2sin (example 3.3)

Percentage Accurate: 41.9% → 99.7%

Time: 10.2s

Alternatives: 6

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 41.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 44.8%

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

   1. sin-sum 65.6%

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

   2. associate--l+ 65.6%

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

4. Applied egg-rr 65.6%

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

   1. +-commutative 65.6%

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

   2. associate-+l- 99.4%

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

   3. *-commutative 99.4%

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

   4. *-rgt-identity 99.4%

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

   5. distribute-lft-out-- 99.4%

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

6. Simplified 99.4%

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

   1. flip-- 99.2%

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

   2. metadata-eval 99.2%

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

   3. 1-sub-cos 99.6%

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

   4. pow2 99.6%

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

8. Applied egg-rr 99.6%

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

9. Taylor expanded in x around inf 99.6%

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

    1. *-lft-identity 99.6%

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

    2. *-commutative 99.6%

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

    3. times-frac 99.6%

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

    4. /-rgt-identity 99.6%

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

    5. unpow2 99.6%

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

    6. associate-*r/ 99.6%

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

    7. hang-0p-tan 99.7%

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

11. Simplified 99.7%

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

    1. associate-*r* 99.7%

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

    2. cancel-sign-sub-inv 99.7%

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

    3. distribute-lft-neg-in 99.7%

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

    4. distribute-rgt-neg-in 99.7%

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

    5. div-inv 99.7%

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

    6. metadata-eval 99.7%

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

13. Applied egg-rr 99.7%

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

    1. distribute-rgt-neg-out 99.7%

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

    2. unsub-neg 99.7%

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

    3. *-commutative 99.7%

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

    4. associate-*r* 99.7%

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

    5. *-commutative 99.7%

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

    6. distribute-lft-out-- 99.7%

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

    7. *-commutative 99.7%

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

15. Simplified 99.7%

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

16. Final simplification 99.7%

    \[\leadsto \sin \varepsilon \cdot \left(\cos x - \sin x \cdot \tan \left(\varepsilon \cdot 0.5\right)\right) \]
17. Add Preprocessing

Alternative 2: 77.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.023 \lor \neg \left(\varepsilon \leq 1.05 \cdot 10^{-5}\right):\\ \;\;\;\;\sin \varepsilon - \sin x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.023) (not (<= eps 1.05e-5)))
   (- (sin eps) (sin x))
   (* eps (cos x))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.023) || !(eps <= 1.05e-5)) {
		tmp = sin(eps) - sin(x);
	} 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) :: tmp
    if ((eps <= (-0.023d0)) .or. (.not. (eps <= 1.05d-5))) then
        tmp = sin(eps) - sin(x)
    else
        tmp = eps * cos(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.023) || !(eps <= 1.05e-5)) {
		tmp = Math.sin(eps) - Math.sin(x);
	} else {
		tmp = eps * Math.cos(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.023) or not (eps <= 1.05e-5):
		tmp = math.sin(eps) - math.sin(x)
	else:
		tmp = eps * math.cos(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.023) || !(eps <= 1.05e-5))
		tmp = Float64(sin(eps) - sin(x));
	else
		tmp = Float64(eps * cos(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.023) || ~((eps <= 1.05e-5)))
		tmp = sin(eps) - sin(x);
	else
		tmp = eps * cos(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.023], N[Not[LessEqual[eps, 1.05e-5]], $MachinePrecision]], N[(N[Sin[eps], $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision], N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -0.023 or 1.04999999999999994e-5 < eps

   1. Initial program 61.0%

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

      1. add-cbrt-cube 60.9%

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

      2. pow3 60.8%

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

   4. Applied egg-rr 60.8%

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

   5. Taylor expanded in x around 0 61.8%

      \[\leadsto \sqrt[3]{\color{blue}{{\sin \varepsilon}^{3}}} - \sin x \]

   6. Taylor expanded in eps around inf 62.2%

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

   if -0.023 < eps < 1.04999999999999994e-5

   1. Initial program 27.6%

      \[\sin \left(x + \varepsilon\right) - \sin x \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0 98.6%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.023 \lor \neg \left(\varepsilon \leq 1.05 \cdot 10^{-5}\right):\\ \;\;\;\;\sin \varepsilon - \sin x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 44.8%

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

   1. add-cbrt-cube 39.5%

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

   2. pow3 39.6%

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

4. Applied egg-rr 39.6%

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

   1. rem-cbrt-cube 44.8%

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

   2. diff-sin 44.4%

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

   3. +-commutative 44.4%

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

   4. +-commutative 44.4%

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

6. Applied egg-rr 44.4%

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

   1. associate-*r* 44.4%

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

   2. associate--l+ 79.3%

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

   3. +-inverses 79.3%

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

   4. associate-+l+ 79.3%

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

   5. count-2 79.3%

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

8. Simplified 79.3%

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

9. Final simplification 79.3%

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

Alternative 4: 76.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.023 \lor \neg \left(\varepsilon \leq 4.8 \cdot 10^{-5}\right):\\ \;\;\;\;\sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.023) (not (<= eps 4.8e-5))) (sin eps) (* eps (cos x))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.023) || !(eps <= 4.8e-5)) {
		tmp = sin(eps);
	} 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) :: tmp
    if ((eps <= (-0.023d0)) .or. (.not. (eps <= 4.8d-5))) then
        tmp = sin(eps)
    else
        tmp = eps * cos(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.023) || !(eps <= 4.8e-5)) {
		tmp = Math.sin(eps);
	} else {
		tmp = eps * Math.cos(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.023) or not (eps <= 4.8e-5):
		tmp = math.sin(eps)
	else:
		tmp = eps * math.cos(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.023) || !(eps <= 4.8e-5))
		tmp = sin(eps);
	else
		tmp = Float64(eps * cos(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.023) || ~((eps <= 4.8e-5)))
		tmp = sin(eps);
	else
		tmp = eps * cos(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.023], N[Not[LessEqual[eps, 4.8e-5]], $MachinePrecision]], N[Sin[eps], $MachinePrecision], N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -0.023 or 4.8000000000000001e-5 < eps

   1. Initial program 61.0%

      \[\sin \left(x + \varepsilon\right) - \sin x \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 61.0%

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

   if -0.023 < eps < 4.8000000000000001e-5

   1. Initial program 27.6%

      \[\sin \left(x + \varepsilon\right) - \sin x \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0 98.6%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.023 \lor \neg \left(\varepsilon \leq 4.8 \cdot 10^{-5}\right):\\ \;\;\;\;\sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 55.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 44.8%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
2. Add Preprocessing

3. Taylor expanded in x around 0 58.5%

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

4. Final simplification 58.5%

   \[\leadsto \sin \varepsilon \]
5. Add Preprocessing

Alternative 6: 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 44.8%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0 49.6%

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

4. Taylor expanded in x around 0 28.8%

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

5. Final simplification 28.8%

   \[\leadsto \varepsilon \]
6. Add Preprocessing

Developer target: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2024020 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64

  :herbie-target
  (fma (sin x) (- (cos eps) 1.0) (* (sin eps) (cos x)))

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