2cos (problem 3.3.5)

Percentage Accurate: 38.4% → 99.0%

Time: 27.5s

Alternatives: 12

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: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\varepsilon \cdot 0.5\right)\\ t_1 := \cos \varepsilon + -1\\ t_2 := \sin x \cdot \sin \varepsilon\\ \mathbf{if}\;x \leq -1.9 \cdot 10^{-8}:\\ \;\;\;\;\cos x \cdot \log \left(e^{t_1}\right) - t_2\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-50}:\\ \;\;\;\;\left(t_0 + x \cdot \cos \left(\varepsilon \cdot 0.5\right)\right) \cdot \left(t_0 \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot t_1 - t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (* eps 0.5)))
        (t_1 (+ (cos eps) -1.0))
        (t_2 (* (sin x) (sin eps))))
   (if (<= x -1.9e-8)
     (- (* (cos x) (log (exp t_1))) t_2)
     (if (<= x 1.6e-50)
       (* (+ t_0 (* x (cos (* eps 0.5)))) (* t_0 -2.0))
       (- (* (cos x) t_1) t_2)))))

C

double code(double x, double eps) {
	double t_0 = sin((eps * 0.5));
	double t_1 = cos(eps) + -1.0;
	double t_2 = sin(x) * sin(eps);
	double tmp;
	if (x <= -1.9e-8) {
		tmp = (cos(x) * log(exp(t_1))) - t_2;
	} else if (x <= 1.6e-50) {
		tmp = (t_0 + (x * cos((eps * 0.5)))) * (t_0 * -2.0);
	} else {
		tmp = (cos(x) * t_1) - t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = sin((eps * 0.5d0))
    t_1 = cos(eps) + (-1.0d0)
    t_2 = sin(x) * sin(eps)
    if (x <= (-1.9d-8)) then
        tmp = (cos(x) * log(exp(t_1))) - t_2
    else if (x <= 1.6d-50) then
        tmp = (t_0 + (x * cos((eps * 0.5d0)))) * (t_0 * (-2.0d0))
    else
        tmp = (cos(x) * t_1) - t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin((eps * 0.5));
	double t_1 = Math.cos(eps) + -1.0;
	double t_2 = Math.sin(x) * Math.sin(eps);
	double tmp;
	if (x <= -1.9e-8) {
		tmp = (Math.cos(x) * Math.log(Math.exp(t_1))) - t_2;
	} else if (x <= 1.6e-50) {
		tmp = (t_0 + (x * Math.cos((eps * 0.5)))) * (t_0 * -2.0);
	} else {
		tmp = (Math.cos(x) * t_1) - t_2;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.sin((eps * 0.5))
	t_1 = math.cos(eps) + -1.0
	t_2 = math.sin(x) * math.sin(eps)
	tmp = 0
	if x <= -1.9e-8:
		tmp = (math.cos(x) * math.log(math.exp(t_1))) - t_2
	elif x <= 1.6e-50:
		tmp = (t_0 + (x * math.cos((eps * 0.5)))) * (t_0 * -2.0)
	else:
		tmp = (math.cos(x) * t_1) - t_2
	return tmp

Julia

function code(x, eps)
	t_0 = sin(Float64(eps * 0.5))
	t_1 = Float64(cos(eps) + -1.0)
	t_2 = Float64(sin(x) * sin(eps))
	tmp = 0.0
	if (x <= -1.9e-8)
		tmp = Float64(Float64(cos(x) * log(exp(t_1))) - t_2);
	elseif (x <= 1.6e-50)
		tmp = Float64(Float64(t_0 + Float64(x * cos(Float64(eps * 0.5)))) * Float64(t_0 * -2.0));
	else
		tmp = Float64(Float64(cos(x) * t_1) - t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = sin((eps * 0.5));
	t_1 = cos(eps) + -1.0;
	t_2 = sin(x) * sin(eps);
	tmp = 0.0;
	if (x <= -1.9e-8)
		tmp = (cos(x) * log(exp(t_1))) - t_2;
	elseif (x <= 1.6e-50)
		tmp = (t_0 + (x * cos((eps * 0.5)))) * (t_0 * -2.0);
	else
		tmp = (cos(x) * t_1) - t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.9e-8], N[(N[(N[Cos[x], $MachinePrecision] * N[Log[N[Exp[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision], If[LessEqual[x, 1.6e-50], N[(N[(t$95$0 + N[(x * N[Cos[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * -2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision] - t$95$2), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.90000000000000014e-8

   1. Initial program 7.9%

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

      1. sub-neg 7.9%

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

      2. +-commutative 7.9%

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

      3. cos-sum 50.2%

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

      4. sub-neg 50.2%

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

      5. associate-+r+ 98.6%

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

      6. distribute-rgt-neg-in 98.6%

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

   3. Applied egg-rr 98.6%

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

   4. Taylor expanded in x around inf 98.6%

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

      1. fma-neg 98.7%

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

      2. fma-def 98.6%

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

      3. neg-mul-1 98.6%

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

      4. distribute-rgt-out 98.7%

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

   6. Simplified 98.7%

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

      1. add-log-exp_binary64 98.7%

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

   8. Applied rewrite-once 98.7%

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

   if -1.90000000000000014e-8 < x < 1.6e-50

   1. Initial program 71.6%

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

      1. diff-cos 92.4%

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

      2. div-inv 92.4%

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

      3. +-commutative 92.4%

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

      4. associate--l+ 99.3%

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

      5. +-inverses 99.3%

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

      6. metadata-eval 99.3%

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

      7. div-inv 99.3%

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

      8. +-commutative 99.3%

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

      9. metadata-eval 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. associate-*r* 99.4%

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

      2. *-commutative 99.4%

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

      3. *-commutative 99.4%

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

      4. associate-+r+ 99.4%

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

      5. +-commutative 99.4%

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

      6. +-rgt-identity 99.4%

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

   5. Simplified 99.4%

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

   6. Taylor expanded in x around 0 99.5%

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

      1. *-commutative 99.5%

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

      2. *-commutative 99.5%

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

   8. Simplified 99.5%

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

   if 1.6e-50 < x

   1. Initial program 13.7%

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

      1. sub-neg 13.7%

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

      2. +-commutative 13.7%

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

      3. cos-sum 46.0%

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

      4. sub-neg 46.0%

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

      5. associate-+r+ 99.3%

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

      6. distribute-rgt-neg-in 99.3%

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

   3. Applied egg-rr 99.3%

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

   4. Taylor expanded in x around inf 99.3%

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

      1. fma-neg 99.3%

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

      2. fma-def 99.3%

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

      3. neg-mul-1 99.3%

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

      4. distribute-rgt-out 99.3%

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

   6. Simplified 99.3%

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

   7. Taylor expanded in x around inf 99.3%

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

      1. +-commutative 99.3%

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

      2. sub-neg 99.3%

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

      3. metadata-eval 99.3%

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

      4. mul-1-neg 99.3%

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

      5. unsub-neg 99.3%

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

   9. Simplified 99.3%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-8}:\\ \;\;\;\;\cos x \cdot \log \left(e^{\cos \varepsilon + -1}\right) - \sin x \cdot \sin \varepsilon\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-50}:\\ \;\;\;\;\left(\sin \left(\varepsilon \cdot 0.5\right) + x \cdot \cos \left(\varepsilon \cdot 0.5\right)\right) \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right) - \sin x \cdot \sin \varepsilon\\ \end{array} \]

Alternative 2: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\varepsilon \cdot 0.5\right)\\ t_1 := \cos \varepsilon + -1\\ \mathbf{if}\;x \leq -3.2 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, t_1, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-55}:\\ \;\;\;\;\left(t_0 + x \cdot \cos \left(\varepsilon \cdot 0.5\right)\right) \cdot \left(t_0 \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot t_1 - \sin x \cdot \sin \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (* eps 0.5))) (t_1 (+ (cos eps) -1.0)))
   (if (<= x -3.2e-10)
     (fma (cos x) t_1 (* (sin eps) (- (sin x))))
     (if (<= x 9.5e-55)
       (* (+ t_0 (* x (cos (* eps 0.5)))) (* t_0 -2.0))
       (- (* (cos x) t_1) (* (sin x) (sin eps)))))))

C

double code(double x, double eps) {
	double t_0 = sin((eps * 0.5));
	double t_1 = cos(eps) + -1.0;
	double tmp;
	if (x <= -3.2e-10) {
		tmp = fma(cos(x), t_1, (sin(eps) * -sin(x)));
	} else if (x <= 9.5e-55) {
		tmp = (t_0 + (x * cos((eps * 0.5)))) * (t_0 * -2.0);
	} else {
		tmp = (cos(x) * t_1) - (sin(x) * sin(eps));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = sin(Float64(eps * 0.5))
	t_1 = Float64(cos(eps) + -1.0)
	tmp = 0.0
	if (x <= -3.2e-10)
		tmp = fma(cos(x), t_1, Float64(sin(eps) * Float64(-sin(x))));
	elseif (x <= 9.5e-55)
		tmp = Float64(Float64(t_0 + Float64(x * cos(Float64(eps * 0.5)))) * Float64(t_0 * -2.0));
	else
		tmp = Float64(Float64(cos(x) * t_1) - Float64(sin(x) * sin(eps)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[x, -3.2e-10], N[(N[Cos[x], $MachinePrecision] * t$95$1 + N[(N[Sin[eps], $MachinePrecision] * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9.5e-55], N[(N[(t$95$0 + N[(x * N[Cos[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * -2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.19999999999999981e-10

   1. Initial program 7.9%

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

      1. sub-neg 7.9%

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

      2. +-commutative 7.9%

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

      3. cos-sum 50.2%

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

      4. sub-neg 50.2%

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

      5. associate-+r+ 98.6%

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

      6. distribute-rgt-neg-in 98.6%

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

   3. Applied egg-rr 98.6%

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

   4. Taylor expanded in x around inf 98.6%

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

      1. fma-neg 98.7%

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

      2. fma-def 98.6%

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

      3. neg-mul-1 98.6%

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

      4. distribute-rgt-out 98.7%

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

   6. Simplified 98.7%

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

   7. Taylor expanded in x around inf 98.7%

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

      1. +-commutative 98.7%

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

      2. sub-neg 98.7%

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

      3. metadata-eval 98.7%

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

      4. associate-*r* 98.7%

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

      5. neg-mul-1 98.7%

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

      6. *-commutative 98.7%

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

      7. fma-def 98.7%

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

   9. Simplified 98.7%

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

   if -3.19999999999999981e-10 < x < 9.5000000000000006e-55

   1. Initial program 71.6%

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

      1. diff-cos 92.4%

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

      2. div-inv 92.4%

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

      3. +-commutative 92.4%

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

      4. associate--l+ 99.3%

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

      5. +-inverses 99.3%

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

      6. metadata-eval 99.3%

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

      7. div-inv 99.3%

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

      8. +-commutative 99.3%

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

      9. metadata-eval 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. associate-*r* 99.4%

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

      2. *-commutative 99.4%

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

      3. *-commutative 99.4%

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

      4. associate-+r+ 99.4%

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

      5. +-commutative 99.4%

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

      6. +-rgt-identity 99.4%

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

   5. Simplified 99.4%

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

   6. Taylor expanded in x around 0 99.5%

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

      1. *-commutative 99.5%

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

      2. *-commutative 99.5%

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

   8. Simplified 99.5%

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

   if 9.5000000000000006e-55 < x

   1. Initial program 13.7%

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

      1. sub-neg 13.7%

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

      2. +-commutative 13.7%

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

      3. cos-sum 46.0%

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

      4. sub-neg 46.0%

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

      5. associate-+r+ 99.3%

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

      6. distribute-rgt-neg-in 99.3%

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

   3. Applied egg-rr 99.3%

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

   4. Taylor expanded in x around inf 99.3%

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

      1. fma-neg 99.3%

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

      2. fma-def 99.3%

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

      3. neg-mul-1 99.3%

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

      4. distribute-rgt-out 99.3%

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

   6. Simplified 99.3%

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

   7. Taylor expanded in x around inf 99.3%

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

      1. +-commutative 99.3%

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

      2. sub-neg 99.3%

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

      3. metadata-eval 99.3%

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

      4. mul-1-neg 99.3%

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

      5. unsub-neg 99.3%

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

   9. Simplified 99.3%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon + -1, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-55}:\\ \;\;\;\;\left(\sin \left(\varepsilon \cdot 0.5\right) + x \cdot \cos \left(\varepsilon \cdot 0.5\right)\right) \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right) - \sin x \cdot \sin \varepsilon\\ \end{array} \]

Alternative 3: 74.9% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- (cos (+ x eps)) (cos x)) -2e-5)
   (* -2.0 (pow (sin (* eps 0.5)) 2.0))
   (- (* (* eps eps) (* (cos x) -0.5)) (* eps (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if ((cos((x + eps)) - cos(x)) <= -2e-5) {
		tmp = -2.0 * pow(sin((eps * 0.5)), 2.0);
	} else {
		tmp = ((eps * eps) * (cos(x) * -0.5)) - (eps * sin(x));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((cos((x + eps)) - cos(x)) <= (-2d-5)) then
        tmp = (-2.0d0) * (sin((eps * 0.5d0)) ** 2.0d0)
    else
        tmp = ((eps * eps) * (cos(x) * (-0.5d0))) - (eps * sin(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((Math.cos((x + eps)) - Math.cos(x)) <= -2e-5) {
		tmp = -2.0 * Math.pow(Math.sin((eps * 0.5)), 2.0);
	} else {
		tmp = ((eps * eps) * (Math.cos(x) * -0.5)) - (eps * Math.sin(x));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (math.cos((x + eps)) - math.cos(x)) <= -2e-5:
		tmp = -2.0 * math.pow(math.sin((eps * 0.5)), 2.0)
	else:
		tmp = ((eps * eps) * (math.cos(x) * -0.5)) - (eps * math.sin(x))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(cos(Float64(x + eps)) - cos(x)) <= -2e-5)
		tmp = Float64(-2.0 * (sin(Float64(eps * 0.5)) ^ 2.0));
	else
		tmp = Float64(Float64(Float64(eps * eps) * Float64(cos(x) * -0.5)) - Float64(eps * sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((cos((x + eps)) - cos(x)) <= -2e-5)
		tmp = -2.0 * (sin((eps * 0.5)) ^ 2.0);
	else
		tmp = ((eps * eps) * (cos(x) * -0.5)) - (eps * sin(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(N[Cos[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], -2e-5], N[(-2.0 * N[Power[N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(eps * eps), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] - N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cos.f64 (+.f64 x eps)) (cos.f64 x)) < -2.00000000000000016e-5

   1. Initial program 83.8%

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

      1. diff-cos 84.4%

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

      2. div-inv 84.4%

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

      3. +-commutative 84.4%

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

      4. associate--l+ 84.0%

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

      5. +-inverses 84.0%

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

      6. metadata-eval 84.0%

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

      7. div-inv 84.0%

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

      8. +-commutative 84.0%

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

      9. metadata-eval 84.0%

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

   3. Applied egg-rr 84.0%

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

      1. associate-*r* 84.0%

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

      2. *-commutative 84.0%

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

      3. *-commutative 84.0%

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

      4. associate-+r+ 84.4%

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

      5. +-commutative 84.4%

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

      6. +-rgt-identity 84.4%

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

   5. Simplified 84.4%

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

   6. Taylor expanded in x around 0 84.5%

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

      1. *-commutative 84.5%

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

   8. Simplified 84.5%

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

   if -2.00000000000000016e-5 < (-.f64 (cos.f64 (+.f64 x eps)) (cos.f64 x))

   1. Initial program 15.4%

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

   2. Taylor expanded in eps around 0 76.4%

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

      1. +-commutative 76.4%

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

      2. mul-1-neg 76.4%

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

      3. unsub-neg 76.4%

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

      4. *-commutative 76.4%

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

      5. associate-*l* 76.4%

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

      6. unpow2 76.4%

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

   4. Simplified 76.4%

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

4. Final simplification 79.2%

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

Alternative 4: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\varepsilon \cdot 0.5\right)\\ \mathbf{if}\;x \leq -1.9 \cdot 10^{-8} \lor \neg \left(x \leq 1.6 \cdot 10^{-50}\right):\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right) - \sin x \cdot \sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 + x \cdot \cos \left(\varepsilon \cdot 0.5\right)\right) \cdot \left(t_0 \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (* eps 0.5))))
   (if (or (<= x -1.9e-8) (not (<= x 1.6e-50)))
     (- (* (cos x) (+ (cos eps) -1.0)) (* (sin x) (sin eps)))
     (* (+ t_0 (* x (cos (* eps 0.5)))) (* t_0 -2.0)))))

C

double code(double x, double eps) {
	double t_0 = sin((eps * 0.5));
	double tmp;
	if ((x <= -1.9e-8) || !(x <= 1.6e-50)) {
		tmp = (cos(x) * (cos(eps) + -1.0)) - (sin(x) * sin(eps));
	} else {
		tmp = (t_0 + (x * cos((eps * 0.5)))) * (t_0 * -2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((eps * 0.5d0))
    if ((x <= (-1.9d-8)) .or. (.not. (x <= 1.6d-50))) then
        tmp = (cos(x) * (cos(eps) + (-1.0d0))) - (sin(x) * sin(eps))
    else
        tmp = (t_0 + (x * cos((eps * 0.5d0)))) * (t_0 * (-2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin((eps * 0.5));
	double tmp;
	if ((x <= -1.9e-8) || !(x <= 1.6e-50)) {
		tmp = (Math.cos(x) * (Math.cos(eps) + -1.0)) - (Math.sin(x) * Math.sin(eps));
	} else {
		tmp = (t_0 + (x * Math.cos((eps * 0.5)))) * (t_0 * -2.0);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.sin((eps * 0.5))
	tmp = 0
	if (x <= -1.9e-8) or not (x <= 1.6e-50):
		tmp = (math.cos(x) * (math.cos(eps) + -1.0)) - (math.sin(x) * math.sin(eps))
	else:
		tmp = (t_0 + (x * math.cos((eps * 0.5)))) * (t_0 * -2.0)
	return tmp

Julia

function code(x, eps)
	t_0 = sin(Float64(eps * 0.5))
	tmp = 0.0
	if ((x <= -1.9e-8) || !(x <= 1.6e-50))
		tmp = Float64(Float64(cos(x) * Float64(cos(eps) + -1.0)) - Float64(sin(x) * sin(eps)));
	else
		tmp = Float64(Float64(t_0 + Float64(x * cos(Float64(eps * 0.5)))) * Float64(t_0 * -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = sin((eps * 0.5));
	tmp = 0.0;
	if ((x <= -1.9e-8) || ~((x <= 1.6e-50)))
		tmp = (cos(x) * (cos(eps) + -1.0)) - (sin(x) * sin(eps));
	else
		tmp = (t_0 + (x * cos((eps * 0.5)))) * (t_0 * -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[x, -1.9e-8], N[Not[LessEqual[x, 1.6e-50]], $MachinePrecision]], N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 + N[(x * N[Cos[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * -2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.90000000000000014e-8 or 1.6e-50 < x

   1. Initial program 10.6%

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

      1. sub-neg 10.6%

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

      2. +-commutative 10.6%

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

      3. cos-sum 48.3%

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

      4. sub-neg 48.3%

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

      5. associate-+r+ 98.9%

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

      6. distribute-rgt-neg-in 98.9%

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

   3. Applied egg-rr 98.9%

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

   4. Taylor expanded in x around inf 98.9%

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

      1. fma-neg 99.0%

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

      2. fma-def 98.9%

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

      3. neg-mul-1 98.9%

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

      4. distribute-rgt-out 99.0%

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

   6. Simplified 99.0%

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

   7. Taylor expanded in x around inf 99.0%

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

      1. +-commutative 99.0%

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

      2. sub-neg 99.0%

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

      3. metadata-eval 99.0%

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

      4. mul-1-neg 99.0%

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

      5. unsub-neg 99.0%

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

   9. Simplified 99.0%

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

   if -1.90000000000000014e-8 < x < 1.6e-50

   1. Initial program 71.6%

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

      1. diff-cos 92.4%

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

      2. div-inv 92.4%

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

      3. +-commutative 92.4%

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

      4. associate--l+ 99.3%

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

      5. +-inverses 99.3%

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

      6. metadata-eval 99.3%

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

      7. div-inv 99.3%

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

      8. +-commutative 99.3%

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

      9. metadata-eval 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. associate-*r* 99.4%

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

      2. *-commutative 99.4%

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

      3. *-commutative 99.4%

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

      4. associate-+r+ 99.4%

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

      5. +-commutative 99.4%

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

      6. +-rgt-identity 99.4%

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

   5. Simplified 99.4%

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

   6. Taylor expanded in x around 0 99.5%

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

      1. *-commutative 99.5%

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

      2. *-commutative 99.5%

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

   8. Simplified 99.5%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-8} \lor \neg \left(x \leq 1.6 \cdot 10^{-50}\right):\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right) - \sin x \cdot \sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\sin \left(\varepsilon \cdot 0.5\right) + x \cdot \cos \left(\varepsilon \cdot 0.5\right)\right) \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot -2\right)\\ \end{array} \]

Alternative 5: 66.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(x + \varepsilon\right) - \cos x \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- (cos (+ x eps)) (cos x)) -2e-16)
   (+ (cos eps) -1.0)
   (* eps (- (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if ((cos((x + eps)) - cos(x)) <= -2e-16) {
		tmp = cos(eps) + -1.0;
	} else {
		tmp = eps * -sin(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((cos((x + eps)) - cos(x)) <= (-2d-16)) then
        tmp = cos(eps) + (-1.0d0)
    else
        tmp = eps * -sin(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((Math.cos((x + eps)) - Math.cos(x)) <= -2e-16) {
		tmp = Math.cos(eps) + -1.0;
	} else {
		tmp = eps * -Math.sin(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (math.cos((x + eps)) - math.cos(x)) <= -2e-16:
		tmp = math.cos(eps) + -1.0
	else:
		tmp = eps * -math.sin(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(cos(Float64(x + eps)) - cos(x)) <= -2e-16)
		tmp = Float64(cos(eps) + -1.0);
	else
		tmp = Float64(eps * Float64(-sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((cos((x + eps)) - cos(x)) <= -2e-16)
		tmp = cos(eps) + -1.0;
	else
		tmp = eps * -sin(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(N[Cos[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], -2e-16], N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision], N[(eps * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cos.f64 (+.f64 x eps)) (cos.f64 x)) < -2e-16

   1. Initial program 83.1%

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

   2. Taylor expanded in x around 0 83.1%

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

   if -2e-16 < (-.f64 (cos.f64 (+.f64 x eps)) (cos.f64 x))

   1. Initial program 15.4%

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

   2. Taylor expanded in eps around 0 62.3%

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

      1. associate-*r* 62.3%

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

      2. *-commutative 62.3%

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

      3. mul-1-neg 62.3%

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

   4. Simplified 62.3%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(x + \varepsilon\right) - \cos x \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \]

Alternative 6: 75.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.9%

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

   1. diff-cos 48.8%

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

   2. div-inv 48.8%

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

   3. +-commutative 48.8%

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

   4. associate--l+ 80.3%

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

   5. +-inverses 80.3%

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

   6. metadata-eval 80.3%

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

   7. div-inv 80.3%

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

   8. +-commutative 80.3%

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

   9. metadata-eval 80.3%

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

3. Applied egg-rr 80.3%

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

   1. associate-*r* 80.3%

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

   2. *-commutative 80.3%

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

   3. *-commutative 80.3%

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

   4. associate-+r+ 80.4%

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

   5. +-commutative 80.4%

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

   6. +-rgt-identity 80.4%

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

5. Simplified 80.4%

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

6. Final simplification 80.4%

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

Alternative 7: 68.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\varepsilon \cdot 0.5\right)\\ \mathbf{if}\;x \leq -3500 \lor \neg \left(x \leq 4.1 \cdot 10^{-49}\right):\\ \;\;\;\;\sin x \cdot \left(t_0 \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {t_0}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (* eps 0.5))))
   (if (or (<= x -3500.0) (not (<= x 4.1e-49)))
     (* (sin x) (* t_0 -2.0))
     (* -2.0 (pow t_0 2.0)))))

C

double code(double x, double eps) {
	double t_0 = sin((eps * 0.5));
	double tmp;
	if ((x <= -3500.0) || !(x <= 4.1e-49)) {
		tmp = sin(x) * (t_0 * -2.0);
	} else {
		tmp = -2.0 * pow(t_0, 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((eps * 0.5d0))
    if ((x <= (-3500.0d0)) .or. (.not. (x <= 4.1d-49))) then
        tmp = sin(x) * (t_0 * (-2.0d0))
    else
        tmp = (-2.0d0) * (t_0 ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin((eps * 0.5));
	double tmp;
	if ((x <= -3500.0) || !(x <= 4.1e-49)) {
		tmp = Math.sin(x) * (t_0 * -2.0);
	} else {
		tmp = -2.0 * Math.pow(t_0, 2.0);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.sin((eps * 0.5))
	tmp = 0
	if (x <= -3500.0) or not (x <= 4.1e-49):
		tmp = math.sin(x) * (t_0 * -2.0)
	else:
		tmp = -2.0 * math.pow(t_0, 2.0)
	return tmp

Julia

function code(x, eps)
	t_0 = sin(Float64(eps * 0.5))
	tmp = 0.0
	if ((x <= -3500.0) || !(x <= 4.1e-49))
		tmp = Float64(sin(x) * Float64(t_0 * -2.0));
	else
		tmp = Float64(-2.0 * (t_0 ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = sin((eps * 0.5));
	tmp = 0.0;
	if ((x <= -3500.0) || ~((x <= 4.1e-49)))
		tmp = sin(x) * (t_0 * -2.0);
	else
		tmp = -2.0 * (t_0 ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[x, -3500.0], N[Not[LessEqual[x, 4.1e-49]], $MachinePrecision]], N[(N[Sin[x], $MachinePrecision] * N[(t$95$0 * -2.0), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -3500 or 4.1000000000000001e-49 < x

   1. Initial program 9.1%

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

      1. diff-cos 9.4%

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

      2. div-inv 9.4%

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

      3. +-commutative 9.4%

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

      4. associate--l+ 63.9%

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

      5. +-inverses 63.9%

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

      6. metadata-eval 63.9%

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

      7. div-inv 63.9%

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

      8. +-commutative 63.9%

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

      9. metadata-eval 63.9%

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

   3. Applied egg-rr 63.9%

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

      1. associate-*r* 63.9%

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

      2. *-commutative 63.9%

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

      3. *-commutative 63.9%

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

      4. associate-+r+ 64.1%

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

      5. +-commutative 64.1%

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

      6. +-rgt-identity 64.1%

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

   5. Simplified 64.1%

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

   6. Taylor expanded in eps around 0 61.9%

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

   if -3500 < x < 4.1000000000000001e-49

   1. Initial program 70.7%

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

      1. diff-cos 90.7%

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

      2. div-inv 90.7%

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

      3. +-commutative 90.7%

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

      4. associate--l+ 97.8%

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

      5. +-inverses 97.8%

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

      6. metadata-eval 97.8%

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

      7. div-inv 97.8%

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

      8. +-commutative 97.8%

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

      9. metadata-eval 97.8%

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

   3. Applied egg-rr 97.8%

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

      1. associate-*r* 97.8%

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

      2. *-commutative 97.8%

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

      3. *-commutative 97.8%

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

      4. associate-+r+ 97.8%

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

      5. +-commutative 97.8%

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

      6. +-rgt-identity 97.8%

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

   5. Simplified 97.8%

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

   6. Taylor expanded in x around 0 89.2%

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

      1. *-commutative 89.2%

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

   8. Simplified 89.2%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3500 \lor \neg \left(x \leq 4.1 \cdot 10^{-49}\right):\\ \;\;\;\;\sin x \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array} \]

Alternative 8: 66.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8200 \lor \neg \left(x \leq 4.8 \cdot 10^{-49}\right):\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -8200.0) (not (<= x 4.8e-49)))
   (* eps (- (sin x)))
   (* -2.0 (pow (sin (* eps 0.5)) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -8200.0) || !(x <= 4.8e-49)) {
		tmp = eps * -sin(x);
	} else {
		tmp = -2.0 * pow(sin((eps * 0.5)), 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= (-8200.0d0)) .or. (.not. (x <= 4.8d-49))) then
        tmp = eps * -sin(x)
    else
        tmp = (-2.0d0) * (sin((eps * 0.5d0)) ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -8200.0) || !(x <= 4.8e-49)) {
		tmp = eps * -Math.sin(x);
	} else {
		tmp = -2.0 * Math.pow(Math.sin((eps * 0.5)), 2.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -8200.0) or not (x <= 4.8e-49):
		tmp = eps * -math.sin(x)
	else:
		tmp = -2.0 * math.pow(math.sin((eps * 0.5)), 2.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -8200.0) || !(x <= 4.8e-49))
		tmp = Float64(eps * Float64(-sin(x)));
	else
		tmp = Float64(-2.0 * (sin(Float64(eps * 0.5)) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -8200.0) || ~((x <= 4.8e-49)))
		tmp = eps * -sin(x);
	else
		tmp = -2.0 * (sin((eps * 0.5)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -8200.0], N[Not[LessEqual[x, 4.8e-49]], $MachinePrecision]], N[(eps * (-N[Sin[x], $MachinePrecision])), $MachinePrecision], N[(-2.0 * N[Power[N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8200 or 4.79999999999999985e-49 < x

   1. Initial program 9.2%

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

   2. Taylor expanded in eps around 0 58.4%

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

      1. associate-*r* 58.4%

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

      2. *-commutative 58.4%

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

      3. mul-1-neg 58.4%

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

   4. Simplified 58.4%

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

   if -8200 < x < 4.79999999999999985e-49

   1. Initial program 70.1%

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

      1. diff-cos 90.1%

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

      2. div-inv 90.1%

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

      3. +-commutative 90.1%

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

      4. associate--l+ 97.1%

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

      5. +-inverses 97.1%

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

      6. metadata-eval 97.1%

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

      7. div-inv 97.1%

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

      8. +-commutative 97.1%

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

      9. metadata-eval 97.1%

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

   3. Applied egg-rr 97.1%

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

      1. associate-*r* 97.2%

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

      2. *-commutative 97.2%

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

      3. *-commutative 97.2%

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

      4. associate-+r+ 97.2%

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

      5. +-commutative 97.2%

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

      6. +-rgt-identity 97.2%

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

   5. Simplified 97.2%

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

   6. Taylor expanded in x around 0 88.6%

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

      1. *-commutative 88.6%

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

   8. Simplified 88.6%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8200 \lor \neg \left(x \leq 4.8 \cdot 10^{-49}\right):\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array} \]

Alternative 9: 52.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000155 \lor \neg \left(\varepsilon \leq 0.0106\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.000155) (not (<= eps 0.0106)))
   (+ (cos eps) -1.0)
   (* eps (- (* eps -0.5) x))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.000155) || !(eps <= 0.0106)) {
		tmp = cos(eps) + -1.0;
	} else {
		tmp = eps * ((eps * -0.5) - x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-0.000155d0)) .or. (.not. (eps <= 0.0106d0))) then
        tmp = cos(eps) + (-1.0d0)
    else
        tmp = eps * ((eps * (-0.5d0)) - x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.000155) || !(eps <= 0.0106)) {
		tmp = Math.cos(eps) + -1.0;
	} else {
		tmp = eps * ((eps * -0.5) - x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.000155) or not (eps <= 0.0106):
		tmp = math.cos(eps) + -1.0
	else:
		tmp = eps * ((eps * -0.5) - x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.000155) || !(eps <= 0.0106))
		tmp = Float64(cos(eps) + -1.0);
	else
		tmp = Float64(eps * Float64(Float64(eps * -0.5) - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.000155) || ~((eps <= 0.0106)))
		tmp = cos(eps) + -1.0;
	else
		tmp = eps * ((eps * -0.5) - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.000155], N[Not[LessEqual[eps, 0.0106]], $MachinePrecision]], N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision], N[(eps * N[(N[(eps * -0.5), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.55e-4 or 0.0106 < eps

   1. Initial program 59.1%

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

   2. Taylor expanded in x around 0 61.1%

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

   if -1.55e-4 < eps < 0.0106

   1. Initial program 18.7%

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

   2. Taylor expanded in x around 0 18.4%

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

      1. associate--l+ 18.4%

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

      2. associate-*r* 18.4%

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

      3. mul-1-neg 18.4%

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

   4. Simplified 18.4%

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

   5. Taylor expanded in eps around 0 49.1%

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

      1. +-commutative 49.1%

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

      2. mul-1-neg 49.1%

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

      3. unsub-neg 49.1%

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

      4. *-commutative 49.1%

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

      5. unpow2 49.1%

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

      6. associate-*l* 49.1%

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

      7. distribute-lft-out-- 49.1%

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

   7. Simplified 49.1%

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

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000155 \lor \neg \left(\varepsilon \leq 0.0106\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5 - x\right)\\ \end{array} \]

Alternative 10: 24.1% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{-132} \lor \neg \left(x \leq 7 \cdot 10^{-71}\right):\\ \;\;\;\;x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -7.2e-132) (not (<= x 7e-71)))
   (* x (- eps))
   (* eps (* eps -0.5))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -7.2e-132) || !(x <= 7e-71)) {
		tmp = x * -eps;
	} else {
		tmp = eps * (eps * -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= (-7.2d-132)) .or. (.not. (x <= 7d-71))) then
        tmp = x * -eps
    else
        tmp = eps * (eps * (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -7.2e-132) || !(x <= 7e-71)) {
		tmp = x * -eps;
	} else {
		tmp = eps * (eps * -0.5);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -7.2e-132) or not (x <= 7e-71):
		tmp = x * -eps
	else:
		tmp = eps * (eps * -0.5)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -7.2e-132) || !(x <= 7e-71))
		tmp = Float64(x * Float64(-eps));
	else
		tmp = Float64(eps * Float64(eps * -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -7.2e-132) || ~((x <= 7e-71)))
		tmp = x * -eps;
	else
		tmp = eps * (eps * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -7.2e-132], N[Not[LessEqual[x, 7e-71]], $MachinePrecision]], N[(x * (-eps)), $MachinePrecision], N[(eps * N[(eps * -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.20000000000000015e-132 or 6.9999999999999998e-71 < x

   1. Initial program 20.1%

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

   2. Taylor expanded in x around 0 19.0%

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

      1. associate--l+ 19.0%

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

      2. associate-*r* 19.0%

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

      3. mul-1-neg 19.0%

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

   4. Simplified 19.0%

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

   5. Taylor expanded in eps around 0 15.1%

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

      1. associate-*r* 15.1%

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

      2. mul-1-neg 15.1%

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

   7. Simplified 15.1%

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

   if -7.20000000000000015e-132 < x < 6.9999999999999998e-71

   1. Initial program 73.0%

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

   2. Taylor expanded in x around 0 73.0%

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

   3. Taylor expanded in eps around 0 41.6%

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

      1. *-commutative 41.6%

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

      2. unpow2 41.6%

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

      3. associate-*l* 41.6%

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

   5. Simplified 41.6%

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

4. Final simplification 24.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{-132} \lor \neg \left(x \leq 7 \cdot 10^{-71}\right):\\ \;\;\;\;x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \end{array} \]

Alternative 11: 26.7% accurate, 29.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.9%

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

2. Taylor expanded in x around 0 38.2%

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

   1. associate--l+ 38.2%

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

   2. associate-*r* 38.2%

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

   3. mul-1-neg 38.2%

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

4. Simplified 38.2%

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

5. Taylor expanded in eps around 0 26.2%

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

   1. +-commutative 26.2%

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

   2. mul-1-neg 26.2%

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

   3. unsub-neg 26.2%

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

   4. *-commutative 26.2%

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

   5. unpow2 26.2%

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

   6. associate-*l* 26.2%

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

   7. distribute-lft-out-- 26.3%

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

7. Simplified 26.3%

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

8. Final simplification 26.3%

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

Alternative 12: 18.2% 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 38.9%

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

2. Taylor expanded in x around 0 38.2%

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

   1. associate--l+ 38.2%

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

   2. associate-*r* 38.2%

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

   3. mul-1-neg 38.2%

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

4. Simplified 38.2%

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

5. Taylor expanded in eps around 0 17.0%

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

   1. associate-*r* 17.0%

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

   2. mul-1-neg 17.0%

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

7. Simplified 17.0%

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

8. Final simplification 17.0%

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

Reproduce

herbie shell --seed 2023297 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))