2cos (problem 3.3.5)

Percentage Accurate: 38.1% → 99.5%

Time: 19.1s

Alternatives: 15

Speedup: 1.9×

Specification

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: 38.1% 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: 99.5% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (fma (cos x) (* (tan (/ eps 2.0)) (- (sin eps))) (* (sin eps) (- (sin x)))))

C

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

Julia

function code(x, eps)
	return fma(cos(x), Float64(tan(Float64(eps / 2.0)) * Float64(-sin(eps))), Float64(sin(eps) * Float64(-sin(x))))
end

Wolfram

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

Derivation

1. Initial program 37.2%

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

   1. cos-sum 64.3%

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

   2. associate--l- 64.4%

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

   3. fma-neg 64.4%

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

   4. fma-def 64.4%

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

3. Applied egg-rr 64.4%

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

4. Taylor expanded in x around inf 64.4%

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

   1. associate--r+ 89.8%

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

   2. *-commutative 89.8%

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

   3. *-rgt-identity 89.8%

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

   4. distribute-lft-out-- 89.8%

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

   5. sub-neg 89.8%

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

   6. metadata-eval 89.8%

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

   7. *-commutative 89.8%

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

   8. fma-neg 89.8%

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

   9. +-commutative 89.8%

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

   10. distribute-rgt-neg-in 89.8%

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

6. Simplified 89.8%

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

   1. flip-+ 89.5%

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

   2. frac-2neg 89.5%

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

   3. metadata-eval 89.5%

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

   4. 1-sub-cos 99.2%

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

   5. add-sqr-sqrt 46.1%

      \[\leadsto \mathsf{fma}\left(\cos x, \frac{-\color{blue}{\left(\sqrt{\sin \varepsilon} \cdot \sqrt{\sin \varepsilon}\right)} \cdot \sin \varepsilon}{-\left(-1 - \cos \varepsilon\right)}, \sin x \cdot \left(-\sin \varepsilon\right)\right) \]

   6. sqrt-unprod 67.7%

      \[\leadsto \mathsf{fma}\left(\cos x, \frac{-\color{blue}{\sqrt{\sin \varepsilon \cdot \sin \varepsilon}} \cdot \sin \varepsilon}{-\left(-1 - \cos \varepsilon\right)}, \sin x \cdot \left(-\sin \varepsilon\right)\right) \]

   7. sqr-neg 67.7%

      \[\leadsto \mathsf{fma}\left(\cos x, \frac{-\sqrt{\color{blue}{\left(-\sin \varepsilon\right) \cdot \left(-\sin \varepsilon\right)}} \cdot \sin \varepsilon}{-\left(-1 - \cos \varepsilon\right)}, \sin x \cdot \left(-\sin \varepsilon\right)\right) \]

   8. sqrt-unprod 21.5%

      \[\leadsto \mathsf{fma}\left(\cos x, \frac{-\color{blue}{\left(\sqrt{-\sin \varepsilon} \cdot \sqrt{-\sin \varepsilon}\right)} \cdot \sin \varepsilon}{-\left(-1 - \cos \varepsilon\right)}, \sin x \cdot \left(-\sin \varepsilon\right)\right) \]

   9. add-sqr-sqrt 40.5%

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

   10. distribute-rgt-neg-out 40.5%

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

   11. sqr-neg 40.5%

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

   12. pow2 40.5%

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

   13. flip-- 3.9%

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

8. Applied egg-rr 99.2%

   \[\leadsto \mathsf{fma}\left(\cos x, \color{blue}{\frac{{\sin \varepsilon}^{2}}{-\left(\cos \varepsilon + 1\right)}}, \sin x \cdot \left(-\sin \varepsilon\right)\right) \]
9. Step-by-step derivation

   1. neg-sub0 99.2%

      \[\leadsto \mathsf{fma}\left(\cos x, \frac{{\sin \varepsilon}^{2}}{\color{blue}{0 - \left(\cos \varepsilon + 1\right)}}, \sin x \cdot \left(-\sin \varepsilon\right)\right) \]

   2. +-commutative 99.2%

      \[\leadsto \mathsf{fma}\left(\cos x, \frac{{\sin \varepsilon}^{2}}{0 - \color{blue}{\left(1 + \cos \varepsilon\right)}}, \sin x \cdot \left(-\sin \varepsilon\right)\right) \]

   3. associate--r+ 99.2%

      \[\leadsto \mathsf{fma}\left(\cos x, \frac{{\sin \varepsilon}^{2}}{\color{blue}{\left(0 - 1\right) - \cos \varepsilon}}, \sin x \cdot \left(-\sin \varepsilon\right)\right) \]

   4. metadata-eval 99.2%

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

10. Simplified 99.2%

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

11. Taylor expanded in eps around inf 99.2%

    \[\leadsto \mathsf{fma}\left(\cos x, \color{blue}{-1 \cdot \frac{{\sin \varepsilon}^{2}}{1 + \cos \varepsilon}}, \sin x \cdot \left(-\sin \varepsilon\right)\right) \]
12. Step-by-step derivation

    1. mul-1-neg 99.2%

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

    2. +-commutative 99.2%

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

    3. unpow2 99.2%

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

    4. associate-*r/ 99.1%

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

    5. distribute-rgt-neg-in 99.1%

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

    6. +-commutative 99.1%

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

    7. hang-0p-tan 99.6%

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

13. Simplified 99.6%

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

14. Final simplification 99.6%

    \[\leadsto \mathsf{fma}\left(\cos x, \tan \left(\frac{\varepsilon}{2}\right) \cdot \left(-\sin \varepsilon\right), \sin \varepsilon \cdot \left(-\sin x\right)\right) \]

Alternative 2: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\varepsilon \cdot 0.5\right)\\ \mathbf{if}\;x \leq -2.95 \cdot 10^{-8} \lor \neg \left(x \leq 2.15 \cdot 10^{-40}\right):\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon + -1, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 + x \cdot \cos \left(\varepsilon \cdot 0.5\right)\right) \cdot \left(t_0 \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (* eps 0.5))))
   (if (or (<= x -2.95e-8) (not (<= x 2.15e-40)))
     (fma (cos x) (+ (cos eps) -1.0) (* (sin eps) (- (sin x))))
     (* (+ t_0 (* x (cos (* eps 0.5)))) (* t_0 -2.0)))))

C

double code(double x, double eps) {
	double t_0 = sin((eps * 0.5));
	double tmp;
	if ((x <= -2.95e-8) || !(x <= 2.15e-40)) {
		tmp = fma(cos(x), (cos(eps) + -1.0), (sin(eps) * -sin(x)));
	} else {
		tmp = (t_0 + (x * cos((eps * 0.5)))) * (t_0 * -2.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = sin(Float64(eps * 0.5))
	tmp = 0.0
	if ((x <= -2.95e-8) || !(x <= 2.15e-40))
		tmp = fma(cos(x), Float64(cos(eps) + -1.0), Float64(sin(eps) * Float64(-sin(x))));
	else
		tmp = Float64(Float64(t_0 + Float64(x * cos(Float64(eps * 0.5)))) * Float64(t_0 * -2.0));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[x, -2.95e-8], N[Not[LessEqual[x, 2.15e-40]], $MachinePrecision]], N[(N[Cos[x], $MachinePrecision] * N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision] + N[(N[Sin[eps], $MachinePrecision] * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 + N[(x * N[Cos[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * -2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.9499999999999999e-8 or 2.1500000000000001e-40 < x

   1. Initial program 9.8%

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

      1. cos-sum 59.2%

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

      2. associate--l- 59.3%

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

      3. fma-neg 59.3%

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

      4. fma-def 59.3%

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

   3. Applied egg-rr 59.3%

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

   4. Taylor expanded in x around inf 59.3%

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

      1. associate--r+ 99.2%

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

      2. *-commutative 99.2%

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

      3. *-rgt-identity 99.2%

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

      4. distribute-lft-out-- 99.1%

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

      5. sub-neg 99.1%

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

      6. metadata-eval 99.1%

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

      7. *-commutative 99.1%

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

      8. fma-neg 99.1%

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

      9. +-commutative 99.1%

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

      10. distribute-rgt-neg-in 99.1%

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

   6. Simplified 99.1%

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

   if -2.9499999999999999e-8 < x < 2.1500000000000001e-40

   1. Initial program 69.8%

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

      1. diff-cos 90.9%

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

      2. div-inv 90.9%

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

      3. associate--l+ 90.9%

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

      4. metadata-eval 90.9%

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

      5. div-inv 90.9%

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

      6. +-commutative 90.9%

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

      7. associate-+l+ 90.9%

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

      8. metadata-eval 90.9%

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

   3. Applied egg-rr 90.9%

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

      1. associate-*r* 90.9%

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

      2. *-commutative 90.9%

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

      3. *-commutative 90.9%

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

      4. +-commutative 90.9%

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

      5. count-2 90.9%

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

      6. fma-def 90.9%

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

      7. sub-neg 90.9%

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

      8. mul-1-neg 90.9%

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

      9. +-commutative 90.9%

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

      10. associate-+r+ 98.9%

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

      11. mul-1-neg 98.9%

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

      12. sub-neg 98.9%

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

      13. +-inverses 98.9%

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

      14. remove-double-neg 98.9%

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

      15. mul-1-neg 98.9%

          \[\leadsto \sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(-2 \cdot \sin \left(\left(0 + \left(-\color{blue}{-1 \cdot \varepsilon}\right)\right) \cdot 0.5\right)\right) \]

      16. sub-neg 98.9%

          \[\leadsto \sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(-2 \cdot \sin \left(\color{blue}{\left(0 - -1 \cdot \varepsilon\right)} \cdot 0.5\right)\right) \]

      17. neg-sub0 98.9%

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

      18. mul-1-neg 98.9%

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

      19. remove-double-neg 98.9%

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

   5. Simplified 98.9%

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

   6. Taylor expanded in x around 0 99.6%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.95 \cdot 10^{-8} \lor \neg \left(x \leq 2.15 \cdot 10^{-40}\right):\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon + -1, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin \left(\varepsilon \cdot 0.5\right) + x \cdot \cos \left(\varepsilon \cdot 0.5\right)\right) \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot -2\right)\\ \end{array} \]

Alternative 3: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\varepsilon \cdot 0.5\right)\\ \mathbf{if}\;x \leq -2.75 \cdot 10^{-8}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \cos x\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{-41}:\\ \;\;\;\;\left(t_0 + x \cdot \cos \left(\varepsilon \cdot 0.5\right)\right) \cdot \left(t_0 \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon + -1, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (* eps 0.5))))
   (if (<= x -2.75e-8)
     (- (- (* (cos x) (cos eps)) (cos x)) (* (sin eps) (sin x)))
     (if (<= x 4.4e-41)
       (* (+ t_0 (* x (cos (* eps 0.5)))) (* t_0 -2.0))
       (fma (cos x) (+ (cos eps) -1.0) (* (sin eps) (- (sin x))))))))

C

double code(double x, double eps) {
	double t_0 = sin((eps * 0.5));
	double tmp;
	if (x <= -2.75e-8) {
		tmp = ((cos(x) * cos(eps)) - cos(x)) - (sin(eps) * sin(x));
	} else if (x <= 4.4e-41) {
		tmp = (t_0 + (x * cos((eps * 0.5)))) * (t_0 * -2.0);
	} else {
		tmp = fma(cos(x), (cos(eps) + -1.0), (sin(eps) * -sin(x)));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = sin(Float64(eps * 0.5))
	tmp = 0.0
	if (x <= -2.75e-8)
		tmp = Float64(Float64(Float64(cos(x) * cos(eps)) - cos(x)) - Float64(sin(eps) * sin(x)));
	elseif (x <= 4.4e-41)
		tmp = Float64(Float64(t_0 + Float64(x * cos(Float64(eps * 0.5)))) * Float64(t_0 * -2.0));
	else
		tmp = fma(cos(x), Float64(cos(eps) + -1.0), Float64(sin(eps) * Float64(-sin(x))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -2.75e-8], N[(N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.4e-41], N[(N[(t$95$0 + N[(x * N[Cos[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * -2.0), $MachinePrecision]), $MachinePrecision], N[(N[Cos[x], $MachinePrecision] * N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision] + N[(N[Sin[eps], $MachinePrecision] * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.7500000000000001e-8

   1. Initial program 6.7%

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

      1. cos-sum 56.6%

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

      2. associate--l- 56.6%

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

      3. fma-neg 56.7%

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

      4. fma-def 56.6%

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

   3. Applied egg-rr 56.6%

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

   4. Taylor expanded in x around inf 56.6%

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

      1. associate--r+ 99.2%

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

      2. *-commutative 99.2%

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

      3. *-rgt-identity 99.2%

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

      4. distribute-lft-out-- 99.1%

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

      5. sub-neg 99.1%

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

      6. metadata-eval 99.1%

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

      7. *-commutative 99.1%

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

      8. fma-neg 99.1%

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

      9. +-commutative 99.1%

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

      10. distribute-rgt-neg-in 99.1%

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

   6. Simplified 99.1%

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

   7. Taylor expanded in x around inf 99.1%

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

      1. +-commutative 99.1%

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

      2. sub-neg 99.1%

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

      3. metadata-eval 99.1%

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

      4. +-commutative 99.1%

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

      5. mul-1-neg 99.1%

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

      6. distribute-lft-neg-in 99.1%

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

      7. cancel-sign-sub-inv 99.1%

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

      8. +-commutative 99.1%

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

   9. Simplified 99.1%

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

       1. distribute-rgt-in 99.2%

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

       2. mul-1-neg 99.2%

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

   11. Applied egg-rr 99.2%

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

   if -2.7500000000000001e-8 < x < 4.4e-41

   1. Initial program 69.8%

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

      1. diff-cos 90.9%

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

      2. div-inv 90.9%

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

      3. associate--l+ 90.9%

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

      4. metadata-eval 90.9%

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

      5. div-inv 90.9%

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

      6. +-commutative 90.9%

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

      7. associate-+l+ 90.9%

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

      8. metadata-eval 90.9%

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

   3. Applied egg-rr 90.9%

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

      1. associate-*r* 90.9%

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

      2. *-commutative 90.9%

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

      3. *-commutative 90.9%

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

      4. +-commutative 90.9%

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

      5. count-2 90.9%

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

      6. fma-def 90.9%

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

      7. sub-neg 90.9%

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

      8. mul-1-neg 90.9%

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

      9. +-commutative 90.9%

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

      10. associate-+r+ 98.9%

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

      11. mul-1-neg 98.9%

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

      12. sub-neg 98.9%

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

      13. +-inverses 98.9%

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

      14. remove-double-neg 98.9%

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

      15. mul-1-neg 98.9%

          \[\leadsto \sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(-2 \cdot \sin \left(\left(0 + \left(-\color{blue}{-1 \cdot \varepsilon}\right)\right) \cdot 0.5\right)\right) \]

      16. sub-neg 98.9%

          \[\leadsto \sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(-2 \cdot \sin \left(\color{blue}{\left(0 - -1 \cdot \varepsilon\right)} \cdot 0.5\right)\right) \]

      17. neg-sub0 98.9%

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

      18. mul-1-neg 98.9%

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

      19. remove-double-neg 98.9%

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

   5. Simplified 98.9%

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

   6. Taylor expanded in x around 0 99.6%

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

   if 4.4e-41 < x

   1. Initial program 12.8%

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

      1. cos-sum 61.6%

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

      2. associate--l- 61.8%

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

      3. fma-neg 61.9%

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

      4. fma-def 61.9%

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

   3. Applied egg-rr 61.9%

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

   4. Taylor expanded in x around inf 61.8%

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

      1. associate--r+ 99.1%

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

      2. *-commutative 99.1%

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

      3. *-rgt-identity 99.1%

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

      4. distribute-lft-out-- 99.1%

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

      5. sub-neg 99.1%

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

      6. metadata-eval 99.1%

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

      7. *-commutative 99.1%

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

      8. fma-neg 99.1%

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

      9. +-commutative 99.1%

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

      10. distribute-rgt-neg-in 99.1%

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

   6. Simplified 99.1%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.75 \cdot 10^{-8}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \cos x\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{-41}:\\ \;\;\;\;\left(\sin \left(\varepsilon \cdot 0.5\right) + x \cdot \cos \left(\varepsilon \cdot 0.5\right)\right) \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon + -1, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \end{array} \]

Alternative 4: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\varepsilon \cdot 0.5\right)\\ \mathbf{if}\;x \leq -3.5 \cdot 10^{-8} \lor \neg \left(x \leq 2.3 \cdot 10^{-40}\right):\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 + x \cdot \cos \left(\varepsilon \cdot 0.5\right)\right) \cdot \left(t_0 \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (* eps 0.5))))
   (if (or (<= x -3.5e-8) (not (<= x 2.3e-40)))
     (- (* (cos x) (+ (cos eps) -1.0)) (* (sin eps) (sin x)))
     (* (+ t_0 (* x (cos (* eps 0.5)))) (* t_0 -2.0)))))

C

double code(double x, double eps) {
	double t_0 = sin((eps * 0.5));
	double tmp;
	if ((x <= -3.5e-8) || !(x <= 2.3e-40)) {
		tmp = (cos(x) * (cos(eps) + -1.0)) - (sin(eps) * sin(x));
	} else {
		tmp = (t_0 + (x * cos((eps * 0.5)))) * (t_0 * -2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((eps * 0.5d0))
    if ((x <= (-3.5d-8)) .or. (.not. (x <= 2.3d-40))) then
        tmp = (cos(x) * (cos(eps) + (-1.0d0))) - (sin(eps) * sin(x))
    else
        tmp = (t_0 + (x * cos((eps * 0.5d0)))) * (t_0 * (-2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin((eps * 0.5));
	double tmp;
	if ((x <= -3.5e-8) || !(x <= 2.3e-40)) {
		tmp = (Math.cos(x) * (Math.cos(eps) + -1.0)) - (Math.sin(eps) * Math.sin(x));
	} else {
		tmp = (t_0 + (x * Math.cos((eps * 0.5)))) * (t_0 * -2.0);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.sin((eps * 0.5))
	tmp = 0
	if (x <= -3.5e-8) or not (x <= 2.3e-40):
		tmp = (math.cos(x) * (math.cos(eps) + -1.0)) - (math.sin(eps) * math.sin(x))
	else:
		tmp = (t_0 + (x * math.cos((eps * 0.5)))) * (t_0 * -2.0)
	return tmp

Julia

function code(x, eps)
	t_0 = sin(Float64(eps * 0.5))
	tmp = 0.0
	if ((x <= -3.5e-8) || !(x <= 2.3e-40))
		tmp = Float64(Float64(cos(x) * Float64(cos(eps) + -1.0)) - Float64(sin(eps) * sin(x)));
	else
		tmp = Float64(Float64(t_0 + Float64(x * cos(Float64(eps * 0.5)))) * Float64(t_0 * -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = sin((eps * 0.5));
	tmp = 0.0;
	if ((x <= -3.5e-8) || ~((x <= 2.3e-40)))
		tmp = (cos(x) * (cos(eps) + -1.0)) - (sin(eps) * sin(x));
	else
		tmp = (t_0 + (x * cos((eps * 0.5)))) * (t_0 * -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[x, -3.5e-8], N[Not[LessEqual[x, 2.3e-40]], $MachinePrecision]], N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 + N[(x * N[Cos[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * -2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.50000000000000024e-8 or 2.3e-40 < x

   1. Initial program 9.8%

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

      1. cos-sum 59.2%

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

      2. associate--l- 59.3%

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

      3. fma-neg 59.3%

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

      4. fma-def 59.3%

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

   3. Applied egg-rr 59.3%

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

   4. Taylor expanded in x around inf 59.3%

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

      1. associate--r+ 99.2%

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

      2. *-commutative 99.2%

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

      3. *-rgt-identity 99.2%

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

      4. distribute-lft-out-- 99.1%

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

      5. sub-neg 99.1%

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

      6. metadata-eval 99.1%

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

      7. +-commutative 99.1%

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

      8. *-commutative 99.1%

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

   6. Simplified 99.1%

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

   if -3.50000000000000024e-8 < x < 2.3e-40

   1. Initial program 69.8%

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

      1. diff-cos 90.9%

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

      2. div-inv 90.9%

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

      3. associate--l+ 90.9%

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

      4. metadata-eval 90.9%

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

      5. div-inv 90.9%

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

      6. +-commutative 90.9%

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

      7. associate-+l+ 90.9%

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

      8. metadata-eval 90.9%

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

   3. Applied egg-rr 90.9%

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

      1. associate-*r* 90.9%

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

      2. *-commutative 90.9%

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

      3. *-commutative 90.9%

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

      4. +-commutative 90.9%

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

      5. count-2 90.9%

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

      6. fma-def 90.9%

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

      7. sub-neg 90.9%

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

      8. mul-1-neg 90.9%

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

      9. +-commutative 90.9%

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

      10. associate-+r+ 98.9%

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

      11. mul-1-neg 98.9%

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

      12. sub-neg 98.9%

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

      13. +-inverses 98.9%

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

      14. remove-double-neg 98.9%

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

      15. mul-1-neg 98.9%

          \[\leadsto \sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(-2 \cdot \sin \left(\left(0 + \left(-\color{blue}{-1 \cdot \varepsilon}\right)\right) \cdot 0.5\right)\right) \]

      16. sub-neg 98.9%

          \[\leadsto \sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(-2 \cdot \sin \left(\color{blue}{\left(0 - -1 \cdot \varepsilon\right)} \cdot 0.5\right)\right) \]

      17. neg-sub0 98.9%

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

      18. mul-1-neg 98.9%

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

      19. remove-double-neg 98.9%

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

   5. Simplified 98.9%

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

   6. Taylor expanded in x around 0 99.6%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-8} \lor \neg \left(x \leq 2.3 \cdot 10^{-40}\right):\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\left(\sin \left(\varepsilon \cdot 0.5\right) + x \cdot \cos \left(\varepsilon \cdot 0.5\right)\right) \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot -2\right)\\ \end{array} \]

Alternative 5: 66.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(x + \varepsilon\right) - \cos x \leq -1 \cdot 10^{-16}:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- (cos (+ x eps)) (cos x)) -1e-16)
   (- (cos eps) (cos x))
   (* eps (- (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if ((cos((x + eps)) - cos(x)) <= -1e-16) {
		tmp = cos(eps) - cos(x);
	} else {
		tmp = eps * -sin(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((cos((x + eps)) - cos(x)) <= (-1d-16)) then
        tmp = cos(eps) - cos(x)
    else
        tmp = eps * -sin(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((Math.cos((x + eps)) - Math.cos(x)) <= -1e-16) {
		tmp = Math.cos(eps) - Math.cos(x);
	} else {
		tmp = eps * -Math.sin(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (math.cos((x + eps)) - math.cos(x)) <= -1e-16:
		tmp = math.cos(eps) - math.cos(x)
	else:
		tmp = eps * -math.sin(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(cos(Float64(x + eps)) - cos(x)) <= -1e-16)
		tmp = Float64(cos(eps) - cos(x));
	else
		tmp = Float64(eps * Float64(-sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((cos((x + eps)) - cos(x)) <= -1e-16)
		tmp = cos(eps) - cos(x);
	else
		tmp = eps * -sin(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(N[Cos[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], -1e-16], N[(N[Cos[eps], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], N[(eps * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cos.f64 (+.f64 x eps)) (cos.f64 x)) < -9.9999999999999998e-17

   1. Initial program 75.9%

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

   2. Taylor expanded in x around 0 76.2%

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

   if -9.9999999999999998e-17 < (-.f64 (cos.f64 (+.f64 x eps)) (cos.f64 x))

   1. Initial program 18.4%

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

   2. Taylor expanded in eps around 0 56.4%

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

      1. mul-1-neg 56.4%

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

      2. *-commutative 56.4%

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

      3. distribute-rgt-neg-in 56.4%

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

   4. Simplified 56.4%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(x + \varepsilon\right) - \cos x \leq -1 \cdot 10^{-16}:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \]

Alternative 6: 76.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 37.2%

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

   1. diff-cos 47.2%

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

   2. div-inv 47.2%

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

   3. associate--l+ 47.2%

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

   4. metadata-eval 47.2%

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

   5. div-inv 47.2%

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

   6. +-commutative 47.2%

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

   7. associate-+l+ 47.3%

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

   8. metadata-eval 47.3%

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

3. Applied egg-rr 47.3%

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

   1. associate-*r* 47.4%

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

   2. *-commutative 47.4%

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

   3. *-commutative 47.4%

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

   4. +-commutative 47.4%

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

   5. count-2 47.4%

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

   6. fma-def 47.4%

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

   7. sub-neg 47.4%

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

   8. mul-1-neg 47.4%

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

   9. +-commutative 47.4%

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

   10. associate-+r+ 74.0%

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

   11. mul-1-neg 74.0%

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

   12. sub-neg 74.0%

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

   13. +-inverses 74.0%

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

   14. remove-double-neg 74.0%

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

   15. mul-1-neg 74.0%

       \[\leadsto \sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(-2 \cdot \sin \left(\left(0 + \left(-\color{blue}{-1 \cdot \varepsilon}\right)\right) \cdot 0.5\right)\right) \]

   16. sub-neg 74.0%

       \[\leadsto \sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(-2 \cdot \sin \left(\color{blue}{\left(0 - -1 \cdot \varepsilon\right)} \cdot 0.5\right)\right) \]

   17. neg-sub0 74.0%

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

   18. mul-1-neg 74.0%

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

   19. remove-double-neg 74.0%

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

5. Simplified 74.0%

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

6. Final simplification 74.0%

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

Alternative 7: 69.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin x \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot -2\right)\\ \mathbf{if}\;x \leq -7.6 \cdot 10^{-9}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -7.2 \cdot 10^{-63}:\\ \;\;\;\;-1 + \left(\cos \varepsilon - x \cdot \sin \varepsilon\right)\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-91}:\\ \;\;\;\;-0.5 \cdot {\varepsilon}^{2} + x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-9}:\\ \;\;\;\;\tan \left(\frac{\varepsilon}{2}\right) \cdot \left(-\sin \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (sin x) (* (sin (* eps 0.5)) -2.0))))
   (if (<= x -7.6e-9)
     t_0
     (if (<= x -7.2e-63)
       (+ -1.0 (- (cos eps) (* x (sin eps))))
       (if (<= x -8.2e-91)
         (+
          (* -0.5 (pow eps 2.0))
          (* x (- (* 0.16666666666666666 (pow eps 3.0)) eps)))
         (if (<= x 1.8e-9) (* (tan (/ eps 2.0)) (- (sin eps))) t_0))))))

C

double code(double x, double eps) {
	double t_0 = sin(x) * (sin((eps * 0.5)) * -2.0);
	double tmp;
	if (x <= -7.6e-9) {
		tmp = t_0;
	} else if (x <= -7.2e-63) {
		tmp = -1.0 + (cos(eps) - (x * sin(eps)));
	} else if (x <= -8.2e-91) {
		tmp = (-0.5 * pow(eps, 2.0)) + (x * ((0.16666666666666666 * pow(eps, 3.0)) - eps));
	} else if (x <= 1.8e-9) {
		tmp = tan((eps / 2.0)) * -sin(eps);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(x) * (sin((eps * 0.5d0)) * (-2.0d0))
    if (x <= (-7.6d-9)) then
        tmp = t_0
    else if (x <= (-7.2d-63)) then
        tmp = (-1.0d0) + (cos(eps) - (x * sin(eps)))
    else if (x <= (-8.2d-91)) then
        tmp = ((-0.5d0) * (eps ** 2.0d0)) + (x * ((0.16666666666666666d0 * (eps ** 3.0d0)) - eps))
    else if (x <= 1.8d-9) then
        tmp = tan((eps / 2.0d0)) * -sin(eps)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin(x) * (Math.sin((eps * 0.5)) * -2.0);
	double tmp;
	if (x <= -7.6e-9) {
		tmp = t_0;
	} else if (x <= -7.2e-63) {
		tmp = -1.0 + (Math.cos(eps) - (x * Math.sin(eps)));
	} else if (x <= -8.2e-91) {
		tmp = (-0.5 * Math.pow(eps, 2.0)) + (x * ((0.16666666666666666 * Math.pow(eps, 3.0)) - eps));
	} else if (x <= 1.8e-9) {
		tmp = Math.tan((eps / 2.0)) * -Math.sin(eps);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.sin(x) * (math.sin((eps * 0.5)) * -2.0)
	tmp = 0
	if x <= -7.6e-9:
		tmp = t_0
	elif x <= -7.2e-63:
		tmp = -1.0 + (math.cos(eps) - (x * math.sin(eps)))
	elif x <= -8.2e-91:
		tmp = (-0.5 * math.pow(eps, 2.0)) + (x * ((0.16666666666666666 * math.pow(eps, 3.0)) - eps))
	elif x <= 1.8e-9:
		tmp = math.tan((eps / 2.0)) * -math.sin(eps)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(sin(x) * Float64(sin(Float64(eps * 0.5)) * -2.0))
	tmp = 0.0
	if (x <= -7.6e-9)
		tmp = t_0;
	elseif (x <= -7.2e-63)
		tmp = Float64(-1.0 + Float64(cos(eps) - Float64(x * sin(eps))));
	elseif (x <= -8.2e-91)
		tmp = Float64(Float64(-0.5 * (eps ^ 2.0)) + Float64(x * Float64(Float64(0.16666666666666666 * (eps ^ 3.0)) - eps)));
	elseif (x <= 1.8e-9)
		tmp = Float64(tan(Float64(eps / 2.0)) * Float64(-sin(eps)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = sin(x) * (sin((eps * 0.5)) * -2.0);
	tmp = 0.0;
	if (x <= -7.6e-9)
		tmp = t_0;
	elseif (x <= -7.2e-63)
		tmp = -1.0 + (cos(eps) - (x * sin(eps)));
	elseif (x <= -8.2e-91)
		tmp = (-0.5 * (eps ^ 2.0)) + (x * ((0.16666666666666666 * (eps ^ 3.0)) - eps));
	elseif (x <= 1.8e-9)
		tmp = tan((eps / 2.0)) * -sin(eps);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Sin[x], $MachinePrecision] * N[(N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7.6e-9], t$95$0, If[LessEqual[x, -7.2e-63], N[(-1.0 + N[(N[Cos[eps], $MachinePrecision] - N[(x * N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -8.2e-91], N[(N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[(0.16666666666666666 * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision] - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.8e-9], N[(N[Tan[N[(eps / 2.0), $MachinePrecision]], $MachinePrecision] * (-N[Sin[eps], $MachinePrecision])), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -7.60000000000000023e-9 or 1.8e-9 < x

   1. Initial program 6.6%

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

      1. diff-cos 7.3%

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

      2. div-inv 7.3%

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

      3. associate--l+ 7.3%

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

      4. metadata-eval 7.3%

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

      5. div-inv 7.3%

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

      6. +-commutative 7.3%

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

      7. associate-+l+ 7.6%

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

      8. metadata-eval 7.6%

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

   3. Applied egg-rr 7.6%

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

      1. associate-*r* 7.6%

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

      2. *-commutative 7.6%

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

      3. *-commutative 7.6%

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

      4. +-commutative 7.6%

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

      5. count-2 7.6%

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

      6. fma-def 7.6%

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

      7. sub-neg 7.6%

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

      8. mul-1-neg 7.6%

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

      9. +-commutative 7.6%

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

      10. associate-+r+ 51.0%

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

      11. mul-1-neg 51.0%

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

      12. sub-neg 51.0%

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

      13. +-inverses 51.0%

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

      14. remove-double-neg 51.0%

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

      15. mul-1-neg 51.0%

          \[\leadsto \sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(-2 \cdot \sin \left(\left(0 + \left(-\color{blue}{-1 \cdot \varepsilon}\right)\right) \cdot 0.5\right)\right) \]

      16. sub-neg 51.0%

          \[\leadsto \sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(-2 \cdot \sin \left(\color{blue}{\left(0 - -1 \cdot \varepsilon\right)} \cdot 0.5\right)\right) \]

      17. neg-sub0 51.0%

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

      18. mul-1-neg 51.0%

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

      19. remove-double-neg 51.0%

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

   5. Simplified 51.0%

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

   6. Taylor expanded in eps around 0 50.8%

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

   if -7.60000000000000023e-9 < x < -7.20000000000000016e-63

   1. Initial program 82.3%

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

      1. log1p-expm1-u 82.0%

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

   3. Applied egg-rr 82.0%

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

   4. Taylor expanded in x around 0 87.7%

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

      1. sub-neg 87.7%

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

      2. metadata-eval 87.7%

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

      3. +-commutative 87.7%

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

      4. mul-1-neg 87.7%

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

      5. unsub-neg 87.7%

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

   6. Simplified 87.7%

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

   if -7.20000000000000016e-63 < x < -8.20000000000000048e-91

   1. Initial program 29.3%

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

   2. Taylor expanded in eps around 0 99.8%

      \[\leadsto \color{blue}{-1 \cdot \left(\varepsilon \cdot \sin x\right) + \left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right)} \]
   3. Step-by-step derivation

      1. associate-+r+ 99.8%

         \[\leadsto \color{blue}{\left(-1 \cdot \left(\varepsilon \cdot \sin x\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)} \]

      2. +-commutative 99.8%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right)} + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right) \]

      3. associate-+l+ 99.8%

         \[\leadsto \color{blue}{-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + \left(-1 \cdot \left(\varepsilon \cdot \sin x\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right)} \]

      4. associate-*r* 99.8%

         \[\leadsto \color{blue}{\left(-0.5 \cdot {\varepsilon}^{2}\right) \cdot \cos x} + \left(-1 \cdot \left(\varepsilon \cdot \sin x\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right) \]

      5. *-commutative 99.8%

         \[\leadsto \color{blue}{\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right)} + \left(-1 \cdot \left(\varepsilon \cdot \sin x\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right) \]

      6. associate-*r* 99.8%

         \[\leadsto \cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) + \left(\color{blue}{\left(-1 \cdot \varepsilon\right) \cdot \sin x} + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right) \]

      7. associate-*r* 99.8%

         \[\leadsto \cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) + \left(\left(-1 \cdot \varepsilon\right) \cdot \sin x + \color{blue}{\left(0.16666666666666666 \cdot {\varepsilon}^{3}\right) \cdot \sin x}\right) \]

      8. distribute-rgt-out 99.8%

         \[\leadsto \cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) + \color{blue}{\sin x \cdot \left(-1 \cdot \varepsilon + 0.16666666666666666 \cdot {\varepsilon}^{3}\right)} \]

      9. mul-1-neg 99.8%

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

   4. Simplified 99.8%

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

   5. Taylor expanded in x around 0 99.8%

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

   if -8.20000000000000048e-91 < x < 1.8e-9

   1. Initial program 72.1%

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

   2. Taylor expanded in x around 0 72.1%

      \[\leadsto \color{blue}{\cos \varepsilon - 1} \]
   3. Step-by-step derivation

      1. flip-- 71.8%

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

      2. frac-2neg 71.8%

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

      3. metadata-eval 71.8%

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

      4. sub-1-cos 92.6%

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

      5. pow2 92.6%

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

   4. Applied egg-rr 92.6%

      \[\leadsto \color{blue}{\frac{-\left(-{\sin \varepsilon}^{2}\right)}{-\left(\cos \varepsilon + 1\right)}} \]
   5. Step-by-step derivation

      1. remove-double-neg 92.6%

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

      2. neg-sub0 92.6%

         \[\leadsto \frac{{\sin \varepsilon}^{2}}{\color{blue}{0 - \left(\cos \varepsilon + 1\right)}} \]

      3. +-commutative 92.6%

         \[\leadsto \frac{{\sin \varepsilon}^{2}}{0 - \color{blue}{\left(1 + \cos \varepsilon\right)}} \]

      4. associate--r+ 92.6%

         \[\leadsto \frac{{\sin \varepsilon}^{2}}{\color{blue}{\left(0 - 1\right) - \cos \varepsilon}} \]

      5. metadata-eval 92.6%

         \[\leadsto \frac{{\sin \varepsilon}^{2}}{\color{blue}{-1} - \cos \varepsilon} \]

   6. Simplified 92.6%

      \[\leadsto \color{blue}{\frac{{\sin \varepsilon}^{2}}{-1 - \cos \varepsilon}} \]

   7. Taylor expanded in eps around inf 92.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{{\sin \varepsilon}^{2}}{1 + \cos \varepsilon}} \]
   8. Step-by-step derivation

      1. mul-1-neg 99.4%

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

      2. +-commutative 99.4%

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

      3. unpow2 99.4%

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

      4. associate-*r/ 99.3%

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

      5. distribute-rgt-neg-in 99.3%

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

      6. +-commutative 99.3%

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

      7. hang-0p-tan 99.8%

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

   9. Simplified 93.0%

      \[\leadsto \color{blue}{\sin \varepsilon \cdot \left(-\tan \left(\frac{\varepsilon}{2}\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{-9}:\\ \;\;\;\;\sin x \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot -2\right)\\ \mathbf{elif}\;x \leq -7.2 \cdot 10^{-63}:\\ \;\;\;\;-1 + \left(\cos \varepsilon - x \cdot \sin \varepsilon\right)\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-91}:\\ \;\;\;\;-0.5 \cdot {\varepsilon}^{2} + x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-9}:\\ \;\;\;\;\tan \left(\frac{\varepsilon}{2}\right) \cdot \left(-\sin \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot -2\right)\\ \end{array} \]

Alternative 8: 75.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return -2.0 * (sin((0.5 * (eps + (x - x)))) * sin((0.5 * (eps + (x + 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 + (x - x)))) * sin((0.5d0 * (eps + (x + x)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.2%

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

   1. log1p-expm1-u 37.1%

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

3. Applied egg-rr 37.1%

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

   1. diff-cos 47.1%

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

   2. log1p-expm1-u 47.2%

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

   3. div-inv 47.2%

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

   4. metadata-eval 47.2%

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

   5. div-inv 47.2%

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

   6. metadata-eval 47.2%

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

5. Applied egg-rr 47.2%

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

   1. *-commutative 47.2%

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

   2. +-commutative 47.2%

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

   3. associate--l+ 73.9%

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

   4. *-commutative 73.9%

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

   5. +-commutative 73.9%

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

   6. associate-+l+ 74.0%

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

7. Simplified 74.0%

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

8. Final simplification 74.0%

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

Alternative 9: 70.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-9} \lor \neg \left(x \leq 1.3 \cdot 10^{-9}\right):\\ \;\;\;\;\sin x \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(\frac{\varepsilon}{2}\right) \cdot \left(-\sin \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -2.8e-9) (not (<= x 1.3e-9)))
   (* (sin x) (* (sin (* eps 0.5)) -2.0))
   (* (tan (/ eps 2.0)) (- (sin eps)))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -2.8e-9) || !(x <= 1.3e-9)) {
		tmp = sin(x) * (sin((eps * 0.5)) * -2.0);
	} else {
		tmp = tan((eps / 2.0)) * -sin(eps);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= (-2.8d-9)) .or. (.not. (x <= 1.3d-9))) then
        tmp = sin(x) * (sin((eps * 0.5d0)) * (-2.0d0))
    else
        tmp = tan((eps / 2.0d0)) * -sin(eps)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -2.8e-9) || !(x <= 1.3e-9)) {
		tmp = Math.sin(x) * (Math.sin((eps * 0.5)) * -2.0);
	} else {
		tmp = Math.tan((eps / 2.0)) * -Math.sin(eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -2.8e-9) or not (x <= 1.3e-9):
		tmp = math.sin(x) * (math.sin((eps * 0.5)) * -2.0)
	else:
		tmp = math.tan((eps / 2.0)) * -math.sin(eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -2.8e-9) || !(x <= 1.3e-9))
		tmp = Float64(sin(x) * Float64(sin(Float64(eps * 0.5)) * -2.0));
	else
		tmp = Float64(tan(Float64(eps / 2.0)) * Float64(-sin(eps)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -2.8e-9) || ~((x <= 1.3e-9)))
		tmp = sin(x) * (sin((eps * 0.5)) * -2.0);
	else
		tmp = tan((eps / 2.0)) * -sin(eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -2.8e-9], N[Not[LessEqual[x, 1.3e-9]], $MachinePrecision]], N[(N[Sin[x], $MachinePrecision] * N[(N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision], N[(N[Tan[N[(eps / 2.0), $MachinePrecision]], $MachinePrecision] * (-N[Sin[eps], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.79999999999999984e-9 or 1.3000000000000001e-9 < x

   1. Initial program 6.6%

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

      1. diff-cos 7.3%

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

      2. div-inv 7.3%

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

      3. associate--l+ 7.3%

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

      4. metadata-eval 7.3%

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

      5. div-inv 7.3%

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

      6. +-commutative 7.3%

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

      7. associate-+l+ 7.6%

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

      8. metadata-eval 7.6%

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

   3. Applied egg-rr 7.6%

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

      1. associate-*r* 7.6%

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

      2. *-commutative 7.6%

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

      3. *-commutative 7.6%

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

      4. +-commutative 7.6%

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

      5. count-2 7.6%

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

      6. fma-def 7.6%

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

      7. sub-neg 7.6%

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

      8. mul-1-neg 7.6%

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

      9. +-commutative 7.6%

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

      10. associate-+r+ 51.0%

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

      11. mul-1-neg 51.0%

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

      12. sub-neg 51.0%

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

      13. +-inverses 51.0%

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

      14. remove-double-neg 51.0%

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

      15. mul-1-neg 51.0%

          \[\leadsto \sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(-2 \cdot \sin \left(\left(0 + \left(-\color{blue}{-1 \cdot \varepsilon}\right)\right) \cdot 0.5\right)\right) \]

      16. sub-neg 51.0%

          \[\leadsto \sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(-2 \cdot \sin \left(\color{blue}{\left(0 - -1 \cdot \varepsilon\right)} \cdot 0.5\right)\right) \]

      17. neg-sub0 51.0%

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

      18. mul-1-neg 51.0%

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

      19. remove-double-neg 51.0%

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

   5. Simplified 51.0%

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

   6. Taylor expanded in eps around 0 50.8%

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

   if -2.79999999999999984e-9 < x < 1.3000000000000001e-9

   1. Initial program 70.4%

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

   2. Taylor expanded in x around 0 70.4%

      \[\leadsto \color{blue}{\cos \varepsilon - 1} \]
   3. Step-by-step derivation

      1. flip-- 70.1%

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

      2. frac-2neg 70.1%

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

      3. metadata-eval 70.1%

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

      4. sub-1-cos 88.7%

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

      5. pow2 88.7%

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

   4. Applied egg-rr 88.7%

      \[\leadsto \color{blue}{\frac{-\left(-{\sin \varepsilon}^{2}\right)}{-\left(\cos \varepsilon + 1\right)}} \]
   5. Step-by-step derivation

      1. remove-double-neg 88.7%

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

      2. neg-sub0 88.7%

         \[\leadsto \frac{{\sin \varepsilon}^{2}}{\color{blue}{0 - \left(\cos \varepsilon + 1\right)}} \]

      3. +-commutative 88.7%

         \[\leadsto \frac{{\sin \varepsilon}^{2}}{0 - \color{blue}{\left(1 + \cos \varepsilon\right)}} \]

      4. associate--r+ 88.7%

         \[\leadsto \frac{{\sin \varepsilon}^{2}}{\color{blue}{\left(0 - 1\right) - \cos \varepsilon}} \]

      5. metadata-eval 88.7%

         \[\leadsto \frac{{\sin \varepsilon}^{2}}{\color{blue}{-1} - \cos \varepsilon} \]

   6. Simplified 88.7%

      \[\leadsto \color{blue}{\frac{{\sin \varepsilon}^{2}}{-1 - \cos \varepsilon}} \]

   7. Taylor expanded in eps around inf 88.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{{\sin \varepsilon}^{2}}{1 + \cos \varepsilon}} \]
   8. Step-by-step derivation

      1. mul-1-neg 99.3%

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

      2. +-commutative 99.3%

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

      3. unpow2 99.3%

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

      4. associate-*r/ 99.3%

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

      5. distribute-rgt-neg-in 99.3%

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

      6. +-commutative 99.3%

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

      7. hang-0p-tan 99.8%

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

   9. Simplified 89.2%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-9} \lor \neg \left(x \leq 1.3 \cdot 10^{-9}\right):\\ \;\;\;\;\sin x \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(\frac{\varepsilon}{2}\right) \cdot \left(-\sin \varepsilon\right)\\ \end{array} \]

Alternative 10: 68.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.75 \cdot 10^{-9} \lor \neg \left(x \leq 3.9 \cdot 10^{-12}\right):\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -2.75e-9) (not (<= x 3.9e-12)))
   (* eps (- (sin x)))
   (* -2.0 (pow (sin (* eps 0.5)) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -2.75e-9) || !(x <= 3.9e-12)) {
		tmp = eps * -sin(x);
	} else {
		tmp = -2.0 * pow(sin((eps * 0.5)), 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= (-2.75d-9)) .or. (.not. (x <= 3.9d-12))) then
        tmp = eps * -sin(x)
    else
        tmp = (-2.0d0) * (sin((eps * 0.5d0)) ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -2.75e-9) || !(x <= 3.9e-12)) {
		tmp = eps * -Math.sin(x);
	} else {
		tmp = -2.0 * Math.pow(Math.sin((eps * 0.5)), 2.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -2.75e-9) or not (x <= 3.9e-12):
		tmp = eps * -math.sin(x)
	else:
		tmp = -2.0 * math.pow(math.sin((eps * 0.5)), 2.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -2.75e-9) || !(x <= 3.9e-12))
		tmp = Float64(eps * Float64(-sin(x)));
	else
		tmp = Float64(-2.0 * (sin(Float64(eps * 0.5)) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -2.75e-9) || ~((x <= 3.9e-12)))
		tmp = eps * -sin(x);
	else
		tmp = -2.0 * (sin((eps * 0.5)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -2.75e-9], N[Not[LessEqual[x, 3.9e-12]], $MachinePrecision]], N[(eps * (-N[Sin[x], $MachinePrecision])), $MachinePrecision], N[(-2.0 * N[Power[N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.7499999999999998e-9 or 3.89999999999999994e-12 < x

   1. Initial program 6.6%

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

   2. Taylor expanded in eps around 0 45.9%

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

      1. mul-1-neg 45.9%

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

      2. *-commutative 45.9%

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

      3. distribute-rgt-neg-in 45.9%

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

   4. Simplified 45.9%

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

   if -2.7499999999999998e-9 < x < 3.89999999999999994e-12

   1. Initial program 70.4%

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

      1. diff-cos 90.4%

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

      2. div-inv 90.4%

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

      3. associate--l+ 90.4%

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

      4. metadata-eval 90.4%

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

      5. div-inv 90.4%

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

      6. +-commutative 90.4%

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

      7. associate-+l+ 90.4%

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

      8. metadata-eval 90.4%

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

   3. Applied egg-rr 90.4%

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

      1. associate-*r* 90.4%

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

      2. *-commutative 90.4%

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

      3. *-commutative 90.4%

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

      4. +-commutative 90.4%

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

      5. count-2 90.4%

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

      6. fma-def 90.4%

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

      7. sub-neg 90.4%

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

      8. mul-1-neg 90.4%

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

      9. +-commutative 90.4%

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

      10. associate-+r+ 98.8%

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

      11. mul-1-neg 98.8%

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

      12. sub-neg 98.8%

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

      13. +-inverses 98.8%

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

      14. remove-double-neg 98.8%

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

      15. mul-1-neg 98.8%

          \[\leadsto \sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(-2 \cdot \sin \left(\left(0 + \left(-\color{blue}{-1 \cdot \varepsilon}\right)\right) \cdot 0.5\right)\right) \]

      16. sub-neg 98.8%

          \[\leadsto \sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(-2 \cdot \sin \left(\color{blue}{\left(0 - -1 \cdot \varepsilon\right)} \cdot 0.5\right)\right) \]

      17. neg-sub0 98.8%

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

      18. mul-1-neg 98.8%

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

      19. remove-double-neg 98.8%

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

   5. Simplified 98.8%

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

   6. Taylor expanded in x around 0 89.0%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.75 \cdot 10^{-9} \lor \neg \left(x \leq 3.9 \cdot 10^{-12}\right):\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array} \]

Alternative 11: 68.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-9} \lor \neg \left(x \leq 1.9 \cdot 10^{-12}\right):\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(\frac{\varepsilon}{2}\right) \cdot \left(-\sin \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -2.7e-9) (not (<= x 1.9e-12)))
   (* eps (- (sin x)))
   (* (tan (/ eps 2.0)) (- (sin eps)))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -2.7e-9) || !(x <= 1.9e-12)) {
		tmp = eps * -sin(x);
	} else {
		tmp = tan((eps / 2.0)) * -sin(eps);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= (-2.7d-9)) .or. (.not. (x <= 1.9d-12))) then
        tmp = eps * -sin(x)
    else
        tmp = tan((eps / 2.0d0)) * -sin(eps)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -2.7e-9) || !(x <= 1.9e-12)) {
		tmp = eps * -Math.sin(x);
	} else {
		tmp = Math.tan((eps / 2.0)) * -Math.sin(eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -2.7e-9) or not (x <= 1.9e-12):
		tmp = eps * -math.sin(x)
	else:
		tmp = math.tan((eps / 2.0)) * -math.sin(eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -2.7e-9) || !(x <= 1.9e-12))
		tmp = Float64(eps * Float64(-sin(x)));
	else
		tmp = Float64(tan(Float64(eps / 2.0)) * Float64(-sin(eps)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -2.7e-9) || ~((x <= 1.9e-12)))
		tmp = eps * -sin(x);
	else
		tmp = tan((eps / 2.0)) * -sin(eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -2.7e-9], N[Not[LessEqual[x, 1.9e-12]], $MachinePrecision]], N[(eps * (-N[Sin[x], $MachinePrecision])), $MachinePrecision], N[(N[Tan[N[(eps / 2.0), $MachinePrecision]], $MachinePrecision] * (-N[Sin[eps], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.7000000000000002e-9 or 1.89999999999999998e-12 < x

   1. Initial program 6.6%

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

   2. Taylor expanded in eps around 0 45.9%

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

      1. mul-1-neg 45.9%

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

      2. *-commutative 45.9%

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

      3. distribute-rgt-neg-in 45.9%

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

   4. Simplified 45.9%

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

   if -2.7000000000000002e-9 < x < 1.89999999999999998e-12

   1. Initial program 70.4%

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

   2. Taylor expanded in x around 0 70.4%

      \[\leadsto \color{blue}{\cos \varepsilon - 1} \]
   3. Step-by-step derivation

      1. flip-- 70.1%

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

      2. frac-2neg 70.1%

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

      3. metadata-eval 70.1%

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

      4. sub-1-cos 88.7%

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

      5. pow2 88.7%

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

   4. Applied egg-rr 88.7%

      \[\leadsto \color{blue}{\frac{-\left(-{\sin \varepsilon}^{2}\right)}{-\left(\cos \varepsilon + 1\right)}} \]
   5. Step-by-step derivation

      1. remove-double-neg 88.7%

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

      2. neg-sub0 88.7%

         \[\leadsto \frac{{\sin \varepsilon}^{2}}{\color{blue}{0 - \left(\cos \varepsilon + 1\right)}} \]

      3. +-commutative 88.7%

         \[\leadsto \frac{{\sin \varepsilon}^{2}}{0 - \color{blue}{\left(1 + \cos \varepsilon\right)}} \]

      4. associate--r+ 88.7%

         \[\leadsto \frac{{\sin \varepsilon}^{2}}{\color{blue}{\left(0 - 1\right) - \cos \varepsilon}} \]

      5. metadata-eval 88.7%

         \[\leadsto \frac{{\sin \varepsilon}^{2}}{\color{blue}{-1} - \cos \varepsilon} \]

   6. Simplified 88.7%

      \[\leadsto \color{blue}{\frac{{\sin \varepsilon}^{2}}{-1 - \cos \varepsilon}} \]

   7. Taylor expanded in eps around inf 88.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{{\sin \varepsilon}^{2}}{1 + \cos \varepsilon}} \]
   8. Step-by-step derivation

      1. mul-1-neg 99.3%

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

      2. +-commutative 99.3%

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

      3. unpow2 99.3%

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

      4. associate-*r/ 99.3%

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

      5. distribute-rgt-neg-in 99.3%

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

      6. +-commutative 99.3%

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

      7. hang-0p-tan 99.8%

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

   9. Simplified 89.2%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-9} \lor \neg \left(x \leq 1.9 \cdot 10^{-12}\right):\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(\frac{\varepsilon}{2}\right) \cdot \left(-\sin \varepsilon\right)\\ \end{array} \]

Alternative 12: 49.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \varepsilon + -1\\ t_1 := -0.5 \cdot {\varepsilon}^{2}\\ \mathbf{if}\;\varepsilon \leq -0.000156:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -2.45 \cdot 10^{-147}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq 3.2 \cdot 10^{-144}:\\ \;\;\;\;\varepsilon \cdot \left(-x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00016:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (cos eps) -1.0)) (t_1 (* -0.5 (pow eps 2.0))))
   (if (<= eps -0.000156)
     t_0
     (if (<= eps -2.45e-147)
       t_1
       (if (<= eps 3.2e-144) (* eps (- x)) (if (<= eps 0.00016) t_1 t_0))))))

C

double code(double x, double eps) {
	double t_0 = cos(eps) + -1.0;
	double t_1 = -0.5 * pow(eps, 2.0);
	double tmp;
	if (eps <= -0.000156) {
		tmp = t_0;
	} else if (eps <= -2.45e-147) {
		tmp = t_1;
	} else if (eps <= 3.2e-144) {
		tmp = eps * -x;
	} else if (eps <= 0.00016) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos(eps) + (-1.0d0)
    t_1 = (-0.5d0) * (eps ** 2.0d0)
    if (eps <= (-0.000156d0)) then
        tmp = t_0
    else if (eps <= (-2.45d-147)) then
        tmp = t_1
    else if (eps <= 3.2d-144) then
        tmp = eps * -x
    else if (eps <= 0.00016d0) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.cos(eps) + -1.0;
	double t_1 = -0.5 * Math.pow(eps, 2.0);
	double tmp;
	if (eps <= -0.000156) {
		tmp = t_0;
	} else if (eps <= -2.45e-147) {
		tmp = t_1;
	} else if (eps <= 3.2e-144) {
		tmp = eps * -x;
	} else if (eps <= 0.00016) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.cos(eps) + -1.0
	t_1 = -0.5 * math.pow(eps, 2.0)
	tmp = 0
	if eps <= -0.000156:
		tmp = t_0
	elif eps <= -2.45e-147:
		tmp = t_1
	elif eps <= 3.2e-144:
		tmp = eps * -x
	elif eps <= 0.00016:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(cos(eps) + -1.0)
	t_1 = Float64(-0.5 * (eps ^ 2.0))
	tmp = 0.0
	if (eps <= -0.000156)
		tmp = t_0;
	elseif (eps <= -2.45e-147)
		tmp = t_1;
	elseif (eps <= 3.2e-144)
		tmp = Float64(eps * Float64(-x));
	elseif (eps <= 0.00016)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = cos(eps) + -1.0;
	t_1 = -0.5 * (eps ^ 2.0);
	tmp = 0.0;
	if (eps <= -0.000156)
		tmp = t_0;
	elseif (eps <= -2.45e-147)
		tmp = t_1;
	elseif (eps <= 3.2e-144)
		tmp = eps * -x;
	elseif (eps <= 0.00016)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.000156], t$95$0, If[LessEqual[eps, -2.45e-147], t$95$1, If[LessEqual[eps, 3.2e-144], N[(eps * (-x)), $MachinePrecision], If[LessEqual[eps, 0.00016], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -1.56e-4 or 1.60000000000000013e-4 < eps

   1. Initial program 48.4%

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

   2. Taylor expanded in x around 0 51.2%

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

   if -1.56e-4 < eps < -2.45000000000000002e-147 or 3.19999999999999973e-144 < eps < 1.60000000000000013e-4

   1. Initial program 6.0%

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

   2. Taylor expanded in x around 0 6.0%

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

   3. Taylor expanded in eps around 0 48.2%

      \[\leadsto \color{blue}{-0.5 \cdot {\varepsilon}^{2}} \]

   if -2.45000000000000002e-147 < eps < 3.19999999999999973e-144

   1. Initial program 39.3%

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

   2. Taylor expanded in eps around 0 98.9%

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

      1. mul-1-neg 98.9%

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

      2. *-commutative 98.9%

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

      3. distribute-rgt-neg-in 98.9%

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

   4. Simplified 98.9%

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

   5. Taylor expanded in x around 0 51.4%

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

      1. mul-1-neg 51.4%

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

      2. distribute-rgt-neg-in 51.4%

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

   7. Simplified 51.4%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000156:\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{elif}\;\varepsilon \leq -2.45 \cdot 10^{-147}:\\ \;\;\;\;-0.5 \cdot {\varepsilon}^{2}\\ \mathbf{elif}\;\varepsilon \leq 3.2 \cdot 10^{-144}:\\ \;\;\;\;\varepsilon \cdot \left(-x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00016:\\ \;\;\;\;-0.5 \cdot {\varepsilon}^{2}\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon + -1\\ \end{array} \]

Alternative 13: 65.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \varepsilon + -1\\ t_1 := \varepsilon \cdot \left(-\sin x\right)\\ \mathbf{if}\;\varepsilon \leq -0.0044:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 3 \cdot 10^{-101}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq 7.5 \cdot 10^{-6}:\\ \;\;\;\;-0.5 \cdot {\varepsilon}^{2}\\ \mathbf{elif}\;\varepsilon \leq 1.9 \cdot 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (cos eps) -1.0)) (t_1 (* eps (- (sin x)))))
   (if (<= eps -0.0044)
     t_0
     (if (<= eps 3e-101)
       t_1
       (if (<= eps 7.5e-6)
         (* -0.5 (pow eps 2.0))
         (if (<= eps 1.9e-5) t_1 t_0))))))

C

double code(double x, double eps) {
	double t_0 = cos(eps) + -1.0;
	double t_1 = eps * -sin(x);
	double tmp;
	if (eps <= -0.0044) {
		tmp = t_0;
	} else if (eps <= 3e-101) {
		tmp = t_1;
	} else if (eps <= 7.5e-6) {
		tmp = -0.5 * pow(eps, 2.0);
	} else if (eps <= 1.9e-5) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos(eps) + (-1.0d0)
    t_1 = eps * -sin(x)
    if (eps <= (-0.0044d0)) then
        tmp = t_0
    else if (eps <= 3d-101) then
        tmp = t_1
    else if (eps <= 7.5d-6) then
        tmp = (-0.5d0) * (eps ** 2.0d0)
    else if (eps <= 1.9d-5) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.cos(eps) + -1.0;
	double t_1 = eps * -Math.sin(x);
	double tmp;
	if (eps <= -0.0044) {
		tmp = t_0;
	} else if (eps <= 3e-101) {
		tmp = t_1;
	} else if (eps <= 7.5e-6) {
		tmp = -0.5 * Math.pow(eps, 2.0);
	} else if (eps <= 1.9e-5) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.cos(eps) + -1.0
	t_1 = eps * -math.sin(x)
	tmp = 0
	if eps <= -0.0044:
		tmp = t_0
	elif eps <= 3e-101:
		tmp = t_1
	elif eps <= 7.5e-6:
		tmp = -0.5 * math.pow(eps, 2.0)
	elif eps <= 1.9e-5:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(cos(eps) + -1.0)
	t_1 = Float64(eps * Float64(-sin(x)))
	tmp = 0.0
	if (eps <= -0.0044)
		tmp = t_0;
	elseif (eps <= 3e-101)
		tmp = t_1;
	elseif (eps <= 7.5e-6)
		tmp = Float64(-0.5 * (eps ^ 2.0));
	elseif (eps <= 1.9e-5)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = cos(eps) + -1.0;
	t_1 = eps * -sin(x);
	tmp = 0.0;
	if (eps <= -0.0044)
		tmp = t_0;
	elseif (eps <= 3e-101)
		tmp = t_1;
	elseif (eps <= 7.5e-6)
		tmp = -0.5 * (eps ^ 2.0);
	elseif (eps <= 1.9e-5)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(eps * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]}, If[LessEqual[eps, -0.0044], t$95$0, If[LessEqual[eps, 3e-101], t$95$1, If[LessEqual[eps, 7.5e-6], N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 1.9e-5], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.00440000000000000027 or 1.9000000000000001e-5 < eps

   1. Initial program 48.8%

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

   2. Taylor expanded in x around 0 51.6%

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

   if -0.00440000000000000027 < eps < 3.0000000000000003e-101 or 7.50000000000000019e-6 < eps < 1.9000000000000001e-5

   1. Initial program 26.8%

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

   2. Taylor expanded in eps around 0 84.8%

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

      1. mul-1-neg 84.8%

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

      2. *-commutative 84.8%

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

      3. distribute-rgt-neg-in 84.8%

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

   4. Simplified 84.8%

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

   if 3.0000000000000003e-101 < eps < 7.50000000000000019e-6

   1. Initial program 4.2%

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

   2. Taylor expanded in x around 0 4.2%

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

   3. Taylor expanded in eps around 0 65.8%

      \[\leadsto \color{blue}{-0.5 \cdot {\varepsilon}^{2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0044:\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{elif}\;\varepsilon \leq 3 \cdot 10^{-101}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{elif}\;\varepsilon \leq 7.5 \cdot 10^{-6}:\\ \;\;\;\;-0.5 \cdot {\varepsilon}^{2}\\ \mathbf{elif}\;\varepsilon \leq 1.9 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon + -1\\ \end{array} \]

Alternative 14: 43.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.1 \cdot 10^{-46} \lor \neg \left(\varepsilon \leq 2.05 \cdot 10^{-71}\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -1.1e-46) (not (<= eps 2.05e-71)))
   (+ (cos eps) -1.0)
   (* eps (- x))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.1e-46) || !(eps <= 2.05e-71)) {
		tmp = cos(eps) + -1.0;
	} else {
		tmp = eps * -x;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-1.1d-46)) .or. (.not. (eps <= 2.05d-71))) then
        tmp = cos(eps) + (-1.0d0)
    else
        tmp = eps * -x
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.1e-46) || !(eps <= 2.05e-71)) {
		tmp = Math.cos(eps) + -1.0;
	} else {
		tmp = eps * -x;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -1.1e-46) or not (eps <= 2.05e-71):
		tmp = math.cos(eps) + -1.0
	else:
		tmp = eps * -x
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -1.1e-46) || !(eps <= 2.05e-71))
		tmp = Float64(cos(eps) + -1.0);
	else
		tmp = Float64(eps * Float64(-x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -1.1e-46) || ~((eps <= 2.05e-71)))
		tmp = cos(eps) + -1.0;
	else
		tmp = eps * -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -1.1e-46], N[Not[LessEqual[eps, 2.05e-71]], $MachinePrecision]], N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision], N[(eps * (-x)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.1e-46 or 2.04999999999999997e-71 < eps

   1. Initial program 43.9%

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

   2. Taylor expanded in x around 0 46.4%

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

   if -1.1e-46 < eps < 2.04999999999999997e-71

   1. Initial program 27.4%

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

   2. Taylor expanded in eps around 0 83.4%

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

      1. mul-1-neg 83.4%

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

      2. *-commutative 83.4%

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

      3. distribute-rgt-neg-in 83.4%

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

   4. Simplified 83.4%

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

   5. Taylor expanded in x around 0 38.6%

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

      1. mul-1-neg 38.6%

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

      2. distribute-rgt-neg-in 38.6%

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

   7. Simplified 38.6%

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

4. Final simplification 43.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.1 \cdot 10^{-46} \lor \neg \left(\varepsilon \leq 2.05 \cdot 10^{-71}\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-x\right)\\ \end{array} \]

Alternative 15: 17.8% accurate, 51.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.2%

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

2. Taylor expanded in eps around 0 39.1%

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

   1. mul-1-neg 39.1%

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

   2. *-commutative 39.1%

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

   3. distribute-rgt-neg-in 39.1%

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

4. Simplified 39.1%

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

5. Taylor expanded in x around 0 17.5%

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

   1. mul-1-neg 17.5%

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

   2. distribute-rgt-neg-in 17.5%

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

7. Simplified 17.5%

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

8. Final simplification 17.5%

   \[\leadsto \varepsilon \cdot \left(-x\right) \]

Reproduce

herbie shell --seed 2023322 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))