2sin (example 3.3)

Percentage Accurate: 41.9% → 99.6%

Time: 14.9s

Alternatives: 9

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: 99.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 44.8%

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

   1. sin-sum 71.0%

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

   2. associate--l+ 71.0%

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

3. Applied egg-rr 71.0%

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

   1. +-commutative 71.0%

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

   2. associate-+l- 99.3%

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

   3. *-commutative 99.3%

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

   4. *-rgt-identity 99.3%

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

   5. distribute-lft-out-- 99.4%

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

5. Simplified 99.4%

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

   1. flip-- 99.3%

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

   2. div-inv 99.3%

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

   3. metadata-eval 99.3%

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

   4. 1-sub-cos 99.6%

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

   5. pow2 99.6%

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

7. Applied egg-rr 99.6%

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

   1. associate-*r/ 99.6%

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

   2. *-rgt-identity 99.6%

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

9. Simplified 99.6%

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

10. Taylor expanded in x around inf 99.6%

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

    1. *-commutative 99.6%

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

    2. *-lft-identity 99.6%

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

    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%

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

12. Simplified 99.7%

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

    1. sub-neg 99.7%

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

    2. *-commutative 99.7%

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

    3. distribute-rgt-neg-in 99.7%

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

    4. div-inv 99.7%

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

    5. metadata-eval 99.7%

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

14. Applied egg-rr 99.7%

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

    1. associate-*l* 99.7%

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

    2. distribute-lft-out 99.7%

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

16. Simplified 99.7%

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

17. Final simplification 99.7%

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

Alternative 2: 76.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\varepsilon + x\right)\\ t_1 := t_0 - \sin x\\ \mathbf{if}\;t_1 \leq -0.005:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\cos x \cdot \left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|t_0\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (+ eps x))) (t_1 (- t_0 (sin x))))
   (if (<= t_1 -0.005)
     (sin eps)
     (if (<= t_1 2e-19) (* (cos x) (* 2.0 (sin (* eps 0.5)))) (fabs t_0)))))

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 <= -0.005) {
		tmp = sin(eps);
	} else if (t_1 <= 2e-19) {
		tmp = cos(x) * (2.0 * sin((eps * 0.5)));
	} else {
		tmp = fabs(t_0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sin((eps + x))
    t_1 = t_0 - sin(x)
    if (t_1 <= (-0.005d0)) then
        tmp = sin(eps)
    else if (t_1 <= 2d-19) then
        tmp = cos(x) * (2.0d0 * sin((eps * 0.5d0)))
    else
        tmp = abs(t_0)
    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 <= -0.005) {
		tmp = Math.sin(eps);
	} else if (t_1 <= 2e-19) {
		tmp = Math.cos(x) * (2.0 * Math.sin((eps * 0.5)));
	} else {
		tmp = Math.abs(t_0);
	}
	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 <= -0.005:
		tmp = math.sin(eps)
	elif t_1 <= 2e-19:
		tmp = math.cos(x) * (2.0 * math.sin((eps * 0.5)))
	else:
		tmp = math.fabs(t_0)
	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 <= -0.005)
		tmp = sin(eps);
	elseif (t_1 <= 2e-19)
		tmp = Float64(cos(x) * Float64(2.0 * sin(Float64(eps * 0.5))));
	else
		tmp = abs(t_0);
	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 <= -0.005)
		tmp = sin(eps);
	elseif (t_1 <= 2e-19)
		tmp = cos(x) * (2.0 * sin((eps * 0.5)));
	else
		tmp = abs(t_0);
	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[LessEqual[t$95$1, -0.005], N[Sin[eps], $MachinePrecision], If[LessEqual[t$95$1, 2e-19], N[(N[Cos[x], $MachinePrecision] * N[(2.0 * N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Abs[t$95$0], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x)) < -0.0050000000000000001

   1. Initial program 63.1%

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

   2. Taylor expanded in x around 0 63.7%

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

   if -0.0050000000000000001 < (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x)) < 2e-19

   1. Initial program 29.6%

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

      1. diff-sin 29.6%

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

      2. div-inv 29.6%

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

      3. associate--l+ 29.6%

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

      4. metadata-eval 29.6%

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

      5. div-inv 29.6%

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

      6. +-commutative 29.6%

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

      7. associate-+l+ 29.6%

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

      8. metadata-eval 29.6%

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

   3. Applied egg-rr 29.6%

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

      1. associate-*r* 29.6%

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

      2. *-commutative 29.6%

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

      3. *-commutative 29.6%

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

      4. +-commutative 29.6%

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

      5. count-2 29.6%

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

      6. fma-def 29.6%

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

      7. sub-neg 29.6%

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

      8. mul-1-neg 29.6%

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

      9. +-commutative 29.6%

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

      10. associate-+r+ 81.2%

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

      11. mul-1-neg 81.2%

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

      12. sub-neg 81.2%

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

      13. +-inverses 81.2%

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

      14. remove-double-neg 81.2%

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

      15. mul-1-neg 81.2%

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

      16. sub-neg 81.2%

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

      17. neg-sub0 81.2%

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

      18. mul-1-neg 81.2%

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

      19. remove-double-neg 81.2%

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

   5. Simplified 81.2%

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

   6. Taylor expanded in eps around 0 81.2%

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

   if 2e-19 < (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x))

   1. Initial program 67.7%

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

      1. add-cube-cbrt 66.2%

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

      2. pow3 66.2%

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

   3. Applied egg-rr 66.2%

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

      1. rem-cube-cbrt 67.7%

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

      2. add-sqr-sqrt 65.1%

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

      3. sqrt-unprod 67.8%

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

      4. pow2 67.8%

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

      5. +-commutative 67.8%

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

   5. Applied egg-rr 67.8%

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

      1. unpow2 67.8%

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

      2. rem-sqrt-square 67.8%

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

   7. Simplified 67.8%

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

   8. Taylor expanded in x around 0 67.8%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin \left(\varepsilon + x\right) - \sin x \leq -0.005:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\sin \left(\varepsilon + x\right) - \sin x \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\cos x \cdot \left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \left(\varepsilon + x\right)\right|\\ \end{array} \]

Alternative 3: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 44.8%

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

   1. sin-sum 71.0%

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

   2. associate--l+ 71.0%

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

3. Applied egg-rr 71.0%

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

   1. +-commutative 71.0%

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

   2. associate-+l- 99.3%

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

   3. *-commutative 99.3%

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

   4. *-rgt-identity 99.3%

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

   5. distribute-lft-out-- 99.4%

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

5. Simplified 99.4%

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

6. Final simplification 99.4%

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

Alternative 4: 76.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 44.8%

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

   1. add-sqr-sqrt 22.2%

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

   2. sqrt-unprod 22.0%

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

   3. pow2 22.0%

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

3. Applied egg-rr 22.0%

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

   1. sqrt-pow1 44.8%

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

   2. metadata-eval 44.8%

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

   3. pow1 44.8%

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

   4. diff-sin 44.2%

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

   5. div-inv 44.2%

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

   6. +-commutative 44.2%

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

   7. metadata-eval 44.2%

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

   8. div-inv 44.2%

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

   9. +-commutative 44.2%

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

   10. metadata-eval 44.2%

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

5. Applied egg-rr 44.2%

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

   1. *-commutative 44.2%

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

   2. associate--l+ 74.2%

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

   3. *-commutative 74.2%

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

   4. +-commutative 74.2%

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

7. Simplified 74.2%

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

8. Final simplification 74.2%

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

Alternative 5: 76.0% 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. Step-by-step derivation

   1. add-cube-cbrt 43.6%

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

   2. pow3 43.5%

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

3. Applied egg-rr 43.5%

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

   1. rem-cube-cbrt 44.8%

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

   2. diff-sin 44.2%

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

   3. +-commutative 44.2%

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

   4. +-commutative 44.2%

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

5. Applied egg-rr 44.2%

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

   1. associate-*r* 44.2%

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

   2. associate--l+ 74.2%

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

   3. +-inverses 74.2%

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

   4. associate-+l+ 74.1%

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

   5. count-2 74.1%

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

7. Simplified 74.1%

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

8. Final simplification 74.1%

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

Alternative 6: 75.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.76 \lor \neg \left(\varepsilon \leq 4.5 \cdot 10^{-6}\right):\\ \;\;\;\;\sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.76) (not (<= eps 4.5e-6))) (sin eps) (* eps (cos x))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.76) || !(eps <= 4.5e-6)) {
		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 <= (-0.76d0)) .or. (.not. (eps <= 4.5d-6))) 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 <= -0.76) || !(eps <= 4.5e-6)) {
		tmp = Math.sin(eps);
	} else {
		tmp = eps * Math.cos(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.76) or not (eps <= 4.5e-6):
		tmp = math.sin(eps)
	else:
		tmp = eps * math.cos(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.76) || !(eps <= 4.5e-6))
		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 <= -0.76) || ~((eps <= 4.5e-6)))
		tmp = sin(eps);
	else
		tmp = eps * cos(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.76], N[Not[LessEqual[eps, 4.5e-6]], $MachinePrecision]], N[Sin[eps], $MachinePrecision], N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -0.76000000000000001 or 4.50000000000000011e-6 < eps

   1. Initial program 52.3%

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

   2. Taylor expanded in x around 0 53.2%

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

   if -0.76000000000000001 < eps < 4.50000000000000011e-6

   1. Initial program 36.0%

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

   2. Taylor expanded in eps around 0 98.5%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.76 \lor \neg \left(\varepsilon \leq 4.5 \cdot 10^{-6}\right):\\ \;\;\;\;\sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \]

Alternative 7: 54.8% 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. Taylor expanded in x around 0 55.6%

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

3. Final simplification 55.6%

   \[\leadsto \sin \varepsilon \]

Alternative 8: 4.2% 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 44.8%

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

   1. add-cube-cbrt 43.6%

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

   2. pow3 43.5%

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

3. Applied egg-rr 43.5%

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

4. Taylor expanded in eps around 0 4.1%

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

   1. pow-base-1 4.1%

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

   2. *-lft-identity 4.1%

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

   3. +-inverses 4.1%

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

6. Simplified 4.1%

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

7. Final simplification 4.1%

   \[\leadsto 0 \]

Alternative 9: 29.0% 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. Taylor expanded in eps around 0 47.5%

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

3. Taylor expanded in x around 0 29.0%

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

4. Final simplification 29.0%

   \[\leadsto \varepsilon \]

Developer target: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2023301 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64

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

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