2cos (problem 3.3.5)

Percentage Accurate: 53.7% → 99.3%

Time: 14.6s

Alternatives: 16

Speedup: 29.3×

Specification

\[\left(\left(-10000 \leq x \land x \leq 10000\right) \land 10^{-16} \cdot \left|x\right| < \varepsilon\right) \land \varepsilon < \left|x\right|\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 56.7%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. diff-cos 84.9%

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

   2. div-inv 84.9%

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

   3. associate--l+ 84.9%

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

   4. metadata-eval 84.9%

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

   5. div-inv 84.9%

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

   6. +-commutative 84.9%

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

   7. associate-+l+ 84.9%

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

   8. metadata-eval 84.9%

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

4. Applied egg-rr 84.9%

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

   1. associate-*r* 84.9%

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

   2. *-commutative 84.9%

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

   3. associate-*l* 84.9%

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

   4. *-commutative 84.9%

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

   5. associate-+r- 84.9%

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

   6. +-commutative 84.9%

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

   7. associate--l+ 99.2%

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

   8. +-inverses 99.2%

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

   9. distribute-lft-in 99.2%

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

   10. metadata-eval 99.2%

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

   11. *-commutative 99.2%

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

   12. +-commutative 99.2%

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

   13. count-2 99.2%

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

   14. fma-define 99.2%

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

6. Simplified 99.2%

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

7. Final simplification 99.2%

   \[\leadsto \left(-2 \cdot \sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right) \]
8. Add Preprocessing

Alternative 2: 98.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.000148:\\ \;\;\;\;\left(0.5 \cdot \varepsilon\right) \cdot \left(-2 \cdot \sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\varepsilon + x\right) - \cos x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 0.000148)
   (* (* 0.5 eps) (* -2.0 (sin (* 0.5 (fma 2.0 x eps)))))
   (- (cos (+ eps x)) (cos x))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 0.000148) {
		tmp = (0.5 * eps) * (-2.0 * sin((0.5 * fma(2.0, x, eps))));
	} else {
		tmp = cos((eps + x)) - cos(x);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 0.000148)
		tmp = Float64(Float64(0.5 * eps) * Float64(-2.0 * sin(Float64(0.5 * fma(2.0, x, eps)))));
	else
		tmp = Float64(cos(Float64(eps + x)) - cos(x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 0.000148], N[(N[(0.5 * eps), $MachinePrecision] * N[(-2.0 * N[Sin[N[(0.5 * N[(2.0 * x + eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(eps + x), $MachinePrecision]], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 1.47999999999999992e-4

   1. Initial program 55.3%

      \[\cos \left(x + \varepsilon\right) - \cos x \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. diff-cos 84.7%

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

      2. div-inv 84.7%

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

      3. associate--l+ 84.7%

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

      4. metadata-eval 84.7%

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

      5. div-inv 84.7%

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

      6. +-commutative 84.7%

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

      7. associate-+l+ 84.7%

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

      8. metadata-eval 84.7%

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

   4. Applied egg-rr 84.7%

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

      1. associate-*r* 84.7%

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

      2. *-commutative 84.7%

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

      3. associate-*l* 84.7%

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

      4. *-commutative 84.7%

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

      5. associate-+r- 84.7%

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

      6. +-commutative 84.7%

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

      7. associate--l+ 99.8%

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

      8. +-inverses 99.8%

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

      9. distribute-lft-in 99.8%

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

      10. metadata-eval 99.8%

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

      11. *-commutative 99.8%

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

      12. +-commutative 99.8%

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

      13. count-2 99.8%

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

      14. fma-define 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in eps around 0 99.7%

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

      1. *-commutative 99.7%

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

   9. Simplified 99.7%

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

   if 1.47999999999999992e-4 < eps

   1. Initial program 82.6%

      \[\cos \left(x + \varepsilon\right) - \cos x \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.000148:\\ \;\;\;\;\left(0.5 \cdot \varepsilon\right) \cdot \left(-2 \cdot \sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\varepsilon + x\right) - \cos x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.7%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. diff-cos 84.9%

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

   2. div-inv 84.9%

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

   3. associate--l+ 84.9%

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

   4. metadata-eval 84.9%

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

   5. div-inv 84.9%

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

   6. +-commutative 84.9%

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

   7. associate-+l+ 84.9%

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

   8. metadata-eval 84.9%

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

4. Applied egg-rr 84.9%

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

   1. associate-*r* 84.9%

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

   2. *-commutative 84.9%

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

   3. associate-*l* 84.9%

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

   4. *-commutative 84.9%

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

   5. associate-+r- 84.9%

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

   6. +-commutative 84.9%

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

   7. associate--l+ 99.2%

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

   8. +-inverses 99.2%

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

   9. distribute-lft-in 99.2%

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

   10. metadata-eval 99.2%

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

   11. *-commutative 99.2%

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

   12. +-commutative 99.2%

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

   13. count-2 99.2%

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

   14. fma-define 99.2%

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

6. Simplified 99.2%

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

7. Taylor expanded in x around -inf 99.2%

   \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(0.5 \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right)} \]
8. Add Preprocessing

Alternative 4: 98.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 0.00016)
   (* -2.0 (* (* 0.5 eps) (sin (* 0.5 (- eps (* -2.0 x))))))
   (- (cos (+ eps x)) (cos x))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 0.00016) {
		tmp = -2.0 * ((0.5 * eps) * sin((0.5 * (eps - (-2.0 * x)))));
	} else {
		tmp = cos((eps + x)) - 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 <= 0.00016d0) then
        tmp = (-2.0d0) * ((0.5d0 * eps) * sin((0.5d0 * (eps - ((-2.0d0) * x)))))
    else
        tmp = cos((eps + x)) - cos(x)
    end if
    code = tmp
end function

Java

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

Python

def code(x, eps):
	tmp = 0
	if eps <= 0.00016:
		tmp = -2.0 * ((0.5 * eps) * math.sin((0.5 * (eps - (-2.0 * x)))))
	else:
		tmp = math.cos((eps + x)) - math.cos(x)
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= 0.00016)
		tmp = -2.0 * ((0.5 * eps) * sin((0.5 * (eps - (-2.0 * x)))));
	else
		tmp = cos((eps + x)) - cos(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 0.00016], N[(-2.0 * N[(N[(0.5 * eps), $MachinePrecision] * N[Sin[N[(0.5 * N[(eps - N[(-2.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(eps + x), $MachinePrecision]], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 1.60000000000000013e-4

   1. Initial program 55.3%

      \[\cos \left(x + \varepsilon\right) - \cos x \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. diff-cos 84.7%

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

      2. div-inv 84.7%

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

      3. associate--l+ 84.7%

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

      4. metadata-eval 84.7%

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

      5. div-inv 84.7%

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

      6. +-commutative 84.7%

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

      7. associate-+l+ 84.7%

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

      8. metadata-eval 84.7%

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

   4. Applied egg-rr 84.7%

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

      1. associate-*r* 84.7%

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

      2. *-commutative 84.7%

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

      3. associate-*l* 84.7%

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

      4. *-commutative 84.7%

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

      5. associate-+r- 84.7%

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

      6. +-commutative 84.7%

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

      7. associate--l+ 99.8%

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

      8. +-inverses 99.8%

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

      9. distribute-lft-in 99.8%

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

      10. metadata-eval 99.8%

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

      11. *-commutative 99.8%

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

      12. +-commutative 99.8%

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

      13. count-2 99.8%

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

      14. fma-define 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in x around -inf 99.7%

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

   8. Taylor expanded in eps around 0 99.6%

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

   if 1.60000000000000013e-4 < eps

   1. Initial program 82.6%

      \[\cos \left(x + \varepsilon\right) - \cos x \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.00016:\\ \;\;\;\;-2 \cdot \left(\left(0.5 \cdot \varepsilon\right) \cdot \sin \left(0.5 \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\varepsilon + x\right) - \cos x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.000186:\\ \;\;\;\;-2 \cdot \left(\left(0.5 \cdot \varepsilon\right) \cdot \sin \left(0.5 \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\varepsilon \cdot \left(\left(2 + \frac{x}{\varepsilon}\right) + -1\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 0.000186)
   (* -2.0 (* (* 0.5 eps) (sin (* 0.5 (- eps (* -2.0 x))))))
   (+ (cos (* eps (+ (+ 2.0 (/ x eps)) -1.0))) -1.0)))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 0.000186) {
		tmp = -2.0 * ((0.5 * eps) * sin((0.5 * (eps - (-2.0 * x)))));
	} else {
		tmp = cos((eps * ((2.0 + (x / eps)) + -1.0))) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= 0.000186d0) then
        tmp = (-2.0d0) * ((0.5d0 * eps) * sin((0.5d0 * (eps - ((-2.0d0) * x)))))
    else
        tmp = cos((eps * ((2.0d0 + (x / eps)) + (-1.0d0)))) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= 0.000186) {
		tmp = -2.0 * ((0.5 * eps) * Math.sin((0.5 * (eps - (-2.0 * x)))));
	} else {
		tmp = Math.cos((eps * ((2.0 + (x / eps)) + -1.0))) + -1.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= 0.000186:
		tmp = -2.0 * ((0.5 * eps) * math.sin((0.5 * (eps - (-2.0 * x)))))
	else:
		tmp = math.cos((eps * ((2.0 + (x / eps)) + -1.0))) + -1.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 0.000186)
		tmp = Float64(-2.0 * Float64(Float64(0.5 * eps) * sin(Float64(0.5 * Float64(eps - Float64(-2.0 * x))))));
	else
		tmp = Float64(cos(Float64(eps * Float64(Float64(2.0 + Float64(x / eps)) + -1.0))) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= 0.000186)
		tmp = -2.0 * ((0.5 * eps) * sin((0.5 * (eps - (-2.0 * x)))));
	else
		tmp = cos((eps * ((2.0 + (x / eps)) + -1.0))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 0.000186], N[(-2.0 * N[(N[(0.5 * eps), $MachinePrecision] * N[Sin[N[(0.5 * N[(eps - N[(-2.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(eps * N[(N[(2.0 + N[(x / eps), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 1.85999999999999995e-4

   1. Initial program 55.3%

      \[\cos \left(x + \varepsilon\right) - \cos x \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. diff-cos 84.7%

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

      2. div-inv 84.7%

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

      3. associate--l+ 84.7%

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

      4. metadata-eval 84.7%

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

      5. div-inv 84.7%

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

      6. +-commutative 84.7%

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

      7. associate-+l+ 84.7%

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

      8. metadata-eval 84.7%

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

   4. Applied egg-rr 84.7%

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

      1. associate-*r* 84.7%

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

      2. *-commutative 84.7%

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

      3. associate-*l* 84.7%

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

      4. *-commutative 84.7%

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

      5. associate-+r- 84.7%

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

      6. +-commutative 84.7%

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

      7. associate--l+ 99.8%

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

      8. +-inverses 99.8%

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

      9. distribute-lft-in 99.8%

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

      10. metadata-eval 99.8%

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

      11. *-commutative 99.8%

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

      12. +-commutative 99.8%

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

      13. count-2 99.8%

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

      14. fma-define 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in x around -inf 99.7%

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

   8. Taylor expanded in eps around 0 99.6%

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

   if 1.85999999999999995e-4 < eps

   1. Initial program 82.6%

      \[\cos \left(x + \varepsilon\right) - \cos x \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 68.1%

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

   4. Taylor expanded in eps around inf 68.1%

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

      1. expm1-log1p-u 68.0%

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

      2. expm1-undefine 68.0%

         \[\leadsto \cos \left(\varepsilon \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 + \frac{x}{\varepsilon}\right)} - 1\right)}\right) - 1 \]

   6. Applied egg-rr 68.0%

      \[\leadsto \cos \left(\varepsilon \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 + \frac{x}{\varepsilon}\right)} - 1\right)}\right) - 1 \]
   7. Step-by-step derivation

      1. sub-neg 68.0%

         \[\leadsto \cos \left(\varepsilon \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 + \frac{x}{\varepsilon}\right)} + \left(-1\right)\right)}\right) - 1 \]

      2. metadata-eval 68.0%

         \[\leadsto \cos \left(\varepsilon \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\varepsilon}\right)} + \color{blue}{-1}\right)\right) - 1 \]

      3. +-commutative 68.0%

         \[\leadsto \cos \left(\varepsilon \cdot \color{blue}{\left(-1 + e^{\mathsf{log1p}\left(1 + \frac{x}{\varepsilon}\right)}\right)}\right) - 1 \]

      4. log1p-undefine 68.0%

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

      5. rem-exp-log 68.0%

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

      6. associate-+r+ 68.1%

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

      7. metadata-eval 68.1%

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

   8. Simplified 68.1%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.000186:\\ \;\;\;\;-2 \cdot \left(\left(0.5 \cdot \varepsilon\right) \cdot \sin \left(0.5 \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\varepsilon \cdot \left(\left(2 + \frac{x}{\varepsilon}\right) + -1\right)\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.000222:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5 - \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\varepsilon \cdot \left(\left(2 + \frac{x}{\varepsilon}\right) + -1\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 0.000222)
   (* eps (- (* eps -0.5) (sin x)))
   (+ (cos (* eps (+ (+ 2.0 (/ x eps)) -1.0))) -1.0)))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 0.000222) {
		tmp = eps * ((eps * -0.5) - sin(x));
	} else {
		tmp = cos((eps * ((2.0 + (x / eps)) + -1.0))) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= 0.000222d0) then
        tmp = eps * ((eps * (-0.5d0)) - sin(x))
    else
        tmp = cos((eps * ((2.0d0 + (x / eps)) + (-1.0d0)))) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= 0.000222) {
		tmp = eps * ((eps * -0.5) - Math.sin(x));
	} else {
		tmp = Math.cos((eps * ((2.0 + (x / eps)) + -1.0))) + -1.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= 0.000222:
		tmp = eps * ((eps * -0.5) - math.sin(x))
	else:
		tmp = math.cos((eps * ((2.0 + (x / eps)) + -1.0))) + -1.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 0.000222)
		tmp = Float64(eps * Float64(Float64(eps * -0.5) - sin(x)));
	else
		tmp = Float64(cos(Float64(eps * Float64(Float64(2.0 + Float64(x / eps)) + -1.0))) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= 0.000222)
		tmp = eps * ((eps * -0.5) - sin(x));
	else
		tmp = cos((eps * ((2.0 + (x / eps)) + -1.0))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 0.000222], N[(eps * N[(N[(eps * -0.5), $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(eps * N[(N[(2.0 + N[(x / eps), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 2.22000000000000002e-4

   1. Initial program 55.3%

      \[\cos \left(x + \varepsilon\right) - \cos x \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0 99.6%

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

      1. associate-*r* 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in x around 0 99.2%

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

   if 2.22000000000000002e-4 < eps

   1. Initial program 82.6%

      \[\cos \left(x + \varepsilon\right) - \cos x \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 68.1%

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

   4. Taylor expanded in eps around inf 68.1%

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

      1. expm1-log1p-u 68.0%

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

      2. expm1-undefine 68.0%

         \[\leadsto \cos \left(\varepsilon \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 + \frac{x}{\varepsilon}\right)} - 1\right)}\right) - 1 \]

   6. Applied egg-rr 68.0%

      \[\leadsto \cos \left(\varepsilon \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 + \frac{x}{\varepsilon}\right)} - 1\right)}\right) - 1 \]
   7. Step-by-step derivation

      1. sub-neg 68.0%

         \[\leadsto \cos \left(\varepsilon \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 + \frac{x}{\varepsilon}\right)} + \left(-1\right)\right)}\right) - 1 \]

      2. metadata-eval 68.0%

         \[\leadsto \cos \left(\varepsilon \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\varepsilon}\right)} + \color{blue}{-1}\right)\right) - 1 \]

      3. +-commutative 68.0%

         \[\leadsto \cos \left(\varepsilon \cdot \color{blue}{\left(-1 + e^{\mathsf{log1p}\left(1 + \frac{x}{\varepsilon}\right)}\right)}\right) - 1 \]

      4. log1p-undefine 68.0%

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

      5. rem-exp-log 68.0%

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

      6. associate-+r+ 68.1%

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

      7. metadata-eval 68.1%

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

   8. Simplified 68.1%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.000222:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5 - \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\varepsilon \cdot \left(\left(2 + \frac{x}{\varepsilon}\right) + -1\right)\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.00018:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5 - \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\varepsilon \cdot \left(\frac{x}{\varepsilon} + 1\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 0.00018)
   (* eps (- (* eps -0.5) (sin x)))
   (+ (cos (* eps (+ (/ x eps) 1.0))) -1.0)))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 0.00018) {
		tmp = eps * ((eps * -0.5) - sin(x));
	} else {
		tmp = cos((eps * ((x / eps) + 1.0))) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= 0.00018d0) then
        tmp = eps * ((eps * (-0.5d0)) - sin(x))
    else
        tmp = cos((eps * ((x / eps) + 1.0d0))) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= 0.00018) {
		tmp = eps * ((eps * -0.5) - Math.sin(x));
	} else {
		tmp = Math.cos((eps * ((x / eps) + 1.0))) + -1.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= 0.00018:
		tmp = eps * ((eps * -0.5) - math.sin(x))
	else:
		tmp = math.cos((eps * ((x / eps) + 1.0))) + -1.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 0.00018)
		tmp = Float64(eps * Float64(Float64(eps * -0.5) - sin(x)));
	else
		tmp = Float64(cos(Float64(eps * Float64(Float64(x / eps) + 1.0))) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= 0.00018)
		tmp = eps * ((eps * -0.5) - sin(x));
	else
		tmp = cos((eps * ((x / eps) + 1.0))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 0.00018], N[(eps * N[(N[(eps * -0.5), $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(eps * N[(N[(x / eps), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 1.80000000000000011e-4

   1. Initial program 55.3%

      \[\cos \left(x + \varepsilon\right) - \cos x \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0 99.6%

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

      1. associate-*r* 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in x around 0 99.2%

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

   if 1.80000000000000011e-4 < eps

   1. Initial program 82.6%

      \[\cos \left(x + \varepsilon\right) - \cos x \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 68.1%

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

   4. Taylor expanded in eps around inf 68.1%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.00018:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5 - \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\varepsilon \cdot \left(\frac{x}{\varepsilon} + 1\right)\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.00026:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5 - \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\varepsilon + x\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 0.00026)
   (* eps (- (* eps -0.5) (sin x)))
   (+ (cos (+ eps x)) -1.0)))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 0.00026) {
		tmp = eps * ((eps * -0.5) - sin(x));
	} else {
		tmp = cos((eps + x)) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= 0.00026d0) then
        tmp = eps * ((eps * (-0.5d0)) - sin(x))
    else
        tmp = cos((eps + x)) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= 0.00026) {
		tmp = eps * ((eps * -0.5) - Math.sin(x));
	} else {
		tmp = Math.cos((eps + x)) + -1.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= 0.00026:
		tmp = eps * ((eps * -0.5) - math.sin(x))
	else:
		tmp = math.cos((eps + x)) + -1.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 0.00026)
		tmp = Float64(eps * Float64(Float64(eps * -0.5) - sin(x)));
	else
		tmp = Float64(cos(Float64(eps + x)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= 0.00026)
		tmp = eps * ((eps * -0.5) - sin(x));
	else
		tmp = cos((eps + x)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 0.00026], N[(eps * N[(N[(eps * -0.5), $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(eps + x), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 2.59999999999999977e-4

   1. Initial program 55.3%

      \[\cos \left(x + \varepsilon\right) - \cos x \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0 99.6%

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

      1. associate-*r* 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in x around 0 99.2%

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

   if 2.59999999999999977e-4 < eps

   1. Initial program 82.6%

      \[\cos \left(x + \varepsilon\right) - \cos x \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 68.1%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.00026:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5 - \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\varepsilon + x\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 96.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 9 \cdot 10^{-6}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5 + x \cdot \left(x \cdot \left(\varepsilon \cdot 0.25 + x \cdot 0.16666666666666666\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\varepsilon + x\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 9e-6)
   (*
    eps
    (+
     (* eps -0.5)
     (* x (+ (* x (+ (* eps 0.25) (* x 0.16666666666666666))) -1.0))))
   (+ (cos (+ eps x)) -1.0)))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 9e-6) {
		tmp = eps * ((eps * -0.5) + (x * ((x * ((eps * 0.25) + (x * 0.16666666666666666))) + -1.0)));
	} else {
		tmp = cos((eps + x)) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= 9d-6) then
        tmp = eps * ((eps * (-0.5d0)) + (x * ((x * ((eps * 0.25d0) + (x * 0.16666666666666666d0))) + (-1.0d0))))
    else
        tmp = cos((eps + x)) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= 9e-6) {
		tmp = eps * ((eps * -0.5) + (x * ((x * ((eps * 0.25) + (x * 0.16666666666666666))) + -1.0)));
	} else {
		tmp = Math.cos((eps + x)) + -1.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= 9e-6:
		tmp = eps * ((eps * -0.5) + (x * ((x * ((eps * 0.25) + (x * 0.16666666666666666))) + -1.0)))
	else:
		tmp = math.cos((eps + x)) + -1.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 9e-6)
		tmp = Float64(eps * Float64(Float64(eps * -0.5) + Float64(x * Float64(Float64(x * Float64(Float64(eps * 0.25) + Float64(x * 0.16666666666666666))) + -1.0))));
	else
		tmp = Float64(cos(Float64(eps + x)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= 9e-6)
		tmp = eps * ((eps * -0.5) + (x * ((x * ((eps * 0.25) + (x * 0.16666666666666666))) + -1.0)));
	else
		tmp = cos((eps + x)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 9e-6], N[(eps * N[(N[(eps * -0.5), $MachinePrecision] + N[(x * N[(N[(x * N[(N[(eps * 0.25), $MachinePrecision] + N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(eps + x), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 9.00000000000000023e-6

   1. Initial program 55.3%

      \[\cos \left(x + \varepsilon\right) - \cos x \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0 99.7%

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

      1. associate-*r* 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in x around 0 99.3%

      \[\leadsto \varepsilon \cdot \color{blue}{\left(-0.5 \cdot \varepsilon + x \cdot \left(x \cdot \left(0.16666666666666666 \cdot x + 0.25 \cdot \varepsilon\right) - 1\right)\right)} \]

   if 9.00000000000000023e-6 < eps

   1. Initial program 80.9%

      \[\cos \left(x + \varepsilon\right) - \cos x \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 64.1%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 9 \cdot 10^{-6}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5 + x \cdot \left(x \cdot \left(\varepsilon \cdot 0.25 + x \cdot 0.16666666666666666\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\varepsilon + x\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 96.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.0027:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5 + x \cdot \left(x \cdot \left(\varepsilon \cdot 0.25 + x \cdot 0.16666666666666666\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 0.0027)
   (*
    eps
    (+
     (* eps -0.5)
     (* x (+ (* x (+ (* eps 0.25) (* x 0.16666666666666666))) -1.0))))
   (+ (cos eps) -1.0)))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 0.0027) {
		tmp = eps * ((eps * -0.5) + (x * ((x * ((eps * 0.25) + (x * 0.16666666666666666))) + -1.0)));
	} else {
		tmp = cos(eps) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= 0.0027d0) then
        tmp = eps * ((eps * (-0.5d0)) + (x * ((x * ((eps * 0.25d0) + (x * 0.16666666666666666d0))) + (-1.0d0))))
    else
        tmp = cos(eps) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= 0.0027) {
		tmp = eps * ((eps * -0.5) + (x * ((x * ((eps * 0.25) + (x * 0.16666666666666666))) + -1.0)));
	} else {
		tmp = Math.cos(eps) + -1.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= 0.0027:
		tmp = eps * ((eps * -0.5) + (x * ((x * ((eps * 0.25) + (x * 0.16666666666666666))) + -1.0)))
	else:
		tmp = math.cos(eps) + -1.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 0.0027)
		tmp = Float64(eps * Float64(Float64(eps * -0.5) + Float64(x * Float64(Float64(x * Float64(Float64(eps * 0.25) + Float64(x * 0.16666666666666666))) + -1.0))));
	else
		tmp = Float64(cos(eps) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= 0.0027)
		tmp = eps * ((eps * -0.5) + (x * ((x * ((eps * 0.25) + (x * 0.16666666666666666))) + -1.0)));
	else
		tmp = cos(eps) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 0.0027], N[(eps * N[(N[(eps * -0.5), $MachinePrecision] + N[(x * N[(N[(x * N[(N[(eps * 0.25), $MachinePrecision] + N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 0.0027000000000000001

   1. Initial program 55.4%

      \[\cos \left(x + \varepsilon\right) - \cos x \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0 99.4%

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

      1. associate-*r* 99.4%

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

   5. Simplified 99.4%

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

   6. Taylor expanded in x around 0 98.7%

      \[\leadsto \varepsilon \cdot \color{blue}{\left(-0.5 \cdot \varepsilon + x \cdot \left(x \cdot \left(0.16666666666666666 \cdot x + 0.25 \cdot \varepsilon\right) - 1\right)\right)} \]

   if 0.0027000000000000001 < eps

   1. Initial program 83.4%

      \[\cos \left(x + \varepsilon\right) - \cos x \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 61.1%

      \[\leadsto \color{blue}{\cos \varepsilon - 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.0027:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5 + x \cdot \left(x \cdot \left(\varepsilon \cdot 0.25 + x \cdot 0.16666666666666666\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 95.0% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\varepsilon \cdot -0.5 + x \cdot \left(x \cdot \left(\varepsilon \cdot 0.25 + x \cdot 0.16666666666666666\right) + -1\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   (* eps -0.5)
   (* x (+ (* x (+ (* eps 0.25) (* x 0.16666666666666666))) -1.0)))))

C

double code(double x, double eps) {
	return eps * ((eps * -0.5) + (x * ((x * ((eps * 0.25) + (x * 0.16666666666666666))) + -1.0)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * ((eps * (-0.5d0)) + (x * ((x * ((eps * 0.25d0) + (x * 0.16666666666666666d0))) + (-1.0d0))))
end function

Java

public static double code(double x, double eps) {
	return eps * ((eps * -0.5) + (x * ((x * ((eps * 0.25) + (x * 0.16666666666666666))) + -1.0)));
}

Python

def code(x, eps):
	return eps * ((eps * -0.5) + (x * ((x * ((eps * 0.25) + (x * 0.16666666666666666))) + -1.0)))

Julia

function code(x, eps)
	return Float64(eps * Float64(Float64(eps * -0.5) + Float64(x * Float64(Float64(x * Float64(Float64(eps * 0.25) + Float64(x * 0.16666666666666666))) + -1.0))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * ((eps * -0.5) + (x * ((x * ((eps * 0.25) + (x * 0.16666666666666666))) + -1.0)));
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(eps * -0.5), $MachinePrecision] + N[(x * N[(N[(x * N[(N[(eps * 0.25), $MachinePrecision] + N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.7%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0 95.9%

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

   1. associate-*r* 95.9%

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

5. Simplified 95.9%

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

6. Taylor expanded in x around 0 95.1%

   \[\leadsto \varepsilon \cdot \color{blue}{\left(-0.5 \cdot \varepsilon + x \cdot \left(x \cdot \left(0.16666666666666666 \cdot x + 0.25 \cdot \varepsilon\right) - 1\right)\right)} \]

7. Final simplification 95.1%

   \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot -0.5 + x \cdot \left(x \cdot \left(\varepsilon \cdot 0.25 + x \cdot 0.16666666666666666\right) + -1\right)\right) \]
8. Add Preprocessing

Alternative 12: 94.9% accurate, 13.7× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\varepsilon \cdot -0.5 + x \cdot \left(x \cdot \left(x \cdot 0.16666666666666666\right) + -1\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* eps (+ (* eps -0.5) (* x (+ (* x (* x 0.16666666666666666)) -1.0)))))

C

double code(double x, double eps) {
	return eps * ((eps * -0.5) + (x * ((x * (x * 0.16666666666666666)) + -1.0)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * ((eps * (-0.5d0)) + (x * ((x * (x * 0.16666666666666666d0)) + (-1.0d0))))
end function

Java

public static double code(double x, double eps) {
	return eps * ((eps * -0.5) + (x * ((x * (x * 0.16666666666666666)) + -1.0)));
}

Python

def code(x, eps):
	return eps * ((eps * -0.5) + (x * ((x * (x * 0.16666666666666666)) + -1.0)))

Julia

function code(x, eps)
	return Float64(eps * Float64(Float64(eps * -0.5) + Float64(x * Float64(Float64(x * Float64(x * 0.16666666666666666)) + -1.0))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * ((eps * -0.5) + (x * ((x * (x * 0.16666666666666666)) + -1.0)));
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(eps * -0.5), $MachinePrecision] + N[(x * N[(N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.7%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0 95.9%

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

   1. associate-*r* 95.9%

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

5. Simplified 95.9%

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

6. Taylor expanded in x around 0 95.1%

   \[\leadsto \varepsilon \cdot \color{blue}{\left(-0.5 \cdot \varepsilon + x \cdot \left(x \cdot \left(0.16666666666666666 \cdot x + 0.25 \cdot \varepsilon\right) - 1\right)\right)} \]

7. Taylor expanded in x around inf 95.1%

   \[\leadsto \varepsilon \cdot \left(-0.5 \cdot \varepsilon + x \cdot \left(x \cdot \color{blue}{\left(0.16666666666666666 \cdot x\right)} - 1\right)\right) \]
8. Step-by-step derivation

   1. *-commutative 95.1%

      \[\leadsto \varepsilon \cdot \left(-0.5 \cdot \varepsilon + x \cdot \left(x \cdot \color{blue}{\left(x \cdot 0.16666666666666666\right)} - 1\right)\right) \]

9. Simplified 95.1%

   \[\leadsto \varepsilon \cdot \left(-0.5 \cdot \varepsilon + x \cdot \left(x \cdot \color{blue}{\left(x \cdot 0.16666666666666666\right)} - 1\right)\right) \]

10. Final simplification 95.1%

    \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot -0.5 + x \cdot \left(x \cdot \left(x \cdot 0.16666666666666666\right) + -1\right)\right) \]
11. Add Preprocessing

Alternative 13: 94.5% accurate, 13.7× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\varepsilon \cdot -0.5 + x \cdot \left(0.25 \cdot \left(\varepsilon \cdot x\right) + -1\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* eps (+ (* eps -0.5) (* x (+ (* 0.25 (* eps x)) -1.0)))))

C

double code(double x, double eps) {
	return eps * ((eps * -0.5) + (x * ((0.25 * (eps * x)) + -1.0)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * ((eps * (-0.5d0)) + (x * ((0.25d0 * (eps * x)) + (-1.0d0))))
end function

Java

public static double code(double x, double eps) {
	return eps * ((eps * -0.5) + (x * ((0.25 * (eps * x)) + -1.0)));
}

Python

def code(x, eps):
	return eps * ((eps * -0.5) + (x * ((0.25 * (eps * x)) + -1.0)))

Julia

function code(x, eps)
	return Float64(eps * Float64(Float64(eps * -0.5) + Float64(x * Float64(Float64(0.25 * Float64(eps * x)) + -1.0))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * ((eps * -0.5) + (x * ((0.25 * (eps * x)) + -1.0)));
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(eps * -0.5), $MachinePrecision] + N[(x * N[(N[(0.25 * N[(eps * x), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.7%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0 95.9%

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

   1. associate-*r* 95.9%

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

5. Simplified 95.9%

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

6. Taylor expanded in x around 0 94.7%

   \[\leadsto \varepsilon \cdot \color{blue}{\left(-0.5 \cdot \varepsilon + x \cdot \left(0.25 \cdot \left(\varepsilon \cdot x\right) - 1\right)\right)} \]

7. Final simplification 94.7%

   \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot -0.5 + x \cdot \left(0.25 \cdot \left(\varepsilon \cdot x\right) + -1\right)\right) \]
8. Add Preprocessing

Alternative 14: 94.4% accurate, 29.3× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 (* eps (- (* eps -0.5) x)))

C

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

Fortran

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

Java

public static double code(double x, double eps) {
	return eps * ((eps * -0.5) - x);
}

Python

def code(x, eps):
	return eps * ((eps * -0.5) - x)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.7%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0 95.9%

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

   1. associate-*r* 95.9%

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

5. Simplified 95.9%

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

6. Taylor expanded in x around 0 94.7%

   \[\leadsto \varepsilon \cdot \color{blue}{\left(-1 \cdot x + -0.5 \cdot \varepsilon\right)} \]
7. Step-by-step derivation

   1. +-commutative 94.7%

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

   2. mul-1-neg 94.7%

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

   3. unsub-neg 94.7%

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

   4. *-commutative 94.7%

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

8. Simplified 94.7%

   \[\leadsto \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot -0.5 - x\right)} \]
9. Add Preprocessing

Alternative 15: 51.2% accurate, 41.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 (* eps (* eps -0.5)))

C

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

Fortran

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

Java

public static double code(double x, double eps) {
	return eps * (eps * -0.5);
}

Python

def code(x, eps):
	return eps * (eps * -0.5)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.7%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0 95.9%

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

   1. associate-*r* 95.9%

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

5. Simplified 95.9%

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

6. Taylor expanded in x around 0 54.4%

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

   1. *-commutative 54.4%

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

8. Simplified 54.4%

   \[\leadsto \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot -0.5\right)} \]
9. Add Preprocessing

Alternative 16: 48.8% 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 56.7%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
2. Add Preprocessing

3. Taylor expanded in x around 0 54.7%

   \[\leadsto \color{blue}{\cos \varepsilon - 1} \]

4. Taylor expanded in eps around 0 51.8%

   \[\leadsto \color{blue}{1} - 1 \]
5. Step-by-step derivation

   1. metadata-eval 51.8%

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

6. Applied egg-rr 51.8%

   \[\leadsto \color{blue}{0} \]
7. Add Preprocessing

Developer Target 1: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return (-2.0 * sin((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) * sin((x + (eps / 2.0d0)))) * sin((eps / 2.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 2: 98.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ {\left(\sqrt[3]{\left(-2 \cdot \sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)}\right)}^{3} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (pow (cbrt (* (* -2.0 (sin (* 0.5 (fma 2.0 x eps)))) (sin (* 0.5 eps)))) 3.0))

C

double code(double x, double eps) {
	return pow(cbrt(((-2.0 * sin((0.5 * fma(2.0, x, eps)))) * sin((0.5 * eps)))), 3.0);
}

Julia

function code(x, eps)
	return cbrt(Float64(Float64(-2.0 * sin(Float64(0.5 * fma(2.0, x, eps)))) * sin(Float64(0.5 * eps)))) ^ 3.0
end

Wolfram

code[x_, eps_] := N[Power[N[Power[N[(N[(-2.0 * N[Sin[N[(0.5 * N[(2.0 * x + eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(0.5 * eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]

Reproduce

herbie shell --seed 2024169 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

  :alt
  (! :herbie-platform default (* -2 (sin (+ x (/ eps 2))) (sin (/ eps 2))))

  :alt
  (! :herbie-platform default (pow (cbrt (* -2 (sin (* 1/2 (fma 2 x eps))) (sin (* 1/2 eps)))) 3))

  (- (cos (+ x eps)) (cos x)))