2sin (example 3.3)

Percentage Accurate: 42.1% → 99.5%

Time: 13.5s

Alternatives: 9

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 42.1% 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: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \sin \varepsilon - \sin x\\ \mathbf{if}\;\varepsilon \leq -0.00018:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + t_0\\ \mathbf{elif}\;\varepsilon \leq 0.00018:\\ \;\;\;\;-0.5 \cdot \left(\sin x \cdot {\varepsilon}^{2}\right) + \cos x \cdot \left(\varepsilon + -0.16666666666666666 \cdot {\varepsilon}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin x, \cos \varepsilon, t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (* (cos x) (sin eps)) (sin x))))
   (if (<= eps -0.00018)
     (+ (* (sin x) (cos eps)) t_0)
     (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) t_0)))))

C

double code(double x, double eps) {
	double t_0 = (cos(x) * sin(eps)) - sin(x);
	double tmp;
	if (eps <= -0.00018) {
		tmp = (sin(x) * cos(eps)) + t_0;
	} 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), t_0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(cos(x) * sin(eps)) - sin(x))
	tmp = 0.0
	if (eps <= -0.00018)
		tmp = Float64(Float64(sin(x) * cos(eps)) + t_0);
	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 = fma(sin(x), cos(eps), t_0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(N[Cos[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.00018], N[(N[(N[Sin[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision] + t$95$0), $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[Sin[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -1.80000000000000011e-4

   1. Initial program 53.5%

      \[\sin \left(x + \varepsilon\right) - \sin x \]
   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. associate--l+ 99.7%

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

   3. Applied egg-rr 99.7%

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

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

   1. Initial program 25.7%

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

   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 47.3%

      \[\sin \left(x + \varepsilon\right) - \sin x \]
   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+ 99.1%

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

      3. fma-def 99.3%

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

   3. Applied egg-rr 99.3%

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

4. Final simplification 99.7%

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

Alternative 2: 76.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\varepsilon + x\right) - \sin x\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-5} \lor \neg \left(t_0 \leq 2 \cdot 10^{-61}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos x \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (sin (+ eps x)) (sin x))))
   (if (or (<= t_0 -1e-5) (not (<= t_0 2e-61)))
     t_0
     (* 2.0 (* (cos x) (sin (* eps 0.5)))))))

C

double code(double x, double eps) {
	double t_0 = sin((eps + x)) - sin(x);
	double tmp;
	if ((t_0 <= -1e-5) || !(t_0 <= 2e-61)) {
		tmp = t_0;
	} else {
		tmp = 2.0 * (cos(x) * 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 <= (-1d-5)) .or. (.not. (t_0 <= 2d-61))) then
        tmp = t_0
    else
        tmp = 2.0d0 * (cos(x) * 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 <= -1e-5) || !(t_0 <= 2e-61)) {
		tmp = t_0;
	} else {
		tmp = 2.0 * (Math.cos(x) * 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 <= -1e-5) or not (t_0 <= 2e-61):
		tmp = t_0
	else:
		tmp = 2.0 * (math.cos(x) * 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 <= -1e-5) || !(t_0 <= 2e-61))
		tmp = t_0;
	else
		tmp = Float64(2.0 * Float64(cos(x) * 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 <= -1e-5) || ~((t_0 <= 2e-61)))
		tmp = t_0;
	else
		tmp = 2.0 * (cos(x) * 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, -1e-5], N[Not[LessEqual[t$95$0, 2e-61]], $MachinePrecision]], t$95$0, N[(2.0 * N[(N[Cos[x], $MachinePrecision] * 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)) < -1.00000000000000008e-5 or 2.0000000000000001e-61 < (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x))

   1. Initial program 68.6%

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

   if -1.00000000000000008e-5 < (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x)) < 2.0000000000000001e-61

   1. Initial program 20.2%

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

      1. add-cube-cbrt 19.5%

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

      2. pow3 19.4%

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

   3. Applied egg-rr 19.4%

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

      1. rem-cube-cbrt 20.2%

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

      2. diff-sin 20.2%

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

   5. Applied egg-rr 20.2%

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

      1. +-commutative 20.2%

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

      2. associate--l+ 83.3%

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

      3. +-commutative 83.3%

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

      4. associate-+l+ 83.3%

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

   7. Simplified 83.3%

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

   8. Taylor expanded in x around -inf 83.3%

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

   9. Taylor expanded in eps around 0 83.3%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin \left(\varepsilon + x\right) - \sin x \leq -1 \cdot 10^{-5} \lor \neg \left(\sin \left(\varepsilon + x\right) - \sin x \leq 2 \cdot 10^{-61}\right):\\ \;\;\;\;\sin \left(\varepsilon + x\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos x \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 3: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \sin \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.000175:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(t_0 - \sin x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.000155:\\ \;\;\;\;-0.5 \cdot \left(\sin x \cdot {\varepsilon}^{2}\right) + \cos x \cdot \left(\varepsilon + -0.16666666666666666 \cdot {\varepsilon}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin x, \cos \varepsilon, t_0\right) - \sin x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (cos x) (sin eps))))
   (if (<= eps -0.000175)
     (+ (* (sin x) (cos eps)) (- t_0 (sin x)))
     (if (<= eps 0.000155)
       (+
        (* -0.5 (* (sin x) (pow eps 2.0)))
        (* (cos x) (+ eps (* -0.16666666666666666 (pow eps 3.0)))))
       (- (fma (sin x) (cos eps) t_0) (sin x))))))

C

double code(double x, double eps) {
	double t_0 = cos(x) * sin(eps);
	double tmp;
	if (eps <= -0.000175) {
		tmp = (sin(x) * cos(eps)) + (t_0 - sin(x));
	} else if (eps <= 0.000155) {
		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), t_0) - sin(x);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(cos(x) * sin(eps))
	tmp = 0.0
	if (eps <= -0.000175)
		tmp = Float64(Float64(sin(x) * cos(eps)) + Float64(t_0 - sin(x)));
	elseif (eps <= 0.000155)
		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), t_0) - sin(x));
	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, -0.000175], N[(N[(N[Sin[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision] + N[(t$95$0 - N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.000155], 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] + t$95$0), $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -1.74999999999999998e-4

   1. Initial program 53.5%

      \[\sin \left(x + \varepsilon\right) - \sin x \]
   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. associate--l+ 99.7%

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

   3. Applied egg-rr 99.7%

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

   if -1.74999999999999998e-4 < eps < 1.55e-4

   1. Initial program 25.7%

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

   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 1.55e-4 < eps

   1. Initial program 47.3%

      \[\sin \left(x + \varepsilon\right) - \sin x \]
   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. fma-def 99.2%

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

   3. Applied egg-rr 99.2%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000175:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.000155:\\ \;\;\;\;-0.5 \cdot \left(\sin x \cdot {\varepsilon}^{2}\right) + \cos x \cdot \left(\varepsilon + -0.16666666666666666 \cdot {\varepsilon}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin x, \cos \varepsilon, \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array} \]

Alternative 4: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000165 \lor \neg \left(\varepsilon \leq 0.000175\right):\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\sin x \cdot {\varepsilon}^{2}\right) + \cos x \cdot \left(\varepsilon + -0.16666666666666666 \cdot {\varepsilon}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.000165) (not (<= eps 0.000175)))
   (+ (* (sin x) (cos eps)) (- (* (cos x) (sin eps)) (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 <= -0.000165) || !(eps <= 0.000175)) {
		tmp = (sin(x) * cos(eps)) + ((cos(x) * sin(eps)) - 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 <= (-0.000165d0)) .or. (.not. (eps <= 0.000175d0))) then
        tmp = (sin(x) * cos(eps)) + ((cos(x) * sin(eps)) - 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 <= -0.000165) || !(eps <= 0.000175)) {
		tmp = (Math.sin(x) * Math.cos(eps)) + ((Math.cos(x) * Math.sin(eps)) - 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 <= -0.000165) or not (eps <= 0.000175):
		tmp = (math.sin(x) * math.cos(eps)) + ((math.cos(x) * math.sin(eps)) - 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 <= -0.000165) || !(eps <= 0.000175))
		tmp = Float64(Float64(sin(x) * cos(eps)) + Float64(Float64(cos(x) * sin(eps)) - 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 <= -0.000165) || ~((eps <= 0.000175)))
		tmp = (sin(x) * cos(eps)) + ((cos(x) * sin(eps)) - 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, -0.000165], N[Not[LessEqual[eps, 0.000175]], $MachinePrecision]], N[(N[(N[Sin[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[x], $MachinePrecision] * N[Sin[eps], $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 < -1.65e-4 or 1.74999999999999998e-4 < eps

   1. Initial program 50.5%

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

      1. sin-sum 99.4%

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

      2. associate--l+ 99.4%

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

   3. Applied egg-rr 99.4%

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

   if -1.65e-4 < eps < 1.74999999999999998e-4

   1. Initial program 25.7%

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

   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 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000165 \lor \neg \left(\varepsilon \leq 0.000175\right):\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\sin x \cdot {\varepsilon}^{2}\right) + \cos x \cdot \left(\varepsilon + -0.16666666666666666 \cdot {\varepsilon}^{3}\right)\\ \end{array} \]

Alternative 5: 75.5% 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 37.0%

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

   1. add-cube-cbrt 36.1%

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

   2. pow3 36.0%

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

3. Applied egg-rr 36.0%

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

   1. rem-cube-cbrt 37.0%

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

   2. diff-sin 36.7%

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

5. Applied egg-rr 36.7%

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

   1. +-commutative 36.7%

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

   2. associate--l+ 78.1%

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

   3. +-commutative 78.1%

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

   4. associate-+l+ 78.1%

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

7. Simplified 78.1%

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

8. Taylor expanded in x around -inf 78.1%

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

9. Final simplification 78.1%

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

Alternative 6: 75.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.25 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 0.00044\right):\\ \;\;\;\;\sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -1.25e-5) (not (<= eps 0.00044))) (sin eps) (* eps (cos x))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.25e-5) || !(eps <= 0.00044)) {
		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 <= (-1.25d-5)) .or. (.not. (eps <= 0.00044d0))) 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 <= -1.25e-5) || !(eps <= 0.00044)) {
		tmp = Math.sin(eps);
	} else {
		tmp = eps * Math.cos(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -1.25e-5) or not (eps <= 0.00044):
		tmp = math.sin(eps)
	else:
		tmp = eps * math.cos(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -1.25e-5) || !(eps <= 0.00044))
		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 <= -1.25e-5) || ~((eps <= 0.00044)))
		tmp = sin(eps);
	else
		tmp = eps * cos(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -1.25e-5], N[Not[LessEqual[eps, 0.00044]], $MachinePrecision]], N[Sin[eps], $MachinePrecision], N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.25000000000000006e-5 or 4.40000000000000016e-4 < eps

   1. Initial program 50.5%

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

   2. Taylor expanded in x around 0 52.1%

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

   if -1.25000000000000006e-5 < eps < 4.40000000000000016e-4

   1. Initial program 25.4%

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

   2. Taylor expanded in eps around 0 99.5%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.25 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 0.00044\right):\\ \;\;\;\;\sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \]

Alternative 7: 55.3% 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 37.0%

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

2. Taylor expanded in x around 0 55.4%

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

3. Final simplification 55.4%

   \[\leadsto \sin \varepsilon \]

Alternative 8: 4.2% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

double code(double x, double eps) {
	return 0.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 0.0d0
end function

Java

public static double code(double x, double eps) {
	return 0.0;
}

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

function tmp = code(x, eps)
	tmp = 0.0;
end

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 37.0%

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

   1. add-cube-cbrt 36.1%

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

   2. pow3 36.0%

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

3. Applied egg-rr 36.0%

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

4. Taylor expanded in eps around 0 4.3%

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

   1. pow-base-1 4.3%

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

   2. *-lft-identity 4.3%

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

   3. +-inverses 4.3%

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

6. Simplified 4.3%

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

7. Final simplification 4.3%

   \[\leadsto 0 \]

Alternative 9: 29.5% 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 37.0%

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

2. Taylor expanded in eps around 0 15.6%

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

3. Taylor expanded in x around 0 32.9%

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

4. Final simplification 32.9%

   \[\leadsto \varepsilon \]

Developer target: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2023330 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64

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

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