2cos (problem 3.3.5)

Percentage Accurate: 37.7% → 99.3%

Time: 17.6s

Alternatives: 18

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: 37.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 + \cos \varepsilon\\ t_1 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0056:\\ \;\;\;\;\cos x \cdot t_0 - \mathsf{expm1}\left(\mathsf{log1p}\left(t_1\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0049:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \cos x, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ -1.0 (cos eps))) (t_1 (* (sin eps) (sin x))))
   (if (<= eps -0.0056)
     (- (* (cos x) t_0) (expm1 (log1p t_1)))
     (if (<= eps 0.0049)
       (-
        (*
         (cos x)
         (+ (* -0.5 (pow eps 2.0)) (* 0.041666666666666664 (pow eps 4.0))))
        t_1)
       (fma t_0 (cos x) (* (sin eps) (- (sin x))))))))

C

double code(double x, double eps) {
	double t_0 = -1.0 + cos(eps);
	double t_1 = sin(eps) * sin(x);
	double tmp;
	if (eps <= -0.0056) {
		tmp = (cos(x) * t_0) - expm1(log1p(t_1));
	} else if (eps <= 0.0049) {
		tmp = (cos(x) * ((-0.5 * pow(eps, 2.0)) + (0.041666666666666664 * pow(eps, 4.0)))) - t_1;
	} else {
		tmp = fma(t_0, cos(x), (sin(eps) * -sin(x)));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(-1.0 + cos(eps))
	t_1 = Float64(sin(eps) * sin(x))
	tmp = 0.0
	if (eps <= -0.0056)
		tmp = Float64(Float64(cos(x) * t_0) - expm1(log1p(t_1)));
	elseif (eps <= 0.0049)
		tmp = Float64(Float64(cos(x) * Float64(Float64(-0.5 * (eps ^ 2.0)) + Float64(0.041666666666666664 * (eps ^ 4.0)))) - t_1);
	else
		tmp = fma(t_0, cos(x), Float64(sin(eps) * Float64(-sin(x))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.0056], N[(N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision] - N[(Exp[N[Log[1 + t$95$1], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.0049], N[(N[(N[Cos[x], $MachinePrecision] * N[(N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.041666666666666664 * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(N[Sin[eps], $MachinePrecision] * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.00559999999999999994

   1. Initial program 50.1%

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

      1. sub-neg 50.1%

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

      2. cos-sum 98.5%

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

      3. associate-+l- 98.6%

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

      4. fma-neg 98.6%

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

   4. Applied egg-rr 98.6%

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

   5. Taylor expanded in x around inf 98.6%

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

      1. associate--r+ 98.6%

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

      2. *-lft-identity 98.6%

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

      3. distribute-rgt-out-- 98.7%

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

      4. sub-neg 98.7%

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

      5. metadata-eval 98.7%

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

      6. +-commutative 98.7%

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

      7. *-commutative 98.7%

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

   7. Simplified 98.7%

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

      1. expm1-log1p-u 98.8%

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

   9. Applied egg-rr 98.8%

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

   if -0.00559999999999999994 < eps < 0.0048999999999999998

   1. Initial program 20.7%

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

      1. sub-neg 20.7%

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

      2. cos-sum 23.0%

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

      3. associate-+l- 23.0%

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

      4. fma-neg 23.0%

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

   4. Applied egg-rr 23.0%

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

   5. Taylor expanded in x around inf 23.0%

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

      1. associate--r+ 81.8%

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

      2. *-lft-identity 81.8%

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

      3. distribute-rgt-out-- 81.8%

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

      4. sub-neg 81.8%

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

      5. metadata-eval 81.8%

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

      6. +-commutative 81.8%

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

      7. *-commutative 81.8%

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

   7. Simplified 81.8%

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

   8. Taylor expanded in eps around 0 99.8%

      \[\leadsto \cos x \cdot \color{blue}{\left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right)} - \sin x \cdot \sin \varepsilon \]

   if 0.0048999999999999998 < eps

   1. Initial program 47.6%

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

      1. sub-neg 47.6%

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

      2. cos-sum 98.5%

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

      3. associate-+l- 98.4%

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

      4. fma-neg 98.5%

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

   4. Applied egg-rr 98.5%

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

   5. Taylor expanded in x around inf 98.4%

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

      1. associate--r+ 98.5%

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

      2. *-lft-identity 98.5%

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

      3. distribute-rgt-out-- 98.5%

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

      4. sub-neg 98.5%

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

      5. metadata-eval 98.5%

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

      6. +-commutative 98.5%

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

      7. *-commutative 98.5%

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

   7. Simplified 98.5%

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

      1. log1p-expm1-u 98.0%

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

   9. Applied egg-rr 98.0%

      \[\leadsto \cos x \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(-1 + \cos \varepsilon\right)\right)} - \sin x \cdot \sin \varepsilon \]
   10. Step-by-step derivation

       1. log1p-expm1-u 98.5%

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

       2. *-commutative 98.5%

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

       3. fma-neg 98.7%

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

       4. *-commutative 98.7%

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

       5. distribute-rgt-neg-in 98.7%

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

   11. Applied egg-rr 98.7%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0056:\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \varepsilon \cdot \sin x\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0049:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1 + \cos \varepsilon, \cos x, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left({\sin \varepsilon}^{2} \cdot \cos x, \frac{1}{-1 - \cos \varepsilon}, \sin \varepsilon \cdot \left(-\sin x\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (* (pow (sin eps) 2.0) (cos x))
  (/ 1.0 (- -1.0 (cos eps)))
  (* (sin eps) (- (sin x)))))

C

double code(double x, double eps) {
	return fma((pow(sin(eps), 2.0) * cos(x)), (1.0 / (-1.0 - cos(eps))), (sin(eps) * -sin(x)));
}

Julia

function code(x, eps)
	return fma(Float64((sin(eps) ^ 2.0) * cos(x)), Float64(1.0 / Float64(-1.0 - cos(eps))), Float64(sin(eps) * Float64(-sin(x))))
end

Wolfram

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

Derivation

1. Initial program 35.4%

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

   1. sub-neg 35.4%

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

   2. cos-sum 62.5%

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

   3. associate-+l- 62.5%

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

   4. fma-neg 62.6%

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

4. Applied egg-rr 62.6%

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

5. Taylor expanded in x around inf 62.5%

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

   1. associate--r+ 90.6%

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

   2. *-lft-identity 90.6%

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

   3. distribute-rgt-out-- 90.6%

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

   4. sub-neg 90.6%

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

   5. metadata-eval 90.6%

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

   6. +-commutative 90.6%

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

   7. *-commutative 90.6%

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

7. Simplified 90.6%

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

   1. +-commutative 90.6%

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

   2. *-commutative 90.6%

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

   3. +-commutative 90.6%

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

   4. flip-+ 90.0%

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

   5. associate-*l/ 90.0%

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

   6. metadata-eval 90.0%

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

   7. 1-sub-cos 98.7%

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

   8. pow2 98.7%

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

9. Applied egg-rr 98.7%

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

    1. div-inv 98.6%

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

    2. fma-neg 98.7%

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

    3. *-commutative 98.7%

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

    4. distribute-rgt-neg-in 98.7%

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

11. Applied egg-rr 98.7%

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

12. Final simplification 98.7%

    \[\leadsto \mathsf{fma}\left({\sin \varepsilon}^{2} \cdot \cos x, \frac{1}{-1 - \cos \varepsilon}, \sin \varepsilon \cdot \left(-\sin x\right)\right) \]
13. Add Preprocessing

Alternative 3: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{{\sin \varepsilon}^{2} \cdot \cos x}{-1 - \cos \varepsilon} - \sin \varepsilon \cdot \sin x \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (-
  (/ (* (pow (sin eps) 2.0) (cos x)) (- -1.0 (cos eps)))
  (* (sin eps) (sin x))))

C

double code(double x, double eps) {
	return ((pow(sin(eps), 2.0) * cos(x)) / (-1.0 - cos(eps))) - (sin(eps) * sin(x));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (((sin(eps) ** 2.0d0) * cos(x)) / ((-1.0d0) - cos(eps))) - (sin(eps) * sin(x))
end function

Java

public static double code(double x, double eps) {
	return ((Math.pow(Math.sin(eps), 2.0) * Math.cos(x)) / (-1.0 - Math.cos(eps))) - (Math.sin(eps) * Math.sin(x));
}

Python

def code(x, eps):
	return ((math.pow(math.sin(eps), 2.0) * math.cos(x)) / (-1.0 - math.cos(eps))) - (math.sin(eps) * math.sin(x))

Julia

function code(x, eps)
	return Float64(Float64(Float64((sin(eps) ^ 2.0) * cos(x)) / Float64(-1.0 - cos(eps))) - Float64(sin(eps) * sin(x)))
end

MATLAB

function tmp = code(x, eps)
	tmp = (((sin(eps) ^ 2.0) * cos(x)) / (-1.0 - cos(eps))) - (sin(eps) * sin(x));
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[Power[N[Sin[eps], $MachinePrecision], 2.0], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[(-1.0 - N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.4%

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

   1. sub-neg 35.4%

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

   2. cos-sum 62.5%

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

   3. associate-+l- 62.5%

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

   4. fma-neg 62.6%

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

4. Applied egg-rr 62.6%

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

5. Taylor expanded in x around inf 62.5%

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

   1. associate--r+ 90.6%

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

   2. *-lft-identity 90.6%

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

   3. distribute-rgt-out-- 90.6%

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

   4. sub-neg 90.6%

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

   5. metadata-eval 90.6%

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

   6. +-commutative 90.6%

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

   7. *-commutative 90.6%

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

7. Simplified 90.6%

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

   1. +-commutative 90.6%

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

   2. *-commutative 90.6%

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

   3. +-commutative 90.6%

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

   4. flip-+ 90.0%

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

   5. associate-*l/ 90.0%

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

   6. metadata-eval 90.0%

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

   7. 1-sub-cos 98.7%

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

   8. pow2 98.7%

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

9. Applied egg-rr 98.7%

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

10. Final simplification 98.7%

    \[\leadsto \frac{{\sin \varepsilon}^{2} \cdot \cos x}{-1 - \cos \varepsilon} - \sin \varepsilon \cdot \sin x \]
11. Add Preprocessing

Alternative 4: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 + \cos \varepsilon\\ t_1 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0056:\\ \;\;\;\;\cos x \cdot t_0 - t_1\\ \mathbf{elif}\;\varepsilon \leq 0.0049:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \cos x, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ -1.0 (cos eps))) (t_1 (* (sin eps) (sin x))))
   (if (<= eps -0.0056)
     (- (* (cos x) t_0) t_1)
     (if (<= eps 0.0049)
       (-
        (*
         (cos x)
         (+ (* -0.5 (pow eps 2.0)) (* 0.041666666666666664 (pow eps 4.0))))
        t_1)
       (fma t_0 (cos x) (* (sin eps) (- (sin x))))))))

C

double code(double x, double eps) {
	double t_0 = -1.0 + cos(eps);
	double t_1 = sin(eps) * sin(x);
	double tmp;
	if (eps <= -0.0056) {
		tmp = (cos(x) * t_0) - t_1;
	} else if (eps <= 0.0049) {
		tmp = (cos(x) * ((-0.5 * pow(eps, 2.0)) + (0.041666666666666664 * pow(eps, 4.0)))) - t_1;
	} else {
		tmp = fma(t_0, cos(x), (sin(eps) * -sin(x)));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(-1.0 + cos(eps))
	t_1 = Float64(sin(eps) * sin(x))
	tmp = 0.0
	if (eps <= -0.0056)
		tmp = Float64(Float64(cos(x) * t_0) - t_1);
	elseif (eps <= 0.0049)
		tmp = Float64(Float64(cos(x) * Float64(Float64(-0.5 * (eps ^ 2.0)) + Float64(0.041666666666666664 * (eps ^ 4.0)))) - t_1);
	else
		tmp = fma(t_0, cos(x), Float64(sin(eps) * Float64(-sin(x))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.0056], N[(N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[eps, 0.0049], N[(N[(N[Cos[x], $MachinePrecision] * N[(N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.041666666666666664 * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(N[Sin[eps], $MachinePrecision] * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.00559999999999999994

   1. Initial program 50.1%

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

      1. sub-neg 50.1%

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

      2. cos-sum 98.5%

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

      3. associate-+l- 98.6%

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

      4. fma-neg 98.6%

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

   4. Applied egg-rr 98.6%

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

   5. Taylor expanded in x around inf 98.6%

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

      1. associate--r+ 98.6%

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

      2. *-lft-identity 98.6%

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

      3. distribute-rgt-out-- 98.7%

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

      4. sub-neg 98.7%

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

      5. metadata-eval 98.7%

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

      6. +-commutative 98.7%

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

      7. *-commutative 98.7%

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

   7. Simplified 98.7%

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

   if -0.00559999999999999994 < eps < 0.0048999999999999998

   1. Initial program 20.7%

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

      1. sub-neg 20.7%

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

      2. cos-sum 23.0%

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

      3. associate-+l- 23.0%

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

      4. fma-neg 23.0%

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

   4. Applied egg-rr 23.0%

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

   5. Taylor expanded in x around inf 23.0%

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

      1. associate--r+ 81.8%

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

      2. *-lft-identity 81.8%

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

      3. distribute-rgt-out-- 81.8%

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

      4. sub-neg 81.8%

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

      5. metadata-eval 81.8%

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

      6. +-commutative 81.8%

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

      7. *-commutative 81.8%

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

   7. Simplified 81.8%

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

   8. Taylor expanded in eps around 0 99.8%

      \[\leadsto \cos x \cdot \color{blue}{\left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right)} - \sin x \cdot \sin \varepsilon \]

   if 0.0048999999999999998 < eps

   1. Initial program 47.6%

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

      1. sub-neg 47.6%

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

      2. cos-sum 98.5%

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

      3. associate-+l- 98.4%

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

      4. fma-neg 98.5%

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

   4. Applied egg-rr 98.5%

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

   5. Taylor expanded in x around inf 98.4%

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

      1. associate--r+ 98.5%

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

      2. *-lft-identity 98.5%

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

      3. distribute-rgt-out-- 98.5%

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

      4. sub-neg 98.5%

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

      5. metadata-eval 98.5%

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

      6. +-commutative 98.5%

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

      7. *-commutative 98.5%

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

   7. Simplified 98.5%

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

      1. log1p-expm1-u 98.0%

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

   9. Applied egg-rr 98.0%

      \[\leadsto \cos x \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(-1 + \cos \varepsilon\right)\right)} - \sin x \cdot \sin \varepsilon \]
   10. Step-by-step derivation

       1. log1p-expm1-u 98.5%

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

       2. *-commutative 98.5%

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

       3. fma-neg 98.7%

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

       4. *-commutative 98.7%

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

       5. distribute-rgt-neg-in 98.7%

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

   11. Applied egg-rr 98.7%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0056:\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{elif}\;\varepsilon \leq 0.0049:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1 + \cos \varepsilon, \cos x, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 + \cos \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.00019:\\ \;\;\;\;\cos x \cdot t_0 - \sin \varepsilon \cdot \sin x\\ \mathbf{elif}\;\varepsilon \leq 0.00019:\\ \;\;\;\;\left(\cos x \cdot \left(\varepsilon \cdot 0.5\right) + \sin x \cdot \left(1 + {\varepsilon}^{2} \cdot -0.125\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \cos x, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ -1.0 (cos eps))))
   (if (<= eps -0.00019)
     (- (* (cos x) t_0) (* (sin eps) (sin x)))
     (if (<= eps 0.00019)
       (*
        (+
         (* (cos x) (* eps 0.5))
         (* (sin x) (+ 1.0 (* (pow eps 2.0) -0.125))))
        (* -2.0 (sin (* eps 0.5))))
       (fma t_0 (cos x) (* (sin eps) (- (sin x))))))))

C

double code(double x, double eps) {
	double t_0 = -1.0 + cos(eps);
	double tmp;
	if (eps <= -0.00019) {
		tmp = (cos(x) * t_0) - (sin(eps) * sin(x));
	} else if (eps <= 0.00019) {
		tmp = ((cos(x) * (eps * 0.5)) + (sin(x) * (1.0 + (pow(eps, 2.0) * -0.125)))) * (-2.0 * sin((eps * 0.5)));
	} else {
		tmp = fma(t_0, cos(x), (sin(eps) * -sin(x)));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(-1.0 + cos(eps))
	tmp = 0.0
	if (eps <= -0.00019)
		tmp = Float64(Float64(cos(x) * t_0) - Float64(sin(eps) * sin(x)));
	elseif (eps <= 0.00019)
		tmp = Float64(Float64(Float64(cos(x) * Float64(eps * 0.5)) + Float64(sin(x) * Float64(1.0 + Float64((eps ^ 2.0) * -0.125)))) * Float64(-2.0 * sin(Float64(eps * 0.5))));
	else
		tmp = fma(t_0, cos(x), Float64(sin(eps) * Float64(-sin(x))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.00019], N[(N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision] - N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.00019], N[(N[(N[(N[Cos[x], $MachinePrecision] * N[(eps * 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(1.0 + N[(N[Power[eps, 2.0], $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-2.0 * N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(N[Sin[eps], $MachinePrecision] * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -1.9000000000000001e-4

   1. Initial program 50.1%

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

      1. sub-neg 50.1%

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

      2. cos-sum 98.5%

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

      3. associate-+l- 98.6%

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

      4. fma-neg 98.6%

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

   4. Applied egg-rr 98.6%

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

   5. Taylor expanded in x around inf 98.6%

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

      1. associate--r+ 98.6%

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

      2. *-lft-identity 98.6%

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

      3. distribute-rgt-out-- 98.7%

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

      4. sub-neg 98.7%

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

      5. metadata-eval 98.7%

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

      6. +-commutative 98.7%

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

      7. *-commutative 98.7%

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

   7. Simplified 98.7%

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

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

   1. Initial program 20.9%

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

      1. diff-cos 38.5%

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

      2. div-inv 38.5%

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

      3. associate--l+ 38.5%

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

      4. metadata-eval 38.5%

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

      5. div-inv 38.5%

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

      6. +-commutative 38.5%

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

      7. associate-+l+ 38.5%

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

      8. metadata-eval 38.5%

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

   4. Applied egg-rr 38.5%

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

      1. associate-*r* 38.5%

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

      2. *-commutative 38.5%

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

      3. *-commutative 38.5%

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

      4. +-commutative 38.5%

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

      5. count-2 38.5%

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

      6. fma-def 38.5%

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

      7. *-commutative 38.5%

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

      8. associate-+r- 38.5%

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

      9. +-commutative 38.5%

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

      10. associate--l+ 98.9%

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

      11. +-inverses 98.9%

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

   6. Simplified 98.9%

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

   7. Taylor expanded in eps around 0 99.6%

      \[\leadsto \color{blue}{\left(\sin x + \left(-0.125 \cdot \left({\varepsilon}^{2} \cdot \sin x\right) + 0.5 \cdot \left(\varepsilon \cdot \cos x\right)\right)\right)} \cdot \left(-2 \cdot \sin \left(0.5 \cdot \left(\varepsilon + 0\right)\right)\right) \]
   8. Step-by-step derivation

      1. associate-+r+ 99.6%

         \[\leadsto \color{blue}{\left(\left(\sin x + -0.125 \cdot \left({\varepsilon}^{2} \cdot \sin x\right)\right) + 0.5 \cdot \left(\varepsilon \cdot \cos x\right)\right)} \cdot \left(-2 \cdot \sin \left(0.5 \cdot \left(\varepsilon + 0\right)\right)\right) \]

      2. +-commutative 99.6%

         \[\leadsto \color{blue}{\left(0.5 \cdot \left(\varepsilon \cdot \cos x\right) + \left(\sin x + -0.125 \cdot \left({\varepsilon}^{2} \cdot \sin x\right)\right)\right)} \cdot \left(-2 \cdot \sin \left(0.5 \cdot \left(\varepsilon + 0\right)\right)\right) \]

      3. associate-*r* 99.6%

         \[\leadsto \left(\color{blue}{\left(0.5 \cdot \varepsilon\right) \cdot \cos x} + \left(\sin x + -0.125 \cdot \left({\varepsilon}^{2} \cdot \sin x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(0.5 \cdot \left(\varepsilon + 0\right)\right)\right) \]

      4. *-lft-identity 99.6%

         \[\leadsto \left(\left(0.5 \cdot \varepsilon\right) \cdot \cos x + \left(\color{blue}{1 \cdot \sin x} + -0.125 \cdot \left({\varepsilon}^{2} \cdot \sin x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(0.5 \cdot \left(\varepsilon + 0\right)\right)\right) \]

      5. associate-*r* 99.6%

         \[\leadsto \left(\left(0.5 \cdot \varepsilon\right) \cdot \cos x + \left(1 \cdot \sin x + \color{blue}{\left(-0.125 \cdot {\varepsilon}^{2}\right) \cdot \sin x}\right)\right) \cdot \left(-2 \cdot \sin \left(0.5 \cdot \left(\varepsilon + 0\right)\right)\right) \]

      6. distribute-rgt-out 99.6%

         \[\leadsto \left(\left(0.5 \cdot \varepsilon\right) \cdot \cos x + \color{blue}{\sin x \cdot \left(1 + -0.125 \cdot {\varepsilon}^{2}\right)}\right) \cdot \left(-2 \cdot \sin \left(0.5 \cdot \left(\varepsilon + 0\right)\right)\right) \]

   9. Simplified 99.6%

      \[\leadsto \color{blue}{\left(\left(0.5 \cdot \varepsilon\right) \cdot \cos x + \sin x \cdot \left(1 + -0.125 \cdot {\varepsilon}^{2}\right)\right)} \cdot \left(-2 \cdot \sin \left(0.5 \cdot \left(\varepsilon + 0\right)\right)\right) \]

   if 1.9000000000000001e-4 < eps

   1. Initial program 47.0%

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

      1. sub-neg 47.0%

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

      2. cos-sum 98.3%

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

      3. associate-+l- 98.3%

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

      4. fma-neg 98.4%

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

   4. Applied egg-rr 98.4%

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

   5. Taylor expanded in x around inf 98.3%

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

      1. associate--r+ 98.4%

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

      2. *-lft-identity 98.4%

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

      3. distribute-rgt-out-- 98.5%

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

      4. sub-neg 98.5%

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

      5. metadata-eval 98.5%

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

      6. +-commutative 98.5%

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

      7. *-commutative 98.5%

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

   7. Simplified 98.5%

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

      1. log1p-expm1-u 98.0%

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

   9. Applied egg-rr 98.0%

      \[\leadsto \cos x \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(-1 + \cos \varepsilon\right)\right)} - \sin x \cdot \sin \varepsilon \]
   10. Step-by-step derivation

       1. log1p-expm1-u 98.5%

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

       2. *-commutative 98.5%

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

       3. fma-neg 98.6%

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

       4. *-commutative 98.6%

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

       5. distribute-rgt-neg-in 98.6%

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

   11. Applied egg-rr 98.6%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00019:\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{elif}\;\varepsilon \leq 0.00019:\\ \;\;\;\;\left(\cos x \cdot \left(\varepsilon \cdot 0.5\right) + \sin x \cdot \left(1 + {\varepsilon}^{2} \cdot -0.125\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1 + \cos \varepsilon, \cos x, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00019 \lor \neg \left(\varepsilon \leq 0.00019\right):\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \left(\varepsilon \cdot 0.5\right) + \sin x \cdot \left(1 + {\varepsilon}^{2} \cdot -0.125\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.00019) (not (<= eps 0.00019)))
   (- (* (cos x) (+ -1.0 (cos eps))) (* (sin eps) (sin x)))
   (*
    (+ (* (cos x) (* eps 0.5)) (* (sin x) (+ 1.0 (* (pow eps 2.0) -0.125))))
    (* -2.0 (sin (* eps 0.5))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.00019) || !(eps <= 0.00019)) {
		tmp = (cos(x) * (-1.0 + cos(eps))) - (sin(eps) * sin(x));
	} else {
		tmp = ((cos(x) * (eps * 0.5)) + (sin(x) * (1.0 + (pow(eps, 2.0) * -0.125)))) * (-2.0 * sin((eps * 0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-0.00019d0)) .or. (.not. (eps <= 0.00019d0))) then
        tmp = (cos(x) * ((-1.0d0) + cos(eps))) - (sin(eps) * sin(x))
    else
        tmp = ((cos(x) * (eps * 0.5d0)) + (sin(x) * (1.0d0 + ((eps ** 2.0d0) * (-0.125d0))))) * ((-2.0d0) * sin((eps * 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.00019) || !(eps <= 0.00019)) {
		tmp = (Math.cos(x) * (-1.0 + Math.cos(eps))) - (Math.sin(eps) * Math.sin(x));
	} else {
		tmp = ((Math.cos(x) * (eps * 0.5)) + (Math.sin(x) * (1.0 + (Math.pow(eps, 2.0) * -0.125)))) * (-2.0 * Math.sin((eps * 0.5)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.00019) or not (eps <= 0.00019):
		tmp = (math.cos(x) * (-1.0 + math.cos(eps))) - (math.sin(eps) * math.sin(x))
	else:
		tmp = ((math.cos(x) * (eps * 0.5)) + (math.sin(x) * (1.0 + (math.pow(eps, 2.0) * -0.125)))) * (-2.0 * math.sin((eps * 0.5)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.00019) || !(eps <= 0.00019))
		tmp = Float64(Float64(cos(x) * Float64(-1.0 + cos(eps))) - Float64(sin(eps) * sin(x)));
	else
		tmp = Float64(Float64(Float64(cos(x) * Float64(eps * 0.5)) + Float64(sin(x) * Float64(1.0 + Float64((eps ^ 2.0) * -0.125)))) * Float64(-2.0 * sin(Float64(eps * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.00019) || ~((eps <= 0.00019)))
		tmp = (cos(x) * (-1.0 + cos(eps))) - (sin(eps) * sin(x));
	else
		tmp = ((cos(x) * (eps * 0.5)) + (sin(x) * (1.0 + ((eps ^ 2.0) * -0.125)))) * (-2.0 * sin((eps * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.00019], N[Not[LessEqual[eps, 0.00019]], $MachinePrecision]], N[(N[(N[Cos[x], $MachinePrecision] * N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[x], $MachinePrecision] * N[(eps * 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(1.0 + N[(N[Power[eps, 2.0], $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-2.0 * N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 48.4%

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

      1. sub-neg 48.4%

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

      2. cos-sum 98.4%

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

      3. associate-+l- 98.4%

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

      4. fma-neg 98.5%

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

   4. Applied egg-rr 98.5%

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

   5. Taylor expanded in x around inf 98.4%

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

      1. associate--r+ 98.5%

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

      2. *-lft-identity 98.5%

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

      3. distribute-rgt-out-- 98.6%

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

      4. sub-neg 98.6%

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

      5. metadata-eval 98.6%

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

      6. +-commutative 98.6%

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

      7. *-commutative 98.6%

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

   7. Simplified 98.6%

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

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

   1. Initial program 20.9%

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

      1. diff-cos 38.5%

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

      2. div-inv 38.5%

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

      3. associate--l+ 38.5%

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

      4. metadata-eval 38.5%

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

      5. div-inv 38.5%

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

      6. +-commutative 38.5%

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

      7. associate-+l+ 38.5%

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

      8. metadata-eval 38.5%

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

   4. Applied egg-rr 38.5%

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

      1. associate-*r* 38.5%

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

      2. *-commutative 38.5%

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

      3. *-commutative 38.5%

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

      4. +-commutative 38.5%

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

      5. count-2 38.5%

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

      6. fma-def 38.5%

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

      7. *-commutative 38.5%

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

      8. associate-+r- 38.5%

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

      9. +-commutative 38.5%

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

      10. associate--l+ 98.9%

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

      11. +-inverses 98.9%

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

   6. Simplified 98.9%

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

   7. Taylor expanded in eps around 0 99.6%

      \[\leadsto \color{blue}{\left(\sin x + \left(-0.125 \cdot \left({\varepsilon}^{2} \cdot \sin x\right) + 0.5 \cdot \left(\varepsilon \cdot \cos x\right)\right)\right)} \cdot \left(-2 \cdot \sin \left(0.5 \cdot \left(\varepsilon + 0\right)\right)\right) \]
   8. Step-by-step derivation

      1. associate-+r+ 99.6%

         \[\leadsto \color{blue}{\left(\left(\sin x + -0.125 \cdot \left({\varepsilon}^{2} \cdot \sin x\right)\right) + 0.5 \cdot \left(\varepsilon \cdot \cos x\right)\right)} \cdot \left(-2 \cdot \sin \left(0.5 \cdot \left(\varepsilon + 0\right)\right)\right) \]

      2. +-commutative 99.6%

         \[\leadsto \color{blue}{\left(0.5 \cdot \left(\varepsilon \cdot \cos x\right) + \left(\sin x + -0.125 \cdot \left({\varepsilon}^{2} \cdot \sin x\right)\right)\right)} \cdot \left(-2 \cdot \sin \left(0.5 \cdot \left(\varepsilon + 0\right)\right)\right) \]

      3. associate-*r* 99.6%

         \[\leadsto \left(\color{blue}{\left(0.5 \cdot \varepsilon\right) \cdot \cos x} + \left(\sin x + -0.125 \cdot \left({\varepsilon}^{2} \cdot \sin x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(0.5 \cdot \left(\varepsilon + 0\right)\right)\right) \]

      4. *-lft-identity 99.6%

         \[\leadsto \left(\left(0.5 \cdot \varepsilon\right) \cdot \cos x + \left(\color{blue}{1 \cdot \sin x} + -0.125 \cdot \left({\varepsilon}^{2} \cdot \sin x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(0.5 \cdot \left(\varepsilon + 0\right)\right)\right) \]

      5. associate-*r* 99.6%

         \[\leadsto \left(\left(0.5 \cdot \varepsilon\right) \cdot \cos x + \left(1 \cdot \sin x + \color{blue}{\left(-0.125 \cdot {\varepsilon}^{2}\right) \cdot \sin x}\right)\right) \cdot \left(-2 \cdot \sin \left(0.5 \cdot \left(\varepsilon + 0\right)\right)\right) \]

      6. distribute-rgt-out 99.6%

         \[\leadsto \left(\left(0.5 \cdot \varepsilon\right) \cdot \cos x + \color{blue}{\sin x \cdot \left(1 + -0.125 \cdot {\varepsilon}^{2}\right)}\right) \cdot \left(-2 \cdot \sin \left(0.5 \cdot \left(\varepsilon + 0\right)\right)\right) \]

   9. Simplified 99.6%

      \[\leadsto \color{blue}{\left(\left(0.5 \cdot \varepsilon\right) \cdot \cos x + \sin x \cdot \left(1 + -0.125 \cdot {\varepsilon}^{2}\right)\right)} \cdot \left(-2 \cdot \sin \left(0.5 \cdot \left(\varepsilon + 0\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00019 \lor \neg \left(\varepsilon \leq 0.00019\right):\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \left(\varepsilon \cdot 0.5\right) + \sin x \cdot \left(1 + {\varepsilon}^{2} \cdot -0.125\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.000115 \lor \neg \left(\varepsilon \leq 0.000125\right):\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (sin eps) (sin x))))
   (if (or (<= eps -0.000115) (not (<= eps 0.000125)))
     (- (* (cos x) (+ -1.0 (cos eps))) t_0)
     (- (* (cos x) (* -0.5 (pow eps 2.0))) t_0))))

C

double code(double x, double eps) {
	double t_0 = sin(eps) * sin(x);
	double tmp;
	if ((eps <= -0.000115) || !(eps <= 0.000125)) {
		tmp = (cos(x) * (-1.0 + cos(eps))) - t_0;
	} else {
		tmp = (cos(x) * (-0.5 * pow(eps, 2.0))) - 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(eps) * sin(x)
    if ((eps <= (-0.000115d0)) .or. (.not. (eps <= 0.000125d0))) then
        tmp = (cos(x) * ((-1.0d0) + cos(eps))) - t_0
    else
        tmp = (cos(x) * ((-0.5d0) * (eps ** 2.0d0))) - t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin(eps) * Math.sin(x);
	double tmp;
	if ((eps <= -0.000115) || !(eps <= 0.000125)) {
		tmp = (Math.cos(x) * (-1.0 + Math.cos(eps))) - t_0;
	} else {
		tmp = (Math.cos(x) * (-0.5 * Math.pow(eps, 2.0))) - t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.sin(eps) * math.sin(x)
	tmp = 0
	if (eps <= -0.000115) or not (eps <= 0.000125):
		tmp = (math.cos(x) * (-1.0 + math.cos(eps))) - t_0
	else:
		tmp = (math.cos(x) * (-0.5 * math.pow(eps, 2.0))) - t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(sin(eps) * sin(x))
	tmp = 0.0
	if ((eps <= -0.000115) || !(eps <= 0.000125))
		tmp = Float64(Float64(cos(x) * Float64(-1.0 + cos(eps))) - t_0);
	else
		tmp = Float64(Float64(cos(x) * Float64(-0.5 * (eps ^ 2.0))) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = sin(eps) * sin(x);
	tmp = 0.0;
	if ((eps <= -0.000115) || ~((eps <= 0.000125)))
		tmp = (cos(x) * (-1.0 + cos(eps))) - t_0;
	else
		tmp = (cos(x) * (-0.5 * (eps ^ 2.0))) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[eps, -0.000115], N[Not[LessEqual[eps, 0.000125]], $MachinePrecision]], N[(N[(N[Cos[x], $MachinePrecision] * N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[Cos[x], $MachinePrecision] * N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.15e-4 or 1.25e-4 < eps

   1. Initial program 48.4%

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

      1. sub-neg 48.4%

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

      2. cos-sum 98.4%

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

      3. associate-+l- 98.4%

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

      4. fma-neg 98.5%

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

   4. Applied egg-rr 98.5%

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

   5. Taylor expanded in x around inf 98.4%

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

      1. associate--r+ 98.5%

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

      2. *-lft-identity 98.5%

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

      3. distribute-rgt-out-- 98.6%

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

      4. sub-neg 98.6%

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

      5. metadata-eval 98.6%

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

      6. +-commutative 98.6%

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

      7. *-commutative 98.6%

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

   7. Simplified 98.6%

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

   if -1.15e-4 < eps < 1.25e-4

   1. Initial program 20.9%

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

      1. sub-neg 20.9%

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

      2. cos-sum 22.5%

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

      3. associate-+l- 22.5%

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

      4. fma-neg 22.5%

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

   4. Applied egg-rr 22.5%

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

   5. Taylor expanded in x around inf 22.5%

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

      1. associate--r+ 81.7%

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

      2. *-lft-identity 81.7%

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

      3. distribute-rgt-out-- 81.7%

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

      4. sub-neg 81.7%

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

      5. metadata-eval 81.7%

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

      6. +-commutative 81.7%

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

      7. *-commutative 81.7%

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

   7. Simplified 81.7%

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

   8. Taylor expanded in eps around 0 99.6%

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

      1. associate-*r* 99.6%

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

   10. Simplified 99.6%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000115 \lor \neg \left(\varepsilon \leq 0.000125\right):\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - \sin \varepsilon \cdot \sin x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.1 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 4.5 \cdot 10^{-5}\right):\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \cdot \left(\sin x + 0.5 \cdot \left(\varepsilon \cdot \cos x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -1.1e-5) (not (<= eps 4.5e-5)))
   (- (* (cos x) (+ -1.0 (cos eps))) (* (sin eps) (sin x)))
   (* (* -2.0 (sin (* eps 0.5))) (+ (sin x) (* 0.5 (* eps (cos x)))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.1e-5) || !(eps <= 4.5e-5)) {
		tmp = (cos(x) * (-1.0 + cos(eps))) - (sin(eps) * sin(x));
	} else {
		tmp = (-2.0 * sin((eps * 0.5))) * (sin(x) + (0.5 * (eps * cos(x))));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-1.1d-5)) .or. (.not. (eps <= 4.5d-5))) then
        tmp = (cos(x) * ((-1.0d0) + cos(eps))) - (sin(eps) * sin(x))
    else
        tmp = ((-2.0d0) * sin((eps * 0.5d0))) * (sin(x) + (0.5d0 * (eps * cos(x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.1e-5) || !(eps <= 4.5e-5)) {
		tmp = (Math.cos(x) * (-1.0 + Math.cos(eps))) - (Math.sin(eps) * Math.sin(x));
	} else {
		tmp = (-2.0 * Math.sin((eps * 0.5))) * (Math.sin(x) + (0.5 * (eps * Math.cos(x))));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -1.1e-5) or not (eps <= 4.5e-5):
		tmp = (math.cos(x) * (-1.0 + math.cos(eps))) - (math.sin(eps) * math.sin(x))
	else:
		tmp = (-2.0 * math.sin((eps * 0.5))) * (math.sin(x) + (0.5 * (eps * math.cos(x))))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -1.1e-5) || !(eps <= 4.5e-5))
		tmp = Float64(Float64(cos(x) * Float64(-1.0 + cos(eps))) - Float64(sin(eps) * sin(x)));
	else
		tmp = Float64(Float64(-2.0 * sin(Float64(eps * 0.5))) * Float64(sin(x) + Float64(0.5 * Float64(eps * cos(x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -1.1e-5) || ~((eps <= 4.5e-5)))
		tmp = (cos(x) * (-1.0 + cos(eps))) - (sin(eps) * sin(x));
	else
		tmp = (-2.0 * sin((eps * 0.5))) * (sin(x) + (0.5 * (eps * cos(x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -1.1e-5], N[Not[LessEqual[eps, 4.5e-5]], $MachinePrecision]], N[(N[(N[Cos[x], $MachinePrecision] * N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] + N[(0.5 * N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.1e-5 or 4.50000000000000028e-5 < eps

   1. Initial program 48.1%

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

      1. sub-neg 48.1%

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

      2. cos-sum 98.3%

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

      3. associate-+l- 98.3%

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

      4. fma-neg 98.3%

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

   4. Applied egg-rr 98.3%

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

   5. Taylor expanded in x around inf 98.3%

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

      1. associate--r+ 98.4%

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

      2. *-lft-identity 98.4%

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

      3. distribute-rgt-out-- 98.5%

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

      4. sub-neg 98.5%

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

      5. metadata-eval 98.5%

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

      6. +-commutative 98.5%

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

      7. *-commutative 98.5%

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

   7. Simplified 98.5%

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

   if -1.1e-5 < eps < 4.50000000000000028e-5

   1. Initial program 21.0%

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

      1. diff-cos 38.8%

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

      2. div-inv 38.8%

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

      3. associate--l+ 38.8%

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

      4. metadata-eval 38.8%

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

      5. div-inv 38.8%

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

      6. +-commutative 38.8%

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

      7. associate-+l+ 38.8%

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

      8. metadata-eval 38.8%

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

   4. Applied egg-rr 38.8%

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

      1. associate-*r* 38.8%

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

      2. *-commutative 38.8%

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

      3. *-commutative 38.8%

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

      4. +-commutative 38.8%

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

      5. count-2 38.8%

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

      6. fma-def 38.8%

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

      7. *-commutative 38.8%

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

      8. associate-+r- 38.8%

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

      9. +-commutative 38.8%

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

      10. associate--l+ 99.3%

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

      11. +-inverses 99.3%

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

   6. Simplified 99.3%

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

   7. Taylor expanded in eps around 0 99.6%

      \[\leadsto \color{blue}{\left(\sin x + 0.5 \cdot \left(\varepsilon \cdot \cos x\right)\right)} \cdot \left(-2 \cdot \sin \left(0.5 \cdot \left(\varepsilon + 0\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.1 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 4.5 \cdot 10^{-5}\right):\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \cdot \left(\sin x + 0.5 \cdot \left(\varepsilon \cdot \cos x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 35.4%

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

   1. diff-cos 43.5%

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

   2. div-inv 43.5%

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

   3. associate--l+ 43.5%

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

   4. metadata-eval 43.5%

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

   5. div-inv 43.5%

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

   6. +-commutative 43.5%

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

   7. associate-+l+ 43.6%

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

   8. metadata-eval 43.6%

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

4. Applied egg-rr 43.6%

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

   1. associate-*r* 43.6%

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

   2. *-commutative 43.6%

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

   3. *-commutative 43.6%

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

   4. +-commutative 43.6%

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

   5. count-2 43.6%

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

   6. fma-def 43.6%

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

   7. *-commutative 43.6%

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

   8. associate-+r- 43.6%

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

   9. +-commutative 43.6%

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

   10. associate--l+ 73.0%

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

   11. +-inverses 73.0%

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

6. Simplified 73.0%

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

7. Final simplification 73.0%

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

Alternative 10: 75.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.4%

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

   1. sub-neg 35.4%

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

   2. cos-sum 62.5%

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

   3. associate-+l- 62.5%

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

   4. fma-neg 62.6%

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

4. Applied egg-rr 62.6%

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

5. Taylor expanded in x around inf 62.5%

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

   1. associate--r+ 90.6%

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

   2. *-lft-identity 90.6%

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

   3. distribute-rgt-out-- 90.6%

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

   4. sub-neg 90.6%

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

   5. metadata-eval 90.6%

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

   6. +-commutative 90.6%

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

   7. *-commutative 90.6%

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

7. Simplified 90.6%

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

   1. cancel-sign-sub-inv 90.6%

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

   2. distribute-lft-in 90.6%

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

   3. associate-+l+ 62.5%

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

   4. *-commutative 62.5%

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

   5. neg-mul-1 62.5%

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

   6. fma-udef 62.6%

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

   7. +-commutative 62.6%

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

   8. sub-neg 62.6%

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

   9. distribute-lft-neg-out 62.6%

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

   10. fma-neg 62.5%

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

   11. cos-sum 35.4%

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

   12. diff-cos 43.5%

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

9. Applied egg-rr 73.0%

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

    1. *-commutative 73.0%

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

    2. +-inverses 73.0%

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

    3. *-commutative 73.0%

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

    4. count-2 73.0%

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

11. Simplified 73.0%

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

12. Final simplification 73.0%

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

Alternative 11: 69.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\varepsilon \cdot 0.5\right)\\ \mathbf{if}\;x \leq -1.75 \cdot 10^{-46} \lor \neg \left(x \leq 0.37\right):\\ \;\;\;\;\sin x \cdot \left(-2 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {t_0}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (* eps 0.5))))
   (if (or (<= x -1.75e-46) (not (<= x 0.37)))
     (* (sin x) (* -2.0 t_0))
     (* -2.0 (pow t_0 2.0)))))

C

double code(double x, double eps) {
	double t_0 = sin((eps * 0.5));
	double tmp;
	if ((x <= -1.75e-46) || !(x <= 0.37)) {
		tmp = sin(x) * (-2.0 * t_0);
	} else {
		tmp = -2.0 * pow(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 <= (-1.75d-46)) .or. (.not. (x <= 0.37d0))) then
        tmp = sin(x) * ((-2.0d0) * t_0)
    else
        tmp = (-2.0d0) * (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 <= -1.75e-46) || !(x <= 0.37)) {
		tmp = Math.sin(x) * (-2.0 * t_0);
	} else {
		tmp = -2.0 * Math.pow(t_0, 2.0);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.sin((eps * 0.5))
	tmp = 0
	if (x <= -1.75e-46) or not (x <= 0.37):
		tmp = math.sin(x) * (-2.0 * t_0)
	else:
		tmp = -2.0 * math.pow(t_0, 2.0)
	return tmp

Julia

function code(x, eps)
	t_0 = sin(Float64(eps * 0.5))
	tmp = 0.0
	if ((x <= -1.75e-46) || !(x <= 0.37))
		tmp = Float64(sin(x) * Float64(-2.0 * t_0));
	else
		tmp = Float64(-2.0 * (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 <= -1.75e-46) || ~((x <= 0.37)))
		tmp = sin(x) * (-2.0 * t_0);
	else
		tmp = -2.0 * (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, -1.75e-46], N[Not[LessEqual[x, 0.37]], $MachinePrecision]], N[(N[Sin[x], $MachinePrecision] * N[(-2.0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.7500000000000001e-46 or 0.37 < x

   1. Initial program 7.9%

      \[\cos \left(x + \varepsilon\right) - \cos x \]
   2. Add Preprocessing
   3. 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%

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

   4. Applied egg-rr 7.6%

      \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\left(x + \left(\varepsilon - x\right)\right) \cdot 0.5\right) \cdot \sin \left(\left(\varepsilon + \left(x + x\right)\right) \cdot 0.5\right)\right)} \]
   5. 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. *-commutative 7.6%

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

      8. associate-+r- 7.6%

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

      9. +-commutative 7.6%

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

      10. associate--l+ 55.4%

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

      11. +-inverses 55.4%

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

   6. Simplified 55.4%

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

   7. Taylor expanded in eps around 0 54.2%

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

   if -1.7500000000000001e-46 < x < 0.37

   1. Initial program 74.9%

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

      1. diff-cos 95.4%

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

      2. div-inv 95.4%

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

      3. associate--l+ 95.4%

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

      4. metadata-eval 95.4%

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

      5. div-inv 95.4%

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

      6. +-commutative 95.4%

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

      7. associate-+l+ 95.4%

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

      8. metadata-eval 95.4%

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

   4. Applied egg-rr 95.4%

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

      1. associate-*r* 95.4%

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

      2. *-commutative 95.4%

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

      3. *-commutative 95.4%

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

      4. +-commutative 95.4%

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

      5. count-2 95.4%

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

      6. fma-def 95.4%

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

      7. *-commutative 95.4%

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

      8. associate-+r- 95.4%

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

      9. +-commutative 95.4%

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

      10. associate--l+ 98.3%

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

      11. +-inverses 98.3%

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

   6. Simplified 98.3%

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

   7. Taylor expanded in x around 0 95.2%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{-46} \lor \neg \left(x \leq 0.37\right):\\ \;\;\;\;\sin x \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 67.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{-46} \lor \neg \left(x \leq 0.37\right):\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -1.75e-46) (not (<= x 0.37)))
   (* eps (- (sin x)))
   (* -2.0 (pow (sin (* eps 0.5)) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -1.75e-46) || !(x <= 0.37)) {
		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 <= (-1.75d-46)) .or. (.not. (x <= 0.37d0))) 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 <= -1.75e-46) || !(x <= 0.37)) {
		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 <= -1.75e-46) or not (x <= 0.37):
		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 <= -1.75e-46) || !(x <= 0.37))
		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 <= -1.75e-46) || ~((x <= 0.37)))
		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, -1.75e-46], N[Not[LessEqual[x, 0.37]], $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 < -1.7500000000000001e-46 or 0.37 < x

   1. Initial program 7.9%

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

   3. Taylor expanded in eps around 0 51.3%

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

      1. mul-1-neg 51.3%

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

      2. *-commutative 51.3%

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

      3. distribute-rgt-neg-in 51.3%

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

   5. Simplified 51.3%

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

   if -1.7500000000000001e-46 < x < 0.37

   1. Initial program 74.9%

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

      1. diff-cos 95.4%

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

      2. div-inv 95.4%

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

      3. associate--l+ 95.4%

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

      4. metadata-eval 95.4%

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

      5. div-inv 95.4%

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

      6. +-commutative 95.4%

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

      7. associate-+l+ 95.4%

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

      8. metadata-eval 95.4%

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

   4. Applied egg-rr 95.4%

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

      1. associate-*r* 95.4%

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

      2. *-commutative 95.4%

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

      3. *-commutative 95.4%

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

      4. +-commutative 95.4%

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

      5. count-2 95.4%

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

      6. fma-def 95.4%

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

      7. *-commutative 95.4%

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

      8. associate-+r- 95.4%

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

      9. +-commutative 95.4%

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

      10. associate--l+ 98.3%

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

      11. +-inverses 98.3%

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

   6. Simplified 98.3%

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

   7. Taylor expanded in x around 0 95.2%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{-46} \lor \neg \left(x \leq 0.37\right):\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 66.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0082 \lor \neg \left(\varepsilon \leq 7.5 \cdot 10^{-8}\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.0082) (not (<= eps 7.5e-8)))
   (- (cos eps) (cos x))
   (* eps (- (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.0082) || !(eps <= 7.5e-8)) {
		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 ((eps <= (-0.0082d0)) .or. (.not. (eps <= 7.5d-8))) 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 ((eps <= -0.0082) || !(eps <= 7.5e-8)) {
		tmp = Math.cos(eps) - Math.cos(x);
	} else {
		tmp = eps * -Math.sin(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.0082) or not (eps <= 7.5e-8):
		tmp = math.cos(eps) - math.cos(x)
	else:
		tmp = eps * -math.sin(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.0082) || !(eps <= 7.5e-8))
		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 ((eps <= -0.0082) || ~((eps <= 7.5e-8)))
		tmp = cos(eps) - cos(x);
	else
		tmp = eps * -sin(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.0082], N[Not[LessEqual[eps, 7.5e-8]], $MachinePrecision]], 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 eps < -0.00820000000000000069 or 7.4999999999999997e-8 < eps

   1. Initial program 48.6%

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

   3. Taylor expanded in x around 0 51.5%

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

   if -0.00820000000000000069 < eps < 7.4999999999999997e-8

   1. Initial program 20.6%

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

   3. Taylor expanded in eps around 0 81.1%

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

      1. mul-1-neg 81.1%

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

      2. *-commutative 81.1%

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

      3. distribute-rgt-neg-in 81.1%

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

   5. Simplified 81.1%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0082 \lor \neg \left(\varepsilon \leq 7.5 \cdot 10^{-8}\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 47.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 + \cos \varepsilon\\ t_1 := -0.5 \cdot {\varepsilon}^{2}\\ \mathbf{if}\;\varepsilon \leq -0.0082:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 5.9 \cdot 10^{-269}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq 7.2 \cdot 10^{-140}:\\ \;\;\;\;-\varepsilon \cdot x\\ \mathbf{elif}\;\varepsilon \leq 0.000125:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ -1.0 (cos eps))) (t_1 (* -0.5 (pow eps 2.0))))
   (if (<= eps -0.0082)
     t_0
     (if (<= eps 5.9e-269)
       t_1
       (if (<= eps 7.2e-140) (- (* eps x)) (if (<= eps 0.000125) t_1 t_0))))))

C

double code(double x, double eps) {
	double t_0 = -1.0 + cos(eps);
	double t_1 = -0.5 * pow(eps, 2.0);
	double tmp;
	if (eps <= -0.0082) {
		tmp = t_0;
	} else if (eps <= 5.9e-269) {
		tmp = t_1;
	} else if (eps <= 7.2e-140) {
		tmp = -(eps * x);
	} else if (eps <= 0.000125) {
		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 = (-1.0d0) + cos(eps)
    t_1 = (-0.5d0) * (eps ** 2.0d0)
    if (eps <= (-0.0082d0)) then
        tmp = t_0
    else if (eps <= 5.9d-269) then
        tmp = t_1
    else if (eps <= 7.2d-140) then
        tmp = -(eps * x)
    else if (eps <= 0.000125d0) 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 = -1.0 + Math.cos(eps);
	double t_1 = -0.5 * Math.pow(eps, 2.0);
	double tmp;
	if (eps <= -0.0082) {
		tmp = t_0;
	} else if (eps <= 5.9e-269) {
		tmp = t_1;
	} else if (eps <= 7.2e-140) {
		tmp = -(eps * x);
	} else if (eps <= 0.000125) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = -1.0 + math.cos(eps)
	t_1 = -0.5 * math.pow(eps, 2.0)
	tmp = 0
	if eps <= -0.0082:
		tmp = t_0
	elif eps <= 5.9e-269:
		tmp = t_1
	elif eps <= 7.2e-140:
		tmp = -(eps * x)
	elif eps <= 0.000125:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(-1.0 + cos(eps))
	t_1 = Float64(-0.5 * (eps ^ 2.0))
	tmp = 0.0
	if (eps <= -0.0082)
		tmp = t_0;
	elseif (eps <= 5.9e-269)
		tmp = t_1;
	elseif (eps <= 7.2e-140)
		tmp = Float64(-Float64(eps * x));
	elseif (eps <= 0.000125)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = -1.0 + cos(eps);
	t_1 = -0.5 * (eps ^ 2.0);
	tmp = 0.0;
	if (eps <= -0.0082)
		tmp = t_0;
	elseif (eps <= 5.9e-269)
		tmp = t_1;
	elseif (eps <= 7.2e-140)
		tmp = -(eps * x);
	elseif (eps <= 0.000125)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.0082], t$95$0, If[LessEqual[eps, 5.9e-269], t$95$1, If[LessEqual[eps, 7.2e-140], (-N[(eps * x), $MachinePrecision]), If[LessEqual[eps, 0.000125], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.00820000000000000069 or 1.25e-4 < eps

   1. Initial program 48.7%

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

   3. Taylor expanded in x around 0 49.7%

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

   if -0.00820000000000000069 < eps < 5.9e-269 or 7.2000000000000001e-140 < eps < 1.25e-4

   1. Initial program 20.4%

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

   3. Taylor expanded in x around 0 20.5%

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

   4. Taylor expanded in eps around 0 41.4%

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

   if 5.9e-269 < eps < 7.2000000000000001e-140

   1. Initial program 22.1%

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

   3. Taylor expanded in eps around 0 22.1%

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

      1. mul-1-neg 22.1%

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

      2. unsub-neg 22.1%

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

   5. Simplified 22.1%

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

   6. Taylor expanded in x around 0 49.8%

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

      1. associate-*r* 49.8%

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

      2. mul-1-neg 49.8%

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

   8. Simplified 49.8%

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

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0082:\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 5.9 \cdot 10^{-269}:\\ \;\;\;\;-0.5 \cdot {\varepsilon}^{2}\\ \mathbf{elif}\;\varepsilon \leq 7.2 \cdot 10^{-140}:\\ \;\;\;\;-\varepsilon \cdot x\\ \mathbf{elif}\;\varepsilon \leq 0.000125:\\ \;\;\;\;-0.5 \cdot {\varepsilon}^{2}\\ \mathbf{else}:\\ \;\;\;\;-1 + \cos \varepsilon\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 66.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0082 \lor \neg \left(\varepsilon \leq 0.00385\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.0082) (not (<= eps 0.00385)))
   (+ -1.0 (cos eps))
   (* eps (- (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.0082) || !(eps <= 0.00385)) {
		tmp = -1.0 + cos(eps);
	} 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 ((eps <= (-0.0082d0)) .or. (.not. (eps <= 0.00385d0))) then
        tmp = (-1.0d0) + cos(eps)
    else
        tmp = eps * -sin(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.0082) || !(eps <= 0.00385)) {
		tmp = -1.0 + Math.cos(eps);
	} else {
		tmp = eps * -Math.sin(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.0082) or not (eps <= 0.00385):
		tmp = -1.0 + math.cos(eps)
	else:
		tmp = eps * -math.sin(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.0082) || !(eps <= 0.00385))
		tmp = Float64(-1.0 + cos(eps));
	else
		tmp = Float64(eps * Float64(-sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.0082) || ~((eps <= 0.00385)))
		tmp = -1.0 + cos(eps);
	else
		tmp = eps * -sin(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.0082], N[Not[LessEqual[eps, 0.00385]], $MachinePrecision]], N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision], N[(eps * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -0.00820000000000000069 or 0.0038500000000000001 < eps

   1. Initial program 49.1%

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

   3. Taylor expanded in x around 0 50.0%

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

   if -0.00820000000000000069 < eps < 0.0038500000000000001

   1. Initial program 20.6%

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

   3. Taylor expanded in eps around 0 80.0%

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

      1. mul-1-neg 80.0%

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

      2. *-commutative 80.0%

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

      3. distribute-rgt-neg-in 80.0%

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

   5. Simplified 80.0%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0082 \lor \neg \left(\varepsilon \leq 0.00385\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 43.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -7.4 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 1.05 \cdot 10^{-8}\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;-\varepsilon \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -7.4e-9) (not (<= eps 1.05e-8)))
   (+ -1.0 (cos eps))
   (- (* eps x))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -7.4e-9) || !(eps <= 1.05e-8)) {
		tmp = -1.0 + cos(eps);
	} 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 <= (-7.4d-9)) .or. (.not. (eps <= 1.05d-8))) then
        tmp = (-1.0d0) + cos(eps)
    else
        tmp = -(eps * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -7.4e-9) || !(eps <= 1.05e-8)) {
		tmp = -1.0 + Math.cos(eps);
	} else {
		tmp = -(eps * x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -7.4e-9) or not (eps <= 1.05e-8):
		tmp = -1.0 + math.cos(eps)
	else:
		tmp = -(eps * x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -7.4e-9) || !(eps <= 1.05e-8))
		tmp = Float64(-1.0 + cos(eps));
	else
		tmp = Float64(-Float64(eps * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -7.4e-9) || ~((eps <= 1.05e-8)))
		tmp = -1.0 + cos(eps);
	else
		tmp = -(eps * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -7.4e-9], N[Not[LessEqual[eps, 1.05e-8]], $MachinePrecision]], N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision], (-N[(eps * x), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if eps < -7.4e-9 or 1.04999999999999997e-8 < eps

   1. Initial program 47.9%

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

   3. Taylor expanded in x around 0 49.0%

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

   if -7.4e-9 < eps < 1.04999999999999997e-8

   1. Initial program 20.7%

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

   3. Taylor expanded in eps around 0 21.7%

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

      1. mul-1-neg 21.7%

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

      2. unsub-neg 21.7%

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

   5. Simplified 21.7%

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

   6. Taylor expanded in x around 0 27.2%

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

      1. associate-*r* 27.2%

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

      2. mul-1-neg 27.2%

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

   8. Simplified 27.2%

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

4. Final simplification 39.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -7.4 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 1.05 \cdot 10^{-8}\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;-\varepsilon \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 18.0% accurate, 51.3× speedup

Math

\[\begin{array}{l} \\ -\varepsilon \cdot x \end{array} \]

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(-Float64(eps * x))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.4%

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

3. Taylor expanded in eps around 0 12.2%

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

   1. mul-1-neg 12.2%

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

   2. unsub-neg 12.2%

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

5. Simplified 12.2%

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

6. Taylor expanded in x around 0 14.2%

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

   1. associate-*r* 14.2%

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

   2. mul-1-neg 14.2%

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

8. Simplified 14.2%

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

9. Final simplification 14.2%

   \[\leadsto -\varepsilon \cdot x \]
10. Add Preprocessing

Alternative 18: 12.8% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

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

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 35.4%

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

   1. add-cube-cbrt 35.2%

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

   2. pow3 35.2%

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

4. Applied egg-rr 35.2%

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

5. Taylor expanded in eps around 0 11.2%

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

   1. pow-base-1 11.2%

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

   2. *-lft-identity 11.2%

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

   3. +-inverses 11.2%

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

7. Simplified 11.2%

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

8. Final simplification 11.2%

   \[\leadsto 0 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024010 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))