2sin (example 3.3)

Percentage Accurate: 42.1% → 99.5%

Time: 15.1s

Alternatives: 13

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: 42.1% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\varepsilon \cdot 0.5\right)\\ t_0 \cdot \left(2 \cdot \mathsf{fma}\left(\cos x, \cos \left(\varepsilon \cdot 0.5\right), -\sin x \cdot t_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (* eps 0.5))))
   (* t_0 (* 2.0 (fma (cos x) (cos (* eps 0.5)) (- (* (sin x) t_0)))))))

C

double code(double x, double eps) {
	double t_0 = sin((eps * 0.5));
	return t_0 * (2.0 * fma(cos(x), cos((eps * 0.5)), -(sin(x) * t_0)));
}

Julia

function code(x, eps)
	t_0 = sin(Float64(eps * 0.5))
	return Float64(t_0 * Float64(2.0 * fma(cos(x), cos(Float64(eps * 0.5)), Float64(-Float64(sin(x) * t_0)))))
end

Wolfram

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

Derivation

1. Initial program 40.2%

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

   1. diff-sin 39.8%

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

   2. div-inv 39.8%

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

   3. metadata-eval 39.8%

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

   4. div-inv 39.8%

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

   5. +-commutative 39.8%

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

   6. metadata-eval 39.8%

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

3. Applied egg-rr 39.8%

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

   1. associate-*r* 39.8%

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

   2. *-commutative 39.8%

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

   3. associate-*l* 39.8%

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

   4. +-commutative 39.8%

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

   5. associate--l+ 75.7%

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

   6. +-inverses 75.7%

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

   7. *-commutative 75.7%

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

   8. associate-+r+ 75.5%

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

   9. +-commutative 75.5%

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

5. Simplified 75.5%

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

   1. distribute-lft-in 75.5%

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

   2. *-commutative 75.5%

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

   3. +-rgt-identity 75.5%

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

   4. cos-sum 99.4%

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

   5. +-rgt-identity 99.4%

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

   6. *-commutative 99.4%

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

   7. +-rgt-identity 99.4%

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

   8. *-commutative 99.4%

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

7. Applied egg-rr 99.4%

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

   1. *-commutative 99.4%

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

   2. count-2 99.4%

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

   3. associate-*r* 99.4%

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

   4. metadata-eval 99.4%

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

   5. *-lft-identity 99.4%

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

   6. *-commutative 99.4%

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

   7. *-commutative 99.4%

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

   8. count-2 99.4%

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

   9. associate-*r* 99.4%

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

   10. metadata-eval 99.4%

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

   11. *-lft-identity 99.4%

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

   12. *-commutative 99.4%

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

9. Simplified 99.4%

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

    1. fma-neg 99.5%

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

11. Applied egg-rr 99.5%

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

12. Final simplification 99.5%

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

Alternative 2: 76.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\varepsilon + x\right)\\ t_1 := t_0 - \sin x\\ \mathbf{if}\;t_1 \leq -0.0512 \lor \neg \left(t_1 \leq 10^{-106}\right):\\ \;\;\;\;t_0 - \left(-1 + \left(\sin x + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\varepsilon \cdot 0.5\right) \cdot \left(2 \cdot \cos x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (+ eps x))) (t_1 (- t_0 (sin x))))
   (if (or (<= t_1 -0.0512) (not (<= t_1 1e-106)))
     (- t_0 (+ -1.0 (+ (sin x) 1.0)))
     (* (sin (* eps 0.5)) (* 2.0 (cos x))))))

C

double code(double x, double eps) {
	double t_0 = sin((eps + x));
	double t_1 = t_0 - sin(x);
	double tmp;
	if ((t_1 <= -0.0512) || !(t_1 <= 1e-106)) {
		tmp = t_0 - (-1.0 + (sin(x) + 1.0));
	} else {
		tmp = sin((eps * 0.5)) * (2.0 * cos(x));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sin((eps + x))
    t_1 = t_0 - sin(x)
    if ((t_1 <= (-0.0512d0)) .or. (.not. (t_1 <= 1d-106))) then
        tmp = t_0 - ((-1.0d0) + (sin(x) + 1.0d0))
    else
        tmp = sin((eps * 0.5d0)) * (2.0d0 * cos(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin((eps + x));
	double t_1 = t_0 - Math.sin(x);
	double tmp;
	if ((t_1 <= -0.0512) || !(t_1 <= 1e-106)) {
		tmp = t_0 - (-1.0 + (Math.sin(x) + 1.0));
	} else {
		tmp = Math.sin((eps * 0.5)) * (2.0 * Math.cos(x));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.sin((eps + x))
	t_1 = t_0 - math.sin(x)
	tmp = 0
	if (t_1 <= -0.0512) or not (t_1 <= 1e-106):
		tmp = t_0 - (-1.0 + (math.sin(x) + 1.0))
	else:
		tmp = math.sin((eps * 0.5)) * (2.0 * math.cos(x))
	return tmp

Julia

function code(x, eps)
	t_0 = sin(Float64(eps + x))
	t_1 = Float64(t_0 - sin(x))
	tmp = 0.0
	if ((t_1 <= -0.0512) || !(t_1 <= 1e-106))
		tmp = Float64(t_0 - Float64(-1.0 + Float64(sin(x) + 1.0)));
	else
		tmp = Float64(sin(Float64(eps * 0.5)) * Float64(2.0 * cos(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = sin((eps + x));
	t_1 = t_0 - sin(x);
	tmp = 0.0;
	if ((t_1 <= -0.0512) || ~((t_1 <= 1e-106)))
		tmp = t_0 - (-1.0 + (sin(x) + 1.0));
	else
		tmp = sin((eps * 0.5)) * (2.0 * cos(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Sin[N[(eps + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[Sin[x], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -0.0512], N[Not[LessEqual[t$95$1, 1e-106]], $MachinePrecision]], N[(t$95$0 - N[(-1.0 + N[(N[Sin[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x)) < -0.0512000000000000025 or 9.99999999999999941e-107 < (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x))

   1. Initial program 67.4%

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

      1. expm1-log1p-u 67.4%

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

   3. Applied egg-rr 67.4%

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

      1. expm1-udef 67.4%

         \[\leadsto \sin \left(x + \varepsilon\right) - \color{blue}{\left(e^{\mathsf{log1p}\left(\sin x\right)} - 1\right)} \]

      2. log1p-udef 67.4%

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

      3. add-exp-log 67.4%

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

   5. Applied egg-rr 67.4%

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

   if -0.0512000000000000025 < (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x)) < 9.99999999999999941e-107

   1. Initial program 22.8%

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

      1. diff-sin 22.7%

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

      2. div-inv 22.7%

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

      3. metadata-eval 22.7%

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

      4. div-inv 22.7%

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

      5. +-commutative 22.7%

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

      6. metadata-eval 22.7%

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

   3. Applied egg-rr 22.7%

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

      1. associate-*r* 22.7%

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

      2. *-commutative 22.7%

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

      3. associate-*l* 22.7%

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

      4. +-commutative 22.7%

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

      5. associate--l+ 81.8%

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

      6. +-inverses 81.8%

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

      7. *-commutative 81.8%

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

      8. associate-+r+ 81.8%

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

      9. +-commutative 81.8%

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

   5. Simplified 81.8%

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

   6. Taylor expanded in eps around 0 82.2%

      \[\leadsto \sin \left(\left(\varepsilon + 0\right) \cdot 0.5\right) \cdot \left(2 \cdot \color{blue}{\cos x}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin \left(\varepsilon + x\right) - \sin x \leq -0.0512 \lor \neg \left(\sin \left(\varepsilon + x\right) - \sin x \leq 10^{-106}\right):\\ \;\;\;\;\sin \left(\varepsilon + x\right) - \left(-1 + \left(\sin x + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\varepsilon \cdot 0.5\right) \cdot \left(2 \cdot \cos x\right)\\ \end{array} \]

Alternative 3: 76.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\varepsilon + x\right) - \sin x\\ \mathbf{if}\;t_0 \leq -0.0512 \lor \neg \left(t_0 \leq 10^{-106}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\varepsilon \cdot 0.5\right) \cdot \left(2 \cdot \cos x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (sin (+ eps x)) (sin x))))
   (if (or (<= t_0 -0.0512) (not (<= t_0 1e-106)))
     t_0
     (* (sin (* eps 0.5)) (* 2.0 (cos x))))))

C

double code(double x, double eps) {
	double t_0 = sin((eps + x)) - sin(x);
	double tmp;
	if ((t_0 <= -0.0512) || !(t_0 <= 1e-106)) {
		tmp = t_0;
	} else {
		tmp = sin((eps * 0.5)) * (2.0 * cos(x));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((eps + x)) - sin(x)
    if ((t_0 <= (-0.0512d0)) .or. (.not. (t_0 <= 1d-106))) then
        tmp = t_0
    else
        tmp = sin((eps * 0.5d0)) * (2.0d0 * cos(x))
    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 <= -0.0512) || !(t_0 <= 1e-106)) {
		tmp = t_0;
	} else {
		tmp = Math.sin((eps * 0.5)) * (2.0 * Math.cos(x));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.sin((eps + x)) - math.sin(x)
	tmp = 0
	if (t_0 <= -0.0512) or not (t_0 <= 1e-106):
		tmp = t_0
	else:
		tmp = math.sin((eps * 0.5)) * (2.0 * math.cos(x))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(sin(Float64(eps + x)) - sin(x))
	tmp = 0.0
	if ((t_0 <= -0.0512) || !(t_0 <= 1e-106))
		tmp = t_0;
	else
		tmp = Float64(sin(Float64(eps * 0.5)) * Float64(2.0 * cos(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = sin((eps + x)) - sin(x);
	tmp = 0.0;
	if ((t_0 <= -0.0512) || ~((t_0 <= 1e-106)))
		tmp = t_0;
	else
		tmp = sin((eps * 0.5)) * (2.0 * cos(x));
	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[Or[LessEqual[t$95$0, -0.0512], N[Not[LessEqual[t$95$0, 1e-106]], $MachinePrecision]], t$95$0, N[(N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x)) < -0.0512000000000000025 or 9.99999999999999941e-107 < (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x))

   1. Initial program 67.4%

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

   if -0.0512000000000000025 < (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x)) < 9.99999999999999941e-107

   1. Initial program 22.8%

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

      1. diff-sin 22.7%

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

      2. div-inv 22.7%

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

      3. metadata-eval 22.7%

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

      4. div-inv 22.7%

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

      5. +-commutative 22.7%

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

      6. metadata-eval 22.7%

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

   3. Applied egg-rr 22.7%

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

      1. associate-*r* 22.7%

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

      2. *-commutative 22.7%

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

      3. associate-*l* 22.7%

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

      4. +-commutative 22.7%

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

      5. associate--l+ 81.8%

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

      6. +-inverses 81.8%

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

      7. *-commutative 81.8%

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

      8. associate-+r+ 81.8%

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

      9. +-commutative 81.8%

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

   5. Simplified 81.8%

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

   6. Taylor expanded in eps around 0 82.2%

      \[\leadsto \sin \left(\left(\varepsilon + 0\right) \cdot 0.5\right) \cdot \left(2 \cdot \color{blue}{\cos x}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin \left(\varepsilon + x\right) - \sin x \leq -0.0512 \lor \neg \left(\sin \left(\varepsilon + x\right) - \sin x \leq 10^{-106}\right):\\ \;\;\;\;\sin \left(\varepsilon + x\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\varepsilon \cdot 0.5\right) \cdot \left(2 \cdot \cos x\right)\\ \end{array} \]

Alternative 4: 76.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\varepsilon + x\right) - \sin x\\ \mathbf{if}\;t_0 \leq -0.02 \lor \neg \left(t_0 \leq 10^{-106}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (sin (+ eps x)) (sin x))))
   (if (or (<= t_0 -0.02) (not (<= t_0 1e-106))) t_0 (* eps (cos x)))))

C

double code(double x, double eps) {
	double t_0 = sin((eps + x)) - sin(x);
	double tmp;
	if ((t_0 <= -0.02) || !(t_0 <= 1e-106)) {
		tmp = t_0;
	} else {
		tmp = eps * cos(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((eps + x)) - sin(x)
    if ((t_0 <= (-0.02d0)) .or. (.not. (t_0 <= 1d-106))) then
        tmp = t_0
    else
        tmp = eps * cos(x)
    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 <= -0.02) || !(t_0 <= 1e-106)) {
		tmp = t_0;
	} else {
		tmp = eps * Math.cos(x);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.sin((eps + x)) - math.sin(x)
	tmp = 0
	if (t_0 <= -0.02) or not (t_0 <= 1e-106):
		tmp = t_0
	else:
		tmp = eps * math.cos(x)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(sin(Float64(eps + x)) - sin(x))
	tmp = 0.0
	if ((t_0 <= -0.02) || !(t_0 <= 1e-106))
		tmp = t_0;
	else
		tmp = Float64(eps * cos(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = sin((eps + x)) - sin(x);
	tmp = 0.0;
	if ((t_0 <= -0.02) || ~((t_0 <= 1e-106)))
		tmp = t_0;
	else
		tmp = eps * cos(x);
	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[Or[LessEqual[t$95$0, -0.02], N[Not[LessEqual[t$95$0, 1e-106]], $MachinePrecision]], t$95$0, N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x)) < -0.0200000000000000004 or 9.99999999999999941e-107 < (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x))

   1. Initial program 65.8%

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

   if -0.0200000000000000004 < (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x)) < 9.99999999999999941e-107

   1. Initial program 23.0%

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

   2. Taylor expanded in eps around 0 82.2%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin \left(\varepsilon + x\right) - \sin x \leq -0.02 \lor \neg \left(\sin \left(\varepsilon + x\right) - \sin x \leq 10^{-106}\right):\\ \;\;\;\;\sin \left(\varepsilon + x\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \]

Alternative 5: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\varepsilon \cdot 0.5\right)\\ 2 \cdot \left(t_0 \cdot \left(\cos x \cdot \cos \left(\varepsilon \cdot 0.5\right) - \sin x \cdot t_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (* eps 0.5))))
   (* 2.0 (* t_0 (- (* (cos x) (cos (* eps 0.5))) (* (sin x) t_0))))))

C

double code(double x, double eps) {
	double t_0 = sin((eps * 0.5));
	return 2.0 * (t_0 * ((cos(x) * cos((eps * 0.5))) - (sin(x) * t_0)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    t_0 = sin((eps * 0.5d0))
    code = 2.0d0 * (t_0 * ((cos(x) * cos((eps * 0.5d0))) - (sin(x) * t_0)))
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin((eps * 0.5));
	return 2.0 * (t_0 * ((Math.cos(x) * Math.cos((eps * 0.5))) - (Math.sin(x) * t_0)));
}

Python

def code(x, eps):
	t_0 = math.sin((eps * 0.5))
	return 2.0 * (t_0 * ((math.cos(x) * math.cos((eps * 0.5))) - (math.sin(x) * t_0)))

Julia

function code(x, eps)
	t_0 = sin(Float64(eps * 0.5))
	return Float64(2.0 * Float64(t_0 * Float64(Float64(cos(x) * cos(Float64(eps * 0.5))) - Float64(sin(x) * t_0))))
end

MATLAB

function tmp = code(x, eps)
	t_0 = sin((eps * 0.5));
	tmp = 2.0 * (t_0 * ((cos(x) * cos((eps * 0.5))) - (sin(x) * t_0)));
end

Wolfram

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

Derivation

1. Initial program 40.2%

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

   1. diff-sin 39.8%

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

   2. div-inv 39.8%

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

   3. metadata-eval 39.8%

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

   4. div-inv 39.8%

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

   5. +-commutative 39.8%

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

   6. metadata-eval 39.8%

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

3. Applied egg-rr 39.8%

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

   1. associate-*r* 39.8%

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

   2. *-commutative 39.8%

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

   3. associate-*l* 39.8%

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

   4. +-commutative 39.8%

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

   5. associate--l+ 75.7%

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

   6. +-inverses 75.7%

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

   7. *-commutative 75.7%

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

   8. associate-+r+ 75.5%

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

   9. +-commutative 75.5%

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

5. Simplified 75.5%

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

   1. distribute-lft-in 75.5%

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

   2. *-commutative 75.5%

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

   3. +-rgt-identity 75.5%

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

   4. cos-sum 99.4%

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

   5. +-rgt-identity 99.4%

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

   6. *-commutative 99.4%

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

   7. +-rgt-identity 99.4%

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

   8. *-commutative 99.4%

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

7. Applied egg-rr 99.4%

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

   1. *-commutative 99.4%

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

   2. count-2 99.4%

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

   3. associate-*r* 99.4%

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

   4. metadata-eval 99.4%

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

   5. *-lft-identity 99.4%

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

   6. *-commutative 99.4%

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

   7. *-commutative 99.4%

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

   8. count-2 99.4%

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

   9. associate-*r* 99.4%

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

   10. metadata-eval 99.4%

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

   11. *-lft-identity 99.4%

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

   12. *-commutative 99.4%

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

9. Simplified 99.4%

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

10. Taylor expanded in eps around inf 99.4%

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

    1. *-commutative 99.4%

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

    2. *-commutative 99.4%

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

    3. *-commutative 99.4%

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

    4. *-commutative 99.4%

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

12. Simplified 99.4%

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

13. Final simplification 99.4%

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

Alternative 6: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\cos x, \sin \varepsilon, \sin x \cdot \left(-1 + \cos \varepsilon\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma (cos x) (sin eps) (* (sin x) (+ -1.0 (cos eps)))))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 40.2%

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

   1. sin-sum 64.6%

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

3. Applied egg-rr 64.6%

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

4. Taylor expanded in x around inf 64.6%

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

   1. associate--l+ 99.4%

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

   2. fma-def 99.4%

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

   3. *-lft-identity 99.4%

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

   4. distribute-rgt-out-- 99.4%

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

   5. sub-neg 99.4%

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

   6. metadata-eval 99.4%

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

   7. +-commutative 99.4%

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

6. Simplified 99.4%

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

7. Final simplification 99.4%

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

Alternative 7: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.2%

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

   1. sin-sum 64.6%

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

3. Applied egg-rr 64.6%

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

4. Taylor expanded in x around inf 64.6%

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

   1. associate--l+ 99.4%

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

   2. fma-def 99.4%

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

   3. *-lft-identity 99.4%

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

   4. distribute-rgt-out-- 99.4%

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

   5. sub-neg 99.4%

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

   6. metadata-eval 99.4%

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

   7. +-commutative 99.4%

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

6. Simplified 99.4%

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

7. Taylor expanded in x around inf 99.4%

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

8. Final simplification 99.4%

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

Alternative 8: 76.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 + (2.0d0 * x)))))
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = 2.0 * (sin((eps * 0.5)) * cos((0.5 * (eps + (2.0 * x)))));
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[(2.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.2%

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

   1. diff-sin 39.8%

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

   2. div-inv 39.8%

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

   3. metadata-eval 39.8%

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

   4. div-inv 39.8%

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

   5. +-commutative 39.8%

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

   6. metadata-eval 39.8%

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

3. Applied egg-rr 39.8%

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

   1. associate-*r* 39.8%

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

   2. *-commutative 39.8%

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

   3. associate-*l* 39.8%

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

   4. +-commutative 39.8%

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

   5. associate--l+ 75.7%

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

   6. +-inverses 75.7%

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

   7. *-commutative 75.7%

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

   8. associate-+r+ 75.5%

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

   9. +-commutative 75.5%

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

5. Simplified 75.5%

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

6. Taylor expanded in eps around inf 75.5%

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

7. Final simplification 75.5%

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

Alternative 9: 76.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.2%

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

   1. expm1-log1p-u 40.2%

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

3. Applied egg-rr 40.2%

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

   1. expm1-log1p-u 40.2%

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

   2. diff-sin 39.8%

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

   3. +-commutative 39.8%

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

   4. +-commutative 39.8%

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

5. Applied egg-rr 39.8%

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

   1. associate-*r* 39.8%

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

   2. *-commutative 39.8%

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

   3. +-commutative 39.8%

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

   4. +-commutative 39.8%

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

   5. associate--l+ 75.7%

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

   6. +-inverses 75.7%

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

   7. +-rgt-identity 75.7%

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

7. Simplified 75.7%

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

8. Final simplification 75.7%

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

Alternative 10: 76.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.5 \cdot 10^{-5}:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 6.9 \cdot 10^{-18}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -4.5e-5)
   (sin eps)
   (if (<= eps 6.9e-18) (* eps (cos x)) (sin eps))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -4.5e-5) {
		tmp = sin(eps);
	} else if (eps <= 6.9e-18) {
		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 <= (-4.5d-5)) then
        tmp = sin(eps)
    else if (eps <= 6.9d-18) 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 <= -4.5e-5) {
		tmp = Math.sin(eps);
	} else if (eps <= 6.9e-18) {
		tmp = eps * Math.cos(x);
	} else {
		tmp = Math.sin(eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -4.5e-5:
		tmp = math.sin(eps)
	elif eps <= 6.9e-18:
		tmp = eps * math.cos(x)
	else:
		tmp = math.sin(eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -4.5e-5)
		tmp = sin(eps);
	elseif (eps <= 6.9e-18)
		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 <= -4.5e-5)
		tmp = sin(eps);
	elseif (eps <= 6.9e-18)
		tmp = eps * cos(x);
	else
		tmp = sin(eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -4.5e-5], N[Sin[eps], $MachinePrecision], If[LessEqual[eps, 6.9e-18], N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision], N[Sin[eps], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < -4.50000000000000028e-5 or 6.9000000000000003e-18 < eps

   1. Initial program 51.3%

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

   2. Taylor expanded in x around 0 51.3%

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

   if -4.50000000000000028e-5 < eps < 6.9000000000000003e-18

   1. Initial program 29.2%

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

   2. Taylor expanded in eps around 0 99.4%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.5 \cdot 10^{-5}:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 6.9 \cdot 10^{-18}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 11: 54.9% 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 40.2%

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

2. Taylor expanded in x around 0 51.4%

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

3. Final simplification 51.4%

   \[\leadsto \sin \varepsilon \]

Alternative 12: 4.3% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

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

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 40.2%

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

   1. add-cube-cbrt 40.0%

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

   2. pow3 39.9%

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

3. Applied egg-rr 39.9%

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

4. Taylor expanded in eps around 0 4.3%

   \[\leadsto \color{blue}{\sin x - {1}^{0.3333333333333333} \cdot \sin x} \]
5. Step-by-step derivation

   1. pow-base-1 4.3%

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

   2. *-lft-identity 4.3%

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

   3. +-inverses 4.3%

      \[\leadsto \color{blue}{0} \]

6. Simplified 4.3%

   \[\leadsto \color{blue}{0} \]

7. Final simplification 4.3%

   \[\leadsto 0 \]

Alternative 13: 28.9% 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 40.2%

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

2. Taylor expanded in eps around 0 52.5%

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

   1. fma-def 52.5%

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

   2. *-commutative 52.5%

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

   3. unpow2 52.5%

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

4. Simplified 52.5%

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

5. Taylor expanded in x around 0 28.3%

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

6. Final simplification 28.3%

   \[\leadsto \varepsilon \]

Developer target: 76.7% 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 2023187 
(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)))