2sin (example 3.3)

Percentage Accurate: 42.4% → 99.6%

Time: 18.9s

Alternatives: 11

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: 42.4% 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

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

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.00017)
   (fma (sin x) (cos eps) (- (* (cos x) (sin eps)) (sin x)))
   (if (<= eps 0.00018)
     (+
      (* -0.5 (* (sin x) (pow eps 2.0)))
      (+ (* -0.16666666666666666 (* (cos x) (pow eps 3.0))) (* eps (cos x))))
     (fma (cos eps) (sin x) (fma (sin eps) (cos x) (- (sin x)))))))

C

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

Julia

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

Wolfram

code[x_, eps_] := If[LessEqual[eps, -0.00017], N[(N[Sin[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + N[(N[(N[Cos[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $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[(-0.16666666666666666 * N[(N[Cos[x], $MachinePrecision] * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision] + N[(N[Sin[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision] + (-N[Sin[x], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -1.7e-4

   1. Initial program 52.2%

      \[\sin \left(x + \varepsilon\right) - \sin x \]
   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+ 99.5%

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

      3. fma-def 99.5%

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

   3. Applied egg-rr 99.5%

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

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

   1. Initial program 32.2%

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

   2. Taylor expanded in eps around 0 99.8%

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

   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 99.2%

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

      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%

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

      1. *-commutative 99.3%

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

      2. fma-udef 99.5%

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

      3. *-commutative 99.5%

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

      4. fma-neg 99.5%

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

   5. Simplified 99.5%

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

4. Final simplification 99.7%

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

Alternative 2: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(0.5 \cdot \varepsilon\right)\\ \mathsf{log1p}\left(\mathsf{expm1}\left(\left(\cos \left(0.5 \cdot \varepsilon\right) \cdot \cos x - t_0 \cdot \sin x\right) \cdot \left(t_0 \cdot 2\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (* 0.5 eps))))
   (log1p
    (expm1
     (* (- (* (cos (* 0.5 eps)) (cos x)) (* t_0 (sin x))) (* t_0 2.0))))))

C

double code(double x, double eps) {
	double t_0 = sin((0.5 * eps));
	return log1p(expm1((((cos((0.5 * eps)) * cos(x)) - (t_0 * sin(x))) * (t_0 * 2.0))));
}

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin((0.5 * eps));
	return Math.log1p(Math.expm1((((Math.cos((0.5 * eps)) * Math.cos(x)) - (t_0 * Math.sin(x))) * (t_0 * 2.0))));
}

Python

def code(x, eps):
	t_0 = math.sin((0.5 * eps))
	return math.log1p(math.expm1((((math.cos((0.5 * eps)) * math.cos(x)) - (t_0 * math.sin(x))) * (t_0 * 2.0))))

Julia

function code(x, eps)
	t_0 = sin(Float64(0.5 * eps))
	return log1p(expm1(Float64(Float64(Float64(cos(Float64(0.5 * eps)) * cos(x)) - Float64(t_0 * sin(x))) * Float64(t_0 * 2.0))))
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Sin[N[(0.5 * eps), $MachinePrecision]], $MachinePrecision]}, N[Log[1 + N[(Exp[N[(N[(N[(N[Cos[N[(0.5 * eps), $MachinePrecision]], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * 2.0), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 40.9%

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

   1. diff-sin 40.5%

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

   2. div-inv 40.5%

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

   3. metadata-eval 40.5%

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

   4. div-inv 40.5%

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

   5. +-commutative 40.5%

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

   6. metadata-eval 40.5%

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

3. Applied egg-rr 40.5%

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

   1. associate-*r* 40.5%

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

   2. *-commutative 40.5%

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

   3. associate-*l* 40.5%

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

   4. +-commutative 40.5%

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

   5. associate--l+ 75.7%

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

   6. +-inverses 75.7%

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

   7. *-commutative 75.7%

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

   8. associate-+r+ 75.7%

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

   9. +-commutative 75.7%

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

5. Simplified 75.7%

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

6. Taylor expanded in eps around inf 75.7%

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

   1. *-commutative 75.7%

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

   2. +-commutative 75.7%

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

   3. *-lft-identity 75.7%

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

   4. metadata-eval 75.7%

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

   5. cancel-sign-sub-inv 75.7%

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

   6. *-commutative 75.7%

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

   7. associate-*r* 75.7%

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

8. Simplified 75.7%

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

   1. log1p-expm1-u 75.7%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(2 \cdot \cos \left(0.5 \cdot \left(\varepsilon - x \cdot -2\right)\right)\right) \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\right)} \]

   2. *-commutative 75.7%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left(\cos \left(0.5 \cdot \left(\varepsilon - x \cdot -2\right)\right) \cdot 2\right)} \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\right) \]

   3. associate-*l* 75.7%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\cos \left(0.5 \cdot \left(\varepsilon - x \cdot -2\right)\right) \cdot \left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)}\right)\right) \]

   4. sub-neg 75.7%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(0.5 \cdot \color{blue}{\left(\varepsilon + \left(-x \cdot -2\right)\right)}\right) \cdot \left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\right)\right) \]

   5. distribute-rgt-neg-in 75.7%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(0.5 \cdot \left(\varepsilon + \color{blue}{x \cdot \left(--2\right)}\right)\right) \cdot \left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\right)\right) \]

   6. metadata-eval 75.7%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(0.5 \cdot \left(\varepsilon + x \cdot \color{blue}{2}\right)\right) \cdot \left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\right)\right) \]

   7. +-rgt-identity 75.7%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(0.5 \cdot \left(\varepsilon + x \cdot 2\right)\right) \cdot \left(2 \cdot \sin \left(\color{blue}{\left(\varepsilon + 0\right)} \cdot 0.5\right)\right)\right)\right) \]

   8. +-rgt-identity 75.7%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(0.5 \cdot \left(\varepsilon + x \cdot 2\right)\right) \cdot \left(2 \cdot \sin \left(\color{blue}{\varepsilon} \cdot 0.5\right)\right)\right)\right) \]

   9. metadata-eval 75.7%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(0.5 \cdot \left(\varepsilon + x \cdot 2\right)\right) \cdot \left(2 \cdot \sin \left(\varepsilon \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

   10. div-inv 75.7%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(0.5 \cdot \left(\varepsilon + x \cdot 2\right)\right) \cdot \left(2 \cdot \sin \color{blue}{\left(\frac{\varepsilon}{2}\right)}\right)\right)\right) \]

   11. +-rgt-identity 75.7%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(0.5 \cdot \left(\varepsilon + x \cdot 2\right)\right) \cdot \left(2 \cdot \sin \left(\frac{\color{blue}{\varepsilon + 0}}{2}\right)\right)\right)\right) \]

   12. +-rgt-identity 75.7%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(0.5 \cdot \left(\varepsilon + x \cdot 2\right)\right) \cdot \left(2 \cdot \sin \left(\frac{\color{blue}{\varepsilon}}{2}\right)\right)\right)\right) \]

   13. div-inv 75.7%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(0.5 \cdot \left(\varepsilon + x \cdot 2\right)\right) \cdot \left(2 \cdot \sin \color{blue}{\left(\varepsilon \cdot \frac{1}{2}\right)}\right)\right)\right) \]

   14. metadata-eval 75.7%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(0.5 \cdot \left(\varepsilon + x \cdot 2\right)\right) \cdot \left(2 \cdot \sin \left(\varepsilon \cdot \color{blue}{0.5}\right)\right)\right)\right) \]

   15. *-commutative 75.7%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(0.5 \cdot \left(\varepsilon + x \cdot 2\right)\right) \cdot \left(2 \cdot \sin \color{blue}{\left(0.5 \cdot \varepsilon\right)}\right)\right)\right) \]

10. Applied egg-rr 75.7%

    \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(0.5 \cdot \left(\varepsilon + x \cdot 2\right)\right) \cdot \left(2 \cdot \sin \left(0.5 \cdot \varepsilon\right)\right)\right)\right)} \]
11. Step-by-step derivation

    1. distribute-lft-in 75.7%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \color{blue}{\left(0.5 \cdot \varepsilon + 0.5 \cdot \left(x \cdot 2\right)\right)} \cdot \left(2 \cdot \sin \left(0.5 \cdot \varepsilon\right)\right)\right)\right) \]

    2. cos-sum 99.5%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left(\cos \left(0.5 \cdot \varepsilon\right) \cdot \cos \left(0.5 \cdot \left(x \cdot 2\right)\right) - \sin \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(0.5 \cdot \left(x \cdot 2\right)\right)\right)} \cdot \left(2 \cdot \sin \left(0.5 \cdot \varepsilon\right)\right)\right)\right) \]

    3. *-commutative 99.5%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\left(\cos \left(0.5 \cdot \varepsilon\right) \cdot \cos \left(0.5 \cdot \color{blue}{\left(2 \cdot x\right)}\right) - \sin \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(0.5 \cdot \left(x \cdot 2\right)\right)\right) \cdot \left(2 \cdot \sin \left(0.5 \cdot \varepsilon\right)\right)\right)\right) \]

    4. associate-*r* 99.5%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\left(\cos \left(0.5 \cdot \varepsilon\right) \cdot \cos \color{blue}{\left(\left(0.5 \cdot 2\right) \cdot x\right)} - \sin \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(0.5 \cdot \left(x \cdot 2\right)\right)\right) \cdot \left(2 \cdot \sin \left(0.5 \cdot \varepsilon\right)\right)\right)\right) \]

    5. metadata-eval 99.5%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\left(\cos \left(0.5 \cdot \varepsilon\right) \cdot \cos \left(\color{blue}{1} \cdot x\right) - \sin \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(0.5 \cdot \left(x \cdot 2\right)\right)\right) \cdot \left(2 \cdot \sin \left(0.5 \cdot \varepsilon\right)\right)\right)\right) \]

    6. *-un-lft-identity 99.5%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\left(\cos \left(0.5 \cdot \varepsilon\right) \cdot \cos \color{blue}{x} - \sin \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(0.5 \cdot \left(x \cdot 2\right)\right)\right) \cdot \left(2 \cdot \sin \left(0.5 \cdot \varepsilon\right)\right)\right)\right) \]

    7. *-commutative 99.5%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\left(\cos \left(0.5 \cdot \varepsilon\right) \cdot \cos x - \sin \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(0.5 \cdot \color{blue}{\left(2 \cdot x\right)}\right)\right) \cdot \left(2 \cdot \sin \left(0.5 \cdot \varepsilon\right)\right)\right)\right) \]

    8. associate-*r* 99.5%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\left(\cos \left(0.5 \cdot \varepsilon\right) \cdot \cos x - \sin \left(0.5 \cdot \varepsilon\right) \cdot \sin \color{blue}{\left(\left(0.5 \cdot 2\right) \cdot x\right)}\right) \cdot \left(2 \cdot \sin \left(0.5 \cdot \varepsilon\right)\right)\right)\right) \]

    9. metadata-eval 99.5%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\left(\cos \left(0.5 \cdot \varepsilon\right) \cdot \cos x - \sin \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(\color{blue}{1} \cdot x\right)\right) \cdot \left(2 \cdot \sin \left(0.5 \cdot \varepsilon\right)\right)\right)\right) \]

    10. *-un-lft-identity 99.5%

        \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\left(\cos \left(0.5 \cdot \varepsilon\right) \cdot \cos x - \sin \left(0.5 \cdot \varepsilon\right) \cdot \sin \color{blue}{x}\right) \cdot \left(2 \cdot \sin \left(0.5 \cdot \varepsilon\right)\right)\right)\right) \]

12. Applied egg-rr 99.5%

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left(\cos \left(0.5 \cdot \varepsilon\right) \cdot \cos x - \sin \left(0.5 \cdot \varepsilon\right) \cdot \sin x\right)} \cdot \left(2 \cdot \sin \left(0.5 \cdot \varepsilon\right)\right)\right)\right) \]

13. Final simplification 99.5%

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\left(\cos \left(0.5 \cdot \varepsilon\right) \cdot \cos x - \sin \left(0.5 \cdot \varepsilon\right) \cdot \sin x\right) \cdot \left(\sin \left(0.5 \cdot \varepsilon\right) \cdot 2\right)\right)\right) \]

Alternative 3: 77.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\varepsilon + x\right) - \sin x\\ \mathbf{if}\;t_0 \leq -0.002 \lor \neg \left(t_0 \leq 10^{-13}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sin \left(0.5 \cdot \varepsilon\right) \cdot \left(\cos x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (sin (+ eps x)) (sin x))))
   (if (or (<= t_0 -0.002) (not (<= t_0 1e-13)))
     t_0
     (* (sin (* 0.5 eps)) (* (cos x) 2.0)))))

C

double code(double x, double eps) {
	double t_0 = sin((eps + x)) - sin(x);
	double tmp;
	if ((t_0 <= -0.002) || !(t_0 <= 1e-13)) {
		tmp = t_0;
	} else {
		tmp = sin((0.5 * eps)) * (cos(x) * 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((eps + x)) - sin(x)
    if ((t_0 <= (-0.002d0)) .or. (.not. (t_0 <= 1d-13))) then
        tmp = t_0
    else
        tmp = sin((0.5d0 * eps)) * (cos(x) * 2.0d0)
    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 <= -0.002) || !(t_0 <= 1e-13)) {
		tmp = t_0;
	} else {
		tmp = Math.sin((0.5 * eps)) * (Math.cos(x) * 2.0);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.sin((eps + x)) - math.sin(x)
	tmp = 0
	if (t_0 <= -0.002) or not (t_0 <= 1e-13):
		tmp = t_0
	else:
		tmp = math.sin((0.5 * eps)) * (math.cos(x) * 2.0)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(sin(Float64(eps + x)) - sin(x))
	tmp = 0.0
	if ((t_0 <= -0.002) || !(t_0 <= 1e-13))
		tmp = t_0;
	else
		tmp = Float64(sin(Float64(0.5 * eps)) * Float64(cos(x) * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = sin((eps + x)) - sin(x);
	tmp = 0.0;
	if ((t_0 <= -0.002) || ~((t_0 <= 1e-13)))
		tmp = t_0;
	else
		tmp = sin((0.5 * eps)) * (cos(x) * 2.0);
	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, -0.002], N[Not[LessEqual[t$95$0, 1e-13]], $MachinePrecision]], t$95$0, N[(N[Sin[N[(0.5 * eps), $MachinePrecision]], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x)) < -2e-3 or 1e-13 < (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x))

   1. Initial program 65.0%

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

   if -2e-3 < (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x)) < 1e-13

   1. Initial program 25.2%

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

      1. diff-sin 25.2%

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

      2. div-inv 25.2%

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

      3. metadata-eval 25.2%

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

      4. div-inv 25.2%

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

      5. +-commutative 25.2%

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

      6. metadata-eval 25.2%

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

   3. Applied egg-rr 25.2%

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

      1. associate-*r* 25.2%

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

      2. *-commutative 25.2%

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

      3. associate-*l* 25.2%

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

      4. +-commutative 25.2%

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

      5. associate--l+ 83.6%

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

      6. +-inverses 83.6%

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

      7. *-commutative 83.6%

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

      8. associate-+r+ 83.6%

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

      9. +-commutative 83.6%

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

   5. Simplified 83.6%

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

   6. Taylor expanded in eps around inf 83.6%

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

      1. *-commutative 83.6%

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

      2. +-commutative 83.6%

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

      3. *-lft-identity 83.6%

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

      4. metadata-eval 83.6%

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

      5. cancel-sign-sub-inv 83.6%

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

      6. *-commutative 83.6%

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

      7. associate-*r* 83.6%

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

   8. Simplified 83.6%

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

   9. Taylor expanded in eps around 0 83.5%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin \left(\varepsilon + x\right) - \sin x \leq -0.002 \lor \neg \left(\sin \left(\varepsilon + x\right) - \sin x \leq 10^{-13}\right):\\ \;\;\;\;\sin \left(\varepsilon + x\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;\sin \left(0.5 \cdot \varepsilon\right) \cdot \left(\cos x \cdot 2\right)\\ \end{array} \]

Alternative 4: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000155 \lor \neg \left(\varepsilon \leq 0.0002\right):\\ \;\;\;\;\mathsf{fma}\left(\sin x, \cos \varepsilon, \cos x \cdot \sin \varepsilon - \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\sin x \cdot {\varepsilon}^{2}\right) + \left(-0.16666666666666666 \cdot \left(\cos x \cdot {\varepsilon}^{3}\right) + \varepsilon \cdot \cos x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.000155) (not (<= eps 0.0002)))
   (fma (sin x) (cos eps) (- (* (cos x) (sin eps)) (sin x)))
   (+
    (* -0.5 (* (sin x) (pow eps 2.0)))
    (+ (* -0.16666666666666666 (* (cos x) (pow eps 3.0))) (* eps (cos x))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.000155) || !(eps <= 0.0002)) {
		tmp = fma(sin(x), cos(eps), ((cos(x) * sin(eps)) - sin(x)));
	} else {
		tmp = (-0.5 * (sin(x) * pow(eps, 2.0))) + ((-0.16666666666666666 * (cos(x) * pow(eps, 3.0))) + (eps * cos(x)));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.000155) || !(eps <= 0.0002))
		tmp = fma(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(Float64(-0.16666666666666666 * Float64(cos(x) * (eps ^ 3.0))) + Float64(eps * cos(x))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.000155], N[Not[LessEqual[eps, 0.0002]], $MachinePrecision]], N[(N[Sin[x], $MachinePrecision] * N[Cos[eps], $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[(-0.16666666666666666 * N[(N[Cos[x], $MachinePrecision] * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.55e-4 or 2.0000000000000001e-4 < eps

   1. Initial program 50.0%

      \[\sin \left(x + \varepsilon\right) - \sin x \]
   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+ 99.4%

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

      3. fma-def 99.5%

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

   3. Applied egg-rr 99.5%

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

   if -1.55e-4 < eps < 2.0000000000000001e-4

   1. Initial program 32.2%

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

   2. Taylor expanded in eps around 0 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000155 \lor \neg \left(\varepsilon \leq 0.0002\right):\\ \;\;\;\;\mathsf{fma}\left(\sin x, \cos \varepsilon, \cos x \cdot \sin \varepsilon - \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\sin x \cdot {\varepsilon}^{2}\right) + \left(-0.16666666666666666 \cdot \left(\cos x \cdot {\varepsilon}^{3}\right) + \varepsilon \cdot \cos x\right)\\ \end{array} \]

Alternative 5: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00019 \lor \neg \left(\varepsilon \leq 0.00018\right):\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\sin x \cdot {\varepsilon}^{2}\right) + \left(-0.16666666666666666 \cdot \left(\cos x \cdot {\varepsilon}^{3}\right) + \varepsilon \cdot \cos x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.00019) (not (<= eps 0.00018)))
   (+ (- (* (cos x) (sin eps)) (sin x)) (* (sin x) (cos eps)))
   (+
    (* -0.5 (* (sin x) (pow eps 2.0)))
    (+ (* -0.16666666666666666 (* (cos x) (pow eps 3.0))) (* eps (cos x))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.00019) || !(eps <= 0.00018)) {
		tmp = ((cos(x) * sin(eps)) - sin(x)) + (sin(x) * cos(eps));
	} else {
		tmp = (-0.5 * (sin(x) * pow(eps, 2.0))) + ((-0.16666666666666666 * (cos(x) * pow(eps, 3.0))) + (eps * cos(x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-0.00019d0)) .or. (.not. (eps <= 0.00018d0))) then
        tmp = ((cos(x) * sin(eps)) - sin(x)) + (sin(x) * cos(eps))
    else
        tmp = ((-0.5d0) * (sin(x) * (eps ** 2.0d0))) + (((-0.16666666666666666d0) * (cos(x) * (eps ** 3.0d0))) + (eps * cos(x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.00019) || !(eps <= 0.00018)) {
		tmp = ((Math.cos(x) * Math.sin(eps)) - Math.sin(x)) + (Math.sin(x) * Math.cos(eps));
	} else {
		tmp = (-0.5 * (Math.sin(x) * Math.pow(eps, 2.0))) + ((-0.16666666666666666 * (Math.cos(x) * Math.pow(eps, 3.0))) + (eps * Math.cos(x)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.00019) or not (eps <= 0.00018):
		tmp = ((math.cos(x) * math.sin(eps)) - math.sin(x)) + (math.sin(x) * math.cos(eps))
	else:
		tmp = (-0.5 * (math.sin(x) * math.pow(eps, 2.0))) + ((-0.16666666666666666 * (math.cos(x) * math.pow(eps, 3.0))) + (eps * math.cos(x)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.00019) || !(eps <= 0.00018))
		tmp = Float64(Float64(Float64(cos(x) * sin(eps)) - sin(x)) + Float64(sin(x) * cos(eps)));
	else
		tmp = Float64(Float64(-0.5 * Float64(sin(x) * (eps ^ 2.0))) + Float64(Float64(-0.16666666666666666 * Float64(cos(x) * (eps ^ 3.0))) + Float64(eps * cos(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.00019) || ~((eps <= 0.00018)))
		tmp = ((cos(x) * sin(eps)) - sin(x)) + (sin(x) * cos(eps));
	else
		tmp = (-0.5 * (sin(x) * (eps ^ 2.0))) + ((-0.16666666666666666 * (cos(x) * (eps ^ 3.0))) + (eps * cos(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.00019], N[Not[LessEqual[eps, 0.00018]], $MachinePrecision]], N[(N[(N[(N[Cos[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(N[Sin[x], $MachinePrecision] * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.16666666666666666 * N[(N[Cos[x], $MachinePrecision] * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.9000000000000001e-4 or 1.80000000000000011e-4 < eps

   1. Initial program 50.0%

      \[\sin \left(x + \varepsilon\right) - \sin x \]
   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+ 99.4%

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

   3. Applied egg-rr 99.4%

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

   if -1.9000000000000001e-4 < eps < 1.80000000000000011e-4

   1. Initial program 32.2%

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

   2. Taylor expanded in eps around 0 99.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00019 \lor \neg \left(\varepsilon \leq 0.00018\right):\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\sin x \cdot {\varepsilon}^{2}\right) + \left(-0.16666666666666666 \cdot \left(\cos x \cdot {\varepsilon}^{3}\right) + \varepsilon \cdot \cos x\right)\\ \end{array} \]

Alternative 6: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00019 \lor \neg \left(\varepsilon \leq 0.0002\right):\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \sin x \cdot \left(\varepsilon \cdot \varepsilon\right), \cos x \cdot \left(\varepsilon + -0.16666666666666666 \cdot {\varepsilon}^{3}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.00019) (not (<= eps 0.0002)))
   (+ (- (* (cos x) (sin eps)) (sin x)) (* (sin x) (cos eps)))
   (fma
    -0.5
    (* (sin x) (* eps eps))
    (* (cos x) (+ eps (* -0.16666666666666666 (pow eps 3.0)))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.00019) || !(eps <= 0.0002)) {
		tmp = ((cos(x) * sin(eps)) - sin(x)) + (sin(x) * cos(eps));
	} else {
		tmp = fma(-0.5, (sin(x) * (eps * eps)), (cos(x) * (eps + (-0.16666666666666666 * pow(eps, 3.0)))));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.00019) || !(eps <= 0.0002))
		tmp = Float64(Float64(Float64(cos(x) * sin(eps)) - sin(x)) + Float64(sin(x) * cos(eps)));
	else
		tmp = fma(-0.5, Float64(sin(x) * Float64(eps * eps)), Float64(cos(x) * Float64(eps + Float64(-0.16666666666666666 * (eps ^ 3.0)))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.00019], N[Not[LessEqual[eps, 0.0002]], $MachinePrecision]], N[(N[(N[(N[Cos[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(N[Sin[x], $MachinePrecision] * N[(eps * eps), $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 < -1.9000000000000001e-4 or 2.0000000000000001e-4 < eps

   1. Initial program 50.0%

      \[\sin \left(x + \varepsilon\right) - \sin x \]
   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+ 99.4%

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

   3. Applied egg-rr 99.4%

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

   if -1.9000000000000001e-4 < eps < 2.0000000000000001e-4

   1. Initial program 32.2%

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

   2. Taylor expanded in eps around 0 99.8%

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

      1. fma-def 99.8%

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

      2. *-commutative 99.8%

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

      3. unpow2 99.8%

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

      4. +-commutative 99.8%

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

      5. associate-*r* 99.8%

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

      6. distribute-rgt-out 99.8%

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

   4. Simplified 99.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00019 \lor \neg \left(\varepsilon \leq 0.0002\right):\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \sin x \cdot \left(\varepsilon \cdot \varepsilon\right), \cos x \cdot \left(\varepsilon + -0.16666666666666666 \cdot {\varepsilon}^{3}\right)\right)\\ \end{array} \]

Alternative 7: 77.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.9%

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

   1. diff-sin 40.5%

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

   2. div-inv 40.5%

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

   3. metadata-eval 40.5%

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

   4. div-inv 40.5%

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

   5. +-commutative 40.5%

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

   6. metadata-eval 40.5%

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

3. Applied egg-rr 40.5%

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

   1. associate-*r* 40.5%

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

   2. *-commutative 40.5%

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

   3. associate-*l* 40.5%

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

   4. +-commutative 40.5%

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

   5. associate--l+ 75.7%

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

   6. +-inverses 75.7%

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

   7. *-commutative 75.7%

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

   8. associate-+r+ 75.7%

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

   9. +-commutative 75.7%

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

5. Simplified 75.7%

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

6. Taylor expanded in eps around inf 75.7%

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

   1. *-commutative 75.7%

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

   2. +-commutative 75.7%

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

   3. *-lft-identity 75.7%

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

   4. metadata-eval 75.7%

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

   5. cancel-sign-sub-inv 75.7%

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

   6. *-commutative 75.7%

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

   7. associate-*r* 75.7%

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

8. Simplified 75.7%

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

9. Final simplification 75.7%

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

Alternative 8: 77.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0007 \lor \neg \left(\varepsilon \leq 0.0011\right):\\ \;\;\;\;\sin \varepsilon - \sin x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.0007) (not (<= eps 0.0011)))
   (- (sin eps) (sin x))
   (* eps (cos x))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.0007) || !(eps <= 0.0011)) {
		tmp = sin(eps) - sin(x);
	} else {
		tmp = eps * cos(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-0.0007d0)) .or. (.not. (eps <= 0.0011d0))) then
        tmp = sin(eps) - sin(x)
    else
        tmp = eps * cos(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.0007) || !(eps <= 0.0011)) {
		tmp = Math.sin(eps) - Math.sin(x);
	} else {
		tmp = eps * Math.cos(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.0007) or not (eps <= 0.0011):
		tmp = math.sin(eps) - math.sin(x)
	else:
		tmp = eps * math.cos(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.0007) || !(eps <= 0.0011))
		tmp = Float64(sin(eps) - sin(x));
	else
		tmp = Float64(eps * cos(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.0007) || ~((eps <= 0.0011)))
		tmp = sin(eps) - sin(x);
	else
		tmp = eps * cos(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.0007], N[Not[LessEqual[eps, 0.0011]], $MachinePrecision]], N[(N[Sin[eps], $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision], N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -6.99999999999999993e-4 or 0.00110000000000000007 < eps

   1. Initial program 50.4%

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

      1. sin-sum 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. sin-sum 50.4%

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

      2. +-commutative 50.4%

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

      3. *-un-lft-identity 50.4%

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

      4. metadata-eval 50.4%

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

      5. associate-*r* 50.4%

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

      6. count-2 50.4%

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

      7. flip-+ 0.0%

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

      8. +-inverses 0.0%

         \[\leadsto \sin \left(\varepsilon + 0.5 \cdot \frac{\color{blue}{0}}{x - x}\right) - \sin x \]

      9. metadata-eval 0.0%

         \[\leadsto \sin \left(\varepsilon + 0.5 \cdot \frac{\color{blue}{0 - 0}}{x - x}\right) - \sin x \]

      10. metadata-eval 0.0%

          \[\leadsto \sin \left(\varepsilon + 0.5 \cdot \frac{\color{blue}{0 \cdot 0} - 0}{x - x}\right) - \sin x \]

      11. metadata-eval 0.0%

          \[\leadsto \sin \left(\varepsilon + 0.5 \cdot \frac{0 \cdot 0 - \color{blue}{0 \cdot 0}}{x - x}\right) - \sin x \]

      12. +-inverses 0.0%

          \[\leadsto \sin \left(\varepsilon + 0.5 \cdot \frac{0 \cdot 0 - 0 \cdot 0}{\color{blue}{0}}\right) - \sin x \]

      13. metadata-eval 0.0%

          \[\leadsto \sin \left(\varepsilon + 0.5 \cdot \frac{0 \cdot 0 - 0 \cdot 0}{\color{blue}{0 + 0}}\right) - \sin x \]

      14. flip-- 52.4%

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

      15. metadata-eval 52.4%

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

      16. metadata-eval 52.4%

          \[\leadsto \sin \left(\varepsilon + \color{blue}{0}\right) - \sin x \]

      17. +-rgt-identity 52.4%

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

      18. add-sqr-sqrt 22.4%

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

      19. sqrt-unprod 26.1%

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

      20. pow2 26.1%

          \[\leadsto \sqrt{\color{blue}{{\sin \varepsilon}^{2}}} - \sin x \]

   5. Applied egg-rr 26.1%

      \[\leadsto \color{blue}{\sqrt{{\sin \varepsilon}^{2}}} - \sin x \]

   6. Taylor expanded in eps around inf 52.4%

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

   if -6.99999999999999993e-4 < eps < 0.00110000000000000007

   1. Initial program 32.0%

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

   2. Taylor expanded in eps around 0 98.5%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0007 \lor \neg \left(\varepsilon \leq 0.0011\right):\\ \;\;\;\;\sin \varepsilon - \sin x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \]

Alternative 9: 77.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -9.8 \cdot 10^{-6}:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.00105:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -9.8e-6)
   (sin eps)
   (if (<= eps 0.00105) (* eps (cos x)) (sin eps))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -9.8e-6) {
		tmp = sin(eps);
	} else if (eps <= 0.00105) {
		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 <= (-9.8d-6)) then
        tmp = sin(eps)
    else if (eps <= 0.00105d0) 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 <= -9.8e-6) {
		tmp = Math.sin(eps);
	} else if (eps <= 0.00105) {
		tmp = eps * Math.cos(x);
	} else {
		tmp = Math.sin(eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -9.8e-6:
		tmp = math.sin(eps)
	elif eps <= 0.00105:
		tmp = eps * math.cos(x)
	else:
		tmp = math.sin(eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -9.8e-6)
		tmp = sin(eps);
	elseif (eps <= 0.00105)
		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 <= -9.8e-6)
		tmp = sin(eps);
	elseif (eps <= 0.00105)
		tmp = eps * cos(x);
	else
		tmp = sin(eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -9.8e-6], N[Sin[eps], $MachinePrecision], If[LessEqual[eps, 0.00105], N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision], N[Sin[eps], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < -9.79999999999999934e-6 or 0.00104999999999999994 < eps

   1. Initial program 50.4%

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

   2. Taylor expanded in x around 0 51.6%

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

   if -9.79999999999999934e-6 < eps < 0.00104999999999999994

   1. Initial program 32.0%

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

   2. Taylor expanded in eps around 0 98.5%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -9.8 \cdot 10^{-6}:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.00105:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 10: 55.7% 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 40.9%

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

2. Taylor expanded in x around 0 53.1%

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

3. Final simplification 53.1%

   \[\leadsto \sin \varepsilon \]

Alternative 11: 29.7% 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 40.9%

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

   1. diff-sin 40.5%

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

   2. div-inv 40.5%

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

   3. metadata-eval 40.5%

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

   4. div-inv 40.5%

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

   5. +-commutative 40.5%

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

   6. metadata-eval 40.5%

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

3. Applied egg-rr 40.5%

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

   1. associate-*r* 40.5%

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

   2. *-commutative 40.5%

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

   3. associate-*l* 40.5%

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

   4. +-commutative 40.5%

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

   5. associate--l+ 75.7%

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

   6. +-inverses 75.7%

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

   7. *-commutative 75.7%

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

   8. associate-+r+ 75.7%

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

   9. +-commutative 75.7%

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

5. Simplified 75.7%

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

6. Taylor expanded in eps around 0 52.3%

   \[\leadsto \color{blue}{\varepsilon \cdot \cos x} \]
7. Step-by-step derivation

   1. *-commutative 52.3%

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

8. Simplified 52.3%

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

9. Taylor expanded in x around 0 29.5%

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

10. Final simplification 29.5%

    \[\leadsto \varepsilon \]

Developer target: 77.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023279 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64

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

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