2sin (example 3.3)

Percentage Accurate: 41.9% → 99.5%

Time: 14.2s

Alternatives: 10

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 41.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.95 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\cos \varepsilon, \sin x, \mathsf{fma}\left(\sin \varepsilon, \cos x, -\sin x\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 1.8 \cdot 10^{-8}:\\ \;\;\;\;2 \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot \left(\cos x + \sin x \cdot \left(\varepsilon \cdot -0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin x, \cos \varepsilon, \sin \varepsilon \cdot \cos x\right) - \sin x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -3.95e-7)
   (fma (cos eps) (sin x) (fma (sin eps) (cos x) (- (sin x))))
   (if (<= eps 1.8e-8)
     (* 2.0 (* (sin (* eps 0.5)) (+ (cos x) (* (sin x) (* eps -0.5)))))
     (- (fma (sin x) (cos eps) (* (sin eps) (cos x))) (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -3.95e-7) {
		tmp = fma(cos(eps), sin(x), fma(sin(eps), cos(x), -sin(x)));
	} else if (eps <= 1.8e-8) {
		tmp = 2.0 * (sin((eps * 0.5)) * (cos(x) + (sin(x) * (eps * -0.5))));
	} else {
		tmp = fma(sin(x), cos(eps), (sin(eps) * cos(x))) - sin(x);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -3.95e-7)
		tmp = fma(cos(eps), sin(x), fma(sin(eps), cos(x), Float64(-sin(x))));
	elseif (eps <= 1.8e-8)
		tmp = Float64(2.0 * Float64(sin(Float64(eps * 0.5)) * Float64(cos(x) + Float64(sin(x) * Float64(eps * -0.5)))));
	else
		tmp = Float64(fma(sin(x), cos(eps), Float64(sin(eps) * cos(x))) - sin(x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -3.95e-7], N[(N[Cos[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision] + N[(N[Sin[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision] + (-N[Sin[x], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 1.8e-8], N[(2.0 * N[(N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(eps * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + N[(N[Sin[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -3.94999999999999977e-7

   1. Initial program 50.5%

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

      1. sin-sum 99.2%

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

      2. associate--l+ 99.2%

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

   3. Applied egg-rr 99.2%

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

      1. *-commutative 99.2%

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

      2. fma-udef 99.3%

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

      3. *-commutative 99.3%

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

      4. fma-neg 99.4%

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

   5. Simplified 99.4%

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

   if -3.94999999999999977e-7 < eps < 1.79999999999999991e-8

   1. Initial program 29.5%

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

      1. diff-sin 29.5%

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

      2. div-inv 29.5%

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

      3. metadata-eval 29.5%

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

      4. div-inv 29.5%

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

      5. +-commutative 29.5%

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

      6. metadata-eval 29.5%

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

   3. Applied egg-rr 29.5%

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

      1. associate-*r* 29.5%

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

      2. *-commutative 29.5%

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

      3. associate-*l* 29.5%

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

      4. +-commutative 29.5%

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

      5. associate--l+ 99.6%

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

      6. +-inverses 99.6%

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

      7. *-commutative 99.6%

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

      8. associate-+r+ 99.6%

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

      9. +-commutative 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in eps around inf 99.6%

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

      1. *-commutative 99.6%

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

      2. +-commutative 99.6%

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

      3. remove-double-neg 99.6%

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

      4. mul-1-neg 99.6%

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

      5. sub-neg 99.6%

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

      6. *-commutative 99.6%

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

      7. cancel-sign-sub-inv 99.6%

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

      8. metadata-eval 99.6%

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

      9. *-lft-identity 99.6%

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

      10. +-commutative 99.6%

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

      11. metadata-eval 99.6%

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

      12. cancel-sign-sub-inv 99.6%

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

      13. *-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in eps around 0 99.8%

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

       1. associate-*r* 99.8%

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

   11. Simplified 99.8%

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

   if 1.79999999999999991e-8 < eps

   1. Initial program 70.4%

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

      1. sin-sum 99.3%

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

      2. fma-def 99.4%

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

   3. Applied egg-rr 99.4%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.95 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\cos \varepsilon, \sin x, \mathsf{fma}\left(\sin \varepsilon, \cos x, -\sin x\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 1.8 \cdot 10^{-8}:\\ \;\;\;\;2 \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot \left(\cos x + \sin x \cdot \left(\varepsilon \cdot -0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin x, \cos \varepsilon, \sin \varepsilon \cdot \cos x\right) - \sin x\\ \end{array} \]

Alternative 2: 76.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\varepsilon + x\right) - \sin x\\ \mathbf{if}\;t_0 \leq -0.01:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;2 \cdot \left(\cos x \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (sin (+ eps x)) (sin x))))
   (if (<= t_0 -0.01)
     t_0
     (if (<= t_0 0.0) (* 2.0 (* (cos x) (sin (* eps 0.5)))) (sin eps)))))

C

double code(double x, double eps) {
	double t_0 = sin((eps + x)) - sin(x);
	double tmp;
	if (t_0 <= -0.01) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = 2.0 * (cos(x) * sin((eps * 0.5)));
	} else {
		tmp = sin(eps);
	}
	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 + x)) - sin(x)
    if (t_0 <= (-0.01d0)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) then
        tmp = 2.0d0 * (cos(x) * sin((eps * 0.5d0)))
    else
        tmp = sin(eps)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin((eps + x)) - Math.sin(x);
	double tmp;
	if (t_0 <= -0.01) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = 2.0 * (Math.cos(x) * Math.sin((eps * 0.5)));
	} else {
		tmp = Math.sin(eps);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.sin((eps + x)) - math.sin(x)
	tmp = 0
	if t_0 <= -0.01:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = 2.0 * (math.cos(x) * math.sin((eps * 0.5)))
	else:
		tmp = math.sin(eps)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(sin(Float64(eps + x)) - sin(x))
	tmp = 0.0
	if (t_0 <= -0.01)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(2.0 * Float64(cos(x) * sin(Float64(eps * 0.5))));
	else
		tmp = sin(eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = sin((eps + x)) - sin(x);
	tmp = 0.0;
	if (t_0 <= -0.01)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = 2.0 * (cos(x) * sin((eps * 0.5)));
	else
		tmp = sin(eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Sin[N[(eps + x), $MachinePrecision]], $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.01], t$95$0, If[LessEqual[t$95$0, 0.0], N[(2.0 * N[(N[Cos[x], $MachinePrecision] * N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[eps], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x)) < -0.0100000000000000002

   1. Initial program 69.2%

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

   if -0.0100000000000000002 < (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x)) < 0.0

   1. Initial program 16.7%

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

      1. diff-sin 16.7%

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

      2. div-inv 16.7%

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

      3. metadata-eval 16.7%

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

      4. div-inv 16.7%

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

      5. +-commutative 16.7%

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

      6. metadata-eval 16.7%

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

   3. Applied egg-rr 16.7%

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

      1. associate-*r* 16.7%

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

      2. *-commutative 16.7%

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

      3. associate-*l* 16.7%

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

      4. +-commutative 16.7%

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

      5. associate--l+ 82.3%

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

      6. +-inverses 82.3%

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

      7. *-commutative 82.3%

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

      8. associate-+r+ 82.3%

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

      9. +-commutative 82.3%

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

   5. Simplified 82.3%

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

   6. Taylor expanded in eps around inf 82.3%

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

      1. *-commutative 82.3%

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

      2. +-commutative 82.3%

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

      3. remove-double-neg 82.3%

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

      4. mul-1-neg 82.3%

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

      5. sub-neg 82.3%

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

      6. *-commutative 82.3%

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

      7. cancel-sign-sub-inv 82.3%

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

      8. metadata-eval 82.3%

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

      9. *-lft-identity 82.3%

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

      10. +-commutative 82.3%

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

      11. metadata-eval 82.3%

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

      12. cancel-sign-sub-inv 82.3%

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

      13. *-commutative 82.3%

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

   8. Simplified 82.3%

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

   9. Taylor expanded in eps around 0 82.3%

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

   if 0.0 < (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x))

   1. Initial program 85.3%

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

   2. Taylor expanded in x around 0 85.5%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin \left(\varepsilon + x\right) - \sin x \leq -0.01:\\ \;\;\;\;\sin \left(\varepsilon + x\right) - \sin x\\ \mathbf{elif}\;\sin \left(\varepsilon + x\right) - \sin x \leq 0:\\ \;\;\;\;2 \cdot \left(\cos x \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 3: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -9.8 \cdot 10^{-7} \lor \neg \left(\varepsilon \leq 1.8 \cdot 10^{-8}\right):\\ \;\;\;\;\mathsf{fma}\left(\sin x, \cos \varepsilon, \sin \varepsilon \cdot \cos x\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot \left(\cos x + \sin x \cdot \left(\varepsilon \cdot -0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -9.8e-7) (not (<= eps 1.8e-8)))
   (- (fma (sin x) (cos eps) (* (sin eps) (cos x))) (sin x))
   (* 2.0 (* (sin (* eps 0.5)) (+ (cos x) (* (sin x) (* eps -0.5)))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -9.8e-7) || !(eps <= 1.8e-8)) {
		tmp = fma(sin(x), cos(eps), (sin(eps) * cos(x))) - sin(x);
	} else {
		tmp = 2.0 * (sin((eps * 0.5)) * (cos(x) + (sin(x) * (eps * -0.5))));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -9.8e-7) || !(eps <= 1.8e-8))
		tmp = Float64(fma(sin(x), cos(eps), Float64(sin(eps) * cos(x))) - sin(x));
	else
		tmp = Float64(2.0 * Float64(sin(Float64(eps * 0.5)) * Float64(cos(x) + Float64(sin(x) * Float64(eps * -0.5)))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -9.8e-7], N[Not[LessEqual[eps, 1.8e-8]], $MachinePrecision]], N[(N[(N[Sin[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + N[(N[Sin[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(eps * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -9.7999999999999993e-7 or 1.79999999999999991e-8 < eps

   1. Initial program 60.1%

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

      1. sin-sum 99.3%

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

      2. fma-def 99.4%

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

   3. Applied egg-rr 99.4%

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

   if -9.7999999999999993e-7 < eps < 1.79999999999999991e-8

   1. Initial program 29.5%

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

      1. diff-sin 29.5%

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

      2. div-inv 29.5%

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

      3. metadata-eval 29.5%

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

      4. div-inv 29.5%

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

      5. +-commutative 29.5%

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

      6. metadata-eval 29.5%

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

   3. Applied egg-rr 29.5%

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

      1. associate-*r* 29.5%

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

      2. *-commutative 29.5%

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

      3. associate-*l* 29.5%

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

      4. +-commutative 29.5%

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

      5. associate--l+ 99.6%

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

      6. +-inverses 99.6%

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

      7. *-commutative 99.6%

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

      8. associate-+r+ 99.6%

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

      9. +-commutative 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in eps around inf 99.6%

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

      1. *-commutative 99.6%

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

      2. +-commutative 99.6%

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

      3. remove-double-neg 99.6%

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

      4. mul-1-neg 99.6%

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

      5. sub-neg 99.6%

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

      6. *-commutative 99.6%

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

      7. cancel-sign-sub-inv 99.6%

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

      8. metadata-eval 99.6%

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

      9. *-lft-identity 99.6%

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

      10. +-commutative 99.6%

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

      11. metadata-eval 99.6%

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

      12. cancel-sign-sub-inv 99.6%

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

      13. *-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in eps around 0 99.8%

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

       1. associate-*r* 99.8%

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

   11. Simplified 99.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -9.8 \cdot 10^{-7} \lor \neg \left(\varepsilon \leq 1.8 \cdot 10^{-8}\right):\\ \;\;\;\;\mathsf{fma}\left(\sin x, \cos \varepsilon, \sin \varepsilon \cdot \cos x\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot \left(\cos x + \sin x \cdot \left(\varepsilon \cdot -0.5\right)\right)\right)\\ \end{array} \]

Alternative 4: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.6 \cdot 10^{-7} \lor \neg \left(\varepsilon \leq 1.8 \cdot 10^{-8}\right):\\ \;\;\;\;\cos \varepsilon \cdot \sin x + \left(\sin \varepsilon \cdot \cos x - \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot \left(\cos x + \sin x \cdot \left(\varepsilon \cdot -0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -4.6e-7) (not (<= eps 1.8e-8)))
   (+ (* (cos eps) (sin x)) (- (* (sin eps) (cos x)) (sin x)))
   (* 2.0 (* (sin (* eps 0.5)) (+ (cos x) (* (sin x) (* eps -0.5)))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -4.6e-7) || !(eps <= 1.8e-8)) {
		tmp = (cos(eps) * sin(x)) + ((sin(eps) * cos(x)) - sin(x));
	} else {
		tmp = 2.0 * (sin((eps * 0.5)) * (cos(x) + (sin(x) * (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 ((eps <= (-4.6d-7)) .or. (.not. (eps <= 1.8d-8))) then
        tmp = (cos(eps) * sin(x)) + ((sin(eps) * cos(x)) - sin(x))
    else
        tmp = 2.0d0 * (sin((eps * 0.5d0)) * (cos(x) + (sin(x) * (eps * (-0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -4.6e-7) || !(eps <= 1.8e-8)) {
		tmp = (Math.cos(eps) * Math.sin(x)) + ((Math.sin(eps) * Math.cos(x)) - Math.sin(x));
	} else {
		tmp = 2.0 * (Math.sin((eps * 0.5)) * (Math.cos(x) + (Math.sin(x) * (eps * -0.5))));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -4.6e-7) or not (eps <= 1.8e-8):
		tmp = (math.cos(eps) * math.sin(x)) + ((math.sin(eps) * math.cos(x)) - math.sin(x))
	else:
		tmp = 2.0 * (math.sin((eps * 0.5)) * (math.cos(x) + (math.sin(x) * (eps * -0.5))))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -4.6e-7) || !(eps <= 1.8e-8))
		tmp = Float64(Float64(cos(eps) * sin(x)) + Float64(Float64(sin(eps) * cos(x)) - sin(x)));
	else
		tmp = Float64(2.0 * Float64(sin(Float64(eps * 0.5)) * Float64(cos(x) + Float64(sin(x) * Float64(eps * -0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -4.6e-7) || ~((eps <= 1.8e-8)))
		tmp = (cos(eps) * sin(x)) + ((sin(eps) * cos(x)) - sin(x));
	else
		tmp = 2.0 * (sin((eps * 0.5)) * (cos(x) + (sin(x) * (eps * -0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -4.6e-7], N[Not[LessEqual[eps, 1.8e-8]], $MachinePrecision]], N[(N[(N[Cos[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sin[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(eps * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -4.5999999999999999e-7 or 1.79999999999999991e-8 < eps

   1. Initial program 60.1%

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

      1. sin-sum 99.3%

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

      2. associate--l+ 99.3%

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

   3. Applied egg-rr 99.3%

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

   if -4.5999999999999999e-7 < eps < 1.79999999999999991e-8

   1. Initial program 29.5%

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

      1. diff-sin 29.5%

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

      2. div-inv 29.5%

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

      3. metadata-eval 29.5%

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

      4. div-inv 29.5%

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

      5. +-commutative 29.5%

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

      6. metadata-eval 29.5%

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

   3. Applied egg-rr 29.5%

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

      1. associate-*r* 29.5%

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

      2. *-commutative 29.5%

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

      3. associate-*l* 29.5%

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

      4. +-commutative 29.5%

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

      5. associate--l+ 99.6%

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

      6. +-inverses 99.6%

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

      7. *-commutative 99.6%

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

      8. associate-+r+ 99.6%

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

      9. +-commutative 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in eps around inf 99.6%

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

      1. *-commutative 99.6%

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

      2. +-commutative 99.6%

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

      3. remove-double-neg 99.6%

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

      4. mul-1-neg 99.6%

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

      5. sub-neg 99.6%

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

      6. *-commutative 99.6%

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

      7. cancel-sign-sub-inv 99.6%

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

      8. metadata-eval 99.6%

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

      9. *-lft-identity 99.6%

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

      10. +-commutative 99.6%

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

      11. metadata-eval 99.6%

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

      12. cancel-sign-sub-inv 99.6%

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

      13. *-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in eps around 0 99.8%

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

       1. associate-*r* 99.8%

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

   11. Simplified 99.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.6 \cdot 10^{-7} \lor \neg \left(\varepsilon \leq 1.8 \cdot 10^{-8}\right):\\ \;\;\;\;\cos \varepsilon \cdot \sin x + \left(\sin \varepsilon \cdot \cos x - \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot \left(\cos x + \sin x \cdot \left(\varepsilon \cdot -0.5\right)\right)\right)\\ \end{array} \]

Alternative 5: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \varepsilon \cdot \cos x\\ t_1 := \cos \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -8 \cdot 10^{-7}:\\ \;\;\;\;\left(t_0 + t_1\right) - \sin x\\ \mathbf{elif}\;\varepsilon \leq 1.8 \cdot 10^{-8}:\\ \;\;\;\;2 \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot \left(\cos x + \sin x \cdot \left(\varepsilon \cdot -0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 + \left(t_0 - \sin x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (sin eps) (cos x))) (t_1 (* (cos eps) (sin x))))
   (if (<= eps -8e-7)
     (- (+ t_0 t_1) (sin x))
     (if (<= eps 1.8e-8)
       (* 2.0 (* (sin (* eps 0.5)) (+ (cos x) (* (sin x) (* eps -0.5)))))
       (+ t_1 (- t_0 (sin x)))))))

C

double code(double x, double eps) {
	double t_0 = sin(eps) * cos(x);
	double t_1 = cos(eps) * sin(x);
	double tmp;
	if (eps <= -8e-7) {
		tmp = (t_0 + t_1) - sin(x);
	} else if (eps <= 1.8e-8) {
		tmp = 2.0 * (sin((eps * 0.5)) * (cos(x) + (sin(x) * (eps * -0.5))));
	} else {
		tmp = t_1 + (t_0 - sin(x));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sin(eps) * cos(x)
    t_1 = cos(eps) * sin(x)
    if (eps <= (-8d-7)) then
        tmp = (t_0 + t_1) - sin(x)
    else if (eps <= 1.8d-8) then
        tmp = 2.0d0 * (sin((eps * 0.5d0)) * (cos(x) + (sin(x) * (eps * (-0.5d0)))))
    else
        tmp = t_1 + (t_0 - sin(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin(eps) * Math.cos(x);
	double t_1 = Math.cos(eps) * Math.sin(x);
	double tmp;
	if (eps <= -8e-7) {
		tmp = (t_0 + t_1) - Math.sin(x);
	} else if (eps <= 1.8e-8) {
		tmp = 2.0 * (Math.sin((eps * 0.5)) * (Math.cos(x) + (Math.sin(x) * (eps * -0.5))));
	} else {
		tmp = t_1 + (t_0 - Math.sin(x));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.sin(eps) * math.cos(x)
	t_1 = math.cos(eps) * math.sin(x)
	tmp = 0
	if eps <= -8e-7:
		tmp = (t_0 + t_1) - math.sin(x)
	elif eps <= 1.8e-8:
		tmp = 2.0 * (math.sin((eps * 0.5)) * (math.cos(x) + (math.sin(x) * (eps * -0.5))))
	else:
		tmp = t_1 + (t_0 - math.sin(x))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(sin(eps) * cos(x))
	t_1 = Float64(cos(eps) * sin(x))
	tmp = 0.0
	if (eps <= -8e-7)
		tmp = Float64(Float64(t_0 + t_1) - sin(x));
	elseif (eps <= 1.8e-8)
		tmp = Float64(2.0 * Float64(sin(Float64(eps * 0.5)) * Float64(cos(x) + Float64(sin(x) * Float64(eps * -0.5)))));
	else
		tmp = Float64(t_1 + Float64(t_0 - sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = sin(eps) * cos(x);
	t_1 = cos(eps) * sin(x);
	tmp = 0.0;
	if (eps <= -8e-7)
		tmp = (t_0 + t_1) - sin(x);
	elseif (eps <= 1.8e-8)
		tmp = 2.0 * (sin((eps * 0.5)) * (cos(x) + (sin(x) * (eps * -0.5))));
	else
		tmp = t_1 + (t_0 - sin(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Sin[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -8e-7], N[(N[(t$95$0 + t$95$1), $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 1.8e-8], N[(2.0 * N[(N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(eps * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(t$95$0 - N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -7.9999999999999996e-7

   1. Initial program 50.5%

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

      1. sin-sum 99.2%

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

   3. Applied egg-rr 99.2%

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

   if -7.9999999999999996e-7 < eps < 1.79999999999999991e-8

   1. Initial program 29.5%

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

      1. diff-sin 29.5%

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

      2. div-inv 29.5%

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

      3. metadata-eval 29.5%

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

      4. div-inv 29.5%

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

      5. +-commutative 29.5%

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

      6. metadata-eval 29.5%

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

   3. Applied egg-rr 29.5%

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

      1. associate-*r* 29.5%

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

      2. *-commutative 29.5%

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

      3. associate-*l* 29.5%

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

      4. +-commutative 29.5%

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

      5. associate--l+ 99.6%

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

      6. +-inverses 99.6%

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

      7. *-commutative 99.6%

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

      8. associate-+r+ 99.6%

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

      9. +-commutative 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in eps around inf 99.6%

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

      1. *-commutative 99.6%

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

      2. +-commutative 99.6%

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

      3. remove-double-neg 99.6%

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

      4. mul-1-neg 99.6%

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

      5. sub-neg 99.6%

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

      6. *-commutative 99.6%

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

      7. cancel-sign-sub-inv 99.6%

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

      8. metadata-eval 99.6%

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

      9. *-lft-identity 99.6%

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

      10. +-commutative 99.6%

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

      11. metadata-eval 99.6%

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

      12. cancel-sign-sub-inv 99.6%

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

      13. *-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in eps around 0 99.8%

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

       1. associate-*r* 99.8%

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

   11. Simplified 99.8%

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

   if 1.79999999999999991e-8 < eps

   1. Initial program 70.4%

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

      1. sin-sum 99.3%

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

      2. associate--l+ 99.4%

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

   3. Applied egg-rr 99.4%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -8 \cdot 10^{-7}:\\ \;\;\;\;\left(\sin \varepsilon \cdot \cos x + \cos \varepsilon \cdot \sin x\right) - \sin x\\ \mathbf{elif}\;\varepsilon \leq 1.8 \cdot 10^{-8}:\\ \;\;\;\;2 \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot \left(\cos x + \sin x \cdot \left(\varepsilon \cdot -0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon \cdot \sin x + \left(\sin \varepsilon \cdot \cos x - \sin x\right)\\ \end{array} \]

Alternative 6: 75.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 45.0%

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

   1. diff-sin 44.6%

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

   2. div-inv 44.6%

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

   3. metadata-eval 44.6%

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

   4. div-inv 44.6%

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

   5. +-commutative 44.6%

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

   6. metadata-eval 44.6%

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

3. Applied egg-rr 44.6%

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

   1. associate-*r* 44.6%

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

   2. *-commutative 44.6%

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

   3. associate-*l* 44.6%

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

   4. +-commutative 44.6%

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

   5. associate--l+ 79.8%

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

   6. +-inverses 79.8%

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

   7. *-commutative 79.8%

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

   8. associate-+r+ 80.0%

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

   9. +-commutative 80.0%

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

5. Simplified 80.0%

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

6. Taylor expanded in eps around inf 80.0%

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

   1. *-commutative 80.0%

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

   2. +-commutative 80.0%

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

   3. remove-double-neg 80.0%

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

   4. mul-1-neg 80.0%

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

   5. sub-neg 80.0%

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

   6. *-commutative 80.0%

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

   7. cancel-sign-sub-inv 80.0%

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

   8. metadata-eval 80.0%

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

   9. *-lft-identity 80.0%

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

   10. +-commutative 80.0%

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

   11. metadata-eval 80.0%

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

   12. cancel-sign-sub-inv 80.0%

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

   13. *-commutative 80.0%

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

8. Simplified 80.0%

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

9. Final simplification 80.0%

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

Alternative 7: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.052 \lor \neg \left(x \leq 0.00052\right):\\ \;\;\;\;2 \cdot \left(\cos x \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon + x \cdot \left(\cos \varepsilon + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -0.052) (not (<= x 0.00052)))
   (* 2.0 (* (cos x) (sin (* eps 0.5))))
   (+ (sin eps) (* x (+ (cos eps) -1.0)))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -0.052) || !(x <= 0.00052)) {
		tmp = 2.0 * (cos(x) * sin((eps * 0.5)));
	} else {
		tmp = sin(eps) + (x * (cos(eps) + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= (-0.052d0)) .or. (.not. (x <= 0.00052d0))) then
        tmp = 2.0d0 * (cos(x) * sin((eps * 0.5d0)))
    else
        tmp = sin(eps) + (x * (cos(eps) + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -0.052) || !(x <= 0.00052)) {
		tmp = 2.0 * (Math.cos(x) * Math.sin((eps * 0.5)));
	} else {
		tmp = Math.sin(eps) + (x * (Math.cos(eps) + -1.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -0.052) or not (x <= 0.00052):
		tmp = 2.0 * (math.cos(x) * math.sin((eps * 0.5)))
	else:
		tmp = math.sin(eps) + (x * (math.cos(eps) + -1.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -0.052) || !(x <= 0.00052))
		tmp = Float64(2.0 * Float64(cos(x) * sin(Float64(eps * 0.5))));
	else
		tmp = Float64(sin(eps) + Float64(x * Float64(cos(eps) + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -0.052) || ~((x <= 0.00052)))
		tmp = 2.0 * (cos(x) * sin((eps * 0.5)));
	else
		tmp = sin(eps) + (x * (cos(eps) + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -0.052], N[Not[LessEqual[x, 0.00052]], $MachinePrecision]], N[(2.0 * N[(N[Cos[x], $MachinePrecision] * N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[eps], $MachinePrecision] + N[(x * N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0519999999999999976 or 5.19999999999999954e-4 < x

   1. Initial program 6.5%

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

      1. diff-sin 5.9%

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

      2. div-inv 5.9%

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

      3. metadata-eval 5.9%

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

      4. div-inv 5.9%

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

      5. +-commutative 5.9%

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

      6. metadata-eval 5.9%

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

   3. Applied egg-rr 5.9%

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

      1. associate-*r* 5.9%

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

      2. *-commutative 5.9%

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

      3. associate-*l* 5.9%

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

      4. +-commutative 5.9%

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

      5. associate--l+ 59.2%

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

      6. +-inverses 59.2%

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

      7. *-commutative 59.2%

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

      8. associate-+r+ 59.5%

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

      9. +-commutative 59.5%

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

   5. Simplified 59.5%

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

   6. Taylor expanded in eps around inf 59.5%

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

      1. *-commutative 59.5%

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

      2. +-commutative 59.5%

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

      3. remove-double-neg 59.5%

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

      4. mul-1-neg 59.5%

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

      5. sub-neg 59.5%

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

      6. *-commutative 59.5%

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

      7. cancel-sign-sub-inv 59.5%

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

      8. metadata-eval 59.5%

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

      9. *-lft-identity 59.5%

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

      10. +-commutative 59.5%

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

      11. metadata-eval 59.5%

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

      12. cancel-sign-sub-inv 59.5%

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

      13. *-commutative 59.5%

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

   8. Simplified 59.5%

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

   9. Taylor expanded in eps around 0 59.9%

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

   if -0.0519999999999999976 < x < 5.19999999999999954e-4

   1. Initial program 79.6%

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

   2. Taylor expanded in x around 0 98.9%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.052 \lor \neg \left(x \leq 0.00052\right):\\ \;\;\;\;2 \cdot \left(\cos x \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon + x \cdot \left(\cos \varepsilon + -1\right)\\ \end{array} \]

Alternative 8: 75.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.9 \cdot 10^{-6}:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.0106:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -1.9e-6)
   (sin eps)
   (if (<= eps 0.0106) (* eps (cos x)) (sin eps))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -1.9e-6) {
		tmp = sin(eps);
	} else if (eps <= 0.0106) {
		tmp = eps * cos(x);
	} else {
		tmp = sin(eps);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= (-1.9d-6)) then
        tmp = sin(eps)
    else if (eps <= 0.0106d0) then
        tmp = eps * cos(x)
    else
        tmp = sin(eps)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -1.9e-6) {
		tmp = Math.sin(eps);
	} else if (eps <= 0.0106) {
		tmp = eps * Math.cos(x);
	} else {
		tmp = Math.sin(eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -1.9e-6:
		tmp = math.sin(eps)
	elif eps <= 0.0106:
		tmp = eps * math.cos(x)
	else:
		tmp = math.sin(eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -1.9e-6)
		tmp = sin(eps);
	elseif (eps <= 0.0106)
		tmp = Float64(eps * cos(x));
	else
		tmp = sin(eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -1.9e-6)
		tmp = sin(eps);
	elseif (eps <= 0.0106)
		tmp = eps * cos(x);
	else
		tmp = sin(eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -1.9e-6], N[Sin[eps], $MachinePrecision], If[LessEqual[eps, 0.0106], N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision], N[Sin[eps], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.9e-6 or 0.0106 < eps

   1. Initial program 60.2%

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

   2. Taylor expanded in x around 0 61.2%

      \[\leadsto \color{blue}{\sin \varepsilon} \]

   if -1.9e-6 < eps < 0.0106

   1. Initial program 29.8%

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

   2. Taylor expanded in eps around 0 99.0%

      \[\leadsto \color{blue}{\varepsilon \cdot \cos x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.9 \cdot 10^{-6}:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.0106:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 9: 55.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sin \varepsilon \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (sin eps))

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return math.sin(eps)

Julia

function code(x, eps)
	return sin(eps)
end

MATLAB

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

Wolfram

code[x_, eps_] := N[Sin[eps], $MachinePrecision]

Derivation

1. Initial program 45.0%

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

2. Taylor expanded in x around 0 57.2%

   \[\leadsto \color{blue}{\sin \varepsilon} \]

3. Final simplification 57.2%

   \[\leadsto \sin \varepsilon \]

Alternative 10: 28.8% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 eps)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps

Julia

function code(x, eps)
	return eps
end

MATLAB

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

Wolfram

code[x_, eps_] := eps

Derivation

1. Initial program 45.0%

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

   1. diff-sin 44.6%

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

   2. div-inv 44.6%

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

   3. metadata-eval 44.6%

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

   4. div-inv 44.6%

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

   5. +-commutative 44.6%

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

   6. metadata-eval 44.6%

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

3. Applied egg-rr 44.6%

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

   1. associate-*r* 44.6%

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

   2. *-commutative 44.6%

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

   3. associate-*l* 44.6%

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

   4. +-commutative 44.6%

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

   5. associate--l+ 79.8%

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

   6. +-inverses 79.8%

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

   7. *-commutative 79.8%

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

   8. associate-+r+ 80.0%

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

   9. +-commutative 80.0%

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

5. Simplified 80.0%

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

6. Taylor expanded in eps around 0 51.1%

   \[\leadsto \color{blue}{\varepsilon \cdot \cos x} \]
7. Step-by-step derivation

   1. *-commutative 51.1%

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

8. Simplified 51.1%

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

9. Taylor expanded in x around 0 28.1%

   \[\leadsto \color{blue}{\varepsilon} \]

10. Final simplification 28.1%

    \[\leadsto \varepsilon \]

Developer target: 75.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023297 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64

  :herbie-target
  (* 2.0 (* (cos (+ x (/ eps 2.0))) (sin (/ eps 2.0))))

  (- (sin (+ x eps)) (sin x)))