simplified 98.3%

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

       1. add-log-exp 98.5%

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

   11. Applied egg-rr 98.5%

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

   if -0.00619999999999999978 < eps < 0.0050000000000000001

   1. Initial program 24.7%

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

      1. cos-sum 27.1%

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

   3. Applied egg-rr 27.1%

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

   4. Taylor expanded in x around inf 27.1%

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

      1. associate--r+ 84.4%

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

      2. *-commutative 84.4%

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

      3. *-commutative 84.4%

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

   6. Simplified 84.4%

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

   7. Taylor expanded in x around inf 84.4%

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

      1. sub-neg 84.4%

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

      2. neg-mul-1 84.4%

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

      3. distribute-rgt-out 84.4%

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

   9. Simplified 84.4%

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

   10. Taylor expanded in eps around 0 99.8%

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

       1. associate-*r* 99.8%

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

       2. associate-*r* 99.8%

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

       3. distribute-rgt-out 99.8%

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

       4. unpow2 99.8%

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

   12. Simplified 99.8%

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

   if 0.0050000000000000001 < eps

   1. Initial program 53.9%

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

      1. cos-sum 98.5%

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

   3. Applied egg-rr 98.5%

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

   4. Taylor expanded in x around inf 98.5%

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

      1. associate--r+ 98.5%

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

      2. *-commutative 98.5%

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

      3. *-commutative 98.5%

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

   6. Simplified 98.5%

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

   7. Taylor expanded in x around inf 98.5%

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

      1. sub-neg 98.5%

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

      2. neg-mul-1 98.5%

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

      3. distribute-rgt-out 98.5%

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

   9. Simplified 98.5%

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

       1. *-commutative 98.5%

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

       2. fma-neg 98.5%

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

   11. Applied egg-rr 98.5%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0062:\\ \;\;\;\;\log \left(e^{\cos x \cdot \left(\cos \varepsilon + -1\right) - \sin x \cdot \sin \varepsilon}\right)\\ \mathbf{elif}\;\varepsilon \leq 0.005:\\ \;\;\;\;\cos x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4} + -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \sin x \cdot \sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos \varepsilon + -1, \cos x, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \end{array} \]

Alternative 4: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \varepsilon + -1\\ t_1 := \sin x \cdot \sin \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.0055:\\ \;\;\;\;\cos x \cdot t_0 - t_1\\ \mathbf{elif}\;\varepsilon \leq 0.005:\\ \;\;\;\;\cos x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4} + -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \cos x, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (cos eps) -1.0)) (t_1 (* (sin x) (sin eps))))
   (if (<= eps -0.0055)
     (- (* (cos x) t_0) t_1)
     (if (<= eps 0.005)
       (-
        (*
         (cos x)
         (+ (* 0.041666666666666664 (pow eps 4.0)) (* -0.5 (* eps eps))))
        t_1)
       (fma t_0 (cos x) (* (sin eps) (- (sin x))))))))

C

double code(double x, double eps) {
	double t_0 = cos(eps) + -1.0;
	double t_1 = sin(x) * sin(eps);
	double tmp;
	if (eps <= -0.0055) {
		tmp = (cos(x) * t_0) - t_1;
	} else if (eps <= 0.005) {
		tmp = (cos(x) * ((0.041666666666666664 * pow(eps, 4.0)) + (-0.5 * (eps * eps)))) - t_1;
	} else {
		tmp = fma(t_0, cos(x), (sin(eps) * -sin(x)));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(cos(eps) + -1.0)
	t_1 = Float64(sin(x) * sin(eps))
	tmp = 0.0
	if (eps <= -0.0055)
		tmp = Float64(Float64(cos(x) * t_0) - t_1);
	elseif (eps <= 0.005)
		tmp = Float64(Float64(cos(x) * Float64(Float64(0.041666666666666664 * (eps ^ 4.0)) + Float64(-0.5 * Float64(eps * eps)))) - t_1);
	else
		tmp = fma(t_0, cos(x), Float64(sin(eps) * Float64(-sin(x))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.0055], N[(N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[eps, 0.005], N[(N[(N[Cos[x], $MachinePrecision] * N[(N[(0.041666666666666664 * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(t$95$0 * 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 < -0.0054999999999999997

   1. Initial program 58.5%

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

      1. cos-sum 98.3%

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

   3. Applied egg-rr 98.3%

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

   4. Taylor expanded in x around inf 98.3%

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

      1. associate--r+ 98.2%

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

      2. *-commutative 98.2%

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

      3. *-commutative 98.2%

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

   6. Simplified 98.2%

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

   7. Taylor expanded in x around inf 98.2%

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

      1. sub-neg 98.2%

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

      2. neg-mul-1 98.2%

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

      3. distribute-rgt-out 98.3%

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

   9. Simplified 98.3%

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

   if -0.0054999999999999997 < eps < 0.0050000000000000001

   1. Initial program 24.8%

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

      1. cos-sum 26.6%

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

   3. Applied egg-rr 26.6%

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

   4. Taylor expanded in x around inf 26.6%

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

      1. associate--r+ 84.3%

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

      2. *-commutative 84.3%

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

      3. *-commutative 84.3%

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

   6. Simplified 84.3%

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

   7. Taylor expanded in x around inf 84.3%

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

      1. sub-neg 84.3%

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

      2. neg-mul-1 84.3%

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

      3. distribute-rgt-out 84.3%

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

   9. Simplified 84.3%

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

   10. Taylor expanded in eps around 0 99.8%

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

       1. associate-*r* 99.8%

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

       2. associate-*r* 99.8%

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

       3. distribute-rgt-out 99.8%

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

       4. unpow2 99.8%

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

   12. Simplified 99.8%

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

   if 0.0050000000000000001 < eps

   1. Initial program 53.9%

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

      1. cos-sum 98.5%

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

   3. Applied egg-rr 98.5%

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

   4. Taylor expanded in x around inf 98.5%

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

      1. associate--r+ 98.5%

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

      2. *-commutative 98.5%

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

      3. *-commutative 98.5%

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

   6. Simplified 98.5%

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

   7. Taylor expanded in x around inf 98.5%

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

      1. sub-neg 98.5%

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

      2. neg-mul-1 98.5%

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

      3. distribute-rgt-out 98.5%

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

   9. Simplified 98.5%

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

       1. *-commutative 98.5%

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

       2. fma-neg 98.5%

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

   11. Applied egg-rr 98.5%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0055:\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right) - \sin x \cdot \sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.005:\\ \;\;\;\;\cos x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4} + -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \sin x \cdot \sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos \varepsilon + -1, \cos x, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \end{array} \]

Alternative 5: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin x \cdot \sin \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.0062:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + t_0\right)\\ \mathbf{elif}\;\varepsilon \leq 0.005:\\ \;\;\;\;\cos x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4} + -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos \varepsilon + -1, \cos x, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (sin x) (sin eps))))
   (if (<= eps -0.0062)
     (- (* (cos x) (cos eps)) (+ (cos x) t_0))
     (if (<= eps 0.005)
       (-
        (*
         (cos x)
         (+ (* 0.041666666666666664 (pow eps 4.0)) (* -0.5 (* eps eps))))
        t_0)
       (fma (+ (cos eps) -1.0) (cos x) (* (sin eps) (- (sin x))))))))

C

double code(double x, double eps) {
	double t_0 = sin(x) * sin(eps);
	double tmp;
	if (eps <= -0.0062) {
		tmp = (cos(x) * cos(eps)) - (cos(x) + t_0);
	} else if (eps <= 0.005) {
		tmp = (cos(x) * ((0.041666666666666664 * pow(eps, 4.0)) + (-0.5 * (eps * eps)))) - t_0;
	} else {
		tmp = fma((cos(eps) + -1.0), cos(x), (sin(eps) * -sin(x)));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(sin(x) * sin(eps))
	tmp = 0.0
	if (eps <= -0.0062)
		tmp = Float64(Float64(cos(x) * cos(eps)) - Float64(cos(x) + t_0));
	elseif (eps <= 0.005)
		tmp = Float64(Float64(cos(x) * Float64(Float64(0.041666666666666664 * (eps ^ 4.0)) + Float64(-0.5 * Float64(eps * eps)))) - t_0);
	else
		tmp = fma(Float64(cos(eps) + -1.0), cos(x), Float64(sin(eps) * Float64(-sin(x))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.0062], N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[x], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.005], N[(N[(N[Cos[x], $MachinePrecision] * N[(N[(0.041666666666666664 * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $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 < -0.00619999999999999978

   1. Initial program 59.5%

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

      1. cos-sum 98.3%

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

   3. Applied egg-rr 98.3%

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

   4. Taylor expanded in x around inf 98.4%

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

   if -0.00619999999999999978 < eps < 0.0050000000000000001

   1. Initial program 24.7%

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

      1. cos-sum 27.1%

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

   3. Applied egg-rr 27.1%

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

   4. Taylor expanded in x around inf 27.1%

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

      1. associate--r+ 84.4%

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

      2. *-commutative 84.4%

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

      3. *-commutative 84.4%

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

   6. Simplified 84.4%

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

   7. Taylor expanded in x around inf 84.4%

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

      1. sub-neg 84.4%

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

      2. neg-mul-1 84.4%

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

      3. distribute-rgt-out 84.4%

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

   9. Simplified 84.4%

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

   10. Taylor expanded in eps around 0 99.8%

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

       1. associate-*r* 99.8%

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

       2. associate-*r* 99.8%

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

       3. distribute-rgt-out 99.8%

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

       4. unpow2 99.8%

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

   12. Simplified 99.8%

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

   if 0.0050000000000000001 < eps

   1. Initial program 53.9%

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

      1. cos-sum 98.5%

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

   3. Applied egg-rr 98.5%

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

   4. Taylor expanded in x around inf 98.5%

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

      1. associate--r+ 98.5%

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

      2. *-commutative 98.5%

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

      3. *-commutative 98.5%

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

   6. Simplified 98.5%

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

   7. Taylor expanded in x around inf 98.5%

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

      1. sub-neg 98.5%

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

      2. neg-mul-1 98.5%

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

      3. distribute-rgt-out 98.5%

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

   9. Simplified 98.5%

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

       1. *-commutative 98.5%

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

       2. fma-neg 98.5%

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

   11. Applied egg-rr 98.5%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0062:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 0.005:\\ \;\;\;\;\cos x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4} + -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \sin x \cdot \sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos \varepsilon + -1, \cos x, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \end{array} \]

Alternative 6: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin x \cdot \sin \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.0055 \lor \neg \left(\varepsilon \leq 0.005\right):\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4} + -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (sin x) (sin eps))))
   (if (or (<= eps -0.0055) (not (<= eps 0.005)))
     (- (* (cos x) (+ (cos eps) -1.0)) t_0)
     (-
      (*
       (cos x)
       (+ (* 0.041666666666666664 (pow eps 4.0)) (* -0.5 (* eps eps))))
      t_0))))

C

double code(double x, double eps) {
	double t_0 = sin(x) * sin(eps);
	double tmp;
	if ((eps <= -0.0055) || !(eps <= 0.005)) {
		tmp = (cos(x) * (cos(eps) + -1.0)) - t_0;
	} else {
		tmp = (cos(x) * ((0.041666666666666664 * pow(eps, 4.0)) + (-0.5 * (eps * eps)))) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(x) * sin(eps)
    if ((eps <= (-0.0055d0)) .or. (.not. (eps <= 0.005d0))) then
        tmp = (cos(x) * (cos(eps) + (-1.0d0))) - t_0
    else
        tmp = (cos(x) * ((0.041666666666666664d0 * (eps ** 4.0d0)) + ((-0.5d0) * (eps * eps)))) - t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin(x) * Math.sin(eps);
	double tmp;
	if ((eps <= -0.0055) || !(eps <= 0.005)) {
		tmp = (Math.cos(x) * (Math.cos(eps) + -1.0)) - t_0;
	} else {
		tmp = (Math.cos(x) * ((0.041666666666666664 * Math.pow(eps, 4.0)) + (-0.5 * (eps * eps)))) - t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.sin(x) * math.sin(eps)
	tmp = 0
	if (eps <= -0.0055) or not (eps <= 0.005):
		tmp = (math.cos(x) * (math.cos(eps) + -1.0)) - t_0
	else:
		tmp = (math.cos(x) * ((0.041666666666666664 * math.pow(eps, 4.0)) + (-0.5 * (eps * eps)))) - t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(sin(x) * sin(eps))
	tmp = 0.0
	if ((eps <= -0.0055) || !(eps <= 0.005))
		tmp = Float64(Float64(cos(x) * Float64(cos(eps) + -1.0)) - t_0);
	else
		tmp = Float64(Float64(cos(x) * Float64(Float64(0.041666666666666664 * (eps ^ 4.0)) + Float64(-0.5 * Float64(eps * eps)))) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = sin(x) * sin(eps);
	tmp = 0.0;
	if ((eps <= -0.0055) || ~((eps <= 0.005)))
		tmp = (cos(x) * (cos(eps) + -1.0)) - t_0;
	else
		tmp = (cos(x) * ((0.041666666666666664 * (eps ^ 4.0)) + (-0.5 * (eps * eps)))) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[eps, -0.0055], N[Not[LessEqual[eps, 0.005]], $MachinePrecision]], N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[Cos[x], $MachinePrecision] * N[(N[(0.041666666666666664 * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < -0.0054999999999999997 or 0.0050000000000000001 < eps

   1. Initial program 56.0%

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

      1. cos-sum 98.4%

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

   3. Applied egg-rr 98.4%

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

   4. Taylor expanded in x around inf 98.4%

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

      1. associate--r+ 98.4%

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

      2. *-commutative 98.4%

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

      3. *-commutative 98.4%

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

   6. Simplified 98.4%

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

   7. Taylor expanded in x around inf 98.4%

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

      1. sub-neg 98.4%

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

      2. neg-mul-1 98.4%

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

      3. distribute-rgt-out 98.4%

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

   9. Simplified 98.4%

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

   if -0.0054999999999999997 < eps < 0.0050000000000000001

   1. Initial program 24.8%

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

      1. cos-sum 26.6%

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

   3. Applied egg-rr 26.6%

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

   4. Taylor expanded in x around inf 26.6%

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

      1. associate--r+ 84.3%

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

      2. *-commutative 84.3%

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

      3. *-commutative 84.3%

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

   6. Simplified 84.3%

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

   7. Taylor expanded in x around inf 84.3%

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

      1. sub-neg 84.3%

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

      2. neg-mul-1 84.3%

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

      3. distribute-rgt-out 84.3%

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

   9. Simplified 84.3%

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

   10. Taylor expanded in eps around 0 99.8%

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

       1. associate-*r* 99.8%

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

       2. associate-*r* 99.8%

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

       3. distribute-rgt-out 99.8%

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

       4. unpow2 99.8%

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

   12. Simplified 99.8%

       \[\leadsto \color{blue}{\cos x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) + 0.041666666666666664 \cdot {\varepsilon}^{4}\right)} - \sin x \cdot \sin \varepsilon \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0055 \lor \neg \left(\varepsilon \leq 0.005\right):\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right) - \sin x \cdot \sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4} + -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \sin x \cdot \sin \varepsilon\\ \end{array} \]

Alternative 7: 75.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\varepsilon + x\right) - \cos x \leq -0.0002:\\ \;\;\;\;\sin \varepsilon \cdot \left(-\tan \left(\frac{\varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) - \varepsilon \cdot \sin x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- (cos (+ eps x)) (cos x)) -0.0002)
   (* (sin eps) (- (tan (/ eps 2.0))))
   (- (* -0.5 (* eps (* eps (cos x)))) (* eps (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if ((cos((eps + x)) - cos(x)) <= -0.0002) {
		tmp = sin(eps) * -tan((eps / 2.0));
	} else {
		tmp = (-0.5 * (eps * (eps * cos(x)))) - (eps * sin(x));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((cos((eps + x)) - cos(x)) <= (-0.0002d0)) then
        tmp = sin(eps) * -tan((eps / 2.0d0))
    else
        tmp = ((-0.5d0) * (eps * (eps * cos(x)))) - (eps * sin(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((Math.cos((eps + x)) - Math.cos(x)) <= -0.0002) {
		tmp = Math.sin(eps) * -Math.tan((eps / 2.0));
	} else {
		tmp = (-0.5 * (eps * (eps * Math.cos(x)))) - (eps * Math.sin(x));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (math.cos((eps + x)) - math.cos(x)) <= -0.0002:
		tmp = math.sin(eps) * -math.tan((eps / 2.0))
	else:
		tmp = (-0.5 * (eps * (eps * math.cos(x)))) - (eps * math.sin(x))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(cos(Float64(eps + x)) - cos(x)) <= -0.0002)
		tmp = Float64(sin(eps) * Float64(-tan(Float64(eps / 2.0))));
	else
		tmp = Float64(Float64(-0.5 * Float64(eps * Float64(eps * cos(x)))) - Float64(eps * sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((cos((eps + x)) - cos(x)) <= -0.0002)
		tmp = sin(eps) * -tan((eps / 2.0));
	else
		tmp = (-0.5 * (eps * (eps * cos(x)))) - (eps * sin(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(N[Cos[N[(eps + x), $MachinePrecision]], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], -0.0002], N[(N[Sin[eps], $MachinePrecision] * (-N[Tan[N[(eps / 2.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[(-0.5 * N[(eps * N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cos.f64 (+.f64 x eps)) (cos.f64 x)) < -2.0000000000000001e-4

   1. Initial program 85.1%

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

   2. Taylor expanded in x around 0 85.1%

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

      1. flip-- 84.4%

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

      2. metadata-eval 84.4%

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

   4. Applied egg-rr 84.4%

      \[\leadsto \color{blue}{\frac{\cos \varepsilon \cdot \cos \varepsilon - 1}{\cos \varepsilon + 1}} \]
   5. Step-by-step derivation

      1. sub-1-cos 85.1%

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

      2. unpow2 85.1%

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

      3. +-commutative 85.1%

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

   6. Simplified 85.1%

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

   7. Taylor expanded in eps around inf 85.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{{\sin \varepsilon}^{2}}{1 + \cos \varepsilon}} \]
   8. Step-by-step derivation

      1. mul-1-neg 85.1%

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

      2. +-commutative 85.1%

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

      3. unpow2 85.1%

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

      4. associate-*r/ 85.2%

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

      5. distribute-lft-neg-in 85.2%

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

      6. +-commutative 85.2%

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

      7. hang-0p-tan 86.2%

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

   9. Simplified 86.2%

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

   if -2.0000000000000001e-4 < (-.f64 (cos.f64 (+.f64 x eps)) (cos.f64 x))

   1. Initial program 20.0%

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

   2. Taylor expanded in eps around 0 75.3%

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

      1. +-commutative 75.3%

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

      2. mul-1-neg 75.3%

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

      3. unsub-neg 75.3%

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

      4. unpow2 75.3%

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

      5. associate-*l* 75.3%

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

   4. Simplified 75.3%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\varepsilon + x\right) - \cos x \leq -0.0002:\\ \;\;\;\;\sin \varepsilon \cdot \left(-\tan \left(\frac{\varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) - \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 8: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin x \cdot \sin \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.00016 \lor \neg \left(\varepsilon \leq 0.00015\right):\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\right) - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (sin x) (sin eps))))
   (if (or (<= eps -0.00016) (not (<= eps 0.00015)))
     (- (* (cos x) (+ (cos eps) -1.0)) t_0)
     (- (* (cos x) (* eps (* eps -0.5))) t_0))))

C

double code(double x, double eps) {
	double t_0 = sin(x) * sin(eps);
	double tmp;
	if ((eps <= -0.00016) || !(eps <= 0.00015)) {
		tmp = (cos(x) * (cos(eps) + -1.0)) - t_0;
	} else {
		tmp = (cos(x) * (eps * (eps * -0.5))) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(x) * sin(eps)
    if ((eps <= (-0.00016d0)) .or. (.not. (eps <= 0.00015d0))) then
        tmp = (cos(x) * (cos(eps) + (-1.0d0))) - t_0
    else
        tmp = (cos(x) * (eps * (eps * (-0.5d0)))) - t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin(x) * Math.sin(eps);
	double tmp;
	if ((eps <= -0.00016) || !(eps <= 0.00015)) {
		tmp = (Math.cos(x) * (Math.cos(eps) + -1.0)) - t_0;
	} else {
		tmp = (Math.cos(x) * (eps * (eps * -0.5))) - t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.sin(x) * math.sin(eps)
	tmp = 0
	if (eps <= -0.00016) or not (eps <= 0.00015):
		tmp = (math.cos(x) * (math.cos(eps) + -1.0)) - t_0
	else:
		tmp = (math.cos(x) * (eps * (eps * -0.5))) - t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(sin(x) * sin(eps))
	tmp = 0.0
	if ((eps <= -0.00016) || !(eps <= 0.00015))
		tmp = Float64(Float64(cos(x) * Float64(cos(eps) + -1.0)) - t_0);
	else
		tmp = Float64(Float64(cos(x) * Float64(eps * Float64(eps * -0.5))) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = sin(x) * sin(eps);
	tmp = 0.0;
	if ((eps <= -0.00016) || ~((eps <= 0.00015)))
		tmp = (cos(x) * (cos(eps) + -1.0)) - t_0;
	else
		tmp = (cos(x) * (eps * (eps * -0.5))) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[eps, -0.00016], N[Not[LessEqual[eps, 0.00015]], $MachinePrecision]], N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[Cos[x], $MachinePrecision] * N[(eps * N[(eps * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.60000000000000013e-4 or 1.49999999999999987e-4 < eps

   1. Initial program 56.0%

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

      1. cos-sum 98.4%

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

   3. Applied egg-rr 98.4%

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

   4. Taylor expanded in x around inf 98.4%

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

      1. associate--r+ 98.4%

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

      2. *-commutative 98.4%

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

      3. *-commutative 98.4%

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

   6. Simplified 98.4%

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

   7. Taylor expanded in x around inf 98.4%

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

      1. sub-neg 98.4%

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

      2. neg-mul-1 98.4%

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

      3. distribute-rgt-out 98.4%

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

   9. Simplified 98.4%

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

   if -1.60000000000000013e-4 < eps < 1.49999999999999987e-4

   1. Initial program 24.8%

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

      1. cos-sum 26.6%

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

   3. Applied egg-rr 26.6%

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

   4. Taylor expanded in x around inf 26.6%

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

      1. associate--r+ 84.3%

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

      2. *-commutative 84.3%

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

      3. *-commutative 84.3%

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

   6. Simplified 84.3%

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

   7. Taylor expanded in eps around 0 99.7%

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

      1. associate-*r* 99.7%

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

      2. *-commutative 99.7%

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

      3. unpow2 99.7%

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

      4. associate-*r* 99.7%

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

   9. Simplified 99.7%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00016 \lor \neg \left(\varepsilon \leq 0.00015\right):\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right) - \sin x \cdot \sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\right) - \sin x \cdot \sin \varepsilon\\ \end{array} \]

Alternative 9: 66.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\varepsilon + x\right) - \cos x \leq -0.0002:\\ \;\;\;\;-1 + \cos x \cdot \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- (cos (+ eps x)) (cos x)) -0.0002)
   (+ -1.0 (* (cos x) (cos eps)))
   (* (sin x) (- eps))))

C

double code(double x, double eps) {
	double tmp;
	if ((cos((eps + x)) - cos(x)) <= -0.0002) {
		tmp = -1.0 + (cos(x) * cos(eps));
	} else {
		tmp = sin(x) * -eps;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((cos((eps + x)) - cos(x)) <= (-0.0002d0)) then
        tmp = (-1.0d0) + (cos(x) * cos(eps))
    else
        tmp = sin(x) * -eps
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((Math.cos((eps + x)) - Math.cos(x)) <= -0.0002) {
		tmp = -1.0 + (Math.cos(x) * Math.cos(eps));
	} else {
		tmp = Math.sin(x) * -eps;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (math.cos((eps + x)) - math.cos(x)) <= -0.0002:
		tmp = -1.0 + (math.cos(x) * math.cos(eps))
	else:
		tmp = math.sin(x) * -eps
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(cos(Float64(eps + x)) - cos(x)) <= -0.0002)
		tmp = Float64(-1.0 + Float64(cos(x) * cos(eps)));
	else
		tmp = Float64(sin(x) * Float64(-eps));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((cos((eps + x)) - cos(x)) <= -0.0002)
		tmp = -1.0 + (cos(x) * cos(eps));
	else
		tmp = sin(x) * -eps;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(N[Cos[N[(eps + x), $MachinePrecision]], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], -0.0002], N[(-1.0 + N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * (-eps)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cos.f64 (+.f64 x eps)) (cos.f64 x)) < -2.0000000000000001e-4

   1. Initial program 85.1%

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

      1. cos-sum 98.4%

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

   3. Applied egg-rr 98.4%

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

   4. Taylor expanded in x around inf 98.4%

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

   5. Taylor expanded in x around 0 85.3%

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

   if -2.0000000000000001e-4 < (-.f64 (cos.f64 (+.f64 x eps)) (cos.f64 x))

   1. Initial program 20.0%

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

   2. Taylor expanded in eps around 0 63.6%

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

      1. mul-1-neg 63.6%

         \[\leadsto \color{blue}{-\varepsilon \cdot \sin x} \]

      2. *-commutative 63.6%

         \[\leadsto -\color{blue}{\sin x \cdot \varepsilon} \]

      3. distribute-rgt-neg-in 63.6%

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

   4. Simplified 63.6%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\varepsilon + x\right) - \cos x \leq -0.0002:\\ \;\;\;\;-1 + \cos x \cdot \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \end{array} \]

Alternative 10: 66.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\varepsilon + x\right) - \cos x \leq -0.0002:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- (cos (+ eps x)) (cos x)) -0.0002)
   (- (cos eps) (cos x))
   (* eps (- (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if ((cos((eps + x)) - cos(x)) <= -0.0002) {
		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 ((cos((eps + x)) - cos(x)) <= (-0.0002d0)) 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 ((Math.cos((eps + x)) - Math.cos(x)) <= -0.0002) {
		tmp = Math.cos(eps) - Math.cos(x);
	} else {
		tmp = eps * -Math.sin(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (math.cos((eps + x)) - math.cos(x)) <= -0.0002:
		tmp = math.cos(eps) - math.cos(x)
	else:
		tmp = eps * -math.sin(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(cos(Float64(eps + x)) - cos(x)) <= -0.0002)
		tmp = Float64(cos(eps) - cos(x));
	else
		tmp = Float64(eps * Float64(-sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((cos((eps + x)) - cos(x)) <= -0.0002)
		tmp = cos(eps) - cos(x);
	else
		tmp = eps * -sin(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(N[Cos[N[(eps + x), $MachinePrecision]], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], -0.0002], 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 (-.f64 (cos.f64 (+.f64 x eps)) (cos.f64 x)) < -2.0000000000000001e-4

   1. Initial program 85.1%

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

   2. Taylor expanded in x around 0 85.2%

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

   if -2.0000000000000001e-4 < (-.f64 (cos.f64 (+.f64 x eps)) (cos.f64 x))

   1. Initial program 20.0%

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

   2. Taylor expanded in eps around 0 63.6%

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

      1. mul-1-neg 63.6%

         \[\leadsto \color{blue}{-\varepsilon \cdot \sin x} \]

      2. *-commutative 63.6%

         \[\leadsto -\color{blue}{\sin x \cdot \varepsilon} \]

      3. distribute-rgt-neg-in 63.6%

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

   4. Simplified 63.6%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\varepsilon + x\right) - \cos x \leq -0.0002:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \]

Alternative 11: 76.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.8%

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

   1. diff-cos 47.9%

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

   2. div-inv 47.9%

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

   3. metadata-eval 47.9%

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

   4. div-inv 47.9%

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

   5. +-commutative 47.9%

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

   6. metadata-eval 47.9%

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

3. Applied egg-rr 47.9%

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

   1. *-commutative 47.9%

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

   2. +-commutative 47.9%

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

   3. associate--l+ 79.0%

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

   4. +-inverses 79.0%

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

   5. distribute-lft-in 79.0%

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

   6. metadata-eval 79.0%

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

   7. *-commutative 79.0%

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

   8. associate-+r+ 78.9%

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

   9. +-commutative 78.9%

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

5. Simplified 78.9%

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

6. Taylor expanded in x around -inf 78.9%

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

7. Final simplification 78.9%

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

Alternative 12: 70.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{-61} \lor \neg \left(x \leq 2.1 \cdot 10^{-49}\right):\\ \;\;\;\;-2 \cdot \left(\sin x \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon \cdot \left(-\tan \left(\frac{\varepsilon}{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -3.8e-61) (not (<= x 2.1e-49)))
   (* -2.0 (* (sin x) (sin (* eps 0.5))))
   (* (sin eps) (- (tan (/ eps 2.0))))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -3.8e-61) || !(x <= 2.1e-49)) {
		tmp = -2.0 * (sin(x) * sin((eps * 0.5)));
	} else {
		tmp = sin(eps) * -tan((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 ((x <= (-3.8d-61)) .or. (.not. (x <= 2.1d-49))) then
        tmp = (-2.0d0) * (sin(x) * sin((eps * 0.5d0)))
    else
        tmp = sin(eps) * -tan((eps / 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -3.8e-61) || !(x <= 2.1e-49)) {
		tmp = -2.0 * (Math.sin(x) * Math.sin((eps * 0.5)));
	} else {
		tmp = Math.sin(eps) * -Math.tan((eps / 2.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -3.8e-61) or not (x <= 2.1e-49):
		tmp = -2.0 * (math.sin(x) * math.sin((eps * 0.5)))
	else:
		tmp = math.sin(eps) * -math.tan((eps / 2.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -3.8e-61) || !(x <= 2.1e-49))
		tmp = Float64(-2.0 * Float64(sin(x) * sin(Float64(eps * 0.5))));
	else
		tmp = Float64(sin(eps) * Float64(-tan(Float64(eps / 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -3.8e-61) || ~((x <= 2.1e-49)))
		tmp = -2.0 * (sin(x) * sin((eps * 0.5)));
	else
		tmp = sin(eps) * -tan((eps / 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -3.8e-61], N[Not[LessEqual[x, 2.1e-49]], $MachinePrecision]], N[(-2.0 * N[(N[Sin[x], $MachinePrecision] * N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[eps], $MachinePrecision] * (-N[Tan[N[(eps / 2.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.7999999999999998e-61 or 2.0999999999999999e-49 < x

   1. Initial program 11.1%

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

      1. diff-cos 11.3%

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

      2. div-inv 11.3%

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

      3. metadata-eval 11.3%

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

      4. div-inv 11.3%

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

      5. +-commutative 11.3%

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

      6. metadata-eval 11.3%

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

   3. Applied egg-rr 11.3%

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

      1. *-commutative 11.3%

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

      2. +-commutative 11.3%

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

      3. associate--l+ 63.8%

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

      4. +-inverses 63.8%

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

      5. distribute-lft-in 63.8%

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

      6. metadata-eval 63.8%

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

      7. *-commutative 63.8%

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

      8. associate-+r+ 63.7%

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

      9. +-commutative 63.7%

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

   5. Simplified 63.7%

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

   6. Taylor expanded in x around -inf 63.7%

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

   7. Taylor expanded in eps around 0 60.8%

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

   if -3.7999999999999998e-61 < x < 2.0999999999999999e-49

   1. Initial program 78.6%

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

   2. Taylor expanded in x around 0 78.6%

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

      1. flip-- 78.1%

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

      2. metadata-eval 78.1%

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

   4. Applied egg-rr 78.1%

      \[\leadsto \color{blue}{\frac{\cos \varepsilon \cdot \cos \varepsilon - 1}{\cos \varepsilon + 1}} \]
   5. Step-by-step derivation

      1. sub-1-cos 96.3%

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

      2. unpow2 96.3%

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

      3. +-commutative 96.3%

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

   6. Simplified 96.3%

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

   7. Taylor expanded in eps around inf 96.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{{\sin \varepsilon}^{2}}{1 + \cos \varepsilon}} \]
   8. Step-by-step derivation

      1. mul-1-neg 96.3%

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

      2. +-commutative 96.3%

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

      3. unpow2 96.3%

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

      4. associate-*r/ 96.3%

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

      5. distribute-lft-neg-in 96.3%

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

      6. +-commutative 96.3%

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

      7. hang-0p-tan 97.0%

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

   9. Simplified 97.0%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{-61} \lor \neg \left(x \leq 2.1 \cdot 10^{-49}\right):\\ \;\;\;\;-2 \cdot \left(\sin x \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon \cdot \left(-\tan \left(\frac{\varepsilon}{2}\right)\right)\\ \end{array} \]

Alternative 13: 68.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{-62} \lor \neg \left(x \leq 2.1 \cdot 10^{-49}\right):\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -6.6e-62) (not (<= x 2.1e-49)))
   (* eps (- (sin x)))
   (* -2.0 (pow (sin (* eps 0.5)) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -6.6e-62) || !(x <= 2.1e-49)) {
		tmp = eps * -sin(x);
	} else {
		tmp = -2.0 * pow(sin((eps * 0.5)), 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= (-6.6d-62)) .or. (.not. (x <= 2.1d-49))) then
        tmp = eps * -sin(x)
    else
        tmp = (-2.0d0) * (sin((eps * 0.5d0)) ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -6.6e-62) || !(x <= 2.1e-49)) {
		tmp = eps * -Math.sin(x);
	} else {
		tmp = -2.0 * Math.pow(Math.sin((eps * 0.5)), 2.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -6.6e-62) or not (x <= 2.1e-49):
		tmp = eps * -math.sin(x)
	else:
		tmp = -2.0 * math.pow(math.sin((eps * 0.5)), 2.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -6.6e-62) || !(x <= 2.1e-49))
		tmp = Float64(eps * Float64(-sin(x)));
	else
		tmp = Float64(-2.0 * (sin(Float64(eps * 0.5)) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -6.6e-62) || ~((x <= 2.1e-49)))
		tmp = eps * -sin(x);
	else
		tmp = -2.0 * (sin((eps * 0.5)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -6.6e-62], N[Not[LessEqual[x, 2.1e-49]], $MachinePrecision]], N[(eps * (-N[Sin[x], $MachinePrecision])), $MachinePrecision], N[(-2.0 * N[Power[N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.60000000000000009e-62 or 2.0999999999999999e-49 < x

   1. Initial program 11.1%

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

   2. Taylor expanded in eps around 0 58.8%

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

      1. mul-1-neg 58.8%

         \[\leadsto \color{blue}{-\varepsilon \cdot \sin x} \]

      2. *-commutative 58.8%

         \[\leadsto -\color{blue}{\sin x \cdot \varepsilon} \]

      3. distribute-rgt-neg-in 58.8%

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

   4. Simplified 58.8%

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

   if -6.60000000000000009e-62 < x < 2.0999999999999999e-49

   1. Initial program 78.6%

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

      1. diff-cos 97.3%

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

      2. div-inv 97.3%

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

      3. metadata-eval 97.3%

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

      4. div-inv 97.3%

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

      5. +-commutative 97.3%

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

      6. metadata-eval 97.3%

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

   3. Applied egg-rr 97.3%

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

      1. *-commutative 97.3%

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

      2. +-commutative 97.3%

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

      3. associate--l+ 99.4%

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

      4. +-inverses 99.4%

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

      5. distribute-lft-in 99.4%

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

      6. metadata-eval 99.4%

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

      7. *-commutative 99.4%

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

      8. associate-+r+ 99.4%

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

      9. +-commutative 99.4%

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

   5. Simplified 99.4%

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

   6. Taylor expanded in x around 0 96.8%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{-62} \lor \neg \left(x \leq 2.1 \cdot 10^{-49}\right):\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array} \]

Alternative 14: 68.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{-60} \lor \neg \left(x \leq 9.5 \cdot 10^{-50}\right):\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon \cdot \left(-\tan \left(\frac{\varepsilon}{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -2.15e-60) (not (<= x 9.5e-50)))
   (* eps (- (sin x)))
   (* (sin eps) (- (tan (/ eps 2.0))))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -2.15e-60) || !(x <= 9.5e-50)) {
		tmp = eps * -sin(x);
	} else {
		tmp = sin(eps) * -tan((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 ((x <= (-2.15d-60)) .or. (.not. (x <= 9.5d-50))) then
        tmp = eps * -sin(x)
    else
        tmp = sin(eps) * -tan((eps / 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -2.15e-60) || !(x <= 9.5e-50)) {
		tmp = eps * -Math.sin(x);
	} else {
		tmp = Math.sin(eps) * -Math.tan((eps / 2.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -2.15e-60) or not (x <= 9.5e-50):
		tmp = eps * -math.sin(x)
	else:
		tmp = math.sin(eps) * -math.tan((eps / 2.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -2.15e-60) || !(x <= 9.5e-50))
		tmp = Float64(eps * Float64(-sin(x)));
	else
		tmp = Float64(sin(eps) * Float64(-tan(Float64(eps / 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -2.15e-60) || ~((x <= 9.5e-50)))
		tmp = eps * -sin(x);
	else
		tmp = sin(eps) * -tan((eps / 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -2.15e-60], N[Not[LessEqual[x, 9.5e-50]], $MachinePrecision]], N[(eps * (-N[Sin[x], $MachinePrecision])), $MachinePrecision], N[(N[Sin[eps], $MachinePrecision] * (-N[Tan[N[(eps / 2.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.15e-60 or 9.4999999999999993e-50 < x

   1. Initial program 11.1%

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

   2. Taylor expanded in eps around 0 58.8%

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

      1. mul-1-neg 58.8%

         \[\leadsto \color{blue}{-\varepsilon \cdot \sin x} \]

      2. *-commutative 58.8%

         \[\leadsto -\color{blue}{\sin x \cdot \varepsilon} \]

      3. distribute-rgt-neg-in 58.8%

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

   4. Simplified 58.8%

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

   if -2.15e-60 < x < 9.4999999999999993e-50

   1. Initial program 78.6%

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

   2. Taylor expanded in x around 0 78.6%

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

      1. flip-- 78.1%

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

      2. metadata-eval 78.1%

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

   4. Applied egg-rr 78.1%

      \[\leadsto \color{blue}{\frac{\cos \varepsilon \cdot \cos \varepsilon - 1}{\cos \varepsilon + 1}} \]
   5. Step-by-step derivation

      1. sub-1-cos 96.3%

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

      2. unpow2 96.3%

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

      3. +-commutative 96.3%

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

   6. Simplified 96.3%

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

   7. Taylor expanded in eps around inf 96.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{{\sin \varepsilon}^{2}}{1 + \cos \varepsilon}} \]
   8. Step-by-step derivation

      1. mul-1-neg 96.3%

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

      2. +-commutative 96.3%

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

      3. unpow2 96.3%

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

      4. associate-*r/ 96.3%

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

      5. distribute-lft-neg-in 96.3%

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

      6. +-commutative 96.3%

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

      7. hang-0p-tan 97.0%

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

   9. Simplified 97.0%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{-60} \lor \neg \left(x \leq 9.5 \cdot 10^{-50}\right):\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon \cdot \left(-\tan \left(\frac{\varepsilon}{2}\right)\right)\\ \end{array} \]

Alternative 15: 67.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0062 \lor \neg \left(\varepsilon \leq 7 \cdot 10^{-5}\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.0062) (not (<= eps 7e-5)))
   (+ (cos eps) -1.0)
   (* (sin x) (- eps))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.0062) || !(eps <= 7e-5)) {
		tmp = cos(eps) + -1.0;
	} else {
		tmp = sin(x) * -eps;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-0.0062d0)) .or. (.not. (eps <= 7d-5))) then
        tmp = cos(eps) + (-1.0d0)
    else
        tmp = sin(x) * -eps
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.0062) || !(eps <= 7e-5)) {
		tmp = Math.cos(eps) + -1.0;
	} else {
		tmp = Math.sin(x) * -eps;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.0062) or not (eps <= 7e-5):
		tmp = math.cos(eps) + -1.0
	else:
		tmp = math.sin(x) * -eps
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.0062) || !(eps <= 7e-5))
		tmp = Float64(cos(eps) + -1.0);
	else
		tmp = Float64(sin(x) * Float64(-eps));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.0062) || ~((eps <= 7e-5)))
		tmp = cos(eps) + -1.0;
	else
		tmp = sin(x) * -eps;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.0062], N[Not[LessEqual[eps, 7e-5]], $MachinePrecision]], N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * (-eps)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -0.00619999999999999978 or 6.9999999999999994e-5 < eps

   1. Initial program 56.5%

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

   2. Taylor expanded in x around 0 58.0%

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

   if -0.00619999999999999978 < eps < 6.9999999999999994e-5

   1. Initial program 24.7%

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

   2. Taylor expanded in eps around 0 83.4%

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

      1. mul-1-neg 83.4%

         \[\leadsto \color{blue}{-\varepsilon \cdot \sin x} \]

      2. *-commutative 83.4%

         \[\leadsto -\color{blue}{\sin x \cdot \varepsilon} \]

      3. distribute-rgt-neg-in 83.4%

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

   4. Simplified 83.4%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0062 \lor \neg \left(\varepsilon \leq 7 \cdot 10^{-5}\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \end{array} \]

Alternative 16: 52.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.8 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 4.2 \cdot 10^{-7}\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) - \varepsilon \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -5.8e-5) (not (<= eps 4.2e-7)))
   (+ (cos eps) -1.0)
   (- (* -0.5 (* eps eps)) (* eps x))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -5.8e-5) || !(eps <= 4.2e-7)) {
		tmp = cos(eps) + -1.0;
	} else {
		tmp = (-0.5 * (eps * eps)) - (eps * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-5.8d-5)) .or. (.not. (eps <= 4.2d-7))) then
        tmp = cos(eps) + (-1.0d0)
    else
        tmp = ((-0.5d0) * (eps * eps)) - (eps * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -5.8e-5) || !(eps <= 4.2e-7)) {
		tmp = Math.cos(eps) + -1.0;
	} else {
		tmp = (-0.5 * (eps * eps)) - (eps * x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -5.8e-5) or not (eps <= 4.2e-7):
		tmp = math.cos(eps) + -1.0
	else:
		tmp = (-0.5 * (eps * eps)) - (eps * x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -5.8e-5) || !(eps <= 4.2e-7))
		tmp = Float64(cos(eps) + -1.0);
	else
		tmp = Float64(Float64(-0.5 * Float64(eps * eps)) - Float64(eps * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -5.8e-5) || ~((eps <= 4.2e-7)))
		tmp = cos(eps) + -1.0;
	else
		tmp = (-0.5 * (eps * eps)) - (eps * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -5.8e-5], N[Not[LessEqual[eps, 4.2e-7]], $MachinePrecision]], N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(-0.5 * N[(eps * eps), $MachinePrecision]), $MachinePrecision] - N[(eps * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -5.8e-5 or 4.2e-7 < eps

   1. Initial program 54.8%

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

   2. Taylor expanded in x around 0 56.5%

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

   if -5.8e-5 < eps < 4.2e-7

   1. Initial program 25.3%

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

   2. Taylor expanded in eps around 0 25.5%

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

      1. +-commutative 25.5%

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

      2. associate-+r+ 25.5%

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

      3. mul-1-neg 25.5%

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

      4. unsub-neg 25.5%

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

      5. associate-*r* 25.5%

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

      6. distribute-rgt1-in 25.5%

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

      7. distribute-lft1-in 25.5%

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

      8. *-lft-identity 25.5%

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

      9. distribute-rgt-out 25.5%

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

      10. unpow2 25.5%

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

   4. Simplified 25.5%

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

   5. Taylor expanded in x around 0 47.8%

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

      1. +-commutative 47.8%

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

      2. mul-1-neg 47.8%

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

      3. unsub-neg 47.8%

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

      4. *-commutative 47.8%

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

      5. unpow2 47.8%

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

      6. *-commutative 47.8%

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

   7. Simplified 47.8%

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

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.8 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 4.2 \cdot 10^{-7}\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) - \varepsilon \cdot x\\ \end{array} \]

Alternative 17: 28.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -360:\\ \;\;\;\;1 - \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) - \varepsilon \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -360.0) (- 1.0 (cos eps)) (- (* -0.5 (* eps eps)) (* eps x))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -360.0) {
		tmp = 1.0 - cos(eps);
	} else {
		tmp = (-0.5 * (eps * eps)) - (eps * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-360.0d0)) then
        tmp = 1.0d0 - cos(eps)
    else
        tmp = ((-0.5d0) * (eps * eps)) - (eps * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -360.0) {
		tmp = 1.0 - Math.cos(eps);
	} else {
		tmp = (-0.5 * (eps * eps)) - (eps * x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -360.0:
		tmp = 1.0 - math.cos(eps)
	else:
		tmp = (-0.5 * (eps * eps)) - (eps * x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -360.0)
		tmp = Float64(1.0 - cos(eps));
	else
		tmp = Float64(Float64(-0.5 * Float64(eps * eps)) - Float64(eps * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -360.0)
		tmp = 1.0 - cos(eps);
	else
		tmp = (-0.5 * (eps * eps)) - (eps * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -360.0], N[(1.0 - N[Cos[eps], $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(eps * eps), $MachinePrecision]), $MachinePrecision] - N[(eps * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -360

   1. Initial program 6.1%

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

   2. Taylor expanded in x around 0 7.0%

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

      1. flip-- 7.0%

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

      2. metadata-eval 7.0%

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

   4. Applied egg-rr 7.0%

      \[\leadsto \color{blue}{\frac{\cos \varepsilon \cdot \cos \varepsilon - 1}{\cos \varepsilon + 1}} \]
   5. Step-by-step derivation

      1. sub-1-cos 7.1%

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

      2. unpow2 7.1%

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

      3. +-commutative 7.1%

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

   6. Simplified 7.1%

      \[\leadsto \color{blue}{\frac{-{\sin \varepsilon}^{2}}{1 + \cos \varepsilon}} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 2.1%

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

      2. sqrt-unprod 9.4%

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

      3. sqr-neg 9.4%

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

      4. sqrt-unprod 9.5%

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

      5. add-sqr-sqrt 9.5%

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

      6. unpow2 9.5%

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

      7. 1-sub-cos 9.2%

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

      8. metadata-eval 9.2%

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

      9. flip-- 9.2%

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

      10. sub-neg 9.2%

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

   8. Applied egg-rr 9.2%

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

      1. sub-neg 9.2%

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

   10. Simplified 9.2%

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

   if -360 < x

   1. Initial program 51.1%

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

   2. Taylor expanded in eps around 0 18.9%

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

      1. +-commutative 18.9%

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

      2. associate-+r+ 18.9%

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

      3. mul-1-neg 18.9%

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

      4. unsub-neg 18.9%

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

      5. associate-*r* 18.9%

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

      6. distribute-rgt1-in 18.9%

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

      7. distribute-lft1-in 18.9%

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

      8. *-lft-identity 18.9%

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

      9. distribute-rgt-out 18.9%

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

      10. unpow2 18.9%

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

   4. Simplified 18.9%

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

   5. Taylor expanded in x around 0 33.8%

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

      1. +-commutative 33.8%

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

      2. mul-1-neg 33.8%

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

      3. unsub-neg 33.8%

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

      4. *-commutative 33.8%

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

      5. unpow2 33.8%

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

      6. *-commutative 33.8%

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

   7. Simplified 33.8%

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

4. Final simplification 27.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -360:\\ \;\;\;\;1 - \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) - \varepsilon \cdot x\\ \end{array} \]

Alternative 18: 27.1% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.8%

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

2. Taylor expanded in eps around 0 15.9%

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

   1. +-commutative 15.9%

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

   2. associate-+r+ 15.9%

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

   3. mul-1-neg 15.9%

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

   4. unsub-neg 15.9%

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

   5. associate-*r* 15.9%

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

   6. distribute-rgt1-in 15.9%

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

   7. distribute-lft1-in 15.9%

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

   8. *-lft-identity 15.9%

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

   9. distribute-rgt-out 15.9%

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

   10. unpow2 15.9%

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

4. Simplified 15.9%

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

5. Taylor expanded in x around 0 26.0%

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

   1. +-commutative 26.0%

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

   2. mul-1-neg 26.0%

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

   3. unsub-neg 26.0%

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

   4. *-commutative 26.0%

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

   5. unpow2 26.0%

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

   6. *-commutative 26.0%

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

7. Simplified 26.0%

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

8. Final simplification 26.0%

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

Alternative 19: 22.0% accurate, 41.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.8%

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

2. Taylor expanded in x around 0 40.7%

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

3. Taylor expanded in eps around 0 22.5%

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

   1. *-commutative 22.5%

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

   2. unpow2 22.5%

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

5. Simplified 22.5%

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

6. Taylor expanded in eps around 0 22.5%

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

   1. *-commutative 22.5%

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

   2. unpow2 22.5%

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

   3. associate-*r* 22.5%

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

8. Simplified 22.5%

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

9. Final simplification 22.5%

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

Reproduce

herbie shell --seed 2023283 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))