2cos (problem 3.3.5)

Percentage Accurate: 38.4% → 99.2%

Time: 20.9s

Alternatives: 14

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: 38.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \cos \varepsilon\\ t_1 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.005:\\ \;\;\;\;t_0 - \left(\cos x + t_1\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0053:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 - t_1\right) - \cos x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (cos x) (cos eps))) (t_1 (* (sin eps) (sin x))))
   (if (<= eps -0.005)
     (- t_0 (+ (cos x) t_1))
     (if (<= eps 0.0053)
       (-
        (*
         (cos x)
         (+ (* -0.5 (pow eps 2.0)) (* 0.041666666666666664 (pow eps 4.0))))
        t_1)
       (- (- t_0 t_1) (cos x))))))

C

double code(double x, double eps) {
	double t_0 = cos(x) * cos(eps);
	double t_1 = sin(eps) * sin(x);
	double tmp;
	if (eps <= -0.005) {
		tmp = t_0 - (cos(x) + t_1);
	} else if (eps <= 0.0053) {
		tmp = (cos(x) * ((-0.5 * pow(eps, 2.0)) + (0.041666666666666664 * pow(eps, 4.0)))) - t_1;
	} else {
		tmp = (t_0 - t_1) - cos(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos(x) * cos(eps)
    t_1 = sin(eps) * sin(x)
    if (eps <= (-0.005d0)) then
        tmp = t_0 - (cos(x) + t_1)
    else if (eps <= 0.0053d0) then
        tmp = (cos(x) * (((-0.5d0) * (eps ** 2.0d0)) + (0.041666666666666664d0 * (eps ** 4.0d0)))) - t_1
    else
        tmp = (t_0 - t_1) - cos(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.cos(x) * Math.cos(eps);
	double t_1 = Math.sin(eps) * Math.sin(x);
	double tmp;
	if (eps <= -0.005) {
		tmp = t_0 - (Math.cos(x) + t_1);
	} else if (eps <= 0.0053) {
		tmp = (Math.cos(x) * ((-0.5 * Math.pow(eps, 2.0)) + (0.041666666666666664 * Math.pow(eps, 4.0)))) - t_1;
	} else {
		tmp = (t_0 - t_1) - Math.cos(x);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.cos(x) * math.cos(eps)
	t_1 = math.sin(eps) * math.sin(x)
	tmp = 0
	if eps <= -0.005:
		tmp = t_0 - (math.cos(x) + t_1)
	elif eps <= 0.0053:
		tmp = (math.cos(x) * ((-0.5 * math.pow(eps, 2.0)) + (0.041666666666666664 * math.pow(eps, 4.0)))) - t_1
	else:
		tmp = (t_0 - t_1) - math.cos(x)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(cos(x) * cos(eps))
	t_1 = Float64(sin(eps) * sin(x))
	tmp = 0.0
	if (eps <= -0.005)
		tmp = Float64(t_0 - Float64(cos(x) + t_1));
	elseif (eps <= 0.0053)
		tmp = Float64(Float64(cos(x) * Float64(Float64(-0.5 * (eps ^ 2.0)) + Float64(0.041666666666666664 * (eps ^ 4.0)))) - t_1);
	else
		tmp = Float64(Float64(t_0 - t_1) - cos(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = cos(x) * cos(eps);
	t_1 = sin(eps) * sin(x);
	tmp = 0.0;
	if (eps <= -0.005)
		tmp = t_0 - (cos(x) + t_1);
	elseif (eps <= 0.0053)
		tmp = (cos(x) * ((-0.5 * (eps ^ 2.0)) + (0.041666666666666664 * (eps ^ 4.0)))) - t_1;
	else
		tmp = (t_0 - t_1) - cos(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.005], N[(t$95$0 - N[(N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.0053], N[(N[(N[Cos[x], $MachinePrecision] * N[(N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.041666666666666664 * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(N[(t$95$0 - t$95$1), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.0050000000000000001

   1. Initial program 55.7%

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

      1. cos-sum 98.8%

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

      2. cancel-sign-sub-inv 98.8%

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

      3. associate--l+ 98.9%

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

      4. *-commutative 98.9%

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

   3. Applied egg-rr 98.9%

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

      1. associate-+r- 98.8%

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

      2. distribute-rgt-neg-out 98.8%

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

      3. sub-neg 98.8%

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

      4. associate--l- 98.9%

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

      5. *-commutative 98.9%

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

      6. add-sqr-sqrt 46.5%

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

      7. sqrt-unprod 78.7%

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

      8. sqr-neg 78.7%

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

      9. sqrt-unprod 32.1%

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

      10. add-sqr-sqrt 56.1%

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

      11. +-commutative 56.1%

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

      12. add-sqr-sqrt 32.1%

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

      13. sqrt-unprod 78.7%

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

   5. Applied egg-rr 98.9%

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

   if -0.0050000000000000001 < eps < 0.00530000000000000002

   1. Initial program 21.5%

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

      1. sub-neg 21.5%

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

      2. cos-sum 22.5%

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

      3. cancel-sign-sub-inv 22.5%

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

      4. associate-+l+ 22.5%

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

      5. *-commutative 22.5%

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

   3. Applied egg-rr 22.5%

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

      1. +-commutative 22.5%

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

      2. distribute-rgt-neg-out 22.5%

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

      3. *-commutative 22.5%

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

      4. unsub-neg 22.5%

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

      5. associate-+r- 75.9%

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

      6. *-commutative 75.9%

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

      7. neg-mul-1 75.9%

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

      8. distribute-rgt-out 75.9%

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

      9. *-commutative 75.9%

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

   5. Simplified 75.9%

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

   6. Taylor expanded in eps around 0 99.8%

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

   if 0.00530000000000000002 < eps

   1. Initial program 44.3%

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

      1. cos-sum 98.8%

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

   3. Applied egg-rr 98.8%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.005:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin \varepsilon \cdot \sin x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0053:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin \varepsilon \cdot \sin x\right) - \cos x\\ \end{array} \]

Alternative 2: 99.0% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.1%

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

   1. sub-neg 37.1%

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

   2. cos-sum 63.6%

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

   3. cancel-sign-sub-inv 63.6%

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

   4. associate-+l+ 63.6%

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

   5. *-commutative 63.6%

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

3. Applied egg-rr 63.6%

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

   1. +-commutative 63.6%

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

   2. distribute-rgt-neg-out 63.6%

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

   3. *-commutative 63.6%

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

   4. unsub-neg 63.6%

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

   5. associate-+r- 88.2%

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

   6. *-commutative 88.2%

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

   7. neg-mul-1 88.2%

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

   8. distribute-rgt-out 88.2%

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

   9. *-commutative 88.2%

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

5. Simplified 88.2%

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

   1. add-exp-log 48.5%

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

   2. *-commutative 48.5%

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

7. Applied egg-rr 48.5%

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

   1. add-exp-log 88.2%

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

   2. flip-+ 87.9%

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

   3. associate-*l/ 87.9%

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

   4. metadata-eval 87.9%

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

   5. sub-1-cos 99.1%

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

   6. pow2 99.1%

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

   7. sub-neg 99.1%

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

   8. metadata-eval 99.1%

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

9. Applied egg-rr 99.1%

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

10. Final simplification 99.1%

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

Alternative 3: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.00017 \lor \neg \left(\varepsilon \leq 0.000155\right):\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + t_0\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (sin eps) (sin x))))
   (if (or (<= eps -0.00017) (not (<= eps 0.000155)))
     (- (* (cos x) (cos eps)) (+ (cos x) t_0))
     (- (* (cos x) (* -0.5 (pow eps 2.0))) t_0))))

C

double code(double x, double eps) {
	double t_0 = sin(eps) * sin(x);
	double tmp;
	if ((eps <= -0.00017) || !(eps <= 0.000155)) {
		tmp = (cos(x) * cos(eps)) - (cos(x) + t_0);
	} else {
		tmp = (cos(x) * (-0.5 * pow(eps, 2.0))) - 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(eps) * sin(x)
    if ((eps <= (-0.00017d0)) .or. (.not. (eps <= 0.000155d0))) then
        tmp = (cos(x) * cos(eps)) - (cos(x) + t_0)
    else
        tmp = (cos(x) * ((-0.5d0) * (eps ** 2.0d0))) - t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin(eps) * Math.sin(x);
	double tmp;
	if ((eps <= -0.00017) || !(eps <= 0.000155)) {
		tmp = (Math.cos(x) * Math.cos(eps)) - (Math.cos(x) + t_0);
	} else {
		tmp = (Math.cos(x) * (-0.5 * Math.pow(eps, 2.0))) - t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.sin(eps) * math.sin(x)
	tmp = 0
	if (eps <= -0.00017) or not (eps <= 0.000155):
		tmp = (math.cos(x) * math.cos(eps)) - (math.cos(x) + t_0)
	else:
		tmp = (math.cos(x) * (-0.5 * math.pow(eps, 2.0))) - t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(sin(eps) * sin(x))
	tmp = 0.0
	if ((eps <= -0.00017) || !(eps <= 0.000155))
		tmp = Float64(Float64(cos(x) * cos(eps)) - Float64(cos(x) + t_0));
	else
		tmp = Float64(Float64(cos(x) * Float64(-0.5 * (eps ^ 2.0))) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = sin(eps) * sin(x);
	tmp = 0.0;
	if ((eps <= -0.00017) || ~((eps <= 0.000155)))
		tmp = (cos(x) * cos(eps)) - (cos(x) + t_0);
	else
		tmp = (cos(x) * (-0.5 * (eps ^ 2.0))) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[eps, -0.00017], N[Not[LessEqual[eps, 0.000155]], $MachinePrecision]], N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[x], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[x], $MachinePrecision] * N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.7e-4 or 1.55e-4 < eps

   1. Initial program 50.4%

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

      1. cos-sum 98.8%

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

      2. cancel-sign-sub-inv 98.8%

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

      3. associate--l+ 98.8%

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

      4. *-commutative 98.8%

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

   3. Applied egg-rr 98.8%

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

      1. associate-+r- 98.8%

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

      2. distribute-rgt-neg-out 98.8%

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

      3. sub-neg 98.8%

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

      4. associate--l- 98.8%

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

      5. *-commutative 98.8%

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

      6. add-sqr-sqrt 42.9%

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

      7. sqrt-unprod 73.7%

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

      8. sqr-neg 73.7%

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

      9. sqrt-unprod 30.8%

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

      10. add-sqr-sqrt 51.0%

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

      11. +-commutative 51.0%

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

      12. add-sqr-sqrt 30.8%

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

      13. sqrt-unprod 73.7%

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

   5. Applied egg-rr 98.8%

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

   if -1.7e-4 < eps < 1.55e-4

   1. Initial program 21.5%

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

      1. sub-neg 21.5%

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

      2. cos-sum 22.5%

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

      3. cancel-sign-sub-inv 22.5%

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

      4. associate-+l+ 22.5%

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

      5. *-commutative 22.5%

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

   3. Applied egg-rr 22.5%

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

      1. +-commutative 22.5%

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

      2. distribute-rgt-neg-out 22.5%

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

      3. *-commutative 22.5%

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

      4. unsub-neg 22.5%

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

      5. associate-+r- 75.9%

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

      6. *-commutative 75.9%

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

      7. neg-mul-1 75.9%

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

      8. distribute-rgt-out 75.9%

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

      9. *-commutative 75.9%

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

   5. Simplified 75.9%

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

   6. Taylor expanded in eps around 0 99.6%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00017 \lor \neg \left(\varepsilon \leq 0.000155\right):\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin \varepsilon \cdot \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - \sin \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 4: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \cos \varepsilon\\ t_1 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.000155:\\ \;\;\;\;t_0 - \left(\cos x + t_1\right)\\ \mathbf{elif}\;\varepsilon \leq 0.000165:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 - t_1\right) - \cos x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (cos x) (cos eps))) (t_1 (* (sin eps) (sin x))))
   (if (<= eps -0.000155)
     (- t_0 (+ (cos x) t_1))
     (if (<= eps 0.000165)
       (- (* (cos x) (* -0.5 (pow eps 2.0))) t_1)
       (- (- t_0 t_1) (cos x))))))

C

double code(double x, double eps) {
	double t_0 = cos(x) * cos(eps);
	double t_1 = sin(eps) * sin(x);
	double tmp;
	if (eps <= -0.000155) {
		tmp = t_0 - (cos(x) + t_1);
	} else if (eps <= 0.000165) {
		tmp = (cos(x) * (-0.5 * pow(eps, 2.0))) - t_1;
	} else {
		tmp = (t_0 - t_1) - cos(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos(x) * cos(eps)
    t_1 = sin(eps) * sin(x)
    if (eps <= (-0.000155d0)) then
        tmp = t_0 - (cos(x) + t_1)
    else if (eps <= 0.000165d0) then
        tmp = (cos(x) * ((-0.5d0) * (eps ** 2.0d0))) - t_1
    else
        tmp = (t_0 - t_1) - cos(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.cos(x) * Math.cos(eps);
	double t_1 = Math.sin(eps) * Math.sin(x);
	double tmp;
	if (eps <= -0.000155) {
		tmp = t_0 - (Math.cos(x) + t_1);
	} else if (eps <= 0.000165) {
		tmp = (Math.cos(x) * (-0.5 * Math.pow(eps, 2.0))) - t_1;
	} else {
		tmp = (t_0 - t_1) - Math.cos(x);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.cos(x) * math.cos(eps)
	t_1 = math.sin(eps) * math.sin(x)
	tmp = 0
	if eps <= -0.000155:
		tmp = t_0 - (math.cos(x) + t_1)
	elif eps <= 0.000165:
		tmp = (math.cos(x) * (-0.5 * math.pow(eps, 2.0))) - t_1
	else:
		tmp = (t_0 - t_1) - math.cos(x)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(cos(x) * cos(eps))
	t_1 = Float64(sin(eps) * sin(x))
	tmp = 0.0
	if (eps <= -0.000155)
		tmp = Float64(t_0 - Float64(cos(x) + t_1));
	elseif (eps <= 0.000165)
		tmp = Float64(Float64(cos(x) * Float64(-0.5 * (eps ^ 2.0))) - t_1);
	else
		tmp = Float64(Float64(t_0 - t_1) - cos(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = cos(x) * cos(eps);
	t_1 = sin(eps) * sin(x);
	tmp = 0.0;
	if (eps <= -0.000155)
		tmp = t_0 - (cos(x) + t_1);
	elseif (eps <= 0.000165)
		tmp = (cos(x) * (-0.5 * (eps ^ 2.0))) - t_1;
	else
		tmp = (t_0 - t_1) - cos(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.000155], N[(t$95$0 - N[(N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.000165], N[(N[(N[Cos[x], $MachinePrecision] * N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(N[(t$95$0 - t$95$1), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -1.55e-4

   1. Initial program 55.7%

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

      1. cos-sum 98.8%

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

      2. cancel-sign-sub-inv 98.8%

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

      3. associate--l+ 98.9%

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

      4. *-commutative 98.9%

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

   3. Applied egg-rr 98.9%

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

      1. associate-+r- 98.8%

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

      2. distribute-rgt-neg-out 98.8%

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

      3. sub-neg 98.8%

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

      4. associate--l- 98.9%

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

      5. *-commutative 98.9%

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

      6. add-sqr-sqrt 46.5%

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

      7. sqrt-unprod 78.7%

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

      8. sqr-neg 78.7%

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

      9. sqrt-unprod 32.1%

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

      10. add-sqr-sqrt 56.1%

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

      11. +-commutative 56.1%

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

      12. add-sqr-sqrt 32.1%

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

      13. sqrt-unprod 78.7%

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

   5. Applied egg-rr 98.9%

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

   if -1.55e-4 < eps < 1.65e-4

   1. Initial program 21.5%

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

      1. sub-neg 21.5%

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

      2. cos-sum 22.5%

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

      3. cancel-sign-sub-inv 22.5%

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

      4. associate-+l+ 22.5%

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

      5. *-commutative 22.5%

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

   3. Applied egg-rr 22.5%

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

      1. +-commutative 22.5%

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

      2. distribute-rgt-neg-out 22.5%

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

      3. *-commutative 22.5%

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

      4. unsub-neg 22.5%

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

      5. associate-+r- 75.9%

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

      6. *-commutative 75.9%

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

      7. neg-mul-1 75.9%

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

      8. distribute-rgt-out 75.9%

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

      9. *-commutative 75.9%

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

   5. Simplified 75.9%

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

   6. Taylor expanded in eps around 0 99.6%

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

   if 1.65e-4 < eps

   1. Initial program 44.3%

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

      1. cos-sum 98.8%

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

   3. Applied egg-rr 98.8%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000155:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin \varepsilon \cdot \sin x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.000165:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin \varepsilon \cdot \sin x\right) - \cos x\\ \end{array} \]

Alternative 5: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000125 \lor \neg \left(\varepsilon \leq 0.00015\right):\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon + -1, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - \sin \varepsilon \cdot \sin x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.000125) (not (<= eps 0.00015)))
   (fma (cos x) (+ (cos eps) -1.0) (* (sin eps) (- (sin x))))
   (- (* (cos x) (* -0.5 (pow eps 2.0))) (* (sin eps) (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.000125) || !(eps <= 0.00015)) {
		tmp = fma(cos(x), (cos(eps) + -1.0), (sin(eps) * -sin(x)));
	} else {
		tmp = (cos(x) * (-0.5 * pow(eps, 2.0))) - (sin(eps) * sin(x));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.000125) || !(eps <= 0.00015))
		tmp = fma(cos(x), Float64(cos(eps) + -1.0), Float64(sin(eps) * Float64(-sin(x))));
	else
		tmp = Float64(Float64(cos(x) * Float64(-0.5 * (eps ^ 2.0))) - Float64(sin(eps) * sin(x)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.000125], N[Not[LessEqual[eps, 0.00015]], $MachinePrecision]], N[(N[Cos[x], $MachinePrecision] * N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision] + N[(N[Sin[eps], $MachinePrecision] * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[x], $MachinePrecision] * N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.25e-4 or 1.49999999999999987e-4 < eps

   1. Initial program 50.4%

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

      1. add-sqr-sqrt 24.4%

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

      2. sqrt-unprod 32.0%

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

      3. pow2 32.0%

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

   3. Applied egg-rr 32.0%

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

      1. unpow2 32.0%

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

      2. rem-sqrt-square 32.0%

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

      3. +-commutative 32.0%

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

   5. Simplified 32.0%

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

      1. sub-neg 32.0%

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

      2. +-commutative 32.0%

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

      3. +-commutative 32.0%

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

      4. rem-cube-cbrt 31.9%

         \[\leadsto \left(-\cos x\right) + \left|\color{blue}{{\left(\sqrt[3]{\cos \left(x + \varepsilon\right)}\right)}^{3}}\right| \]

      5. sqr-pow 24.2%

         \[\leadsto \left(-\cos x\right) + \left|\color{blue}{{\left(\sqrt[3]{\cos \left(x + \varepsilon\right)}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{\cos \left(x + \varepsilon\right)}\right)}^{\left(\frac{3}{2}\right)}}\right| \]

      6. fabs-sqr 24.2%

         \[\leadsto \left(-\cos x\right) + \color{blue}{{\left(\sqrt[3]{\cos \left(x + \varepsilon\right)}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{\cos \left(x + \varepsilon\right)}\right)}^{\left(\frac{3}{2}\right)}} \]

      7. sqr-pow 50.4%

         \[\leadsto \left(-\cos x\right) + \color{blue}{{\left(\sqrt[3]{\cos \left(x + \varepsilon\right)}\right)}^{3}} \]

      8. rem-cube-cbrt 50.4%

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

      9. +-commutative 50.4%

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

      10. cos-sum 98.8%

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

      11. cancel-sign-sub-inv 98.8%

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

      12. associate-+r+ 98.7%

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

      13. neg-mul-1 98.7%

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

      14. distribute-rgt-in 98.8%

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

      15. +-commutative 98.8%

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

   7. Applied egg-rr 98.8%

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

      1. distribute-lft-neg-in 98.8%

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

      2. fma-def 98.8%

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

      3. +-commutative 98.8%

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

      4. distribute-lft-neg-in 98.8%

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

      5. *-commutative 98.8%

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

   9. Simplified 98.8%

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

   if -1.25e-4 < eps < 1.49999999999999987e-4

   1. Initial program 21.5%

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

      1. sub-neg 21.5%

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

      2. cos-sum 22.5%

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

      3. cancel-sign-sub-inv 22.5%

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

      4. associate-+l+ 22.5%

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

      5. *-commutative 22.5%

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

   3. Applied egg-rr 22.5%

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

      1. +-commutative 22.5%

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

      2. distribute-rgt-neg-out 22.5%

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

      3. *-commutative 22.5%

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

      4. unsub-neg 22.5%

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

      5. associate-+r- 75.9%

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

      6. *-commutative 75.9%

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

      7. neg-mul-1 75.9%

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

      8. distribute-rgt-out 75.9%

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

      9. *-commutative 75.9%

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

   5. Simplified 75.9%

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

   6. Taylor expanded in eps around 0 99.6%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000125 \lor \neg \left(\varepsilon \leq 0.00015\right):\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon + -1, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - \sin \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 6: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.000125 \lor \neg \left(\varepsilon \leq 0.00015\right):\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (sin eps) (sin x))))
   (if (or (<= eps -0.000125) (not (<= eps 0.00015)))
     (- (* (cos x) (+ (cos eps) -1.0)) t_0)
     (- (* (cos x) (* -0.5 (pow eps 2.0))) t_0))))

C

double code(double x, double eps) {
	double t_0 = sin(eps) * sin(x);
	double tmp;
	if ((eps <= -0.000125) || !(eps <= 0.00015)) {
		tmp = (cos(x) * (cos(eps) + -1.0)) - t_0;
	} else {
		tmp = (cos(x) * (-0.5 * pow(eps, 2.0))) - 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(eps) * sin(x)
    if ((eps <= (-0.000125d0)) .or. (.not. (eps <= 0.00015d0))) then
        tmp = (cos(x) * (cos(eps) + (-1.0d0))) - t_0
    else
        tmp = (cos(x) * ((-0.5d0) * (eps ** 2.0d0))) - t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin(eps) * Math.sin(x);
	double tmp;
	if ((eps <= -0.000125) || !(eps <= 0.00015)) {
		tmp = (Math.cos(x) * (Math.cos(eps) + -1.0)) - t_0;
	} else {
		tmp = (Math.cos(x) * (-0.5 * Math.pow(eps, 2.0))) - t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.sin(eps) * math.sin(x)
	tmp = 0
	if (eps <= -0.000125) or not (eps <= 0.00015):
		tmp = (math.cos(x) * (math.cos(eps) + -1.0)) - t_0
	else:
		tmp = (math.cos(x) * (-0.5 * math.pow(eps, 2.0))) - t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(sin(eps) * sin(x))
	tmp = 0.0
	if ((eps <= -0.000125) || !(eps <= 0.00015))
		tmp = Float64(Float64(cos(x) * Float64(cos(eps) + -1.0)) - t_0);
	else
		tmp = Float64(Float64(cos(x) * Float64(-0.5 * (eps ^ 2.0))) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = sin(eps) * sin(x);
	tmp = 0.0;
	if ((eps <= -0.000125) || ~((eps <= 0.00015)))
		tmp = (cos(x) * (cos(eps) + -1.0)) - t_0;
	else
		tmp = (cos(x) * (-0.5 * (eps ^ 2.0))) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[eps, -0.000125], 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[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.25e-4 or 1.49999999999999987e-4 < eps

   1. Initial program 50.4%

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

      1. sub-neg 50.4%

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

      2. cos-sum 98.8%

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

      3. cancel-sign-sub-inv 98.8%

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

      4. associate-+l+ 98.8%

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

      5. *-commutative 98.8%

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

   3. Applied egg-rr 98.8%

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

      1. +-commutative 98.8%

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

      2. distribute-rgt-neg-out 98.8%

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

      3. *-commutative 98.8%

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

      4. unsub-neg 98.8%

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

      5. associate-+r- 98.7%

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

      6. *-commutative 98.7%

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

      7. neg-mul-1 98.7%

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

      8. distribute-rgt-out 98.8%

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

      9. *-commutative 98.8%

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

   5. Simplified 98.8%

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

   if -1.25e-4 < eps < 1.49999999999999987e-4

   1. Initial program 21.5%

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

      1. sub-neg 21.5%

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

      2. cos-sum 22.5%

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

      3. cancel-sign-sub-inv 22.5%

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

      4. associate-+l+ 22.5%

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

      5. *-commutative 22.5%

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

   3. Applied egg-rr 22.5%

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

      1. +-commutative 22.5%

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

      2. distribute-rgt-neg-out 22.5%

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

      3. *-commutative 22.5%

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

      4. unsub-neg 22.5%

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

      5. associate-+r- 75.9%

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

      6. *-commutative 75.9%

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

      7. neg-mul-1 75.9%

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

      8. distribute-rgt-out 75.9%

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

      9. *-commutative 75.9%

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

   5. Simplified 75.9%

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

   6. Taylor expanded in eps around 0 99.6%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000125 \lor \neg \left(\varepsilon \leq 0.00015\right):\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - \sin \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 7: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.4 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 2.4 \cdot 10^{-5}\right):\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - \varepsilon \cdot \sin x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -2.4e-5) (not (<= eps 2.4e-5)))
   (- (* (cos x) (+ (cos eps) -1.0)) (* (sin eps) (sin x)))
   (- (* (cos x) (* -0.5 (pow eps 2.0))) (* eps (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -2.4e-5) || !(eps <= 2.4e-5)) {
		tmp = (cos(x) * (cos(eps) + -1.0)) - (sin(eps) * sin(x));
	} else {
		tmp = (cos(x) * (-0.5 * pow(eps, 2.0))) - (eps * sin(x));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-2.4d-5)) .or. (.not. (eps <= 2.4d-5))) then
        tmp = (cos(x) * (cos(eps) + (-1.0d0))) - (sin(eps) * sin(x))
    else
        tmp = (cos(x) * ((-0.5d0) * (eps ** 2.0d0))) - (eps * sin(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -2.4e-5) || !(eps <= 2.4e-5)) {
		tmp = (Math.cos(x) * (Math.cos(eps) + -1.0)) - (Math.sin(eps) * Math.sin(x));
	} else {
		tmp = (Math.cos(x) * (-0.5 * Math.pow(eps, 2.0))) - (eps * Math.sin(x));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -2.4e-5) or not (eps <= 2.4e-5):
		tmp = (math.cos(x) * (math.cos(eps) + -1.0)) - (math.sin(eps) * math.sin(x))
	else:
		tmp = (math.cos(x) * (-0.5 * math.pow(eps, 2.0))) - (eps * math.sin(x))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -2.4e-5) || !(eps <= 2.4e-5))
		tmp = Float64(Float64(cos(x) * Float64(cos(eps) + -1.0)) - Float64(sin(eps) * sin(x)));
	else
		tmp = Float64(Float64(cos(x) * Float64(-0.5 * (eps ^ 2.0))) - Float64(eps * sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -2.4e-5) || ~((eps <= 2.4e-5)))
		tmp = (cos(x) * (cos(eps) + -1.0)) - (sin(eps) * sin(x));
	else
		tmp = (cos(x) * (-0.5 * (eps ^ 2.0))) - (eps * sin(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -2.4e-5], N[Not[LessEqual[eps, 2.4e-5]], $MachinePrecision]], N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[x], $MachinePrecision] * N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 50.4%

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

      1. sub-neg 50.4%

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

      2. cos-sum 98.8%

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

      3. cancel-sign-sub-inv 98.8%

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

      4. associate-+l+ 98.8%

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

      5. *-commutative 98.8%

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

   3. Applied egg-rr 98.8%

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

      1. +-commutative 98.8%

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

      2. distribute-rgt-neg-out 98.8%

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

      3. *-commutative 98.8%

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

      4. unsub-neg 98.8%

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

      5. associate-+r- 98.7%

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

      6. *-commutative 98.7%

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

      7. neg-mul-1 98.7%

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

      8. distribute-rgt-out 98.8%

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

      9. *-commutative 98.8%

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

   5. Simplified 98.8%

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

   if -2.4000000000000001e-5 < eps < 2.4000000000000001e-5

   1. Initial program 21.5%

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

   2. Taylor expanded in eps around 0 99.6%

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

      1. +-commutative 99.6%

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

      2. mul-1-neg 99.6%

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

      3. unsub-neg 99.6%

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

      4. associate-*r* 99.6%

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

      5. *-commutative 99.6%

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

   4. Simplified 99.6%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.4 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 2.4 \cdot 10^{-5}\right):\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 8: 76.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.1%

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

   1. diff-cos 47.7%

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

   2. div-inv 47.7%

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

   3. +-commutative 47.7%

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

   4. associate--l+ 73.6%

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

   5. *-un-lft-identity 73.6%

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

   6. *-un-lft-identity 73.6%

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

   7. distribute-rgt-out-- 73.6%

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

   8. metadata-eval 73.6%

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

   9. metadata-eval 73.6%

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

   10. div-inv 73.6%

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

   11. +-commutative 73.6%

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

   12. associate-+l+ 73.6%

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

   13. count-2 73.6%

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

   14. *-commutative 73.6%

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

   15. metadata-eval 73.6%

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

3. Applied egg-rr 73.6%

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

   1. associate-*r* 73.6%

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

   2. *-commutative 73.6%

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

   3. *-commutative 73.6%

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

   4. *-commutative 73.6%

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

   5. mul0-rgt 73.6%

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

5. Simplified 73.6%

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

6. Final simplification 73.6%

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

Alternative 9: 67.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.76:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{elif}\;\varepsilon \leq 4.8 \cdot 10^{-6}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.76)
   (- (cos eps) (cos x))
   (if (<= eps 4.8e-6) (* (sin x) (- eps)) (+ (* (cos x) (cos eps)) -1.0))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.76) {
		tmp = cos(eps) - cos(x);
	} else if (eps <= 4.8e-6) {
		tmp = sin(x) * -eps;
	} else {
		tmp = (cos(x) * cos(eps)) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= (-0.76d0)) then
        tmp = cos(eps) - cos(x)
    else if (eps <= 4.8d-6) then
        tmp = sin(x) * -eps
    else
        tmp = (cos(x) * cos(eps)) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -0.76) {
		tmp = Math.cos(eps) - Math.cos(x);
	} else if (eps <= 4.8e-6) {
		tmp = Math.sin(x) * -eps;
	} else {
		tmp = (Math.cos(x) * Math.cos(eps)) + -1.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -0.76:
		tmp = math.cos(eps) - math.cos(x)
	elif eps <= 4.8e-6:
		tmp = math.sin(x) * -eps
	else:
		tmp = (math.cos(x) * math.cos(eps)) + -1.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -0.76)
		tmp = Float64(cos(eps) - cos(x));
	elseif (eps <= 4.8e-6)
		tmp = Float64(sin(x) * Float64(-eps));
	else
		tmp = Float64(Float64(cos(x) * cos(eps)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -0.76)
		tmp = cos(eps) - cos(x);
	elseif (eps <= 4.8e-6)
		tmp = sin(x) * -eps;
	else
		tmp = (cos(x) * cos(eps)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -0.76], N[(N[Cos[eps], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 4.8e-6], N[(N[Sin[x], $MachinePrecision] * (-eps)), $MachinePrecision], N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.76000000000000001

   1. Initial program 56.4%

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

   2. Taylor expanded in x around 0 57.7%

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

   if -0.76000000000000001 < eps < 4.7999999999999998e-6

   1. Initial program 21.1%

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

   2. Taylor expanded in eps around 0 75.5%

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

      1. associate-*r* 75.5%

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

      2. mul-1-neg 75.5%

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

   4. Simplified 75.5%

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

   if 4.7999999999999998e-6 < eps

   1. Initial program 44.3%

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

      1. cos-sum 98.1%

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

      2. cancel-sign-sub-inv 98.1%

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

      3. associate--l+ 98.0%

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

      4. *-commutative 98.0%

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

   3. Applied egg-rr 98.0%

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

   4. Taylor expanded in x around 0 47.6%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.76:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{elif}\;\varepsilon \leq 4.8 \cdot 10^{-6}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon + -1\\ \end{array} \]

Alternative 10: 67.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.76:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{elif}\;\varepsilon \leq 1.7 \cdot 10^{-7}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.76)
   (- (cos eps) (cos x))
   (if (<= eps 1.7e-7) (* (sin x) (- eps)) (+ (cos eps) -1.0))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.76) {
		tmp = cos(eps) - cos(x);
	} else if (eps <= 1.7e-7) {
		tmp = sin(x) * -eps;
	} else {
		tmp = cos(eps) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= (-0.76d0)) then
        tmp = cos(eps) - cos(x)
    else if (eps <= 1.7d-7) then
        tmp = sin(x) * -eps
    else
        tmp = cos(eps) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -0.76) {
		tmp = Math.cos(eps) - Math.cos(x);
	} else if (eps <= 1.7e-7) {
		tmp = Math.sin(x) * -eps;
	} else {
		tmp = Math.cos(eps) + -1.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -0.76:
		tmp = math.cos(eps) - math.cos(x)
	elif eps <= 1.7e-7:
		tmp = math.sin(x) * -eps
	else:
		tmp = math.cos(eps) + -1.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -0.76)
		tmp = Float64(cos(eps) - cos(x));
	elseif (eps <= 1.7e-7)
		tmp = Float64(sin(x) * Float64(-eps));
	else
		tmp = Float64(cos(eps) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -0.76)
		tmp = cos(eps) - cos(x);
	elseif (eps <= 1.7e-7)
		tmp = sin(x) * -eps;
	else
		tmp = cos(eps) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -0.76], N[(N[Cos[eps], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 1.7e-7], N[(N[Sin[x], $MachinePrecision] * (-eps)), $MachinePrecision], N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.76000000000000001

   1. Initial program 56.4%

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

   2. Taylor expanded in x around 0 57.7%

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

   if -0.76000000000000001 < eps < 1.69999999999999987e-7

   1. Initial program 21.1%

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

   2. Taylor expanded in eps around 0 75.5%

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

      1. associate-*r* 75.5%

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

      2. mul-1-neg 75.5%

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

   4. Simplified 75.5%

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

   if 1.69999999999999987e-7 < eps

   1. Initial program 44.3%

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

   2. Taylor expanded in x around 0 47.4%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.76:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{elif}\;\varepsilon \leq 1.7 \cdot 10^{-7}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon + -1\\ \end{array} \]

Alternative 11: 67.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.16 \cdot 10^{-6} \lor \neg \left(\varepsilon \leq 3.15 \cdot 10^{-6}\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -1.16e-6) (not (<= eps 3.15e-6)))
   (+ (cos eps) -1.0)
   (* (sin x) (- eps))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.16e-6) || !(eps <= 3.15e-6)) {
		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 <= (-1.16d-6)) .or. (.not. (eps <= 3.15d-6))) 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 <= -1.16e-6) || !(eps <= 3.15e-6)) {
		tmp = Math.cos(eps) + -1.0;
	} else {
		tmp = Math.sin(x) * -eps;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -1.16e-6) or not (eps <= 3.15e-6):
		tmp = math.cos(eps) + -1.0
	else:
		tmp = math.sin(x) * -eps
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -1.16e-6) || !(eps <= 3.15e-6))
		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 <= -1.16e-6) || ~((eps <= 3.15e-6)))
		tmp = cos(eps) + -1.0;
	else
		tmp = sin(x) * -eps;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -1.16e-6], N[Not[LessEqual[eps, 3.15e-6]], $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 < -1.1599999999999999e-6 or 3.14999999999999991e-6 < eps

   1. Initial program 50.4%

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

   2. Taylor expanded in x around 0 51.8%

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

   if -1.1599999999999999e-6 < eps < 3.14999999999999991e-6

   1. Initial program 21.3%

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

   2. Taylor expanded in eps around 0 76.0%

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

      1. associate-*r* 76.0%

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

      2. mul-1-neg 76.0%

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

   4. Simplified 76.0%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.16 \cdot 10^{-6} \lor \neg \left(\varepsilon \leq 3.15 \cdot 10^{-6}\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \end{array} \]

Alternative 12: 43.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.3 \cdot 10^{-73} \lor \neg \left(\varepsilon \leq 1.05 \cdot 10^{-8}\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-\varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -1.3e-73) (not (<= eps 1.05e-8)))
   (+ (cos eps) -1.0)
   (* x (- eps))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.3e-73) || !(eps <= 1.05e-8)) {
		tmp = cos(eps) + -1.0;
	} else {
		tmp = 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 <= (-1.3d-73)) .or. (.not. (eps <= 1.05d-8))) then
        tmp = cos(eps) + (-1.0d0)
    else
        tmp = x * -eps
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.3e-73) || !(eps <= 1.05e-8)) {
		tmp = Math.cos(eps) + -1.0;
	} else {
		tmp = x * -eps;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -1.3e-73) or not (eps <= 1.05e-8):
		tmp = math.cos(eps) + -1.0
	else:
		tmp = x * -eps
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -1.3e-73) || !(eps <= 1.05e-8))
		tmp = Float64(cos(eps) + -1.0);
	else
		tmp = Float64(x * Float64(-eps));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -1.3e-73) || ~((eps <= 1.05e-8)))
		tmp = cos(eps) + -1.0;
	else
		tmp = x * -eps;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -1.3e-73], N[Not[LessEqual[eps, 1.05e-8]], $MachinePrecision]], N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision], N[(x * (-eps)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.3e-73 or 1.04999999999999997e-8 < eps

   1. Initial program 45.0%

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

   2. Taylor expanded in x around 0 46.3%

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

   if -1.3e-73 < eps < 1.04999999999999997e-8

   1. Initial program 24.5%

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

      1. sub-neg 24.5%

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

      2. cos-sum 25.0%

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

      3. cancel-sign-sub-inv 25.0%

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

      4. associate-+l+ 25.0%

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

      5. *-commutative 25.0%

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

   3. Applied egg-rr 25.0%

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

      1. +-commutative 25.0%

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

      2. distribute-rgt-neg-out 25.0%

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

      3. *-commutative 25.0%

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

      4. unsub-neg 25.0%

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

      5. associate-+r- 83.9%

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

      6. *-commutative 83.9%

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

      7. neg-mul-1 83.9%

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

      8. distribute-rgt-out 83.9%

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

      9. *-commutative 83.9%

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

   5. Simplified 83.9%

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

   6. Taylor expanded in eps around 0 83.9%

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

      1. associate-*r* 83.9%

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

      2. mul-1-neg 83.9%

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

      3. *-commutative 83.9%

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

   8. Simplified 83.9%

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

   9. Taylor expanded in x around 0 37.0%

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

       1. associate-*r* 37.0%

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

       2. mul-1-neg 37.0%

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

   11. Simplified 37.0%

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

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.3 \cdot 10^{-73} \lor \neg \left(\varepsilon \leq 1.05 \cdot 10^{-8}\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-\varepsilon\right)\\ \end{array} \]

Alternative 13: 18.4% accurate, 51.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.1%

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

   1. sub-neg 37.1%

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

   2. cos-sum 63.6%

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

   3. cancel-sign-sub-inv 63.6%

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

   4. associate-+l+ 63.6%

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

   5. *-commutative 63.6%

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

3. Applied egg-rr 63.6%

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

   1. +-commutative 63.6%

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

   2. distribute-rgt-neg-out 63.6%

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

   3. *-commutative 63.6%

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

   4. unsub-neg 63.6%

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

   5. associate-+r- 88.2%

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

   6. *-commutative 88.2%

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

   7. neg-mul-1 88.2%

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

   8. distribute-rgt-out 88.2%

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

   9. *-commutative 88.2%

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

5. Simplified 88.2%

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

6. Taylor expanded in eps around 0 36.6%

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

   1. associate-*r* 36.6%

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

   2. mul-1-neg 36.6%

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

   3. *-commutative 36.6%

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

8. Simplified 36.6%

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

9. Taylor expanded in x around 0 16.0%

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

    1. associate-*r* 16.0%

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

    2. mul-1-neg 16.0%

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

11. Simplified 16.0%

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

12. Final simplification 16.0%

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

Alternative 14: 13.1% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

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

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 37.1%

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

   1. diff-cos 47.7%

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

   2. sin-mult 36.7%

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

   3. associate-*r/ 36.7%

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

3. Applied egg-rr 36.9%

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

   1. *-commutative 36.9%

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

   2. associate-/l* 36.9%

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

   3. associate--l+ 36.9%

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

   4. mul0-rgt 36.9%

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

   5. mul0-rgt 36.9%

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

   6. metadata-eval 36.9%

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

5. Simplified 36.9%

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

   1. sub-neg 36.9%

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

7. Applied egg-rr 11.4%

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

   1. sub-neg 11.4%

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

   2. +-inverses 11.4%

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

9. Simplified 11.4%

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

10. Final simplification 11.4%

    \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023301 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))