2sin (example 3.3)

Percentage Accurate: 41.7% → 99.6%

Time: 16.4s

Alternatives: 12

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 41.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00016:\\ \;\;\;\;\mathsf{fma}\left(\cos \varepsilon, \sin x, \mathsf{fma}\left(\sin \varepsilon, \cos x, -\sin x\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00018:\\ \;\;\;\;-0.5 \cdot \left(\sin x \cdot {\varepsilon}^{2}\right) + \cos x \cdot \left(\varepsilon + -0.16666666666666666 \cdot {\varepsilon}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin x, \cos \varepsilon, \sin \varepsilon \cdot \cos x\right) - \sin x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.00016)
   (fma (cos eps) (sin x) (fma (sin eps) (cos x) (- (sin x))))
   (if (<= eps 0.00018)
     (+
      (* -0.5 (* (sin x) (pow eps 2.0)))
      (* (cos x) (+ eps (* -0.16666666666666666 (pow eps 3.0)))))
     (- (fma (sin x) (cos eps) (* (sin eps) (cos x))) (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.00016) {
		tmp = fma(cos(eps), sin(x), fma(sin(eps), cos(x), -sin(x)));
	} else if (eps <= 0.00018) {
		tmp = (-0.5 * (sin(x) * pow(eps, 2.0))) + (cos(x) * (eps + (-0.16666666666666666 * pow(eps, 3.0))));
	} else {
		tmp = fma(sin(x), cos(eps), (sin(eps) * cos(x))) - sin(x);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -0.00016)
		tmp = fma(cos(eps), sin(x), fma(sin(eps), cos(x), Float64(-sin(x))));
	elseif (eps <= 0.00018)
		tmp = Float64(Float64(-0.5 * Float64(sin(x) * (eps ^ 2.0))) + Float64(cos(x) * Float64(eps + Float64(-0.16666666666666666 * (eps ^ 3.0)))));
	else
		tmp = Float64(fma(sin(x), cos(eps), Float64(sin(eps) * cos(x))) - sin(x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -0.00016], N[(N[Cos[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision] + N[(N[Sin[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision] + (-N[Sin[x], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.00018], 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[(-0.16666666666666666 * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + N[(N[Sin[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -1.60000000000000013e-4

   1. Initial program 53.1%

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

      1. sin-sum 99.1%

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

      2. associate--l+ 99.0%

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

   3. Applied egg-rr 99.0%

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

      1. *-commutative 99.0%

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

      2. fma-udef 99.1%

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

      3. *-commutative 99.1%

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

      4. fma-neg 99.2%

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

   5. Simplified 99.2%

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

   if -1.60000000000000013e-4 < eps < 1.80000000000000011e-4

   1. Initial program 25.3%

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

   2. Taylor expanded in eps around 0 99.9%

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

      1. fma-def 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-*r* 99.9%

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

      4. distribute-rgt-out 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in x around inf 99.9%

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

   if 1.80000000000000011e-4 < eps

   1. Initial program 52.3%

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

      1. sin-sum 99.6%

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

      2. fma-def 99.6%

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

   3. Applied egg-rr 99.6%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00016:\\ \;\;\;\;\mathsf{fma}\left(\cos \varepsilon, \sin x, \mathsf{fma}\left(\sin \varepsilon, \cos x, -\sin x\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00018:\\ \;\;\;\;-0.5 \cdot \left(\sin x \cdot {\varepsilon}^{2}\right) + \cos x \cdot \left(\varepsilon + -0.16666666666666666 \cdot {\varepsilon}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin x, \cos \varepsilon, \sin \varepsilon \cdot \cos x\right) - \sin x\\ \end{array} \]

Alternative 2: 77.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\varepsilon + x\right) - \sin x\\ \mathbf{if}\;t_0 \leq -0.02 \lor \neg \left(t_0 \leq 10^{-51}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (sin (+ eps x)) (sin x))))
   (if (or (<= t_0 -0.02) (not (<= t_0 1e-51)))
     t_0
     (* (cos x) (* 2.0 (sin (* eps 0.5)))))))

C

double code(double x, double eps) {
	double t_0 = sin((eps + x)) - sin(x);
	double tmp;
	if ((t_0 <= -0.02) || !(t_0 <= 1e-51)) {
		tmp = t_0;
	} else {
		tmp = cos(x) * (2.0 * sin((eps * 0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((eps + x)) - sin(x)
    if ((t_0 <= (-0.02d0)) .or. (.not. (t_0 <= 1d-51))) then
        tmp = t_0
    else
        tmp = cos(x) * (2.0d0 * sin((eps * 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin((eps + x)) - Math.sin(x);
	double tmp;
	if ((t_0 <= -0.02) || !(t_0 <= 1e-51)) {
		tmp = t_0;
	} else {
		tmp = Math.cos(x) * (2.0 * Math.sin((eps * 0.5)));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.sin((eps + x)) - math.sin(x)
	tmp = 0
	if (t_0 <= -0.02) or not (t_0 <= 1e-51):
		tmp = t_0
	else:
		tmp = math.cos(x) * (2.0 * math.sin((eps * 0.5)))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(sin(Float64(eps + x)) - sin(x))
	tmp = 0.0
	if ((t_0 <= -0.02) || !(t_0 <= 1e-51))
		tmp = t_0;
	else
		tmp = Float64(cos(x) * Float64(2.0 * sin(Float64(eps * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = sin((eps + x)) - sin(x);
	tmp = 0.0;
	if ((t_0 <= -0.02) || ~((t_0 <= 1e-51)))
		tmp = t_0;
	else
		tmp = cos(x) * (2.0 * sin((eps * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Sin[N[(eps + x), $MachinePrecision]], $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -0.02], N[Not[LessEqual[t$95$0, 1e-51]], $MachinePrecision]], t$95$0, N[(N[Cos[x], $MachinePrecision] * N[(2.0 * N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x)) < -0.0200000000000000004 or 1e-51 < (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x))

   1. Initial program 64.8%

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

   if -0.0200000000000000004 < (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x)) < 1e-51

   1. Initial program 21.0%

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

      1. diff-sin 21.0%

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

      2. div-inv 21.0%

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

      3. associate--l+ 21.0%

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

      4. metadata-eval 21.0%

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

      5. div-inv 21.0%

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

      6. +-commutative 21.0%

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

      7. associate-+l+ 21.0%

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

      8. metadata-eval 21.0%

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

   3. Applied egg-rr 21.0%

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

      1. associate-*r* 21.0%

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

      2. *-commutative 21.0%

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

      3. *-commutative 21.0%

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

      4. +-commutative 21.0%

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

      5. count-2 21.0%

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

      6. fma-def 21.0%

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

      7. associate-+r- 21.0%

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

      8. +-commutative 21.0%

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

      9. associate--l+ 86.5%

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

      10. +-inverses 86.5%

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

   5. Simplified 86.5%

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

   6. Taylor expanded in x around inf 86.5%

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

      1. associate-*r* 86.5%

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

      2. *-commutative 86.5%

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

      3. associate-*r* 86.5%

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

      4. *-commutative 86.5%

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

   8. Simplified 86.5%

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

   9. Taylor expanded in eps around 0 86.5%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin \left(\varepsilon + x\right) - \sin x \leq -0.02 \lor \neg \left(\sin \left(\varepsilon + x\right) - \sin x \leq 10^{-51}\right):\\ \;\;\;\;\sin \left(\varepsilon + x\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 3: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00016 \lor \neg \left(\varepsilon \leq 0.000165\right):\\ \;\;\;\;\mathsf{fma}\left(\sin x, \cos \varepsilon, \sin \varepsilon \cdot \cos x\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\sin x \cdot {\varepsilon}^{2}\right) + \cos x \cdot \left(\varepsilon + -0.16666666666666666 \cdot {\varepsilon}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.00016) (not (<= eps 0.000165)))
   (- (fma (sin x) (cos eps) (* (sin eps) (cos x))) (sin x))
   (+
    (* -0.5 (* (sin x) (pow eps 2.0)))
    (* (cos x) (+ eps (* -0.16666666666666666 (pow eps 3.0)))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.00016) || !(eps <= 0.000165)) {
		tmp = fma(sin(x), cos(eps), (sin(eps) * cos(x))) - sin(x);
	} else {
		tmp = (-0.5 * (sin(x) * pow(eps, 2.0))) + (cos(x) * (eps + (-0.16666666666666666 * pow(eps, 3.0))));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.00016) || !(eps <= 0.000165))
		tmp = Float64(fma(sin(x), cos(eps), Float64(sin(eps) * cos(x))) - sin(x));
	else
		tmp = Float64(Float64(-0.5 * Float64(sin(x) * (eps ^ 2.0))) + Float64(cos(x) * Float64(eps + Float64(-0.16666666666666666 * (eps ^ 3.0)))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.00016], N[Not[LessEqual[eps, 0.000165]], $MachinePrecision]], N[(N[(N[Sin[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + N[(N[Sin[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision], 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[(-0.16666666666666666 * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.60000000000000013e-4 or 1.65e-4 < eps

   1. Initial program 52.7%

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

      1. sin-sum 99.3%

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

      2. fma-def 99.4%

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

   3. Applied egg-rr 99.4%

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

   if -1.60000000000000013e-4 < eps < 1.65e-4

   1. Initial program 25.3%

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

   2. Taylor expanded in eps around 0 99.9%

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

      1. fma-def 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-*r* 99.9%

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

      4. distribute-rgt-out 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in x around inf 99.9%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00016 \lor \neg \left(\varepsilon \leq 0.000165\right):\\ \;\;\;\;\mathsf{fma}\left(\sin x, \cos \varepsilon, \sin \varepsilon \cdot \cos x\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\sin x \cdot {\varepsilon}^{2}\right) + \cos x \cdot \left(\varepsilon + -0.16666666666666666 \cdot {\varepsilon}^{3}\right)\\ \end{array} \]

Alternative 4: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00017 \lor \neg \left(\varepsilon \leq 0.00018\right):\\ \;\;\;\;\cos \varepsilon \cdot \sin x + \left(\sin \varepsilon \cdot \cos x - \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\sin x \cdot {\varepsilon}^{2}\right) + \cos x \cdot \left(\varepsilon + -0.16666666666666666 \cdot {\varepsilon}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.00017) (not (<= eps 0.00018)))
   (+ (* (cos eps) (sin x)) (- (* (sin eps) (cos x)) (sin x)))
   (+
    (* -0.5 (* (sin x) (pow eps 2.0)))
    (* (cos x) (+ eps (* -0.16666666666666666 (pow eps 3.0)))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.00017) || !(eps <= 0.00018)) {
		tmp = (cos(eps) * sin(x)) + ((sin(eps) * cos(x)) - sin(x));
	} else {
		tmp = (-0.5 * (sin(x) * pow(eps, 2.0))) + (cos(x) * (eps + (-0.16666666666666666 * pow(eps, 3.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-0.00017d0)) .or. (.not. (eps <= 0.00018d0))) then
        tmp = (cos(eps) * sin(x)) + ((sin(eps) * cos(x)) - sin(x))
    else
        tmp = ((-0.5d0) * (sin(x) * (eps ** 2.0d0))) + (cos(x) * (eps + ((-0.16666666666666666d0) * (eps ** 3.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.00017) || !(eps <= 0.00018)) {
		tmp = (Math.cos(eps) * Math.sin(x)) + ((Math.sin(eps) * Math.cos(x)) - Math.sin(x));
	} else {
		tmp = (-0.5 * (Math.sin(x) * Math.pow(eps, 2.0))) + (Math.cos(x) * (eps + (-0.16666666666666666 * Math.pow(eps, 3.0))));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.00017) or not (eps <= 0.00018):
		tmp = (math.cos(eps) * math.sin(x)) + ((math.sin(eps) * math.cos(x)) - math.sin(x))
	else:
		tmp = (-0.5 * (math.sin(x) * math.pow(eps, 2.0))) + (math.cos(x) * (eps + (-0.16666666666666666 * math.pow(eps, 3.0))))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.00017) || !(eps <= 0.00018))
		tmp = Float64(Float64(cos(eps) * sin(x)) + Float64(Float64(sin(eps) * cos(x)) - sin(x)));
	else
		tmp = Float64(Float64(-0.5 * Float64(sin(x) * (eps ^ 2.0))) + Float64(cos(x) * Float64(eps + Float64(-0.16666666666666666 * (eps ^ 3.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.00017) || ~((eps <= 0.00018)))
		tmp = (cos(eps) * sin(x)) + ((sin(eps) * cos(x)) - sin(x));
	else
		tmp = (-0.5 * (sin(x) * (eps ^ 2.0))) + (cos(x) * (eps + (-0.16666666666666666 * (eps ^ 3.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.00017], N[Not[LessEqual[eps, 0.00018]], $MachinePrecision]], N[(N[(N[Cos[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sin[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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[(-0.16666666666666666 * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.7e-4 or 1.80000000000000011e-4 < eps

   1. Initial program 52.7%

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

      1. sin-sum 99.3%

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

      2. associate--l+ 99.3%

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

   3. Applied egg-rr 99.3%

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

   if -1.7e-4 < eps < 1.80000000000000011e-4

   1. Initial program 25.3%

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

   2. Taylor expanded in eps around 0 99.9%

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

      1. fma-def 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-*r* 99.9%

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

      4. distribute-rgt-out 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in x around inf 99.9%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00017 \lor \neg \left(\varepsilon \leq 0.00018\right):\\ \;\;\;\;\cos \varepsilon \cdot \sin x + \left(\sin \varepsilon \cdot \cos x - \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\sin x \cdot {\varepsilon}^{2}\right) + \cos x \cdot \left(\varepsilon + -0.16666666666666666 \cdot {\varepsilon}^{3}\right)\\ \end{array} \]

Alternative 5: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00017 \lor \neg \left(\varepsilon \leq 0.000195\right):\\ \;\;\;\;\left(\sin \varepsilon \cdot \cos x + \cos \varepsilon \cdot \sin x\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\sin x \cdot {\varepsilon}^{2}\right) + \cos x \cdot \left(\varepsilon + -0.16666666666666666 \cdot {\varepsilon}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.00017) (not (<= eps 0.000195)))
   (- (+ (* (sin eps) (cos x)) (* (cos eps) (sin x))) (sin x))
   (+
    (* -0.5 (* (sin x) (pow eps 2.0)))
    (* (cos x) (+ eps (* -0.16666666666666666 (pow eps 3.0)))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.00017) || !(eps <= 0.000195)) {
		tmp = ((sin(eps) * cos(x)) + (cos(eps) * sin(x))) - sin(x);
	} else {
		tmp = (-0.5 * (sin(x) * pow(eps, 2.0))) + (cos(x) * (eps + (-0.16666666666666666 * pow(eps, 3.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-0.00017d0)) .or. (.not. (eps <= 0.000195d0))) then
        tmp = ((sin(eps) * cos(x)) + (cos(eps) * sin(x))) - sin(x)
    else
        tmp = ((-0.5d0) * (sin(x) * (eps ** 2.0d0))) + (cos(x) * (eps + ((-0.16666666666666666d0) * (eps ** 3.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.00017) || !(eps <= 0.000195)) {
		tmp = ((Math.sin(eps) * Math.cos(x)) + (Math.cos(eps) * Math.sin(x))) - Math.sin(x);
	} else {
		tmp = (-0.5 * (Math.sin(x) * Math.pow(eps, 2.0))) + (Math.cos(x) * (eps + (-0.16666666666666666 * Math.pow(eps, 3.0))));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.00017) or not (eps <= 0.000195):
		tmp = ((math.sin(eps) * math.cos(x)) + (math.cos(eps) * math.sin(x))) - math.sin(x)
	else:
		tmp = (-0.5 * (math.sin(x) * math.pow(eps, 2.0))) + (math.cos(x) * (eps + (-0.16666666666666666 * math.pow(eps, 3.0))))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.00017) || !(eps <= 0.000195))
		tmp = Float64(Float64(Float64(sin(eps) * cos(x)) + Float64(cos(eps) * sin(x))) - sin(x));
	else
		tmp = Float64(Float64(-0.5 * Float64(sin(x) * (eps ^ 2.0))) + Float64(cos(x) * Float64(eps + Float64(-0.16666666666666666 * (eps ^ 3.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.00017) || ~((eps <= 0.000195)))
		tmp = ((sin(eps) * cos(x)) + (cos(eps) * sin(x))) - sin(x);
	else
		tmp = (-0.5 * (sin(x) * (eps ^ 2.0))) + (cos(x) * (eps + (-0.16666666666666666 * (eps ^ 3.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.00017], N[Not[LessEqual[eps, 0.000195]], $MachinePrecision]], N[(N[(N[(N[Sin[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision], 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[(-0.16666666666666666 * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.7e-4 or 1.94999999999999996e-4 < eps

   1. Initial program 52.7%

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

      1. sin-sum 99.3%

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

   3. Applied egg-rr 99.3%

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

   if -1.7e-4 < eps < 1.94999999999999996e-4

   1. Initial program 25.3%

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

   2. Taylor expanded in eps around 0 99.9%

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

      1. fma-def 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-*r* 99.9%

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

      4. distribute-rgt-out 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in x around inf 99.9%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00017 \lor \neg \left(\varepsilon \leq 0.000195\right):\\ \;\;\;\;\left(\sin \varepsilon \cdot \cos x + \cos \varepsilon \cdot \sin x\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\sin x \cdot {\varepsilon}^{2}\right) + \cos x \cdot \left(\varepsilon + -0.16666666666666666 \cdot {\varepsilon}^{3}\right)\\ \end{array} \]

Alternative 6: 76.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.5%

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

   1. add-sqr-sqrt 19.5%

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

   2. sqrt-unprod 20.0%

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

   3. pow2 20.0%

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

3. Applied egg-rr 20.0%

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

   1. sqrt-pow1 38.5%

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

   2. metadata-eval 38.5%

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

   3. pow1 38.5%

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

   4. diff-sin 38.2%

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

   5. associate-*r* 38.2%

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

   6. div-inv 38.2%

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

   7. associate--l+ 38.1%

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

   8. metadata-eval 38.1%

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

   9. div-inv 38.1%

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

   10. associate-+l+ 38.1%

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

   11. +-commutative 38.1%

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

   12. metadata-eval 38.1%

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

5. Applied egg-rr 38.1%

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

6. Taylor expanded in x around -inf 77.9%

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

7. Final simplification 77.9%

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

Alternative 7: 76.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.5%

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

   1. add-sqr-sqrt 19.5%

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

   2. sqrt-unprod 20.0%

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

   3. pow2 20.0%

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

3. Applied egg-rr 20.0%

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

   1. sqrt-pow1 38.5%

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

   2. metadata-eval 38.5%

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

   3. pow1 38.5%

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

   4. diff-sin 38.2%

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

   5. associate-*r* 38.2%

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

   6. div-inv 38.2%

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

   7. associate--l+ 38.1%

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

   8. metadata-eval 38.1%

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

   9. div-inv 38.1%

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

   10. associate-+l+ 38.1%

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

   11. +-commutative 38.1%

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

   12. metadata-eval 38.1%

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

5. Applied egg-rr 38.1%

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

6. Taylor expanded in x around 0 77.9%

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

7. Final simplification 77.9%

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

Alternative 8: 76.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{-6} \lor \neg \left(x \leq 0.00024\right):\\ \;\;\;\;\cos x \cdot \left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon + x \cdot \left(\cos \varepsilon + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -7.6e-6) (not (<= x 0.00024)))
   (* (cos x) (* 2.0 (sin (* eps 0.5))))
   (+ (sin eps) (* x (+ (cos eps) -1.0)))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -7.6e-6) || !(x <= 0.00024)) {
		tmp = cos(x) * (2.0 * sin((eps * 0.5)));
	} else {
		tmp = sin(eps) + (x * (cos(eps) + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= (-7.6d-6)) .or. (.not. (x <= 0.00024d0))) then
        tmp = cos(x) * (2.0d0 * sin((eps * 0.5d0)))
    else
        tmp = sin(eps) + (x * (cos(eps) + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -7.6e-6) || !(x <= 0.00024)) {
		tmp = Math.cos(x) * (2.0 * Math.sin((eps * 0.5)));
	} else {
		tmp = Math.sin(eps) + (x * (Math.cos(eps) + -1.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -7.6e-6) or not (x <= 0.00024):
		tmp = math.cos(x) * (2.0 * math.sin((eps * 0.5)))
	else:
		tmp = math.sin(eps) + (x * (math.cos(eps) + -1.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -7.6e-6) || !(x <= 0.00024))
		tmp = Float64(cos(x) * Float64(2.0 * sin(Float64(eps * 0.5))));
	else
		tmp = Float64(sin(eps) + Float64(x * Float64(cos(eps) + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -7.6e-6) || ~((x <= 0.00024)))
		tmp = cos(x) * (2.0 * sin((eps * 0.5)));
	else
		tmp = sin(eps) + (x * (cos(eps) + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -7.6e-6], N[Not[LessEqual[x, 0.00024]], $MachinePrecision]], N[(N[Cos[x], $MachinePrecision] * N[(2.0 * N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[eps], $MachinePrecision] + N[(x * N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.6000000000000001e-6 or 2.40000000000000006e-4 < x

   1. Initial program 7.1%

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

      1. diff-sin 6.7%

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

      2. div-inv 6.7%

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

      3. associate--l+ 6.6%

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

      4. metadata-eval 6.6%

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

      5. div-inv 6.6%

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

      6. +-commutative 6.6%

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

      7. associate-+l+ 6.5%

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

      8. metadata-eval 6.5%

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

   3. Applied egg-rr 6.5%

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

      1. associate-*r* 6.5%

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

      2. *-commutative 6.5%

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

      3. *-commutative 6.5%

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

      4. +-commutative 6.5%

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

      5. count-2 6.5%

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

      6. fma-def 6.5%

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

      7. associate-+r- 6.6%

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

      8. +-commutative 6.6%

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

      9. associate--l+ 56.1%

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

      10. +-inverses 56.1%

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

   5. Simplified 56.1%

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

   6. Taylor expanded in x around inf 56.1%

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

      1. associate-*r* 56.1%

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

      2. *-commutative 56.1%

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

      3. associate-*r* 56.1%

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

      4. *-commutative 56.1%

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

   8. Simplified 56.1%

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

   9. Taylor expanded in eps around 0 55.8%

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

   if -7.6000000000000001e-6 < x < 2.40000000000000006e-4

   1. Initial program 69.4%

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

   2. Taylor expanded in x around 0 99.9%

      \[\leadsto \color{blue}{\sin \varepsilon + x \cdot \left(\cos \varepsilon - 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{-6} \lor \neg \left(x \leq 0.00024\right):\\ \;\;\;\;\cos x \cdot \left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon + x \cdot \left(\cos \varepsilon + -1\right)\\ \end{array} \]

Alternative 9: 76.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0004:\\ \;\;\;\;\sin \left(\varepsilon + x\right) - \sin x\\ \mathbf{elif}\;\varepsilon \leq 5.8 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.0004)
   (- (sin (+ eps x)) (sin x))
   (if (<= eps 5.8e-5) (* eps (cos x)) (sin eps))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.0004) {
		tmp = sin((eps + x)) - sin(x);
	} else if (eps <= 5.8e-5) {
		tmp = eps * cos(x);
	} else {
		tmp = sin(eps);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= (-0.0004d0)) then
        tmp = sin((eps + x)) - sin(x)
    else if (eps <= 5.8d-5) then
        tmp = eps * cos(x)
    else
        tmp = sin(eps)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -0.0004) {
		tmp = Math.sin((eps + x)) - Math.sin(x);
	} else if (eps <= 5.8e-5) {
		tmp = eps * Math.cos(x);
	} else {
		tmp = Math.sin(eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -0.0004:
		tmp = math.sin((eps + x)) - math.sin(x)
	elif eps <= 5.8e-5:
		tmp = eps * math.cos(x)
	else:
		tmp = math.sin(eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -0.0004)
		tmp = Float64(sin(Float64(eps + x)) - sin(x));
	elseif (eps <= 5.8e-5)
		tmp = Float64(eps * cos(x));
	else
		tmp = sin(eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -0.0004)
		tmp = sin((eps + x)) - sin(x);
	elseif (eps <= 5.8e-5)
		tmp = eps * cos(x);
	else
		tmp = sin(eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -0.0004], N[(N[Sin[N[(eps + x), $MachinePrecision]], $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 5.8e-5], N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision], N[Sin[eps], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -4.00000000000000019e-4

   1. Initial program 53.1%

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

   if -4.00000000000000019e-4 < eps < 5.8e-5

   1. Initial program 25.3%

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

   2. Taylor expanded in eps around 0 99.6%

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

   if 5.8e-5 < eps

   1. Initial program 52.3%

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

   2. Taylor expanded in x around 0 53.1%

      \[\leadsto \color{blue}{\sin \varepsilon} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0004:\\ \;\;\;\;\sin \left(\varepsilon + x\right) - \sin x\\ \mathbf{elif}\;\varepsilon \leq 5.8 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 10: 76.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0021 \lor \neg \left(\varepsilon \leq 3.8 \cdot 10^{-5}\right):\\ \;\;\;\;\sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.0021) (not (<= eps 3.8e-5))) (sin eps) (* eps (cos x))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.0021) || !(eps <= 3.8e-5)) {
		tmp = sin(eps);
	} else {
		tmp = eps * cos(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-0.0021d0)) .or. (.not. (eps <= 3.8d-5))) then
        tmp = sin(eps)
    else
        tmp = eps * cos(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.0021) || !(eps <= 3.8e-5)) {
		tmp = Math.sin(eps);
	} else {
		tmp = eps * Math.cos(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.0021) or not (eps <= 3.8e-5):
		tmp = math.sin(eps)
	else:
		tmp = eps * math.cos(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.0021) || !(eps <= 3.8e-5))
		tmp = sin(eps);
	else
		tmp = Float64(eps * cos(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.0021) || ~((eps <= 3.8e-5)))
		tmp = sin(eps);
	else
		tmp = eps * cos(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.0021], N[Not[LessEqual[eps, 3.8e-5]], $MachinePrecision]], N[Sin[eps], $MachinePrecision], N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -0.00209999999999999987 or 3.8000000000000002e-5 < eps

   1. Initial program 52.7%

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

   2. Taylor expanded in x around 0 52.8%

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

   if -0.00209999999999999987 < eps < 3.8000000000000002e-5

   1. Initial program 25.3%

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

   2. Taylor expanded in eps around 0 99.6%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0021 \lor \neg \left(\varepsilon \leq 3.8 \cdot 10^{-5}\right):\\ \;\;\;\;\sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \]

Alternative 11: 55.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.5%

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

2. Taylor expanded in x around 0 55.3%

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

3. Final simplification 55.3%

   \[\leadsto \sin \varepsilon \]

Alternative 12: 29.6% 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 38.5%

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

2. Taylor expanded in eps around 0 15.0%

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

3. Taylor expanded in x around 0 31.6%

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

4. Final simplification 31.6%

   \[\leadsto \varepsilon \]

Developer target: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2023339 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64

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

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