2sin (example 3.3)

Percentage Accurate: 61.5% → 98.5%

Time: 20.5s

Alternatives: 16

Speedup: 1.8×

Specification

\[\left(\left(\left|x\right| > \left|\varepsilon\right| \land \left|\varepsilon\right| > {10}^{-16} \cdot \left|x\right|\right) \land 1 < x\right) \land x < 1000\]

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: 61.5% 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: 98.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \sin \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.0013:\\ \;\;\;\;\mathsf{fma}\left(\sin x, \cos \varepsilon, t_0 - \sin x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00162:\\ \;\;\;\;\left(2 \cdot \sin \left(\frac{\varepsilon + \left(x - x\right)}{2}\right)\right) \cdot \left(\mathsf{fma}\left(0.020833333333333332, \sin x \cdot {\varepsilon}^{3}, \cos x \cdot \left(-0.125 \cdot {\varepsilon}^{2}\right)\right) + \mathsf{fma}\left(\sin x, \varepsilon \cdot -0.5, \cos x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 - \sin x \cdot \left(1 - \cos \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (cos x) (sin eps))))
   (if (<= eps -0.0013)
     (fma (sin x) (cos eps) (- t_0 (sin x)))
     (if (<= eps 0.00162)
       (*
        (* 2.0 (sin (/ (+ eps (- x x)) 2.0)))
        (+
         (fma
          0.020833333333333332
          (* (sin x) (pow eps 3.0))
          (* (cos x) (* -0.125 (pow eps 2.0))))
         (fma (sin x) (* eps -0.5) (cos x))))
       (- t_0 (* (sin x) (- 1.0 (cos eps))))))))

C

double code(double x, double eps) {
	double t_0 = cos(x) * sin(eps);
	double tmp;
	if (eps <= -0.0013) {
		tmp = fma(sin(x), cos(eps), (t_0 - sin(x)));
	} else if (eps <= 0.00162) {
		tmp = (2.0 * sin(((eps + (x - x)) / 2.0))) * (fma(0.020833333333333332, (sin(x) * pow(eps, 3.0)), (cos(x) * (-0.125 * pow(eps, 2.0)))) + fma(sin(x), (eps * -0.5), cos(x)));
	} else {
		tmp = t_0 - (sin(x) * (1.0 - cos(eps)));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(cos(x) * sin(eps))
	tmp = 0.0
	if (eps <= -0.0013)
		tmp = fma(sin(x), cos(eps), Float64(t_0 - sin(x)));
	elseif (eps <= 0.00162)
		tmp = Float64(Float64(2.0 * sin(Float64(Float64(eps + Float64(x - x)) / 2.0))) * Float64(fma(0.020833333333333332, Float64(sin(x) * (eps ^ 3.0)), Float64(cos(x) * Float64(-0.125 * (eps ^ 2.0)))) + fma(sin(x), Float64(eps * -0.5), cos(x))));
	else
		tmp = Float64(t_0 - Float64(sin(x) * Float64(1.0 - cos(eps))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.0013], N[(N[Sin[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + N[(t$95$0 - N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.00162], N[(N[(2.0 * N[Sin[N[(N[(eps + N[(x - x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(0.020833333333333332 * N[(N[Sin[x], $MachinePrecision] * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(-0.125 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(eps * -0.5), $MachinePrecision] + N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(N[Sin[x], $MachinePrecision] * N[(1.0 - N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.0012999999999999999

   1. Initial program 91.1%

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

      1. sin-sum 96.6%

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

      2. associate--l+ 96.4%

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

      3. fma-def 97.1%

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

   3. Applied egg-rr 97.1%

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

   if -0.0012999999999999999 < eps < 0.0016199999999999999

   1. Initial program 46.8%

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

      1. add-cube-cbrt 45.7%

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

      2. pow3 45.6%

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

   3. Applied egg-rr 45.6%

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

      1. rem-cube-cbrt 46.8%

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

      2. diff-sin 47.5%

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

   5. Applied egg-rr 47.5%

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

      1. associate-*r* 47.5%

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

      2. +-commutative 47.5%

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

      3. associate--l+ 94.1%

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

      4. +-commutative 94.1%

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

      5. associate-+l+ 94.3%

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

   7. Simplified 94.3%

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

   8. Taylor expanded in eps around 0 99.3%

      \[\leadsto \left(2 \cdot \sin \left(\frac{\varepsilon + \left(x - x\right)}{2}\right)\right) \cdot \color{blue}{\left(\cos x + \left(-0.5 \cdot \left(\varepsilon \cdot \sin x\right) + \left(-0.125 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + 0.020833333333333332 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. associate-+r+ 99.3%

         \[\leadsto \left(2 \cdot \sin \left(\frac{\varepsilon + \left(x - x\right)}{2}\right)\right) \cdot \color{blue}{\left(\left(\cos x + -0.5 \cdot \left(\varepsilon \cdot \sin x\right)\right) + \left(-0.125 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + 0.020833333333333332 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right)\right)} \]

      2. +-commutative 99.3%

         \[\leadsto \left(2 \cdot \sin \left(\frac{\varepsilon + \left(x - x\right)}{2}\right)\right) \cdot \left(\color{blue}{\left(-0.5 \cdot \left(\varepsilon \cdot \sin x\right) + \cos x\right)} + \left(-0.125 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + 0.020833333333333332 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right)\right) \]

      3. fma-udef 99.3%

         \[\leadsto \left(2 \cdot \sin \left(\frac{\varepsilon + \left(x - x\right)}{2}\right)\right) \cdot \left(\color{blue}{\mathsf{fma}\left(-0.5, \varepsilon \cdot \sin x, \cos x\right)} + \left(-0.125 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + 0.020833333333333332 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right)\right) \]

      4. +-commutative 99.3%

         \[\leadsto \left(2 \cdot \sin \left(\frac{\varepsilon + \left(x - x\right)}{2}\right)\right) \cdot \color{blue}{\left(\left(-0.125 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + 0.020833333333333332 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right) + \mathsf{fma}\left(-0.5, \varepsilon \cdot \sin x, \cos x\right)\right)} \]

      5. +-commutative 99.3%

         \[\leadsto \left(2 \cdot \sin \left(\frac{\varepsilon + \left(x - x\right)}{2}\right)\right) \cdot \left(\color{blue}{\left(0.020833333333333332 \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + -0.125 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right)} + \mathsf{fma}\left(-0.5, \varepsilon \cdot \sin x, \cos x\right)\right) \]

      6. fma-def 99.3%

         \[\leadsto \left(2 \cdot \sin \left(\frac{\varepsilon + \left(x - x\right)}{2}\right)\right) \cdot \left(\color{blue}{\mathsf{fma}\left(0.020833333333333332, {\varepsilon}^{3} \cdot \sin x, -0.125 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right)} + \mathsf{fma}\left(-0.5, \varepsilon \cdot \sin x, \cos x\right)\right) \]

      7. *-commutative 99.3%

         \[\leadsto \left(2 \cdot \sin \left(\frac{\varepsilon + \left(x - x\right)}{2}\right)\right) \cdot \left(\mathsf{fma}\left(0.020833333333333332, \color{blue}{\sin x \cdot {\varepsilon}^{3}}, -0.125 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right) + \mathsf{fma}\left(-0.5, \varepsilon \cdot \sin x, \cos x\right)\right) \]

      8. associate-*r* 99.3%

         \[\leadsto \left(2 \cdot \sin \left(\frac{\varepsilon + \left(x - x\right)}{2}\right)\right) \cdot \left(\mathsf{fma}\left(0.020833333333333332, \sin x \cdot {\varepsilon}^{3}, \color{blue}{\left(-0.125 \cdot {\varepsilon}^{2}\right) \cdot \cos x}\right) + \mathsf{fma}\left(-0.5, \varepsilon \cdot \sin x, \cos x\right)\right) \]

      9. fma-udef 99.3%

         \[\leadsto \left(2 \cdot \sin \left(\frac{\varepsilon + \left(x - x\right)}{2}\right)\right) \cdot \left(\mathsf{fma}\left(0.020833333333333332, \sin x \cdot {\varepsilon}^{3}, \left(-0.125 \cdot {\varepsilon}^{2}\right) \cdot \cos x\right) + \color{blue}{\left(-0.5 \cdot \left(\varepsilon \cdot \sin x\right) + \cos x\right)}\right) \]

      10. associate-*r* 99.3%

          \[\leadsto \left(2 \cdot \sin \left(\frac{\varepsilon + \left(x - x\right)}{2}\right)\right) \cdot \left(\mathsf{fma}\left(0.020833333333333332, \sin x \cdot {\varepsilon}^{3}, \left(-0.125 \cdot {\varepsilon}^{2}\right) \cdot \cos x\right) + \left(\color{blue}{\left(-0.5 \cdot \varepsilon\right) \cdot \sin x} + \cos x\right)\right) \]

      11. *-commutative 99.3%

          \[\leadsto \left(2 \cdot \sin \left(\frac{\varepsilon + \left(x - x\right)}{2}\right)\right) \cdot \left(\mathsf{fma}\left(0.020833333333333332, \sin x \cdot {\varepsilon}^{3}, \left(-0.125 \cdot {\varepsilon}^{2}\right) \cdot \cos x\right) + \left(\color{blue}{\left(\varepsilon \cdot -0.5\right)} \cdot \sin x + \cos x\right)\right) \]

      12. *-commutative 99.3%

          \[\leadsto \left(2 \cdot \sin \left(\frac{\varepsilon + \left(x - x\right)}{2}\right)\right) \cdot \left(\mathsf{fma}\left(0.020833333333333332, \sin x \cdot {\varepsilon}^{3}, \left(-0.125 \cdot {\varepsilon}^{2}\right) \cdot \cos x\right) + \left(\color{blue}{\sin x \cdot \left(\varepsilon \cdot -0.5\right)} + \cos x\right)\right) \]

      13. fma-def 99.3%

          \[\leadsto \left(2 \cdot \sin \left(\frac{\varepsilon + \left(x - x\right)}{2}\right)\right) \cdot \left(\mathsf{fma}\left(0.020833333333333332, \sin x \cdot {\varepsilon}^{3}, \left(-0.125 \cdot {\varepsilon}^{2}\right) \cdot \cos x\right) + \color{blue}{\mathsf{fma}\left(\sin x, \varepsilon \cdot -0.5, \cos x\right)}\right) \]

      14. *-commutative 99.3%

          \[\leadsto \left(2 \cdot \sin \left(\frac{\varepsilon + \left(x - x\right)}{2}\right)\right) \cdot \left(\mathsf{fma}\left(0.020833333333333332, \sin x \cdot {\varepsilon}^{3}, \left(-0.125 \cdot {\varepsilon}^{2}\right) \cdot \cos x\right) + \mathsf{fma}\left(\sin x, \color{blue}{-0.5 \cdot \varepsilon}, \cos x\right)\right) \]

   10. Simplified 99.3%

       \[\leadsto \left(2 \cdot \sin \left(\frac{\varepsilon + \left(x - x\right)}{2}\right)\right) \cdot \color{blue}{\left(\mathsf{fma}\left(0.020833333333333332, \sin x \cdot {\varepsilon}^{3}, \left(-0.125 \cdot {\varepsilon}^{2}\right) \cdot \cos x\right) + \mathsf{fma}\left(\sin x, -0.5 \cdot \varepsilon, \cos x\right)\right)} \]

   if 0.0016199999999999999 < eps

   1. Initial program 89.3%

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

      1. sin-sum 95.1%

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

      2. associate--l+ 95.0%

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

   3. Applied egg-rr 95.0%

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

      1. +-commutative 95.0%

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

      2. associate-+l- 95.1%

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

      3. *-commutative 95.1%

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

      4. *-rgt-identity 95.1%

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

      5. distribute-lft-out-- 95.6%

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

   5. Simplified 95.6%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0013:\\ \;\;\;\;\mathsf{fma}\left(\sin x, \cos \varepsilon, \cos x \cdot \sin \varepsilon - \sin x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00162:\\ \;\;\;\;\left(2 \cdot \sin \left(\frac{\varepsilon + \left(x - x\right)}{2}\right)\right) \cdot \left(\mathsf{fma}\left(0.020833333333333332, \sin x \cdot {\varepsilon}^{3}, \cos x \cdot \left(-0.125 \cdot {\varepsilon}^{2}\right)\right) + \mathsf{fma}\left(\sin x, \varepsilon \cdot -0.5, \cos x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \sin \varepsilon - \sin x \cdot \left(1 - \cos \varepsilon\right)\\ \end{array} \]

Alternative 2: 98.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \sin \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.0013:\\ \;\;\;\;\mathsf{fma}\left(\sin x, \cos \varepsilon, t_0 - \sin x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00162:\\ \;\;\;\;\left(\cos x + \left(-0.5 \cdot \left(\varepsilon \cdot \sin x\right) + \left(-0.125 \cdot \left(\cos x \cdot {\varepsilon}^{2}\right) + 0.020833333333333332 \cdot \left(\sin x \cdot {\varepsilon}^{3}\right)\right)\right)\right) \cdot \left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 - \sin x \cdot \left(1 - \cos \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (cos x) (sin eps))))
   (if (<= eps -0.0013)
     (fma (sin x) (cos eps) (- t_0 (sin x)))
     (if (<= eps 0.00162)
       (*
        (+
         (cos x)
         (+
          (* -0.5 (* eps (sin x)))
          (+
           (* -0.125 (* (cos x) (pow eps 2.0)))
           (* 0.020833333333333332 (* (sin x) (pow eps 3.0))))))
        (* 2.0 (sin (* eps 0.5))))
       (- t_0 (* (sin x) (- 1.0 (cos eps))))))))

C

double code(double x, double eps) {
	double t_0 = cos(x) * sin(eps);
	double tmp;
	if (eps <= -0.0013) {
		tmp = fma(sin(x), cos(eps), (t_0 - sin(x)));
	} else if (eps <= 0.00162) {
		tmp = (cos(x) + ((-0.5 * (eps * sin(x))) + ((-0.125 * (cos(x) * pow(eps, 2.0))) + (0.020833333333333332 * (sin(x) * pow(eps, 3.0)))))) * (2.0 * sin((eps * 0.5)));
	} else {
		tmp = t_0 - (sin(x) * (1.0 - cos(eps)));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(cos(x) * sin(eps))
	tmp = 0.0
	if (eps <= -0.0013)
		tmp = fma(sin(x), cos(eps), Float64(t_0 - sin(x)));
	elseif (eps <= 0.00162)
		tmp = Float64(Float64(cos(x) + Float64(Float64(-0.5 * Float64(eps * sin(x))) + Float64(Float64(-0.125 * Float64(cos(x) * (eps ^ 2.0))) + Float64(0.020833333333333332 * Float64(sin(x) * (eps ^ 3.0)))))) * Float64(2.0 * sin(Float64(eps * 0.5))));
	else
		tmp = Float64(t_0 - Float64(sin(x) * Float64(1.0 - cos(eps))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.0013], N[(N[Sin[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + N[(t$95$0 - N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.00162], N[(N[(N[Cos[x], $MachinePrecision] + N[(N[(-0.5 * N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.125 * N[(N[Cos[x], $MachinePrecision] * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.020833333333333332 * N[(N[Sin[x], $MachinePrecision] * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(N[Sin[x], $MachinePrecision] * N[(1.0 - N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.0012999999999999999

   1. Initial program 91.1%

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

      1. sin-sum 96.6%

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

      2. associate--l+ 96.4%

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

      3. fma-def 97.1%

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

   3. Applied egg-rr 97.1%

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

   if -0.0012999999999999999 < eps < 0.0016199999999999999

   1. Initial program 46.8%

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

      1. diff-sin 47.5%

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

      2. div-inv 47.5%

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

      3. associate--l+ 47.5%

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

      4. metadata-eval 47.5%

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

      5. div-inv 47.5%

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

      6. +-commutative 47.5%

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

      7. associate-+l+ 47.5%

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

      8. metadata-eval 47.5%

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

   3. Applied egg-rr 47.5%

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

      1. associate-*r* 47.5%

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

      2. *-commutative 47.5%

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

      3. *-commutative 47.5%

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

      4. +-commutative 47.5%

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

      5. count-2 47.5%

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

      6. fma-def 47.5%

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

      7. sub-neg 47.5%

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

      8. mul-1-neg 47.5%

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

      9. +-commutative 47.5%

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

      10. associate-+r+ 94.3%

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

      11. mul-1-neg 94.3%

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

      12. sub-neg 94.3%

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

      13. +-inverses 94.3%

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

      14. remove-double-neg 94.3%

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

      15. mul-1-neg 94.3%

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

      16. sub-neg 94.3%

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

      17. neg-sub0 94.3%

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

      18. mul-1-neg 94.3%

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

      19. remove-double-neg 94.3%

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

   5. Simplified 94.3%

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

   6. Taylor expanded in eps around 0 99.3%

      \[\leadsto \color{blue}{\left(\cos x + \left(-0.5 \cdot \left(\varepsilon \cdot \sin x\right) + \left(-0.125 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + 0.020833333333333332 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right)\right)\right)} \cdot \left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]

   if 0.0016199999999999999 < eps

   1. Initial program 89.3%

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

      1. sin-sum 95.1%

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

      2. associate--l+ 95.0%

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

   3. Applied egg-rr 95.0%

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

      1. +-commutative 95.0%

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

      2. associate-+l- 95.1%

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

      3. *-commutative 95.1%

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

      4. *-rgt-identity 95.1%

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

      5. distribute-lft-out-- 95.6%

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

   5. Simplified 95.6%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0013:\\ \;\;\;\;\mathsf{fma}\left(\sin x, \cos \varepsilon, \cos x \cdot \sin \varepsilon - \sin x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00162:\\ \;\;\;\;\left(\cos x + \left(-0.5 \cdot \left(\varepsilon \cdot \sin x\right) + \left(-0.125 \cdot \left(\cos x \cdot {\varepsilon}^{2}\right) + 0.020833333333333332 \cdot \left(\sin x \cdot {\varepsilon}^{3}\right)\right)\right)\right) \cdot \left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \sin \varepsilon - \sin x \cdot \left(1 - \cos \varepsilon\right)\\ \end{array} \]

Alternative 3: 98.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \sin \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.0013:\\ \;\;\;\;\mathsf{fma}\left(\sin x, \cos \varepsilon, t_0 - \sin x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00115:\\ \;\;\;\;-0.5 \cdot \left(\sin x \cdot {\varepsilon}^{2}\right) + \left(-0.16666666666666666 \cdot \left(\cos x \cdot {\varepsilon}^{3}\right) + \left(0.041666666666666664 \cdot \left(\sin x \cdot {\varepsilon}^{4}\right) + \varepsilon \cdot \cos x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 - \sin x \cdot \left(1 - \cos \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (cos x) (sin eps))))
   (if (<= eps -0.0013)
     (fma (sin x) (cos eps) (- t_0 (sin x)))
     (if (<= eps 0.00115)
       (+
        (* -0.5 (* (sin x) (pow eps 2.0)))
        (+
         (* -0.16666666666666666 (* (cos x) (pow eps 3.0)))
         (+
          (* 0.041666666666666664 (* (sin x) (pow eps 4.0)))
          (* eps (cos x)))))
       (- t_0 (* (sin x) (- 1.0 (cos eps))))))))

C

double code(double x, double eps) {
	double t_0 = cos(x) * sin(eps);
	double tmp;
	if (eps <= -0.0013) {
		tmp = fma(sin(x), cos(eps), (t_0 - sin(x)));
	} else if (eps <= 0.00115) {
		tmp = (-0.5 * (sin(x) * pow(eps, 2.0))) + ((-0.16666666666666666 * (cos(x) * pow(eps, 3.0))) + ((0.041666666666666664 * (sin(x) * pow(eps, 4.0))) + (eps * cos(x))));
	} else {
		tmp = t_0 - (sin(x) * (1.0 - cos(eps)));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(cos(x) * sin(eps))
	tmp = 0.0
	if (eps <= -0.0013)
		tmp = fma(sin(x), cos(eps), Float64(t_0 - sin(x)));
	elseif (eps <= 0.00115)
		tmp = Float64(Float64(-0.5 * Float64(sin(x) * (eps ^ 2.0))) + Float64(Float64(-0.16666666666666666 * Float64(cos(x) * (eps ^ 3.0))) + Float64(Float64(0.041666666666666664 * Float64(sin(x) * (eps ^ 4.0))) + Float64(eps * cos(x)))));
	else
		tmp = Float64(t_0 - Float64(sin(x) * Float64(1.0 - cos(eps))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.0013], N[(N[Sin[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + N[(t$95$0 - N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.00115], N[(N[(-0.5 * N[(N[Sin[x], $MachinePrecision] * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.16666666666666666 * N[(N[Cos[x], $MachinePrecision] * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.041666666666666664 * N[(N[Sin[x], $MachinePrecision] * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(N[Sin[x], $MachinePrecision] * N[(1.0 - N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.0012999999999999999

   1. Initial program 91.1%

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

      1. sin-sum 96.6%

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

      2. associate--l+ 96.4%

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

      3. fma-def 97.1%

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

   3. Applied egg-rr 97.1%

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

   if -0.0012999999999999999 < eps < 0.00115

   1. Initial program 46.4%

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

   2. Taylor expanded in eps around 0 99.4%

      \[\leadsto \color{blue}{-0.5 \cdot \left({\varepsilon}^{2} \cdot \sin x\right) + \left(-0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \sin x\right) + \varepsilon \cdot \cos x\right)\right)} \]

   if 0.00115 < eps

   1. Initial program 88.7%

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

      1. sin-sum 94.7%

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

      2. associate--l+ 94.6%

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

   3. Applied egg-rr 94.6%

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

      1. +-commutative 94.6%

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

      2. associate-+l- 94.6%

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

      3. *-commutative 94.6%

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

      4. *-rgt-identity 94.6%

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

      5. distribute-lft-out-- 95.2%

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

   5. Simplified 95.2%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0013:\\ \;\;\;\;\mathsf{fma}\left(\sin x, \cos \varepsilon, \cos x \cdot \sin \varepsilon - \sin x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00115:\\ \;\;\;\;-0.5 \cdot \left(\sin x \cdot {\varepsilon}^{2}\right) + \left(-0.16666666666666666 \cdot \left(\cos x \cdot {\varepsilon}^{3}\right) + \left(0.041666666666666664 \cdot \left(\sin x \cdot {\varepsilon}^{4}\right) + \varepsilon \cdot \cos x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \sin \varepsilon - \sin x \cdot \left(1 - \cos \varepsilon\right)\\ \end{array} \]

Alternative 4: 98.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \sin \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.0013:\\ \;\;\;\;\mathsf{fma}\left(\sin x, \cos \varepsilon, t_0 - \sin x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0012:\\ \;\;\;\;\sin x \cdot \left({\varepsilon}^{2} \cdot -0.5 + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) + \mathsf{fma}\left(2, {\varepsilon}^{3} \cdot \left(\cos x \cdot -0.08333333333333333\right), \varepsilon \cdot \cos x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 - \sin x \cdot \left(1 - \cos \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (cos x) (sin eps))))
   (if (<= eps -0.0013)
     (fma (sin x) (cos eps) (- t_0 (sin x)))
     (if (<= eps 0.0012)
       (+
        (*
         (sin x)
         (+ (* (pow eps 2.0) -0.5) (* 0.041666666666666664 (pow eps 4.0))))
        (fma
         2.0
         (* (pow eps 3.0) (* (cos x) -0.08333333333333333))
         (* eps (cos x))))
       (- t_0 (* (sin x) (- 1.0 (cos eps))))))))

C

double code(double x, double eps) {
	double t_0 = cos(x) * sin(eps);
	double tmp;
	if (eps <= -0.0013) {
		tmp = fma(sin(x), cos(eps), (t_0 - sin(x)));
	} else if (eps <= 0.0012) {
		tmp = (sin(x) * ((pow(eps, 2.0) * -0.5) + (0.041666666666666664 * pow(eps, 4.0)))) + fma(2.0, (pow(eps, 3.0) * (cos(x) * -0.08333333333333333)), (eps * cos(x)));
	} else {
		tmp = t_0 - (sin(x) * (1.0 - cos(eps)));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(cos(x) * sin(eps))
	tmp = 0.0
	if (eps <= -0.0013)
		tmp = fma(sin(x), cos(eps), Float64(t_0 - sin(x)));
	elseif (eps <= 0.0012)
		tmp = Float64(Float64(sin(x) * Float64(Float64((eps ^ 2.0) * -0.5) + Float64(0.041666666666666664 * (eps ^ 4.0)))) + fma(2.0, Float64((eps ^ 3.0) * Float64(cos(x) * -0.08333333333333333)), Float64(eps * cos(x))));
	else
		tmp = Float64(t_0 - Float64(sin(x) * Float64(1.0 - cos(eps))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.0013], N[(N[Sin[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + N[(t$95$0 - N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.0012], N[(N[(N[Sin[x], $MachinePrecision] * N[(N[(N[Power[eps, 2.0], $MachinePrecision] * -0.5), $MachinePrecision] + N[(0.041666666666666664 * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[Power[eps, 3.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision] + N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(N[Sin[x], $MachinePrecision] * N[(1.0 - N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.0012999999999999999

   1. Initial program 91.1%

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

      1. sin-sum 96.6%

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

      2. associate--l+ 96.4%

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

      3. fma-def 97.1%

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

   3. Applied egg-rr 97.1%

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

   if -0.0012999999999999999 < eps < 0.00119999999999999989

   1. Initial program 46.4%

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

      1. diff-sin 47.2%

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

      2. div-inv 47.2%

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

      3. associate--l+ 47.2%

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

      4. metadata-eval 47.2%

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

      5. div-inv 47.2%

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

      6. +-commutative 47.2%

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

      7. associate-+l+ 47.2%

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

      8. metadata-eval 47.2%

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

   3. Applied egg-rr 47.2%

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

      1. associate-*r* 47.2%

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

      2. *-commutative 47.2%

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

      3. *-commutative 47.2%

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

      4. +-commutative 47.2%

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

      5. count-2 47.2%

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

      6. fma-def 47.2%

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

      7. sub-neg 47.2%

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

      8. mul-1-neg 47.2%

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

      9. +-commutative 47.2%

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

      10. associate-+r+ 94.3%

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

      11. mul-1-neg 94.3%

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

      12. sub-neg 94.3%

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

      13. +-inverses 94.3%

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

      14. remove-double-neg 94.3%

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

      15. mul-1-neg 94.3%

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

      16. sub-neg 94.3%

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

      17. neg-sub0 94.3%

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

      18. mul-1-neg 94.3%

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

      19. remove-double-neg 94.3%

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

   5. Simplified 94.3%

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

      1. add-log-exp 94.3%

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

   7. Applied egg-rr 94.3%

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

   8. Taylor expanded in eps around 0 99.4%

      \[\leadsto \color{blue}{-0.5 \cdot \left({\varepsilon}^{2} \cdot \sin x\right) + \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \sin x\right) + \left(2 \cdot \left({\varepsilon}^{3} \cdot \left(-0.0625 \cdot \cos x + -0.020833333333333332 \cdot \cos x\right)\right) + \varepsilon \cdot \cos x\right)\right)} \]
   9. Step-by-step derivation

      1. associate-+r+ 99.4%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \sin x\right) + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \sin x\right)\right) + \left(2 \cdot \left({\varepsilon}^{3} \cdot \left(-0.0625 \cdot \cos x + -0.020833333333333332 \cdot \cos x\right)\right) + \varepsilon \cdot \cos x\right)} \]

      2. associate-*r* 99.4%

         \[\leadsto \left(\color{blue}{\left(-0.5 \cdot {\varepsilon}^{2}\right) \cdot \sin x} + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \sin x\right)\right) + \left(2 \cdot \left({\varepsilon}^{3} \cdot \left(-0.0625 \cdot \cos x + -0.020833333333333332 \cdot \cos x\right)\right) + \varepsilon \cdot \cos x\right) \]

      3. associate-*r* 99.4%

         \[\leadsto \left(\left(-0.5 \cdot {\varepsilon}^{2}\right) \cdot \sin x + \color{blue}{\left(0.041666666666666664 \cdot {\varepsilon}^{4}\right) \cdot \sin x}\right) + \left(2 \cdot \left({\varepsilon}^{3} \cdot \left(-0.0625 \cdot \cos x + -0.020833333333333332 \cdot \cos x\right)\right) + \varepsilon \cdot \cos x\right) \]

      4. distribute-rgt-out 99.4%

         \[\leadsto \color{blue}{\sin x \cdot \left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right)} + \left(2 \cdot \left({\varepsilon}^{3} \cdot \left(-0.0625 \cdot \cos x + -0.020833333333333332 \cdot \cos x\right)\right) + \varepsilon \cdot \cos x\right) \]

      5. fma-def 99.4%

         \[\leadsto \sin x \cdot \left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) + \color{blue}{\mathsf{fma}\left(2, {\varepsilon}^{3} \cdot \left(-0.0625 \cdot \cos x + -0.020833333333333332 \cdot \cos x\right), \varepsilon \cdot \cos x\right)} \]

      6. distribute-rgt-out 99.4%

         \[\leadsto \sin x \cdot \left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) + \mathsf{fma}\left(2, {\varepsilon}^{3} \cdot \color{blue}{\left(\cos x \cdot \left(-0.0625 + -0.020833333333333332\right)\right)}, \varepsilon \cdot \cos x\right) \]

      7. metadata-eval 99.4%

         \[\leadsto \sin x \cdot \left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) + \mathsf{fma}\left(2, {\varepsilon}^{3} \cdot \left(\cos x \cdot \color{blue}{-0.08333333333333333}\right), \varepsilon \cdot \cos x\right) \]

   10. Simplified 99.4%

       \[\leadsto \color{blue}{\sin x \cdot \left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) + \mathsf{fma}\left(2, {\varepsilon}^{3} \cdot \left(\cos x \cdot -0.08333333333333333\right), \varepsilon \cdot \cos x\right)} \]

   if 0.00119999999999999989 < eps

   1. Initial program 88.7%

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

      1. sin-sum 94.7%

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

      2. associate--l+ 94.6%

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

   3. Applied egg-rr 94.6%

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

      1. +-commutative 94.6%

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

      2. associate-+l- 94.6%

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

      3. *-commutative 94.6%

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

      4. *-rgt-identity 94.6%

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

      5. distribute-lft-out-- 95.2%

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

   5. Simplified 95.2%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0013:\\ \;\;\;\;\mathsf{fma}\left(\sin x, \cos \varepsilon, \cos x \cdot \sin \varepsilon - \sin x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0012:\\ \;\;\;\;\sin x \cdot \left({\varepsilon}^{2} \cdot -0.5 + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) + \mathsf{fma}\left(2, {\varepsilon}^{3} \cdot \left(\cos x \cdot -0.08333333333333333\right), \varepsilon \cdot \cos x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \sin \varepsilon - \sin x \cdot \left(1 - \cos \varepsilon\right)\\ \end{array} \]

Alternative 5: 98.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \sin \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.0012:\\ \;\;\;\;\mathsf{fma}\left(\sin x, \cos \varepsilon, t_0 - \sin x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00125:\\ \;\;\;\;\left(-0.5 \cdot \left(\sin x \cdot {\varepsilon}^{2}\right) + \cos x \cdot \left(\varepsilon + {\varepsilon}^{3} \cdot -0.16666666666666666\right)\right) + {\varepsilon}^{4} \cdot \left(\sin x \cdot 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 - \sin x \cdot \left(1 - \cos \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (cos x) (sin eps))))
   (if (<= eps -0.0012)
     (fma (sin x) (cos eps) (- t_0 (sin x)))
     (if (<= eps 0.00125)
       (+
        (+
         (* -0.5 (* (sin x) (pow eps 2.0)))
         (* (cos x) (+ eps (* (pow eps 3.0) -0.16666666666666666))))
        (* (pow eps 4.0) (* (sin x) 0.041666666666666664)))
       (- t_0 (* (sin x) (- 1.0 (cos eps))))))))

C

double code(double x, double eps) {
	double t_0 = cos(x) * sin(eps);
	double tmp;
	if (eps <= -0.0012) {
		tmp = fma(sin(x), cos(eps), (t_0 - sin(x)));
	} else if (eps <= 0.00125) {
		tmp = ((-0.5 * (sin(x) * pow(eps, 2.0))) + (cos(x) * (eps + (pow(eps, 3.0) * -0.16666666666666666)))) + (pow(eps, 4.0) * (sin(x) * 0.041666666666666664));
	} else {
		tmp = t_0 - (sin(x) * (1.0 - cos(eps)));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(cos(x) * sin(eps))
	tmp = 0.0
	if (eps <= -0.0012)
		tmp = fma(sin(x), cos(eps), Float64(t_0 - sin(x)));
	elseif (eps <= 0.00125)
		tmp = Float64(Float64(Float64(-0.5 * Float64(sin(x) * (eps ^ 2.0))) + Float64(cos(x) * Float64(eps + Float64((eps ^ 3.0) * -0.16666666666666666)))) + Float64((eps ^ 4.0) * Float64(sin(x) * 0.041666666666666664)));
	else
		tmp = Float64(t_0 - Float64(sin(x) * Float64(1.0 - cos(eps))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.0012], N[(N[Sin[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + N[(t$95$0 - N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.00125], N[(N[(N[(-0.5 * N[(N[Sin[x], $MachinePrecision] * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(eps + N[(N[Power[eps, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[eps, 4.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(N[Sin[x], $MachinePrecision] * N[(1.0 - N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.00119999999999999989

   1. Initial program 91.1%

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

      1. sin-sum 96.6%

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

      2. associate--l+ 96.4%

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

      3. fma-def 97.1%

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

   3. Applied egg-rr 97.1%

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

   if -0.00119999999999999989 < eps < 0.00125000000000000003

   1. Initial program 46.4%

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

   2. Taylor expanded in eps around 0 99.4%

      \[\leadsto \color{blue}{-0.5 \cdot \left({\varepsilon}^{2} \cdot \sin x\right) + \left(-0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \sin x\right) + \varepsilon \cdot \cos x\right)\right)} \]
   3. Step-by-step derivation

      1. +-commutative 99.4%

         \[\leadsto -0.5 \cdot \left({\varepsilon}^{2} \cdot \sin x\right) + \left(-0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + \color{blue}{\left(\varepsilon \cdot \cos x + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \sin x\right)\right)}\right) \]

      2. associate-+r+ 99.4%

         \[\leadsto -0.5 \cdot \left({\varepsilon}^{2} \cdot \sin x\right) + \color{blue}{\left(\left(-0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + \varepsilon \cdot \cos x\right) + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \sin x\right)\right)} \]

      3. associate-+r+ 99.4%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \sin x\right) + \left(-0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + \varepsilon \cdot \cos x\right)\right) + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \sin x\right)} \]

      4. fma-def 99.4%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, {\varepsilon}^{2} \cdot \sin x, -0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + \varepsilon \cdot \cos x\right)} + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \sin x\right) \]

      5. *-commutative 99.4%

         \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{\sin x \cdot {\varepsilon}^{2}}, -0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + \varepsilon \cdot \cos x\right) + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \sin x\right) \]

      6. +-commutative 99.4%

         \[\leadsto \mathsf{fma}\left(-0.5, \sin x \cdot {\varepsilon}^{2}, \color{blue}{\varepsilon \cdot \cos x + -0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \cos x\right)}\right) + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \sin x\right) \]

      7. associate-*r* 99.4%

         \[\leadsto \mathsf{fma}\left(-0.5, \sin x \cdot {\varepsilon}^{2}, \varepsilon \cdot \cos x + \color{blue}{\left(-0.16666666666666666 \cdot {\varepsilon}^{3}\right) \cdot \cos x}\right) + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \sin x\right) \]

      8. distribute-rgt-out 99.3%

         \[\leadsto \mathsf{fma}\left(-0.5, \sin x \cdot {\varepsilon}^{2}, \color{blue}{\cos x \cdot \left(\varepsilon + -0.16666666666666666 \cdot {\varepsilon}^{3}\right)}\right) + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \sin x\right) \]

      9. *-commutative 99.3%

         \[\leadsto \mathsf{fma}\left(-0.5, \sin x \cdot {\varepsilon}^{2}, \cos x \cdot \left(\varepsilon + -0.16666666666666666 \cdot {\varepsilon}^{3}\right)\right) + 0.041666666666666664 \cdot \color{blue}{\left(\sin x \cdot {\varepsilon}^{4}\right)} \]

      10. associate-*r* 99.3%

          \[\leadsto \mathsf{fma}\left(-0.5, \sin x \cdot {\varepsilon}^{2}, \cos x \cdot \left(\varepsilon + -0.16666666666666666 \cdot {\varepsilon}^{3}\right)\right) + \color{blue}{\left(0.041666666666666664 \cdot \sin x\right) \cdot {\varepsilon}^{4}} \]

   4. Simplified 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \sin x \cdot {\varepsilon}^{2}, \cos x \cdot \left(\varepsilon + -0.16666666666666666 \cdot {\varepsilon}^{3}\right)\right) + \left(0.041666666666666664 \cdot \sin x\right) \cdot {\varepsilon}^{4}} \]

   5. Taylor expanded in x around inf 99.3%

      \[\leadsto \color{blue}{\left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \sin x\right) + \cos x \cdot \left(\varepsilon + -0.16666666666666666 \cdot {\varepsilon}^{3}\right)\right)} + \left(0.041666666666666664 \cdot \sin x\right) \cdot {\varepsilon}^{4} \]

   if 0.00125000000000000003 < eps

   1. Initial program 88.7%

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

      1. sin-sum 94.7%

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

      2. associate--l+ 94.6%

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

   3. Applied egg-rr 94.6%

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

      1. +-commutative 94.6%

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

      2. associate-+l- 94.6%

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

      3. *-commutative 94.6%

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

      4. *-rgt-identity 94.6%

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

      5. distribute-lft-out-- 95.2%

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

   5. Simplified 95.2%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0012:\\ \;\;\;\;\mathsf{fma}\left(\sin x, \cos \varepsilon, \cos x \cdot \sin \varepsilon - \sin x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00125:\\ \;\;\;\;\left(-0.5 \cdot \left(\sin x \cdot {\varepsilon}^{2}\right) + \cos x \cdot \left(\varepsilon + {\varepsilon}^{3} \cdot -0.16666666666666666\right)\right) + {\varepsilon}^{4} \cdot \left(\sin x \cdot 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \sin \varepsilon - \sin x \cdot \left(1 - \cos \varepsilon\right)\\ \end{array} \]

Alternative 6: 97.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \sin \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.00019:\\ \;\;\;\;\mathsf{fma}\left(\sin x, \cos \varepsilon, t_0 - \sin x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00017:\\ \;\;\;\;\left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \cdot \left(\cos x + \left(-0.5 \cdot \left(\varepsilon \cdot \sin x\right) + -0.125 \cdot \left(\cos x \cdot {\varepsilon}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 - \sin x \cdot \left(1 - \cos \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (cos x) (sin eps))))
   (if (<= eps -0.00019)
     (fma (sin x) (cos eps) (- t_0 (sin x)))
     (if (<= eps 0.00017)
       (*
        (* 2.0 (sin (* eps 0.5)))
        (+
         (cos x)
         (+ (* -0.5 (* eps (sin x))) (* -0.125 (* (cos x) (pow eps 2.0))))))
       (- t_0 (* (sin x) (- 1.0 (cos eps))))))))

C

double code(double x, double eps) {
	double t_0 = cos(x) * sin(eps);
	double tmp;
	if (eps <= -0.00019) {
		tmp = fma(sin(x), cos(eps), (t_0 - sin(x)));
	} else if (eps <= 0.00017) {
		tmp = (2.0 * sin((eps * 0.5))) * (cos(x) + ((-0.5 * (eps * sin(x))) + (-0.125 * (cos(x) * pow(eps, 2.0)))));
	} else {
		tmp = t_0 - (sin(x) * (1.0 - cos(eps)));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(cos(x) * sin(eps))
	tmp = 0.0
	if (eps <= -0.00019)
		tmp = fma(sin(x), cos(eps), Float64(t_0 - sin(x)));
	elseif (eps <= 0.00017)
		tmp = Float64(Float64(2.0 * sin(Float64(eps * 0.5))) * Float64(cos(x) + Float64(Float64(-0.5 * Float64(eps * sin(x))) + Float64(-0.125 * Float64(cos(x) * (eps ^ 2.0))))));
	else
		tmp = Float64(t_0 - Float64(sin(x) * Float64(1.0 - cos(eps))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.00019], N[(N[Sin[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + N[(t$95$0 - N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.00017], N[(N[(2.0 * N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + N[(N[(-0.5 * N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.125 * N[(N[Cos[x], $MachinePrecision] * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(N[Sin[x], $MachinePrecision] * N[(1.0 - N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -1.9000000000000001e-4

   1. Initial program 90.1%

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

      1. sin-sum 95.7%

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

      2. associate--l+ 95.6%

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

      3. fma-def 96.1%

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

   3. Applied egg-rr 96.1%

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

   if -1.9000000000000001e-4 < eps < 1.7e-4

   1. Initial program 45.2%

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

      1. diff-sin 46.0%

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

      2. div-inv 46.0%

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

      3. associate--l+ 46.0%

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

      4. metadata-eval 46.0%

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

      5. div-inv 46.0%

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

      6. +-commutative 46.0%

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

      7. associate-+l+ 46.0%

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

      8. metadata-eval 46.0%

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

   3. Applied egg-rr 46.0%

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

      1. associate-*r* 46.0%

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

      2. *-commutative 46.0%

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

      3. *-commutative 46.0%

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

      4. +-commutative 46.0%

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

      5. count-2 46.0%

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

      6. fma-def 46.0%

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

      7. sub-neg 46.0%

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

      8. mul-1-neg 46.0%

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

      9. +-commutative 46.0%

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

      10. associate-+r+ 94.4%

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

      11. mul-1-neg 94.4%

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

      12. sub-neg 94.4%

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

      13. +-inverses 94.4%

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

      14. remove-double-neg 94.4%

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

      15. mul-1-neg 94.4%

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

      16. sub-neg 94.4%

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

      17. neg-sub0 94.4%

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

      18. mul-1-neg 94.4%

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

      19. remove-double-neg 94.4%

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

   5. Simplified 94.4%

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

   6. Taylor expanded in eps around 0 99.1%

      \[\leadsto \color{blue}{\left(\cos x + \left(-0.5 \cdot \left(\varepsilon \cdot \sin x\right) + -0.125 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right)\right)} \cdot \left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]

   if 1.7e-4 < eps

   1. Initial program 87.3%

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

      1. sin-sum 93.5%

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

      2. associate--l+ 93.4%

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

   3. Applied egg-rr 93.4%

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

      1. +-commutative 93.4%

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

      2. associate-+l- 93.6%

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

      3. *-commutative 93.6%

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

      4. *-rgt-identity 93.6%

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

      5. distribute-lft-out-- 94.2%

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

   5. Simplified 94.2%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00019:\\ \;\;\;\;\mathsf{fma}\left(\sin x, \cos \varepsilon, \cos x \cdot \sin \varepsilon - \sin x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00017:\\ \;\;\;\;\left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \cdot \left(\cos x + \left(-0.5 \cdot \left(\varepsilon \cdot \sin x\right) + -0.125 \cdot \left(\cos x \cdot {\varepsilon}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \sin \varepsilon - \sin x \cdot \left(1 - \cos \varepsilon\right)\\ \end{array} \]

Alternative 7: 97.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000155 \lor \neg \left(\varepsilon \leq 0.00017\right):\\ \;\;\;\;\cos x \cdot \sin \varepsilon - \sin x \cdot \left(1 - \cos \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \cdot \left(\cos x + \left(-0.5 \cdot \left(\varepsilon \cdot \sin x\right) + -0.125 \cdot \left(\cos x \cdot {\varepsilon}^{2}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.000155) (not (<= eps 0.00017)))
   (- (* (cos x) (sin eps)) (* (sin x) (- 1.0 (cos eps))))
   (*
    (* 2.0 (sin (* eps 0.5)))
    (+
     (cos x)
     (+ (* -0.5 (* eps (sin x))) (* -0.125 (* (cos x) (pow eps 2.0))))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.000155) || !(eps <= 0.00017)) {
		tmp = (cos(x) * sin(eps)) - (sin(x) * (1.0 - cos(eps)));
	} else {
		tmp = (2.0 * sin((eps * 0.5))) * (cos(x) + ((-0.5 * (eps * sin(x))) + (-0.125 * (cos(x) * pow(eps, 2.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-0.000155d0)) .or. (.not. (eps <= 0.00017d0))) then
        tmp = (cos(x) * sin(eps)) - (sin(x) * (1.0d0 - cos(eps)))
    else
        tmp = (2.0d0 * sin((eps * 0.5d0))) * (cos(x) + (((-0.5d0) * (eps * sin(x))) + ((-0.125d0) * (cos(x) * (eps ** 2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.000155) || !(eps <= 0.00017)) {
		tmp = (Math.cos(x) * Math.sin(eps)) - (Math.sin(x) * (1.0 - Math.cos(eps)));
	} else {
		tmp = (2.0 * Math.sin((eps * 0.5))) * (Math.cos(x) + ((-0.5 * (eps * Math.sin(x))) + (-0.125 * (Math.cos(x) * Math.pow(eps, 2.0)))));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.000155) or not (eps <= 0.00017):
		tmp = (math.cos(x) * math.sin(eps)) - (math.sin(x) * (1.0 - math.cos(eps)))
	else:
		tmp = (2.0 * math.sin((eps * 0.5))) * (math.cos(x) + ((-0.5 * (eps * math.sin(x))) + (-0.125 * (math.cos(x) * math.pow(eps, 2.0)))))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.000155) || !(eps <= 0.00017))
		tmp = Float64(Float64(cos(x) * sin(eps)) - Float64(sin(x) * Float64(1.0 - cos(eps))));
	else
		tmp = Float64(Float64(2.0 * sin(Float64(eps * 0.5))) * Float64(cos(x) + Float64(Float64(-0.5 * Float64(eps * sin(x))) + Float64(-0.125 * Float64(cos(x) * (eps ^ 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.000155) || ~((eps <= 0.00017)))
		tmp = (cos(x) * sin(eps)) - (sin(x) * (1.0 - cos(eps)));
	else
		tmp = (2.0 * sin((eps * 0.5))) * (cos(x) + ((-0.5 * (eps * sin(x))) + (-0.125 * (cos(x) * (eps ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.000155], N[Not[LessEqual[eps, 0.00017]], $MachinePrecision]], 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], N[(N[(2.0 * N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + N[(N[(-0.5 * N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.125 * N[(N[Cos[x], $MachinePrecision] * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.55e-4 or 1.7e-4 < eps

   1. Initial program 88.8%

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

      1. sin-sum 94.7%

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

      2. associate--l+ 94.6%

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

   3. Applied egg-rr 94.6%

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

      1. +-commutative 94.6%

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

      2. associate-+l- 94.7%

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

      3. *-commutative 94.7%

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

      4. *-rgt-identity 94.7%

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

      5. distribute-lft-out-- 95.1%

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

   5. Simplified 95.1%

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

   if -1.55e-4 < eps < 1.7e-4

   1. Initial program 45.2%

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

      1. diff-sin 46.0%

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

      2. div-inv 46.0%

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

      3. associate--l+ 46.0%

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

      4. metadata-eval 46.0%

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

      5. div-inv 46.0%

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

      6. +-commutative 46.0%

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

      7. associate-+l+ 46.0%

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

      8. metadata-eval 46.0%

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

   3. Applied egg-rr 46.0%

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

      1. associate-*r* 46.0%

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

      2. *-commutative 46.0%

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

      3. *-commutative 46.0%

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

      4. +-commutative 46.0%

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

      5. count-2 46.0%

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

      6. fma-def 46.0%

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

      7. sub-neg 46.0%

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

      8. mul-1-neg 46.0%

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

      9. +-commutative 46.0%

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

      10. associate-+r+ 94.4%

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

      11. mul-1-neg 94.4%

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

      12. sub-neg 94.4%

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

      13. +-inverses 94.4%

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

      14. remove-double-neg 94.4%

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

      15. mul-1-neg 94.4%

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

      16. sub-neg 94.4%

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

      17. neg-sub0 94.4%

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

      18. mul-1-neg 94.4%

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

      19. remove-double-neg 94.4%

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

   5. Simplified 94.4%

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

   6. Taylor expanded in eps around 0 99.1%

      \[\leadsto \color{blue}{\left(\cos x + \left(-0.5 \cdot \left(\varepsilon \cdot \sin x\right) + -0.125 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right)\right)} \cdot \left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000155 \lor \neg \left(\varepsilon \leq 0.00017\right):\\ \;\;\;\;\cos x \cdot \sin \varepsilon - \sin x \cdot \left(1 - \cos \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \cdot \left(\cos x + \left(-0.5 \cdot \left(\varepsilon \cdot \sin x\right) + -0.125 \cdot \left(\cos x \cdot {\varepsilon}^{2}\right)\right)\right)\\ \end{array} \]

Alternative 8: 97.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00012 \lor \neg \left(\varepsilon \leq 0.00016\right):\\ \;\;\;\;\cos x \cdot \sin \varepsilon - \sin x \cdot \left(1 - \cos \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\sin x \cdot {\varepsilon}^{2}\right) + \left(-0.16666666666666666 \cdot \left(\cos x \cdot {\varepsilon}^{3}\right) + \varepsilon \cdot \cos x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.00012) (not (<= eps 0.00016)))
   (- (* (cos x) (sin eps)) (* (sin x) (- 1.0 (cos eps))))
   (+
    (* -0.5 (* (sin x) (pow eps 2.0)))
    (+ (* -0.16666666666666666 (* (cos x) (pow eps 3.0))) (* eps (cos x))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.00012) || !(eps <= 0.00016)) {
		tmp = (cos(x) * sin(eps)) - (sin(x) * (1.0 - cos(eps)));
	} else {
		tmp = (-0.5 * (sin(x) * pow(eps, 2.0))) + ((-0.16666666666666666 * (cos(x) * pow(eps, 3.0))) + (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 <= (-0.00012d0)) .or. (.not. (eps <= 0.00016d0))) then
        tmp = (cos(x) * sin(eps)) - (sin(x) * (1.0d0 - cos(eps)))
    else
        tmp = ((-0.5d0) * (sin(x) * (eps ** 2.0d0))) + (((-0.16666666666666666d0) * (cos(x) * (eps ** 3.0d0))) + (eps * cos(x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.00012) || !(eps <= 0.00016)) {
		tmp = (Math.cos(x) * Math.sin(eps)) - (Math.sin(x) * (1.0 - Math.cos(eps)));
	} else {
		tmp = (-0.5 * (Math.sin(x) * Math.pow(eps, 2.0))) + ((-0.16666666666666666 * (Math.cos(x) * Math.pow(eps, 3.0))) + (eps * Math.cos(x)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.00012) or not (eps <= 0.00016):
		tmp = (math.cos(x) * math.sin(eps)) - (math.sin(x) * (1.0 - math.cos(eps)))
	else:
		tmp = (-0.5 * (math.sin(x) * math.pow(eps, 2.0))) + ((-0.16666666666666666 * (math.cos(x) * math.pow(eps, 3.0))) + (eps * math.cos(x)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.00012) || !(eps <= 0.00016))
		tmp = Float64(Float64(cos(x) * sin(eps)) - Float64(sin(x) * Float64(1.0 - cos(eps))));
	else
		tmp = Float64(Float64(-0.5 * Float64(sin(x) * (eps ^ 2.0))) + Float64(Float64(-0.16666666666666666 * Float64(cos(x) * (eps ^ 3.0))) + Float64(eps * cos(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.00012) || ~((eps <= 0.00016)))
		tmp = (cos(x) * sin(eps)) - (sin(x) * (1.0 - cos(eps)));
	else
		tmp = (-0.5 * (sin(x) * (eps ^ 2.0))) + ((-0.16666666666666666 * (cos(x) * (eps ^ 3.0))) + (eps * cos(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.00012], N[Not[LessEqual[eps, 0.00016]], $MachinePrecision]], 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], N[(N[(-0.5 * N[(N[Sin[x], $MachinePrecision] * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.16666666666666666 * N[(N[Cos[x], $MachinePrecision] * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.20000000000000003e-4 or 1.60000000000000013e-4 < eps

   1. Initial program 88.5%

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

      1. sin-sum 94.5%

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

      2. associate--l+ 94.4%

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

   3. Applied egg-rr 94.4%

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

      1. +-commutative 94.4%

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

      2. associate-+l- 94.6%

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

      3. *-commutative 94.6%

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

      4. *-rgt-identity 94.6%

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

      5. distribute-lft-out-- 95.0%

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

   5. Simplified 95.0%

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

   if -1.20000000000000003e-4 < eps < 1.60000000000000013e-4

   1. Initial program 45.1%

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

   2. Taylor expanded in eps around 0 99.0%

      \[\leadsto \color{blue}{-0.5 \cdot \left({\varepsilon}^{2} \cdot \sin x\right) + \left(-0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + \varepsilon \cdot \cos x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00012 \lor \neg \left(\varepsilon \leq 0.00016\right):\\ \;\;\;\;\cos x \cdot \sin \varepsilon - \sin x \cdot \left(1 - \cos \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\sin x \cdot {\varepsilon}^{2}\right) + \left(-0.16666666666666666 \cdot \left(\cos x \cdot {\varepsilon}^{3}\right) + \varepsilon \cdot \cos x\right)\\ \end{array} \]

Alternative 9: 97.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00012 \lor \neg \left(\varepsilon \leq 0.00016\right):\\ \;\;\;\;\cos x \cdot \sin \varepsilon - \sin x \cdot \left(1 - \cos \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \sin x \cdot {\varepsilon}^{2}, \cos x \cdot \left(\varepsilon + {\varepsilon}^{3} \cdot -0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.00012) (not (<= eps 0.00016)))
   (- (* (cos x) (sin eps)) (* (sin x) (- 1.0 (cos eps))))
   (fma
    -0.5
    (* (sin x) (pow eps 2.0))
    (* (cos x) (+ eps (* (pow eps 3.0) -0.16666666666666666))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.00012) || !(eps <= 0.00016)) {
		tmp = (cos(x) * sin(eps)) - (sin(x) * (1.0 - cos(eps)));
	} else {
		tmp = fma(-0.5, (sin(x) * pow(eps, 2.0)), (cos(x) * (eps + (pow(eps, 3.0) * -0.16666666666666666))));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.00012) || !(eps <= 0.00016))
		tmp = Float64(Float64(cos(x) * sin(eps)) - Float64(sin(x) * Float64(1.0 - cos(eps))));
	else
		tmp = fma(-0.5, Float64(sin(x) * (eps ^ 2.0)), Float64(cos(x) * Float64(eps + Float64((eps ^ 3.0) * -0.16666666666666666))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.00012], N[Not[LessEqual[eps, 0.00016]], $MachinePrecision]], 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], N[(-0.5 * N[(N[Sin[x], $MachinePrecision] * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(eps + N[(N[Power[eps, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.20000000000000003e-4 or 1.60000000000000013e-4 < eps

   1. Initial program 88.5%

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

      1. sin-sum 94.5%

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

      2. associate--l+ 94.4%

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

   3. Applied egg-rr 94.4%

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

      1. +-commutative 94.4%

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

      2. associate-+l- 94.6%

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

      3. *-commutative 94.6%

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

      4. *-rgt-identity 94.6%

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

      5. distribute-lft-out-- 95.0%

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

   5. Simplified 95.0%

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

   if -1.20000000000000003e-4 < eps < 1.60000000000000013e-4

   1. Initial program 45.1%

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

   2. Taylor expanded in eps around 0 99.0%

      \[\leadsto \color{blue}{-0.5 \cdot \left({\varepsilon}^{2} \cdot \sin x\right) + \left(-0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + \varepsilon \cdot \cos x\right)} \]
   3. Step-by-step derivation

      1. fma-def 99.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, {\varepsilon}^{2} \cdot \sin x, -0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + \varepsilon \cdot \cos x\right)} \]

      2. *-commutative 99.0%

         \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{\sin x \cdot {\varepsilon}^{2}}, -0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + \varepsilon \cdot \cos x\right) \]

      3. +-commutative 99.0%

         \[\leadsto \mathsf{fma}\left(-0.5, \sin x \cdot {\varepsilon}^{2}, \color{blue}{\varepsilon \cdot \cos x + -0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \cos x\right)}\right) \]

      4. associate-*r* 99.0%

         \[\leadsto \mathsf{fma}\left(-0.5, \sin x \cdot {\varepsilon}^{2}, \varepsilon \cdot \cos x + \color{blue}{\left(-0.16666666666666666 \cdot {\varepsilon}^{3}\right) \cdot \cos x}\right) \]

      5. distribute-rgt-out 99.0%

         \[\leadsto \mathsf{fma}\left(-0.5, \sin x \cdot {\varepsilon}^{2}, \color{blue}{\cos x \cdot \left(\varepsilon + -0.16666666666666666 \cdot {\varepsilon}^{3}\right)}\right) \]

   4. Simplified 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \sin x \cdot {\varepsilon}^{2}, \cos x \cdot \left(\varepsilon + -0.16666666666666666 \cdot {\varepsilon}^{3}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00012 \lor \neg \left(\varepsilon \leq 0.00016\right):\\ \;\;\;\;\cos x \cdot \sin \varepsilon - \sin x \cdot \left(1 - \cos \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \sin x \cdot {\varepsilon}^{2}, \cos x \cdot \left(\varepsilon + {\varepsilon}^{3} \cdot -0.16666666666666666\right)\right)\\ \end{array} \]

Alternative 10: 94.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \cdot \log \left(e^{\cos \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right)}\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* (* 2.0 (sin (* eps 0.5))) (log (exp (cos (* 0.5 (fma 2.0 x eps)))))))

C

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

Julia

function code(x, eps)
	return Float64(Float64(2.0 * sin(Float64(eps * 0.5))) * log(exp(cos(Float64(0.5 * fma(2.0, x, eps))))))
end

Wolfram

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

Derivation

1. Initial program 58.1%

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

   1. diff-sin 58.5%

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

   2. div-inv 58.5%

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

   3. associate--l+ 58.5%

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

   4. metadata-eval 58.5%

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

   5. div-inv 58.5%

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

   6. +-commutative 58.5%

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

   7. associate-+l+ 58.5%

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

   8. metadata-eval 58.5%

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

3. Applied egg-rr 58.5%

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

   1. associate-*r* 58.5%

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

   2. *-commutative 58.5%

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

   3. *-commutative 58.5%

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

   4. +-commutative 58.5%

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

   5. count-2 58.5%

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

   6. fma-def 58.5%

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

   7. sub-neg 58.5%

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

   8. mul-1-neg 58.5%

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

   9. +-commutative 58.5%

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

   10. associate-+r+ 94.2%

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

   11. mul-1-neg 94.2%

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

   12. sub-neg 94.2%

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

   13. +-inverses 94.2%

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

   14. remove-double-neg 94.2%

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

   15. mul-1-neg 94.2%

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

   16. sub-neg 94.2%

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

   17. neg-sub0 94.2%

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

   18. mul-1-neg 94.2%

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

   19. remove-double-neg 94.2%

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

5. Simplified 94.2%

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

   1. add-log-exp 94.2%

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

7. Applied egg-rr 94.2%

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

8. Final simplification 94.2%

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

Alternative 11: 94.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.1%

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

   1. add-cube-cbrt 57.2%

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

   2. pow3 57.1%

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

3. Applied egg-rr 57.1%

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

   1. rem-cube-cbrt 58.1%

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

   2. diff-sin 58.5%

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

5. Applied egg-rr 58.5%

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

   1. associate-*r* 58.5%

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

   2. +-commutative 58.5%

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

   3. associate--l+ 93.9%

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

   4. +-commutative 93.9%

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

   5. associate-+l+ 94.2%

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

7. Simplified 94.2%

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

8. Taylor expanded in eps around inf 94.2%

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

   1. *-commutative 94.2%

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

10. Simplified 94.2%

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

11. Final simplification 94.2%

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

Alternative 12: 90.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.45 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 2.6 \cdot 10^{-5}\right):\\ \;\;\;\;\sin \left(\varepsilon + x\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\frac{\varepsilon + \left(x + x\right)}{2}\right) \cdot \left(2 \cdot \left(\varepsilon \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -2.45e-5) (not (<= eps 2.6e-5)))
   (- (sin (+ eps x)) (sin x))
   (* (cos (/ (+ eps (+ x x)) 2.0)) (* 2.0 (* eps 0.5)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -2.45e-5) || !(eps <= 2.6e-5)) {
		tmp = sin((eps + x)) - sin(x);
	} else {
		tmp = cos(((eps + (x + x)) / 2.0)) * (2.0 * (eps * 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-2.45d-5)) .or. (.not. (eps <= 2.6d-5))) then
        tmp = sin((eps + x)) - sin(x)
    else
        tmp = cos(((eps + (x + x)) / 2.0d0)) * (2.0d0 * (eps * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -2.45e-5) || !(eps <= 2.6e-5)) {
		tmp = Math.sin((eps + x)) - Math.sin(x);
	} else {
		tmp = Math.cos(((eps + (x + x)) / 2.0)) * (2.0 * (eps * 0.5));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -2.45e-5) or not (eps <= 2.6e-5):
		tmp = math.sin((eps + x)) - math.sin(x)
	else:
		tmp = math.cos(((eps + (x + x)) / 2.0)) * (2.0 * (eps * 0.5))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -2.45e-5) || !(eps <= 2.6e-5))
		tmp = Float64(sin(Float64(eps + x)) - sin(x));
	else
		tmp = Float64(cos(Float64(Float64(eps + Float64(x + x)) / 2.0)) * Float64(2.0 * Float64(eps * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -2.45e-5) || ~((eps <= 2.6e-5)))
		tmp = sin((eps + x)) - sin(x);
	else
		tmp = cos(((eps + (x + x)) / 2.0)) * (2.0 * (eps * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -2.45e-5], N[Not[LessEqual[eps, 2.6e-5]], $MachinePrecision]], N[(N[Sin[N[(eps + x), $MachinePrecision]], $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(N[(eps + N[(x + x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[(eps * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -2.45e-5 or 2.59999999999999984e-5 < eps

   1. Initial program 86.8%

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

   if -2.45e-5 < eps < 2.59999999999999984e-5

   1. Initial program 43.7%

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

      1. add-cube-cbrt 42.6%

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

      2. pow3 42.5%

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

   3. Applied egg-rr 42.5%

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

      1. rem-cube-cbrt 43.7%

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

      2. diff-sin 44.4%

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

   5. Applied egg-rr 44.4%

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

      1. associate-*r* 44.4%

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

      2. +-commutative 44.4%

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

      3. associate--l+ 94.3%

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

      4. +-commutative 94.3%

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

      5. associate-+l+ 94.5%

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

   7. Simplified 94.5%

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

   8. Taylor expanded in eps around 0 91.9%

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

      1. *-commutative 91.9%

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

   10. Simplified 91.9%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.45 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 2.6 \cdot 10^{-5}\right):\\ \;\;\;\;\sin \left(\varepsilon + x\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\frac{\varepsilon + \left(x + x\right)}{2}\right) \cdot \left(2 \cdot \left(\varepsilon \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 13: 72.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.1%

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

   1. add-cube-cbrt 57.2%

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

   2. pow3 57.1%

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

3. Applied egg-rr 57.1%

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

   1. rem-cube-cbrt 58.1%

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

   2. diff-sin 58.5%

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

5. Applied egg-rr 58.5%

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

   1. associate-*r* 58.5%

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

   2. +-commutative 58.5%

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

   3. associate--l+ 93.9%

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

   4. +-commutative 93.9%

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

   5. associate-+l+ 94.2%

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

7. Simplified 94.2%

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

8. Taylor expanded in eps around 0 73.9%

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

   1. *-commutative 73.9%

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

10. Simplified 73.9%

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

11. Final simplification 73.9%

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

Alternative 14: 52.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \cos x \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* eps (cos x)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.1%

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

2. Taylor expanded in eps around 0 55.6%

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

3. Final simplification 55.6%

   \[\leadsto \varepsilon \cdot \cos x \]

Alternative 15: 3.2% 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 58.1%

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

   1. add-cube-cbrt 57.2%

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

   2. pow3 57.1%

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

3. Applied egg-rr 57.1%

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

4. Taylor expanded in eps around 0 3.2%

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

   1. pow-base-1 3.2%

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

   2. *-lft-identity 3.2%

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

   3. +-inverses 3.2%

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

6. Simplified 3.2%

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

7. Final simplification 3.2%

   \[\leadsto 0 \]

Alternative 16: 10.6% 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 58.1%

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

2. Taylor expanded in eps around 0 55.6%

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

3. Taylor expanded in x around 0 11.8%

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

4. Final simplification 11.8%

   \[\leadsto \varepsilon \]

Developer target: 94.6% accurate, 1.0× speedup

Math

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

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(Float64(2.0 * 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[(N[(2.0 * N[Cos[N[(x + N[(eps / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(eps / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023310 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64
  :pre (and (and (and (> (fabs x) (fabs eps)) (> (fabs eps) (* (pow 10.0 -16.0) (fabs x)))) (< 1.0 x)) (< x 1000.0))

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

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