2cos (problem 3.3.5)

Percentage Accurate: 37.6% → 99.1%

Time: 15.2s

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: 37.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.65 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \left(-\sin x\right) \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -2.65e-5)
   (- (fma (cos x) (cos eps) (* (- (sin x)) (sin eps))) (cos x))
   (if (<= eps 5e-5)
     (- (* (cos x) (* -0.5 (pow eps 2.0))) (* eps (sin x)))
     (- (- (* (cos x) (cos eps)) (* (sin x) (sin eps))) (cos x)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -2.65e-5) {
		tmp = fma(cos(x), cos(eps), (-sin(x) * sin(eps))) - cos(x);
	} else if (eps <= 5e-5) {
		tmp = (cos(x) * (-0.5 * pow(eps, 2.0))) - (eps * sin(x));
	} else {
		tmp = ((cos(x) * cos(eps)) - (sin(x) * sin(eps))) - cos(x);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -2.65e-5)
		tmp = Float64(fma(cos(x), cos(eps), Float64(Float64(-sin(x)) * sin(eps))) - cos(x));
	elseif (eps <= 5e-5)
		tmp = Float64(Float64(cos(x) * Float64(-0.5 * (eps ^ 2.0))) - Float64(eps * sin(x)));
	else
		tmp = Float64(Float64(Float64(cos(x) * cos(eps)) - Float64(sin(x) * sin(eps))) - cos(x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -2.65e-5], N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + N[((-N[Sin[x], $MachinePrecision]) * N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 5e-5], N[(N[(N[Cos[x], $MachinePrecision] * N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -2.65e-5

   1. Initial program 53.5%

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

      1. cos-sum 98.9%

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

      2. cancel-sign-sub-inv 98.9%

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

      3. fma-def 98.9%

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

   3. Applied egg-rr 98.9%

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

   if -2.65e-5 < eps < 5.00000000000000024e-5

   1. Initial program 25.2%

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

   2. Taylor expanded in eps around 0 99.8%

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

      1. +-commutative 99.8%

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

      2. mul-1-neg 99.8%

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

      3. unsub-neg 99.8%

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

      4. associate-*r* 99.8%

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

      5. *-commutative 99.8%

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

   4. Simplified 99.8%

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

   if 5.00000000000000024e-5 < eps

   1. Initial program 51.1%

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

      1. cos-sum 98.7%

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

   3. Applied egg-rr 98.7%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.65 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \left(-\sin x\right) \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array} \]

Alternative 2: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.6 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 5 \cdot 10^{-5}\right):\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - \varepsilon \cdot \sin x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -4.6e-5) (not (<= eps 5e-5)))
   (- (* (cos x) (cos eps)) (+ (cos x) (* (sin x) (sin eps))))
   (- (* (cos x) (* -0.5 (pow eps 2.0))) (* eps (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -4.6e-5) || !(eps <= 5e-5)) {
		tmp = (cos(x) * cos(eps)) - (cos(x) + (sin(x) * sin(eps)));
	} else {
		tmp = (cos(x) * (-0.5 * pow(eps, 2.0))) - (eps * sin(x));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-4.6d-5)) .or. (.not. (eps <= 5d-5))) then
        tmp = (cos(x) * cos(eps)) - (cos(x) + (sin(x) * sin(eps)))
    else
        tmp = (cos(x) * ((-0.5d0) * (eps ** 2.0d0))) - (eps * sin(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -4.6e-5) || !(eps <= 5e-5)) {
		tmp = (Math.cos(x) * Math.cos(eps)) - (Math.cos(x) + (Math.sin(x) * Math.sin(eps)));
	} else {
		tmp = (Math.cos(x) * (-0.5 * Math.pow(eps, 2.0))) - (eps * Math.sin(x));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -4.6e-5) or not (eps <= 5e-5):
		tmp = (math.cos(x) * math.cos(eps)) - (math.cos(x) + (math.sin(x) * math.sin(eps)))
	else:
		tmp = (math.cos(x) * (-0.5 * math.pow(eps, 2.0))) - (eps * math.sin(x))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -4.6e-5) || !(eps <= 5e-5))
		tmp = Float64(Float64(cos(x) * cos(eps)) - Float64(cos(x) + Float64(sin(x) * sin(eps))));
	else
		tmp = Float64(Float64(cos(x) * Float64(-0.5 * (eps ^ 2.0))) - Float64(eps * sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -4.6e-5) || ~((eps <= 5e-5)))
		tmp = (cos(x) * cos(eps)) - (cos(x) + (sin(x) * sin(eps)));
	else
		tmp = (cos(x) * (-0.5 * (eps ^ 2.0))) - (eps * sin(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -4.6e-5], N[Not[LessEqual[eps, 5e-5]], $MachinePrecision]], N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[x], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[x], $MachinePrecision] * N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -4.6e-5 or 5.00000000000000024e-5 < eps

   1. Initial program 52.3%

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

      1. expm1-log1p-u 25.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(x + \varepsilon\right) - \cos x\right)\right)} \]

   3. Applied egg-rr 25.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(x + \varepsilon\right) - \cos x\right)\right)} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 52.3%

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

      2. cos-sum 98.8%

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

      3. associate--l- 98.6%

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

      4. *-commutative 98.6%

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

   5. Applied egg-rr 98.6%

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

   if -4.6e-5 < eps < 5.00000000000000024e-5

   1. Initial program 25.2%

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

   2. Taylor expanded in eps around 0 99.8%

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

      1. +-commutative 99.8%

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

      2. mul-1-neg 99.8%

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

      3. unsub-neg 99.8%

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

      4. associate-*r* 99.8%

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

      5. *-commutative 99.8%

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

   4. Simplified 99.8%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.6 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 5 \cdot 10^{-5}\right):\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 3: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.95 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 3.7 \cdot 10^{-5}\right):\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - \varepsilon \cdot \sin x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -2.95e-5) (not (<= eps 3.7e-5)))
   (- (- (* (cos x) (cos eps)) (* (sin x) (sin eps))) (cos x))
   (- (* (cos x) (* -0.5 (pow eps 2.0))) (* eps (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -2.95e-5) || !(eps <= 3.7e-5)) {
		tmp = ((cos(x) * cos(eps)) - (sin(x) * sin(eps))) - cos(x);
	} else {
		tmp = (cos(x) * (-0.5 * pow(eps, 2.0))) - (eps * sin(x));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-2.95d-5)) .or. (.not. (eps <= 3.7d-5))) then
        tmp = ((cos(x) * cos(eps)) - (sin(x) * sin(eps))) - cos(x)
    else
        tmp = (cos(x) * ((-0.5d0) * (eps ** 2.0d0))) - (eps * sin(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -2.95e-5) || !(eps <= 3.7e-5)) {
		tmp = ((Math.cos(x) * Math.cos(eps)) - (Math.sin(x) * Math.sin(eps))) - Math.cos(x);
	} else {
		tmp = (Math.cos(x) * (-0.5 * Math.pow(eps, 2.0))) - (eps * Math.sin(x));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -2.95e-5) or not (eps <= 3.7e-5):
		tmp = ((math.cos(x) * math.cos(eps)) - (math.sin(x) * math.sin(eps))) - math.cos(x)
	else:
		tmp = (math.cos(x) * (-0.5 * math.pow(eps, 2.0))) - (eps * math.sin(x))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -2.95e-5) || !(eps <= 3.7e-5))
		tmp = Float64(Float64(Float64(cos(x) * cos(eps)) - Float64(sin(x) * sin(eps))) - cos(x));
	else
		tmp = Float64(Float64(cos(x) * Float64(-0.5 * (eps ^ 2.0))) - Float64(eps * sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -2.95e-5) || ~((eps <= 3.7e-5)))
		tmp = ((cos(x) * cos(eps)) - (sin(x) * sin(eps))) - cos(x);
	else
		tmp = (cos(x) * (-0.5 * (eps ^ 2.0))) - (eps * sin(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -2.95e-5], N[Not[LessEqual[eps, 3.7e-5]], $MachinePrecision]], N[(N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[x], $MachinePrecision] * N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -2.9499999999999999e-5 or 3.69999999999999981e-5 < eps

   1. Initial program 52.3%

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

      1. cos-sum 98.8%

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

   3. Applied egg-rr 98.8%

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

   if -2.9499999999999999e-5 < eps < 3.69999999999999981e-5

   1. Initial program 25.2%

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

   2. Taylor expanded in eps around 0 99.8%

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

      1. +-commutative 99.8%

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

      2. mul-1-neg 99.8%

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

      3. unsub-neg 99.8%

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

      4. associate-*r* 99.8%

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

      5. *-commutative 99.8%

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

   4. Simplified 99.8%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.95 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 3.7 \cdot 10^{-5}\right):\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 4: 67.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\varepsilon \cdot 0.5\right)\\ \mathbf{if}\;\cos \left(\varepsilon + x\right) - \cos x \leq -1 \cdot 10^{-6}:\\ \;\;\;\;-2 \cdot {t_0}^{2}\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(-2 \cdot t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (* eps 0.5))))
   (if (<= (- (cos (+ eps x)) (cos x)) -1e-6)
     (* -2.0 (pow t_0 2.0))
     (* (sin x) (* -2.0 t_0)))))

C

double code(double x, double eps) {
	double t_0 = sin((eps * 0.5));
	double tmp;
	if ((cos((eps + x)) - cos(x)) <= -1e-6) {
		tmp = -2.0 * pow(t_0, 2.0);
	} else {
		tmp = sin(x) * (-2.0 * t_0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((eps * 0.5d0))
    if ((cos((eps + x)) - cos(x)) <= (-1d-6)) then
        tmp = (-2.0d0) * (t_0 ** 2.0d0)
    else
        tmp = sin(x) * ((-2.0d0) * t_0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin((eps * 0.5));
	double tmp;
	if ((Math.cos((eps + x)) - Math.cos(x)) <= -1e-6) {
		tmp = -2.0 * Math.pow(t_0, 2.0);
	} else {
		tmp = Math.sin(x) * (-2.0 * t_0);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.sin((eps * 0.5))
	tmp = 0
	if (math.cos((eps + x)) - math.cos(x)) <= -1e-6:
		tmp = -2.0 * math.pow(t_0, 2.0)
	else:
		tmp = math.sin(x) * (-2.0 * t_0)
	return tmp

Julia

function code(x, eps)
	t_0 = sin(Float64(eps * 0.5))
	tmp = 0.0
	if (Float64(cos(Float64(eps + x)) - cos(x)) <= -1e-6)
		tmp = Float64(-2.0 * (t_0 ^ 2.0));
	else
		tmp = Float64(sin(x) * Float64(-2.0 * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = sin((eps * 0.5));
	tmp = 0.0;
	if ((cos((eps + x)) - cos(x)) <= -1e-6)
		tmp = -2.0 * (t_0 ^ 2.0);
	else
		tmp = sin(x) * (-2.0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(eps + x), $MachinePrecision]], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], -1e-6], N[(-2.0 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * N[(-2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cos.f64 (+.f64 x eps)) (cos.f64 x)) < -9.99999999999999955e-7

   1. Initial program 80.1%

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

      1. diff-cos 80.7%

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

      2. div-inv 80.7%

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

      3. associate--l+ 80.7%

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

      4. metadata-eval 80.7%

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

      5. div-inv 80.7%

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

      6. +-commutative 80.7%

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

      7. associate-+l+ 80.5%

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

      8. metadata-eval 80.5%

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

   3. Applied egg-rr 80.5%

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

      1. associate-*r* 80.5%

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

      2. *-commutative 80.5%

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

      3. *-commutative 80.5%

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

      4. +-commutative 80.5%

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

      5. count-2 80.5%

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

      6. fma-def 80.5%

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

      7. sub-neg 80.5%

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

      8. mul-1-neg 80.5%

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

      9. +-commutative 80.5%

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

      10. associate-+r+ 80.5%

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

      11. mul-1-neg 80.5%

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

      12. sub-neg 80.5%

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

      13. +-inverses 80.5%

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

      14. remove-double-neg 80.5%

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

      15. mul-1-neg 80.5%

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

      16. sub-neg 80.5%

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

      17. neg-sub0 80.5%

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

      18. mul-1-neg 80.5%

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

      19. remove-double-neg 80.5%

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

   5. Simplified 80.5%

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

   6. Taylor expanded in x around 0 80.7%

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

   if -9.99999999999999955e-7 < (-.f64 (cos.f64 (+.f64 x eps)) (cos.f64 x))

   1. Initial program 20.8%

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

      1. diff-cos 31.5%

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

      2. div-inv 31.5%

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

      3. associate--l+ 31.5%

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

      4. metadata-eval 31.5%

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

      5. div-inv 31.5%

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

      6. +-commutative 31.5%

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

      7. associate-+l+ 31.4%

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

      8. metadata-eval 31.4%

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

   3. Applied egg-rr 31.4%

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

      1. associate-*r* 31.4%

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

      2. *-commutative 31.4%

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

      3. *-commutative 31.4%

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

      4. +-commutative 31.4%

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

      5. count-2 31.4%

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

      6. fma-def 31.4%

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

      7. sub-neg 31.4%

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

      8. mul-1-neg 31.4%

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

      9. +-commutative 31.4%

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

      10. associate-+r+ 75.5%

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

      11. mul-1-neg 75.5%

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

      12. sub-neg 75.5%

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

      13. +-inverses 75.5%

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

      14. remove-double-neg 75.5%

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

      15. mul-1-neg 75.5%

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

      16. sub-neg 75.5%

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

      17. neg-sub0 75.5%

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

      18. mul-1-neg 75.5%

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

      19. remove-double-neg 75.5%

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

   5. Simplified 75.5%

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

   6. Taylor expanded in eps around 0 64.5%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\varepsilon + x\right) - \cos x \leq -1 \cdot 10^{-6}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 5: 66.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\varepsilon + x\right) - \cos x \leq -1 \cdot 10^{-6}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- (cos (+ eps x)) (cos x)) -1e-6)
   (* -2.0 (pow (sin (* eps 0.5)) 2.0))
   (* (sin x) (- eps))))

C

double code(double x, double eps) {
	double tmp;
	if ((cos((eps + x)) - cos(x)) <= -1e-6) {
		tmp = -2.0 * pow(sin((eps * 0.5)), 2.0);
	} else {
		tmp = sin(x) * -eps;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((cos((eps + x)) - cos(x)) <= (-1d-6)) then
        tmp = (-2.0d0) * (sin((eps * 0.5d0)) ** 2.0d0)
    else
        tmp = sin(x) * -eps
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((Math.cos((eps + x)) - Math.cos(x)) <= -1e-6) {
		tmp = -2.0 * Math.pow(Math.sin((eps * 0.5)), 2.0);
	} else {
		tmp = Math.sin(x) * -eps;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (math.cos((eps + x)) - math.cos(x)) <= -1e-6:
		tmp = -2.0 * math.pow(math.sin((eps * 0.5)), 2.0)
	else:
		tmp = math.sin(x) * -eps
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(cos(Float64(eps + x)) - cos(x)) <= -1e-6)
		tmp = Float64(-2.0 * (sin(Float64(eps * 0.5)) ^ 2.0));
	else
		tmp = Float64(sin(x) * Float64(-eps));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((cos((eps + x)) - cos(x)) <= -1e-6)
		tmp = -2.0 * (sin((eps * 0.5)) ^ 2.0);
	else
		tmp = sin(x) * -eps;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(N[Cos[N[(eps + x), $MachinePrecision]], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], -1e-6], N[(-2.0 * N[Power[N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * (-eps)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cos.f64 (+.f64 x eps)) (cos.f64 x)) < -9.99999999999999955e-7

   1. Initial program 80.1%

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

      1. diff-cos 80.7%

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

      2. div-inv 80.7%

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

      3. associate--l+ 80.7%

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

      4. metadata-eval 80.7%

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

      5. div-inv 80.7%

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

      6. +-commutative 80.7%

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

      7. associate-+l+ 80.5%

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

      8. metadata-eval 80.5%

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

   3. Applied egg-rr 80.5%

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

      1. associate-*r* 80.5%

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

      2. *-commutative 80.5%

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

      3. *-commutative 80.5%

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

      4. +-commutative 80.5%

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

      5. count-2 80.5%

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

      6. fma-def 80.5%

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

      7. sub-neg 80.5%

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

      8. mul-1-neg 80.5%

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

      9. +-commutative 80.5%

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

      10. associate-+r+ 80.5%

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

      11. mul-1-neg 80.5%

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

      12. sub-neg 80.5%

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

      13. +-inverses 80.5%

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

      14. remove-double-neg 80.5%

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

      15. mul-1-neg 80.5%

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

      16. sub-neg 80.5%

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

      17. neg-sub0 80.5%

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

      18. mul-1-neg 80.5%

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

      19. remove-double-neg 80.5%

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

   5. Simplified 80.5%

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

   6. Taylor expanded in x around 0 80.7%

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

   if -9.99999999999999955e-7 < (-.f64 (cos.f64 (+.f64 x eps)) (cos.f64 x))

   1. Initial program 20.8%

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

   2. Taylor expanded in eps around 0 62.8%

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

      1. mul-1-neg 62.8%

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

      2. *-commutative 62.8%

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

      3. distribute-rgt-neg-in 62.8%

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

   4. Simplified 62.8%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\varepsilon + x\right) - \cos x \leq -1 \cdot 10^{-6}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \end{array} \]

Alternative 6: 66.5% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- (cos (+ eps x)) (cos x)) -1e-6)
   (- (cos eps) (cos x))
   (* (sin x) (- eps))))

C

double code(double x, double eps) {
	double tmp;
	if ((cos((eps + x)) - cos(x)) <= -1e-6) {
		tmp = cos(eps) - cos(x);
	} else {
		tmp = sin(x) * -eps;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((cos((eps + x)) - cos(x)) <= (-1d-6)) then
        tmp = cos(eps) - cos(x)
    else
        tmp = sin(x) * -eps
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((Math.cos((eps + x)) - Math.cos(x)) <= -1e-6) {
		tmp = Math.cos(eps) - Math.cos(x);
	} else {
		tmp = Math.sin(x) * -eps;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (math.cos((eps + x)) - math.cos(x)) <= -1e-6:
		tmp = math.cos(eps) - math.cos(x)
	else:
		tmp = math.sin(x) * -eps
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(cos(Float64(eps + x)) - cos(x)) <= -1e-6)
		tmp = Float64(cos(eps) - cos(x));
	else
		tmp = Float64(sin(x) * Float64(-eps));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((cos((eps + x)) - cos(x)) <= -1e-6)
		tmp = cos(eps) - cos(x);
	else
		tmp = sin(x) * -eps;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(N[Cos[N[(eps + x), $MachinePrecision]], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], -1e-6], N[(N[Cos[eps], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * (-eps)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cos.f64 (+.f64 x eps)) (cos.f64 x)) < -9.99999999999999955e-7

   1. Initial program 80.1%

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

   2. Taylor expanded in x around 0 80.3%

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

   if -9.99999999999999955e-7 < (-.f64 (cos.f64 (+.f64 x eps)) (cos.f64 x))

   1. Initial program 20.8%

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

   2. Taylor expanded in eps around 0 62.8%

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

      1. mul-1-neg 62.8%

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

      2. *-commutative 62.8%

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

      3. distribute-rgt-neg-in 62.8%

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

   4. Simplified 62.8%

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

4. Final simplification 67.9%

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

Alternative 7: 77.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0105 \lor \neg \left(\varepsilon \leq 0.00145\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - \varepsilon \cdot \sin x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.0105) (not (<= eps 0.00145)))
   (- (cos eps) (cos x))
   (- (* (cos x) (* -0.5 (pow eps 2.0))) (* eps (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.0105) || !(eps <= 0.00145)) {
		tmp = cos(eps) - cos(x);
	} else {
		tmp = (cos(x) * (-0.5 * pow(eps, 2.0))) - (eps * sin(x));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-0.0105d0)) .or. (.not. (eps <= 0.00145d0))) then
        tmp = cos(eps) - cos(x)
    else
        tmp = (cos(x) * ((-0.5d0) * (eps ** 2.0d0))) - (eps * sin(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.0105) || !(eps <= 0.00145)) {
		tmp = Math.cos(eps) - Math.cos(x);
	} else {
		tmp = (Math.cos(x) * (-0.5 * Math.pow(eps, 2.0))) - (eps * Math.sin(x));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.0105) or not (eps <= 0.00145):
		tmp = math.cos(eps) - math.cos(x)
	else:
		tmp = (math.cos(x) * (-0.5 * math.pow(eps, 2.0))) - (eps * math.sin(x))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.0105) || !(eps <= 0.00145))
		tmp = Float64(cos(eps) - cos(x));
	else
		tmp = Float64(Float64(cos(x) * Float64(-0.5 * (eps ^ 2.0))) - Float64(eps * sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.0105) || ~((eps <= 0.00145)))
		tmp = cos(eps) - cos(x);
	else
		tmp = (cos(x) * (-0.5 * (eps ^ 2.0))) - (eps * sin(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.0105], N[Not[LessEqual[eps, 0.00145]], $MachinePrecision]], N[(N[Cos[eps], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[x], $MachinePrecision] * N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -0.0105000000000000007 or 0.00145 < eps

   1. Initial program 52.3%

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

   2. Taylor expanded in x around 0 54.5%

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

   if -0.0105000000000000007 < eps < 0.00145

   1. Initial program 25.2%

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

   2. Taylor expanded in eps around 0 99.8%

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

      1. +-commutative 99.8%

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

      2. mul-1-neg 99.8%

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

      3. unsub-neg 99.8%

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

      4. associate-*r* 99.8%

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

      5. *-commutative 99.8%

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

   4. Simplified 99.8%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0105 \lor \neg \left(\varepsilon \leq 0.00145\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 8: 76.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 38.2%

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

   1. diff-cos 45.9%

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

   2. div-inv 45.9%

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

   3. associate--l+ 45.9%

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

   4. metadata-eval 45.9%

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

   5. div-inv 45.9%

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

   6. +-commutative 45.9%

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

   7. associate-+l+ 45.8%

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

   8. metadata-eval 45.8%

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

3. Applied egg-rr 45.8%

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

   1. associate-*r* 45.8%

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

   2. *-commutative 45.8%

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

   3. *-commutative 45.8%

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

   4. +-commutative 45.8%

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

   5. count-2 45.8%

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

   6. fma-def 45.8%

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

   7. sub-neg 45.8%

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

   8. mul-1-neg 45.8%

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

   9. +-commutative 45.8%

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

   10. associate-+r+ 77.0%

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

   11. mul-1-neg 77.0%

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

   12. sub-neg 77.0%

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

   13. +-inverses 77.0%

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

   14. remove-double-neg 77.0%

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

   15. mul-1-neg 77.0%

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

   16. sub-neg 77.0%

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

   17. neg-sub0 77.0%

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

   18. mul-1-neg 77.0%

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

   19. remove-double-neg 77.0%

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

5. Simplified 77.0%

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

6. Final simplification 77.0%

   \[\leadsto \sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]

Alternative 9: 48.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \varepsilon + -1\\ t_1 := -0.5 \cdot {\varepsilon}^{2}\\ \mathbf{if}\;\varepsilon \leq -0.00017:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -2.15 \cdot 10^{-160}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq 7.2 \cdot 10^{-75}:\\ \;\;\;\;\varepsilon \cdot \left(-x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00019:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (cos eps) -1.0)) (t_1 (* -0.5 (pow eps 2.0))))
   (if (<= eps -0.00017)
     t_0
     (if (<= eps -2.15e-160)
       t_1
       (if (<= eps 7.2e-75) (* eps (- x)) (if (<= eps 0.00019) t_1 t_0))))))

C

double code(double x, double eps) {
	double t_0 = cos(eps) + -1.0;
	double t_1 = -0.5 * pow(eps, 2.0);
	double tmp;
	if (eps <= -0.00017) {
		tmp = t_0;
	} else if (eps <= -2.15e-160) {
		tmp = t_1;
	} else if (eps <= 7.2e-75) {
		tmp = eps * -x;
	} else if (eps <= 0.00019) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos(eps) + (-1.0d0)
    t_1 = (-0.5d0) * (eps ** 2.0d0)
    if (eps <= (-0.00017d0)) then
        tmp = t_0
    else if (eps <= (-2.15d-160)) then
        tmp = t_1
    else if (eps <= 7.2d-75) then
        tmp = eps * -x
    else if (eps <= 0.00019d0) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.cos(eps) + -1.0;
	double t_1 = -0.5 * Math.pow(eps, 2.0);
	double tmp;
	if (eps <= -0.00017) {
		tmp = t_0;
	} else if (eps <= -2.15e-160) {
		tmp = t_1;
	} else if (eps <= 7.2e-75) {
		tmp = eps * -x;
	} else if (eps <= 0.00019) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.cos(eps) + -1.0
	t_1 = -0.5 * math.pow(eps, 2.0)
	tmp = 0
	if eps <= -0.00017:
		tmp = t_0
	elif eps <= -2.15e-160:
		tmp = t_1
	elif eps <= 7.2e-75:
		tmp = eps * -x
	elif eps <= 0.00019:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(cos(eps) + -1.0)
	t_1 = Float64(-0.5 * (eps ^ 2.0))
	tmp = 0.0
	if (eps <= -0.00017)
		tmp = t_0;
	elseif (eps <= -2.15e-160)
		tmp = t_1;
	elseif (eps <= 7.2e-75)
		tmp = Float64(eps * Float64(-x));
	elseif (eps <= 0.00019)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = cos(eps) + -1.0;
	t_1 = -0.5 * (eps ^ 2.0);
	tmp = 0.0;
	if (eps <= -0.00017)
		tmp = t_0;
	elseif (eps <= -2.15e-160)
		tmp = t_1;
	elseif (eps <= 7.2e-75)
		tmp = eps * -x;
	elseif (eps <= 0.00019)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.00017], t$95$0, If[LessEqual[eps, -2.15e-160], t$95$1, If[LessEqual[eps, 7.2e-75], N[(eps * (-x)), $MachinePrecision], If[LessEqual[eps, 0.00019], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -1.7e-4 or 1.9000000000000001e-4 < eps

   1. Initial program 52.3%

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

   2. Taylor expanded in x around 0 52.0%

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

   if -1.7e-4 < eps < -2.15000000000000007e-160 or 7.2000000000000001e-75 < eps < 1.9000000000000001e-4

   1. Initial program 4.3%

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

   2. Taylor expanded in x around 0 4.5%

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

   3. Taylor expanded in eps around 0 37.0%

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

   if -2.15000000000000007e-160 < eps < 7.2000000000000001e-75

   1. Initial program 35.9%

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

   2. Taylor expanded in eps around 0 95.9%

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

      1. mul-1-neg 95.9%

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

      2. *-commutative 95.9%

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

      3. distribute-rgt-neg-in 95.9%

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

   4. Simplified 95.9%

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

   5. Taylor expanded in x around 0 52.3%

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

      1. mul-1-neg 52.3%

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

      2. *-commutative 52.3%

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

      3. distribute-rgt-neg-in 52.3%

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

   7. Simplified 52.3%

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

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00017:\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{elif}\;\varepsilon \leq -2.15 \cdot 10^{-160}:\\ \;\;\;\;-0.5 \cdot {\varepsilon}^{2}\\ \mathbf{elif}\;\varepsilon \leq 7.2 \cdot 10^{-75}:\\ \;\;\;\;\varepsilon \cdot \left(-x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00019:\\ \;\;\;\;-0.5 \cdot {\varepsilon}^{2}\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon + -1\\ \end{array} \]

Alternative 10: 67.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.8 \cdot 10^{-7} \lor \neg \left(\varepsilon \leq 2.2 \cdot 10^{-6}\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -5.8e-7) (not (<= eps 2.2e-6)))
   (+ (cos eps) -1.0)
   (* (sin x) (- eps))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -5.8e-7) || !(eps <= 2.2e-6)) {
		tmp = cos(eps) + -1.0;
	} else {
		tmp = sin(x) * -eps;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-5.8d-7)) .or. (.not. (eps <= 2.2d-6))) then
        tmp = cos(eps) + (-1.0d0)
    else
        tmp = sin(x) * -eps
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -5.8e-7) || !(eps <= 2.2e-6)) {
		tmp = Math.cos(eps) + -1.0;
	} else {
		tmp = Math.sin(x) * -eps;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -5.8e-7) or not (eps <= 2.2e-6):
		tmp = math.cos(eps) + -1.0
	else:
		tmp = math.sin(x) * -eps
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -5.8e-7) || !(eps <= 2.2e-6))
		tmp = Float64(cos(eps) + -1.0);
	else
		tmp = Float64(sin(x) * Float64(-eps));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -5.8e-7) || ~((eps <= 2.2e-6)))
		tmp = cos(eps) + -1.0;
	else
		tmp = sin(x) * -eps;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -5.8e-7], N[Not[LessEqual[eps, 2.2e-6]], $MachinePrecision]], N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * (-eps)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -5.7999999999999995e-7 or 2.2000000000000001e-6 < eps

   1. Initial program 52.3%

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

   2. Taylor expanded in x around 0 52.0%

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

   if -5.7999999999999995e-7 < eps < 2.2000000000000001e-6

   1. Initial program 25.2%

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

   2. Taylor expanded in eps around 0 84.4%

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

      1. mul-1-neg 84.4%

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

      2. *-commutative 84.4%

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

      3. distribute-rgt-neg-in 84.4%

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

   4. Simplified 84.4%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.8 \cdot 10^{-7} \lor \neg \left(\varepsilon \leq 2.2 \cdot 10^{-6}\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \end{array} \]

Alternative 11: 43.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.1 \cdot 10^{-41} \lor \neg \left(\varepsilon \leq 1.1 \cdot 10^{-26}\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -3.1e-41) (not (<= eps 1.1e-26)))
   (+ (cos eps) -1.0)
   (* eps (- x))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.1e-41) || !(eps <= 1.1e-26)) {
		tmp = cos(eps) + -1.0;
	} else {
		tmp = eps * -x;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-3.1d-41)) .or. (.not. (eps <= 1.1d-26))) then
        tmp = cos(eps) + (-1.0d0)
    else
        tmp = eps * -x
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.1e-41) || !(eps <= 1.1e-26)) {
		tmp = Math.cos(eps) + -1.0;
	} else {
		tmp = eps * -x;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -3.1e-41) or not (eps <= 1.1e-26):
		tmp = math.cos(eps) + -1.0
	else:
		tmp = eps * -x
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -3.1e-41) || !(eps <= 1.1e-26))
		tmp = Float64(cos(eps) + -1.0);
	else
		tmp = Float64(eps * Float64(-x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -3.1e-41) || ~((eps <= 1.1e-26)))
		tmp = cos(eps) + -1.0;
	else
		tmp = eps * -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -3.1e-41], N[Not[LessEqual[eps, 1.1e-26]], $MachinePrecision]], N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision], N[(eps * (-x)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -3.10000000000000001e-41 or 1.1e-26 < eps

   1. Initial program 49.0%

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

   2. Taylor expanded in x around 0 48.8%

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

   if -3.10000000000000001e-41 < eps < 1.1e-26

   1. Initial program 26.8%

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

   2. Taylor expanded in eps around 0 86.0%

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

      1. mul-1-neg 86.0%

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

      2. *-commutative 86.0%

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

      3. distribute-rgt-neg-in 86.0%

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

   4. Simplified 86.0%

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

   5. Taylor expanded in x around 0 41.5%

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

      1. mul-1-neg 41.5%

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

      2. *-commutative 41.5%

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

      3. distribute-rgt-neg-in 41.5%

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

   7. Simplified 41.5%

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

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.1 \cdot 10^{-41} \lor \neg \left(\varepsilon \leq 1.1 \cdot 10^{-26}\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-x\right)\\ \end{array} \]

Alternative 12: 18.3% accurate, 51.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.2%

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

2. Taylor expanded in eps around 0 45.4%

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

   1. mul-1-neg 45.4%

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

   2. *-commutative 45.4%

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

   3. distribute-rgt-neg-in 45.4%

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

4. Simplified 45.4%

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

5. Taylor expanded in x around 0 21.7%

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

   1. mul-1-neg 21.7%

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

   2. *-commutative 21.7%

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

   3. distribute-rgt-neg-in 21.7%

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

7. Simplified 21.7%

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

8. Final simplification 21.7%

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

Reproduce

herbie shell --seed 2023339 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))