2cos (problem 3.3.5)

Percentage Accurate: 52.8% → 99.3%

Time: 16.5s

Alternatives: 9

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: 52.8% 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 54.9%

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

   1. diff-cos 82.5%

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

   2. div-inv 82.5%

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

   3. associate--l+ 82.5%

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

   4. metadata-eval 82.5%

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

   5. div-inv 82.5%

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

   6. +-commutative 82.5%

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

   7. associate-+l+ 82.5%

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

   8. metadata-eval 82.5%

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

4. Applied egg-rr 82.5%

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

   1. associate-*r* 82.5%

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

   2. *-commutative 82.5%

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

   3. associate-*l* 82.5%

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

   4. *-commutative 82.5%

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

   5. associate-+r- 82.5%

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

   6. +-commutative 82.5%

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

   7. associate--l+ 99.4%

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

   8. +-inverses 99.4%

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

   9. distribute-lft-in 99.4%

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

   10. metadata-eval 99.4%

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

   11. *-commutative 99.4%

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

   12. +-commutative 99.4%

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

   13. count-2 99.4%

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

   14. fma-define 99.4%

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

6. Simplified 99.4%

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

7. Final simplification 99.4%

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

Alternative 2: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 0.00016) {
		tmp = eps * ((cos(x) * (eps * -0.5)) - sin(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 = eps * ((cos(x) * (eps * (-0.5d0))) - sin(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 = eps * ((Math.cos(x) * (eps * -0.5)) - Math.sin(x));
	} else {
		tmp = Math.cos((eps + x)) - Math.cos(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= 0.00016:
		tmp = eps * ((math.cos(x) * (eps * -0.5)) - math.sin(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(eps * Float64(Float64(cos(x) * Float64(eps * -0.5)) - sin(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 = eps * ((cos(x) * (eps * -0.5)) - sin(x));
	else
		tmp = cos((eps + x)) - cos(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 0.00016], N[(eps * N[(N[(N[Cos[x], $MachinePrecision] * N[(eps * -0.5), $MachinePrecision]), $MachinePrecision] - N[Sin[x], $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 53.3%

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

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

   if 1.60000000000000013e-4 < eps

   1. Initial program 83.7%

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

4. Final simplification 98.7%

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

Alternative 3: 99.2% 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 54.9%

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

   1. diff-cos 82.5%

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

   2. div-inv 82.5%

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

   3. associate--l+ 82.5%

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

   4. metadata-eval 82.5%

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

   5. div-inv 82.5%

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

   6. +-commutative 82.5%

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

   7. associate-+l+ 82.5%

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

   8. metadata-eval 82.5%

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

4. Applied egg-rr 82.5%

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

   1. associate-*r* 82.5%

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

   2. *-commutative 82.5%

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

   3. associate-*l* 82.5%

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

   4. *-commutative 82.5%

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

   5. associate-+r- 82.5%

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

   6. +-commutative 82.5%

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

   7. associate--l+ 99.4%

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

   8. +-inverses 99.4%

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

   9. distribute-lft-in 99.4%

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

   10. metadata-eval 99.4%

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

   11. *-commutative 99.4%

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

   12. +-commutative 99.4%

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

   13. count-2 99.4%

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

   14. fma-define 99.4%

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

6. Simplified 99.4%

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

7. Taylor expanded in x around -inf 99.4%

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

Alternative 4: 98.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 0.000165)
   (* -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 <= 0.000165) {
		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 <= 0.000165d0) 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 <= 0.000165) {
		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 <= 0.000165:
		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 <= 0.000165)
		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 <= 0.000165)
		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, 0.000165], 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 < 1.65e-4

   1. Initial program 53.3%

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

      1. diff-cos 81.9%

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

      2. div-inv 81.9%

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

      3. associate--l+ 81.9%

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

      4. metadata-eval 81.9%

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

      5. div-inv 81.9%

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

      6. +-commutative 81.9%

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

      7. associate-+l+ 81.9%

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

      8. metadata-eval 81.9%

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

   4. Applied egg-rr 81.9%

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

      1. associate-*r* 81.9%

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

      2. *-commutative 81.9%

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

      3. associate-*l* 81.9%

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

      4. *-commutative 81.9%

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

      5. associate-+r- 81.9%

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

      6. +-commutative 81.9%

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

      7. associate--l+ 99.7%

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

      8. +-inverses 99.7%

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

      9. distribute-lft-in 99.7%

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

      10. metadata-eval 99.7%

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

      11. *-commutative 99.7%

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

      12. +-commutative 99.7%

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

      13. count-2 99.7%

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

      14. fma-define 99.7%

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

   6. Simplified 99.7%

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

   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 99.5%

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

   if 1.65e-4 < eps

   1. Initial program 83.7%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.000165:\\ \;\;\;\;-2 \cdot \left(\left(0.5 \cdot \varepsilon\right) \cdot \sin \left(0.5 \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\varepsilon + x\right) - \cos x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.0019:\\ \;\;\;\;-2 \cdot \left(\left(0.5 \cdot \varepsilon\right) \cdot \sin \left(0.5 \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\varepsilon + x\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 0.0019)
   (* -2.0 (* (* 0.5 eps) (sin (* 0.5 (- eps (* -2.0 x))))))
   (+ (cos (+ eps x)) -1.0)))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 0.0019) {
		tmp = -2.0 * ((0.5 * eps) * sin((0.5 * (eps - (-2.0 * 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 <= 0.0019d0) then
        tmp = (-2.0d0) * ((0.5d0 * eps) * sin((0.5d0 * (eps - ((-2.0d0) * 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 <= 0.0019) {
		tmp = -2.0 * ((0.5 * eps) * Math.sin((0.5 * (eps - (-2.0 * x)))));
	} else {
		tmp = Math.cos((eps + x)) + -1.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= 0.0019:
		tmp = -2.0 * ((0.5 * eps) * math.sin((0.5 * (eps - (-2.0 * x)))))
	else:
		tmp = math.cos((eps + x)) + -1.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 0.0019)
		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)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= 0.0019)
		tmp = -2.0 * ((0.5 * eps) * sin((0.5 * (eps - (-2.0 * x)))));
	else
		tmp = cos((eps + x)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 0.0019], 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] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 0.0019

   1. Initial program 53.5%

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

      1. diff-cos 82.0%

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

      2. div-inv 82.0%

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

      3. associate--l+ 82.0%

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

      4. metadata-eval 82.0%

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

      5. div-inv 82.0%

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

      6. +-commutative 82.0%

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

      7. associate-+l+ 82.0%

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

      8. metadata-eval 82.0%

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

   4. Applied egg-rr 82.0%

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

      1. associate-*r* 82.1%

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

      2. *-commutative 82.1%

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

      3. associate-*l* 82.1%

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

      4. *-commutative 82.1%

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

      5. associate-+r- 82.0%

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

      6. +-commutative 82.0%

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

      7. associate--l+ 99.7%

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

      8. +-inverses 99.7%

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

      9. distribute-lft-in 99.7%

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

      10. metadata-eval 99.7%

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

      11. *-commutative 99.7%

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

      12. +-commutative 99.7%

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

      13. count-2 99.7%

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

      14. fma-define 99.7%

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

   6. Simplified 99.7%

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

   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 99.0%

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

   if 0.0019 < eps

   1. Initial program 85.6%

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

   3. Taylor expanded in x around 0 75.1%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.0019:\\ \;\;\;\;-2 \cdot \left(\left(0.5 \cdot \varepsilon\right) \cdot \sin \left(0.5 \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\varepsilon + x\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.9% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot 0.041666666666666664 + x \cdot 0.16666666666666666\right) - 0.5\right) - x\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (-
   (*
    eps
    (- (* eps (+ (* eps 0.041666666666666664) (* x 0.16666666666666666))) 0.5))
   x)))

C

double code(double x, double eps) {
	return eps * ((eps * ((eps * ((eps * 0.041666666666666664) + (x * 0.16666666666666666))) - 0.5)) - x);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * ((eps * ((eps * ((eps * 0.041666666666666664d0) + (x * 0.16666666666666666d0))) - 0.5d0)) - x)
end function

Java

public static double code(double x, double eps) {
	return eps * ((eps * ((eps * ((eps * 0.041666666666666664) + (x * 0.16666666666666666))) - 0.5)) - x);
}

Python

def code(x, eps):
	return eps * ((eps * ((eps * ((eps * 0.041666666666666664) + (x * 0.16666666666666666))) - 0.5)) - x)

Julia

function code(x, eps)
	return Float64(eps * Float64(Float64(eps * Float64(Float64(eps * Float64(Float64(eps * 0.041666666666666664) + Float64(x * 0.16666666666666666))) - 0.5)) - x))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * ((eps * ((eps * ((eps * 0.041666666666666664) + (x * 0.16666666666666666))) - 0.5)) - x);
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(eps * N[(N[(eps * N[(N[(eps * 0.041666666666666664), $MachinePrecision] + N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.9%

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

3. Taylor expanded in x around 0 53.7%

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

   1. associate--l+ 53.7%

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

   2. associate-*r* 53.7%

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

   3. mul-1-neg 53.7%

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

5. Simplified 53.7%

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

6. Taylor expanded in eps around 0 95.3%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(-1 \cdot x + \varepsilon \cdot \left(\varepsilon \cdot \left(0.041666666666666664 \cdot \varepsilon + 0.16666666666666666 \cdot x\right) - 0.5\right)\right)} \]

7. Final simplification 95.3%

   \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot 0.041666666666666664 + x \cdot 0.16666666666666666\right) - 0.5\right) - x\right) \]
8. Add Preprocessing

Alternative 7: 94.3% 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 54.9%

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

3. Taylor expanded in x around 0 53.7%

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

   1. associate--l+ 53.7%

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

   2. associate-*r* 53.7%

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

   3. mul-1-neg 53.7%

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

5. Simplified 53.7%

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

6. Taylor expanded in eps around 0 94.6%

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

   1. neg-mul-1 94.6%

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

   2. +-commutative 94.6%

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

   3. unsub-neg 94.6%

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

   4. *-commutative 94.6%

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

8. Simplified 94.6%

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

Alternative 8: 50.1% 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 54.9%

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

3. Taylor expanded in eps around 0 96.3%

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

4. Taylor expanded in x around 0 52.3%

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

   1. *-commutative 52.3%

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

6. Simplified 52.3%

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

Alternative 9: 47.8% 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 54.9%

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

3. Taylor expanded in x around 0 52.9%

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

4. Taylor expanded in eps around 0 49.6%

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

   1. metadata-eval 49.6%

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

6. Applied egg-rr 49.6%

   \[\leadsto \color{blue}{0} \]
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 2024170 
(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)))