2sin (example 3.3)

Percentage Accurate: 42.6% → 99.7%

Time: 15.9s

Alternatives: 8

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: 42.6% 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: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 45.1%

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

   1. sin-sum 67.3%

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

   2. associate--l+ 67.3%

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

3. Applied egg-rr 67.3%

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

4. Taylor expanded in x around inf 67.3%

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

   1. associate--l+ 67.3%

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

   2. *-commutative 67.3%

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

   3. +-commutative 67.3%

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

   4. associate-+l- 99.5%

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

   5. *-rgt-identity 99.5%

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

   6. distribute-lft-out-- 99.5%

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

6. Simplified 99.5%

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

   1. flip-- 99.3%

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

   2. associate-*r/ 99.4%

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

   3. metadata-eval 99.4%

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

   4. 1-sub-cos 99.6%

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

   5. pow2 99.6%

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

8. Applied egg-rr 99.6%

   \[\leadsto \cos x \cdot \sin \varepsilon - \color{blue}{\frac{\sin x \cdot {\sin \varepsilon}^{2}}{1 + \cos \varepsilon}} \]
9. Step-by-step derivation

   1. *-commutative 99.6%

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

   2. associate-/l* 99.6%

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

10. Simplified 99.6%

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

11. Taylor expanded in eps around inf 99.6%

    \[\leadsto \cos x \cdot \sin \varepsilon - \color{blue}{\frac{{\sin \varepsilon}^{2} \cdot \sin x}{1 + \cos \varepsilon}} \]
12. Step-by-step derivation

    1. unpow2 99.6%

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

    2. associate-*r* 99.6%

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

    3. /-rgt-identity 99.6%

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

    4. associate-/r/ 99.6%

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

    5. associate-*l/ 99.6%

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

    6. associate-/r/ 99.6%

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

    7. /-rgt-identity 99.6%

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

    8. hang-0p-tan 99.8%

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

13. Simplified 99.8%

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

14. Final simplification 99.8%

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

Alternative 2: 76.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(x + \varepsilon\right) - \sin x\\ \mathbf{if}\;t_0 \leq -0.1 \lor \neg \left(t_0 \leq 0.04\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\varepsilon \cdot 0.5\right) \cdot \left(\cos x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (sin (+ x eps)) (sin x))))
   (if (or (<= t_0 -0.1) (not (<= t_0 0.04)))
     t_0
     (* (sin (* eps 0.5)) (* (cos x) 2.0)))))

C

double code(double x, double eps) {
	double t_0 = sin((x + eps)) - sin(x);
	double tmp;
	if ((t_0 <= -0.1) || !(t_0 <= 0.04)) {
		tmp = t_0;
	} else {
		tmp = sin((eps * 0.5)) * (cos(x) * 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((x + eps)) - sin(x)
    if ((t_0 <= (-0.1d0)) .or. (.not. (t_0 <= 0.04d0))) then
        tmp = t_0
    else
        tmp = sin((eps * 0.5d0)) * (cos(x) * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin((x + eps)) - Math.sin(x);
	double tmp;
	if ((t_0 <= -0.1) || !(t_0 <= 0.04)) {
		tmp = t_0;
	} else {
		tmp = Math.sin((eps * 0.5)) * (Math.cos(x) * 2.0);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.sin((x + eps)) - math.sin(x)
	tmp = 0
	if (t_0 <= -0.1) or not (t_0 <= 0.04):
		tmp = t_0
	else:
		tmp = math.sin((eps * 0.5)) * (math.cos(x) * 2.0)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(sin(Float64(x + eps)) - sin(x))
	tmp = 0.0
	if ((t_0 <= -0.1) || !(t_0 <= 0.04))
		tmp = t_0;
	else
		tmp = Float64(sin(Float64(eps * 0.5)) * Float64(cos(x) * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = sin((x + eps)) - sin(x);
	tmp = 0.0;
	if ((t_0 <= -0.1) || ~((t_0 <= 0.04)))
		tmp = t_0;
	else
		tmp = sin((eps * 0.5)) * (cos(x) * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Sin[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -0.1], N[Not[LessEqual[t$95$0, 0.04]], $MachinePrecision]], t$95$0, N[(N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x)) < -0.10000000000000001 or 0.0400000000000000008 < (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x))

   1. Initial program 81.2%

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

   if -0.10000000000000001 < (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x)) < 0.0400000000000000008

   1. Initial program 24.6%

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

      1. diff-sin 24.4%

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

      2. div-inv 24.4%

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

      3. metadata-eval 24.4%

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

      4. div-inv 24.4%

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

      5. +-commutative 24.4%

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

      6. metadata-eval 24.4%

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

   3. Applied egg-rr 24.4%

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

      1. associate-*r* 24.4%

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

      2. *-commutative 24.4%

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

      3. associate-*l* 24.4%

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

      4. +-commutative 24.4%

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

      5. associate--l+ 76.7%

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

      6. +-inverses 76.7%

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

      7. *-commutative 76.7%

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

      8. associate-+r+ 76.7%

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

      9. +-commutative 76.7%

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

   5. Simplified 76.7%

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

   6. Taylor expanded in eps around 0 77.3%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin \left(x + \varepsilon\right) - \sin x \leq -0.1 \lor \neg \left(\sin \left(x + \varepsilon\right) - \sin x \leq 0.04\right):\\ \;\;\;\;\sin \left(x + \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\varepsilon \cdot 0.5\right) \cdot \left(\cos x \cdot 2\right)\\ \end{array} \]

Alternative 3: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 45.1%

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

   1. sin-sum 67.3%

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

   2. associate--l+ 67.3%

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

3. Applied egg-rr 67.3%

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

4. Taylor expanded in x around inf 67.3%

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

   1. associate--l+ 67.3%

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

   2. *-commutative 67.3%

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

   3. +-commutative 67.3%

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

   4. associate-+l- 99.5%

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

   5. *-rgt-identity 99.5%

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

   6. distribute-lft-out-- 99.5%

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

6. Simplified 99.5%

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

   1. sub-neg 99.5%

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

   2. distribute-lft-in 99.5%

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

   3. *-commutative 99.5%

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

   4. *-un-lft-identity 99.5%

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

8. Applied egg-rr 99.5%

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

9. Final simplification 99.5%

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

Alternative 4: 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 45.1%

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

   1. sin-sum 67.3%

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

   2. associate--l+ 67.3%

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

3. Applied egg-rr 67.3%

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

4. Taylor expanded in x around inf 67.3%

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

   1. associate--l+ 67.3%

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

   2. *-commutative 67.3%

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

   3. +-commutative 67.3%

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

   4. associate-+l- 99.5%

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

   5. *-rgt-identity 99.5%

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

   6. distribute-lft-out-- 99.5%

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

6. Simplified 99.5%

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

7. Final simplification 99.5%

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

Alternative 5: 77.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 45.1%

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

   1. diff-sin 44.8%

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

   2. div-inv 44.8%

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

   3. metadata-eval 44.8%

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

   4. div-inv 44.8%

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

   5. +-commutative 44.8%

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

   6. metadata-eval 44.8%

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

3. Applied egg-rr 44.8%

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

   1. associate-*r* 44.8%

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

   2. *-commutative 44.8%

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

   3. associate-*l* 44.8%

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

   4. +-commutative 44.8%

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

   5. associate--l+ 78.1%

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

   6. +-inverses 78.1%

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

   7. *-commutative 78.1%

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

   8. associate-+r+ 78.0%

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

   9. +-commutative 78.0%

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

5. Simplified 78.0%

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

6. Final simplification 78.0%

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

Alternative 6: 77.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -7 \cdot 10^{-5}:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.038:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -7e-5) (sin eps) (if (<= eps 0.038) (* (cos x) eps) (sin eps))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -7e-5) {
		tmp = sin(eps);
	} else if (eps <= 0.038) {
		tmp = cos(x) * eps;
	} 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 <= (-7d-5)) then
        tmp = sin(eps)
    else if (eps <= 0.038d0) then
        tmp = cos(x) * eps
    else
        tmp = sin(eps)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -7e-5) {
		tmp = Math.sin(eps);
	} else if (eps <= 0.038) {
		tmp = Math.cos(x) * eps;
	} else {
		tmp = Math.sin(eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -7e-5:
		tmp = math.sin(eps)
	elif eps <= 0.038:
		tmp = math.cos(x) * eps
	else:
		tmp = math.sin(eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -7e-5)
		tmp = sin(eps);
	elseif (eps <= 0.038)
		tmp = Float64(cos(x) * eps);
	else
		tmp = sin(eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -7e-5)
		tmp = sin(eps);
	elseif (eps <= 0.038)
		tmp = cos(x) * eps;
	else
		tmp = sin(eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -7e-5], N[Sin[eps], $MachinePrecision], If[LessEqual[eps, 0.038], N[(N[Cos[x], $MachinePrecision] * eps), $MachinePrecision], N[Sin[eps], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < -6.9999999999999994e-5 or 0.0379999999999999991 < eps

   1. Initial program 58.4%

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

   2. Taylor expanded in x around 0 59.9%

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

   if -6.9999999999999994e-5 < eps < 0.0379999999999999991

   1. Initial program 31.0%

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

   2. Taylor expanded in eps around 0 98.4%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -7 \cdot 10^{-5}:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.038:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 7: 55.1% 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 45.1%

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

2. Taylor expanded in x around 0 60.2%

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

3. Final simplification 60.2%

   \[\leadsto \sin \varepsilon \]

Alternative 8: 29.2% 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 45.1%

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

   1. diff-sin 44.8%

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

   2. div-inv 44.8%

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

   3. metadata-eval 44.8%

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

   4. div-inv 44.8%

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

   5. +-commutative 44.8%

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

   6. metadata-eval 44.8%

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

3. Applied egg-rr 44.8%

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

   1. associate-*r* 44.8%

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

   2. *-commutative 44.8%

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

   3. associate-*l* 44.8%

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

   4. +-commutative 44.8%

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

   5. associate--l+ 78.1%

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

   6. +-inverses 78.1%

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

   7. *-commutative 78.1%

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

   8. associate-+r+ 78.0%

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

   9. +-commutative 78.0%

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

5. Simplified 78.0%

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

6. Taylor expanded in eps around 0 49.5%

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

   1. *-commutative 49.5%

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

8. Simplified 49.5%

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

9. Taylor expanded in x around 0 31.1%

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

10. Final simplification 31.1%

    \[\leadsto \varepsilon \]

Developer target: 77.2% 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 2023285 
(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)))