2cos (problem 3.3.5)

Percentage Accurate: 37.7% → 99.3%

Time: 18.6s

Alternatives: 15

Speedup: 2.0×

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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0055:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \cos x\right) - t_0\\ \mathbf{elif}\;\varepsilon \leq 0.0045:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin \varepsilon \cdot \left(-\sin x\right)\right) - \cos x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (sin eps) (sin x))))
   (if (<= eps -0.0055)
     (- (- (* (cos x) (cos eps)) (cos x)) t_0)
     (if (<= eps 0.0045)
       (-
        (*
         (cos x)
         (+ (* -0.5 (pow eps 2.0)) (* 0.041666666666666664 (pow eps 4.0))))
        t_0)
       (- (fma (cos x) (cos eps) (* (sin eps) (- (sin x)))) (cos x))))))

C

double code(double x, double eps) {
	double t_0 = sin(eps) * sin(x);
	double tmp;
	if (eps <= -0.0055) {
		tmp = ((cos(x) * cos(eps)) - cos(x)) - t_0;
	} else if (eps <= 0.0045) {
		tmp = (cos(x) * ((-0.5 * pow(eps, 2.0)) + (0.041666666666666664 * pow(eps, 4.0)))) - t_0;
	} else {
		tmp = fma(cos(x), cos(eps), (sin(eps) * -sin(x))) - cos(x);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(sin(eps) * sin(x))
	tmp = 0.0
	if (eps <= -0.0055)
		tmp = Float64(Float64(Float64(cos(x) * cos(eps)) - cos(x)) - t_0);
	elseif (eps <= 0.0045)
		tmp = Float64(Float64(cos(x) * Float64(Float64(-0.5 * (eps ^ 2.0)) + Float64(0.041666666666666664 * (eps ^ 4.0)))) - t_0);
	else
		tmp = Float64(fma(cos(x), cos(eps), Float64(sin(eps) * Float64(-sin(x)))) - cos(x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.0055], N[(N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[eps, 0.0045], 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$0), $MachinePrecision], N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + N[(N[Sin[eps], $MachinePrecision] * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.0054999999999999997

   1. Initial program 62.6%

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

      1. cos-sum 98.7%

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

      2. cancel-sign-sub-inv 98.7%

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

      3. fma-def 98.7%

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

   4. Applied egg-rr 98.7%

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

   5. Taylor expanded in x around inf 98.7%

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

      1. neg-mul-1 98.7%

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

      2. +-commutative 98.7%

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

      3. sub-neg 98.7%

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

      4. *-commutative 98.7%

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

      5. associate--r+ 98.7%

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

      6. +-commutative 98.7%

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

      7. associate--r+ 98.8%

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

      8. *-rgt-identity 98.8%

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

      9. distribute-lft-out-- 98.8%

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

      10. sub-neg 98.8%

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

      11. metadata-eval 98.8%

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

      12. +-commutative 98.8%

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

      13. *-commutative 98.8%

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

   7. Simplified 98.8%

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

      1. +-commutative 98.8%

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

      2. distribute-rgt-in 98.8%

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

      3. neg-mul-1 98.8%

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

   9. Applied egg-rr 98.8%

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

   10. Taylor expanded in eps around inf 98.8%

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

   if -0.0054999999999999997 < eps < 0.00449999999999999966

   1. Initial program 24.9%

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

      1. cos-sum 26.1%

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

      2. cancel-sign-sub-inv 26.1%

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

      3. fma-def 26.1%

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

   4. Applied egg-rr 26.1%

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

   5. Taylor expanded in x around inf 26.1%

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

      1. neg-mul-1 26.1%

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

      2. +-commutative 26.1%

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

      3. sub-neg 26.1%

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

      4. *-commutative 26.1%

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

      5. associate--r+ 26.1%

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

      6. +-commutative 26.1%

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

      7. associate--r+ 83.4%

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

      8. *-rgt-identity 83.4%

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

      9. distribute-lft-out-- 83.4%

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

      10. sub-neg 83.4%

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

      11. metadata-eval 83.4%

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

      12. +-commutative 83.4%

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

      13. *-commutative 83.4%

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

   7. Simplified 83.4%

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

   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 0.00449999999999999966 < eps

   1. Initial program 49.8%

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

      1. cos-sum 99.1%

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

      2. cancel-sign-sub-inv 99.1%

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

      3. fma-def 99.1%

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

   4. Applied egg-rr 99.1%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0055:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \cos x\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{elif}\;\varepsilon \leq 0.0045:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin \varepsilon \cdot \left(-\sin x\right)\right) - \cos x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 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 40.1%

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

   1. cos-sum 63.1%

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

   2. cancel-sign-sub-inv 63.1%

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

   3. fma-def 63.1%

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

4. Applied egg-rr 63.1%

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

5. Taylor expanded in x around inf 63.1%

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

   1. neg-mul-1 63.1%

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

   2. +-commutative 63.1%

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

   3. sub-neg 63.1%

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

   4. *-commutative 63.1%

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

   5. associate--r+ 63.1%

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

   6. +-commutative 63.1%

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

   7. associate--r+ 91.3%

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

   8. *-rgt-identity 91.3%

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

   9. distribute-lft-out-- 91.3%

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

   10. sub-neg 91.3%

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

   11. metadata-eval 91.3%

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

   12. +-commutative 91.3%

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

   13. *-commutative 91.3%

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

7. Simplified 91.3%

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

   1. *-commutative 91.3%

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

   2. flip-+ 90.8%

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

   3. associate-*l/ 90.8%

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

   4. metadata-eval 90.8%

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

   5. 1-sub-cos 98.8%

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

   6. pow2 98.8%

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

9. Applied egg-rr 98.8%

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

10. Final simplification 98.8%

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

Alternative 3: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \cos \varepsilon\\ t_1 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0055:\\ \;\;\;\;\left(t_0 - \cos x\right) - t_1\\ \mathbf{elif}\;\varepsilon \leq 0.0052:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 - \left(\cos x + t_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (cos x) (cos eps))) (t_1 (* (sin eps) (sin x))))
   (if (<= eps -0.0055)
     (- (- t_0 (cos x)) t_1)
     (if (<= eps 0.0052)
       (-
        (*
         (cos x)
         (+ (* -0.5 (pow eps 2.0)) (* 0.041666666666666664 (pow eps 4.0))))
        t_1)
       (- t_0 (+ (cos x) t_1))))))

C

double code(double x, double eps) {
	double t_0 = cos(x) * cos(eps);
	double t_1 = sin(eps) * sin(x);
	double tmp;
	if (eps <= -0.0055) {
		tmp = (t_0 - cos(x)) - t_1;
	} else if (eps <= 0.0052) {
		tmp = (cos(x) * ((-0.5 * pow(eps, 2.0)) + (0.041666666666666664 * pow(eps, 4.0)))) - t_1;
	} else {
		tmp = t_0 - (cos(x) + t_1);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos(x) * cos(eps)
    t_1 = sin(eps) * sin(x)
    if (eps <= (-0.0055d0)) then
        tmp = (t_0 - cos(x)) - t_1
    else if (eps <= 0.0052d0) then
        tmp = (cos(x) * (((-0.5d0) * (eps ** 2.0d0)) + (0.041666666666666664d0 * (eps ** 4.0d0)))) - t_1
    else
        tmp = t_0 - (cos(x) + t_1)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.cos(x) * Math.cos(eps);
	double t_1 = Math.sin(eps) * Math.sin(x);
	double tmp;
	if (eps <= -0.0055) {
		tmp = (t_0 - Math.cos(x)) - t_1;
	} else if (eps <= 0.0052) {
		tmp = (Math.cos(x) * ((-0.5 * Math.pow(eps, 2.0)) + (0.041666666666666664 * Math.pow(eps, 4.0)))) - t_1;
	} else {
		tmp = t_0 - (Math.cos(x) + t_1);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.cos(x) * math.cos(eps)
	t_1 = math.sin(eps) * math.sin(x)
	tmp = 0
	if eps <= -0.0055:
		tmp = (t_0 - math.cos(x)) - t_1
	elif eps <= 0.0052:
		tmp = (math.cos(x) * ((-0.5 * math.pow(eps, 2.0)) + (0.041666666666666664 * math.pow(eps, 4.0)))) - t_1
	else:
		tmp = t_0 - (math.cos(x) + t_1)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(cos(x) * cos(eps))
	t_1 = Float64(sin(eps) * sin(x))
	tmp = 0.0
	if (eps <= -0.0055)
		tmp = Float64(Float64(t_0 - cos(x)) - t_1);
	elseif (eps <= 0.0052)
		tmp = Float64(Float64(cos(x) * Float64(Float64(-0.5 * (eps ^ 2.0)) + Float64(0.041666666666666664 * (eps ^ 4.0)))) - t_1);
	else
		tmp = Float64(t_0 - Float64(cos(x) + t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = cos(x) * cos(eps);
	t_1 = sin(eps) * sin(x);
	tmp = 0.0;
	if (eps <= -0.0055)
		tmp = (t_0 - cos(x)) - t_1;
	elseif (eps <= 0.0052)
		tmp = (cos(x) * ((-0.5 * (eps ^ 2.0)) + (0.041666666666666664 * (eps ^ 4.0)))) - t_1;
	else
		tmp = t_0 - (cos(x) + t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.0055], N[(N[(t$95$0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[eps, 0.0052], 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[(N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.0054999999999999997

   1. Initial program 62.6%

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

      1. cos-sum 98.7%

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

      2. cancel-sign-sub-inv 98.7%

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

      3. fma-def 98.7%

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

   4. Applied egg-rr 98.7%

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

   5. Taylor expanded in x around inf 98.7%

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

      1. neg-mul-1 98.7%

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

      2. +-commutative 98.7%

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

      3. sub-neg 98.7%

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

      4. *-commutative 98.7%

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

      5. associate--r+ 98.7%

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

      6. +-commutative 98.7%

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

      7. associate--r+ 98.8%

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

      8. *-rgt-identity 98.8%

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

      9. distribute-lft-out-- 98.8%

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

      10. sub-neg 98.8%

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

      11. metadata-eval 98.8%

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

      12. +-commutative 98.8%

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

      13. *-commutative 98.8%

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

   7. Simplified 98.8%

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

      1. +-commutative 98.8%

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

      2. distribute-rgt-in 98.8%

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

      3. neg-mul-1 98.8%

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

   9. Applied egg-rr 98.8%

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

   10. Taylor expanded in eps around inf 98.8%

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

   if -0.0054999999999999997 < eps < 0.0051999999999999998

   1. Initial program 24.9%

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

      1. cos-sum 26.1%

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

      2. cancel-sign-sub-inv 26.1%

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

      3. fma-def 26.1%

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

   4. Applied egg-rr 26.1%

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

   5. Taylor expanded in x around inf 26.1%

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

      1. neg-mul-1 26.1%

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

      2. +-commutative 26.1%

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

      3. sub-neg 26.1%

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

      4. *-commutative 26.1%

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

      5. associate--r+ 26.1%

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

      6. +-commutative 26.1%

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

      7. associate--r+ 83.4%

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

      8. *-rgt-identity 83.4%

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

      9. distribute-lft-out-- 83.4%

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

      10. sub-neg 83.4%

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

      11. metadata-eval 83.4%

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

      12. +-commutative 83.4%

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

      13. *-commutative 83.4%

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

   7. Simplified 83.4%

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

   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 0.0051999999999999998 < eps

   1. Initial program 49.8%

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

      1. add-log-exp 49.8%

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

   4. Applied egg-rr 49.8%

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

      1. rem-log-exp 49.8%

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

      2. cos-sum 99.1%

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

      3. associate--l- 99.1%

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

      4. add-sqr-sqrt 48.2%

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

      5. sqrt-unprod 75.4%

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

      6. sqr-neg 75.4%

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

      7. sqrt-unprod 27.2%

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

      8. add-sqr-sqrt 51.6%

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

      9. *-commutative 51.6%

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

      10. add-sqr-sqrt 27.2%

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

      11. sqrt-unprod 75.4%

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

      12. sqr-neg 75.4%

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

      13. sqrt-unprod 48.2%

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

      14. add-sqr-sqrt 99.1%

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

   6. Applied egg-rr 99.1%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0055:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \cos x\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{elif}\;\varepsilon \leq 0.0052:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin \varepsilon \cdot \sin x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.4 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(-1 + \cos \varepsilon, \cos x, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-33}:\\ \;\;\;\;\left(\sin \left(\left(\varepsilon + \left(x - x\right)\right) \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \cos x\right) - \sin \varepsilon \cdot \sin x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -6.4e-15)
   (fma (+ -1.0 (cos eps)) (cos x) (* (sin eps) (- (sin x))))
   (if (<= x 4.8e-33)
     (* (* (sin (* (+ eps (- x x)) 0.5)) (sin (* 0.5 (+ eps (+ x x))))) -2.0)
     (- (- (* (cos x) (cos eps)) (cos x)) (* (sin eps) (sin x))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -6.4e-15) {
		tmp = fma((-1.0 + cos(eps)), cos(x), (sin(eps) * -sin(x)));
	} else if (x <= 4.8e-33) {
		tmp = (sin(((eps + (x - x)) * 0.5)) * sin((0.5 * (eps + (x + x))))) * -2.0;
	} else {
		tmp = ((cos(x) * cos(eps)) - cos(x)) - (sin(eps) * sin(x));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -6.4e-15)
		tmp = fma(Float64(-1.0 + cos(eps)), cos(x), Float64(sin(eps) * Float64(-sin(x))));
	elseif (x <= 4.8e-33)
		tmp = Float64(Float64(sin(Float64(Float64(eps + Float64(x - x)) * 0.5)) * sin(Float64(0.5 * Float64(eps + Float64(x + x))))) * -2.0);
	else
		tmp = Float64(Float64(Float64(cos(x) * cos(eps)) - cos(x)) - Float64(sin(eps) * sin(x)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -6.4e-15], N[(N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[Sin[eps], $MachinePrecision] * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.8e-33], N[(N[(N[Sin[N[(N[(eps + N[(x - x), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * N[(eps + N[(x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.3999999999999999e-15

   1. Initial program 9.2%

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

      1. cos-sum 50.7%

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

      2. cancel-sign-sub-inv 50.7%

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

      3. fma-def 50.7%

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

   4. Applied egg-rr 50.7%

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

   5. Taylor expanded in x around inf 50.7%

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

      1. neg-mul-1 50.7%

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

      2. +-commutative 50.7%

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

      3. sub-neg 50.7%

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

      4. *-commutative 50.7%

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

      5. associate--r+ 50.6%

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

      6. +-commutative 50.6%

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

      7. associate--r+ 99.5%

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

      8. *-rgt-identity 99.5%

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

      9. distribute-lft-out-- 99.5%

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

      10. sub-neg 99.5%

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

      11. metadata-eval 99.5%

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

      12. +-commutative 99.5%

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

      13. *-commutative 99.5%

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

   7. Simplified 99.5%

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

      1. *-commutative 99.5%

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

      2. fma-neg 99.5%

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

      3. distribute-rgt-neg-in 99.5%

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

   9. Applied egg-rr 99.5%

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

   if -6.3999999999999999e-15 < x < 4.8e-33

   1. Initial program 77.0%

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

      1. cos-sum 77.0%

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

      2. cancel-sign-sub-inv 77.0%

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

      3. fma-def 77.0%

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

   4. Applied egg-rr 77.0%

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

   5. Taylor expanded in x around inf 77.0%

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

      1. neg-mul-1 77.0%

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

      2. +-commutative 77.0%

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

      3. sub-neg 77.0%

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

      4. *-commutative 77.0%

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

      5. associate--r+ 77.0%

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

      6. +-commutative 77.0%

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

      7. associate--r+ 81.7%

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

      8. *-rgt-identity 81.7%

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

      9. distribute-lft-out-- 81.7%

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

      10. sub-neg 81.7%

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

      11. metadata-eval 81.7%

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

      12. +-commutative 81.7%

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

      13. *-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. Step-by-step derivation

      1. distribute-lft-in 81.7%

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

      2. associate--l+ 77.0%

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

      3. *-commutative 77.0%

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

      4. neg-mul-1 77.0%

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

      5. cos-sum 77.0%

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

      6. +-commutative 77.0%

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

      7. sub-neg 77.0%

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

      8. diff-cos 94.9%

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

      9. *-commutative 94.9%

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

   9. Applied egg-rr 99.6%

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

   if 4.8e-33 < x

   1. Initial program 11.0%

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

      1. cos-sum 53.0%

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

      2. cancel-sign-sub-inv 53.0%

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

      3. fma-def 53.0%

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

   4. Applied egg-rr 53.0%

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

   5. Taylor expanded in x around inf 53.0%

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

      1. neg-mul-1 53.0%

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

      2. +-commutative 53.0%

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

      3. sub-neg 53.0%

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

      4. *-commutative 53.0%

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

      5. associate--r+ 53.0%

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

      6. +-commutative 53.0%

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

      7. associate--r+ 98.8%

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

      8. *-rgt-identity 98.8%

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

      9. distribute-lft-out-- 98.7%

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

      10. sub-neg 98.7%

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

      11. metadata-eval 98.7%

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

      12. +-commutative 98.7%

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

      13. *-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. +-commutative 98.7%

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

      2. distribute-rgt-in 98.8%

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

      3. neg-mul-1 98.8%

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

   9. Applied egg-rr 98.8%

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

   10. Taylor expanded in eps around inf 98.8%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.4 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(-1 + \cos \varepsilon, \cos x, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-33}:\\ \;\;\;\;\left(\sin \left(\left(\varepsilon + \left(x - x\right)\right) \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \cos x\right) - \sin \varepsilon \cdot \sin x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 + \cos \varepsilon\\ \mathbf{if}\;x \leq -2.05 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \cos x, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-34}:\\ \;\;\;\;\left(\sin \left(\left(\varepsilon + \left(x - x\right)\right) \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot t_0 - \sin \varepsilon \cdot \sin x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ -1.0 (cos eps))))
   (if (<= x -2.05e-14)
     (fma t_0 (cos x) (* (sin eps) (- (sin x))))
     (if (<= x 1.6e-34)
       (* (* (sin (* (+ eps (- x x)) 0.5)) (sin (* 0.5 (+ eps (+ x x))))) -2.0)
       (- (* (cos x) t_0) (* (sin eps) (sin x)))))))

C

double code(double x, double eps) {
	double t_0 = -1.0 + cos(eps);
	double tmp;
	if (x <= -2.05e-14) {
		tmp = fma(t_0, cos(x), (sin(eps) * -sin(x)));
	} else if (x <= 1.6e-34) {
		tmp = (sin(((eps + (x - x)) * 0.5)) * sin((0.5 * (eps + (x + x))))) * -2.0;
	} else {
		tmp = (cos(x) * t_0) - (sin(eps) * sin(x));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(-1.0 + cos(eps))
	tmp = 0.0
	if (x <= -2.05e-14)
		tmp = fma(t_0, cos(x), Float64(sin(eps) * Float64(-sin(x))));
	elseif (x <= 1.6e-34)
		tmp = Float64(Float64(sin(Float64(Float64(eps + Float64(x - x)) * 0.5)) * sin(Float64(0.5 * Float64(eps + Float64(x + x))))) * -2.0);
	else
		tmp = Float64(Float64(cos(x) * t_0) - Float64(sin(eps) * sin(x)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.05e-14], N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(N[Sin[eps], $MachinePrecision] * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.6e-34], N[(N[(N[Sin[N[(N[(eps + N[(x - x), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * N[(eps + N[(x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision] - N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.0500000000000001e-14

   1. Initial program 9.2%

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

      1. cos-sum 50.7%

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

      2. cancel-sign-sub-inv 50.7%

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

      3. fma-def 50.7%

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

   4. Applied egg-rr 50.7%

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

   5. Taylor expanded in x around inf 50.7%

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

      1. neg-mul-1 50.7%

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

      2. +-commutative 50.7%

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

      3. sub-neg 50.7%

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

      4. *-commutative 50.7%

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

      5. associate--r+ 50.6%

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

      6. +-commutative 50.6%

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

      7. associate--r+ 99.5%

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

      8. *-rgt-identity 99.5%

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

      9. distribute-lft-out-- 99.5%

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

      10. sub-neg 99.5%

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

      11. metadata-eval 99.5%

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

      12. +-commutative 99.5%

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

      13. *-commutative 99.5%

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

   7. Simplified 99.5%

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

      1. *-commutative 99.5%

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

      2. fma-neg 99.5%

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

      3. distribute-rgt-neg-in 99.5%

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

   9. Applied egg-rr 99.5%

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

   if -2.0500000000000001e-14 < x < 1.60000000000000001e-34

   1. Initial program 77.0%

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

      1. cos-sum 77.0%

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

      2. cancel-sign-sub-inv 77.0%

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

      3. fma-def 77.0%

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

   4. Applied egg-rr 77.0%

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

   5. Taylor expanded in x around inf 77.0%

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

      1. neg-mul-1 77.0%

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

      2. +-commutative 77.0%

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

      3. sub-neg 77.0%

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

      4. *-commutative 77.0%

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

      5. associate--r+ 77.0%

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

      6. +-commutative 77.0%

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

      7. associate--r+ 81.7%

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

      8. *-rgt-identity 81.7%

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

      9. distribute-lft-out-- 81.7%

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

      10. sub-neg 81.7%

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

      11. metadata-eval 81.7%

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

      12. +-commutative 81.7%

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

      13. *-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. Step-by-step derivation

      1. distribute-lft-in 81.7%

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

      2. associate--l+ 77.0%

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

      3. *-commutative 77.0%

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

      4. neg-mul-1 77.0%

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

      5. cos-sum 77.0%

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

      6. +-commutative 77.0%

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

      7. sub-neg 77.0%

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

      8. diff-cos 94.9%

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

      9. *-commutative 94.9%

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

   9. Applied egg-rr 99.6%

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

   if 1.60000000000000001e-34 < x

   1. Initial program 11.0%

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

      1. cos-sum 53.0%

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

      2. cancel-sign-sub-inv 53.0%

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

      3. fma-def 53.0%

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

   4. Applied egg-rr 53.0%

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

   5. Taylor expanded in x around inf 53.0%

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

      1. neg-mul-1 53.0%

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

      2. +-commutative 53.0%

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

      3. sub-neg 53.0%

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

      4. *-commutative 53.0%

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

      5. associate--r+ 53.0%

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

      6. +-commutative 53.0%

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

      7. associate--r+ 98.8%

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

      8. *-rgt-identity 98.8%

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

      9. distribute-lft-out-- 98.7%

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

      10. sub-neg 98.7%

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

      11. metadata-eval 98.7%

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

      12. +-commutative 98.7%

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

      13. *-commutative 98.7%

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

   7. Simplified 98.7%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(-1 + \cos \varepsilon, \cos x, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-34}:\\ \;\;\;\;\left(\sin \left(\left(\varepsilon + \left(x - x\right)\right) \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - \sin \varepsilon \cdot \sin x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-16} \lor \neg \left(x \leq 1.6 \cdot 10^{-31}\right):\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\left(\sin \left(\left(\varepsilon + \left(x - x\right)\right) \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)\right) \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -1.25e-16) (not (<= x 1.6e-31)))
   (- (* (cos x) (+ -1.0 (cos eps))) (* (sin eps) (sin x)))
   (* (* (sin (* (+ eps (- x x)) 0.5)) (sin (* 0.5 (+ eps (+ x x))))) -2.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -1.25e-16) || !(x <= 1.6e-31)) {
		tmp = (cos(x) * (-1.0 + cos(eps))) - (sin(eps) * sin(x));
	} else {
		tmp = (sin(((eps + (x - x)) * 0.5)) * sin((0.5 * (eps + (x + x))))) * -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 <= (-1.25d-16)) .or. (.not. (x <= 1.6d-31))) then
        tmp = (cos(x) * ((-1.0d0) + cos(eps))) - (sin(eps) * sin(x))
    else
        tmp = (sin(((eps + (x - x)) * 0.5d0)) * sin((0.5d0 * (eps + (x + x))))) * (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -1.25e-16) || !(x <= 1.6e-31)) {
		tmp = (Math.cos(x) * (-1.0 + Math.cos(eps))) - (Math.sin(eps) * Math.sin(x));
	} else {
		tmp = (Math.sin(((eps + (x - x)) * 0.5)) * Math.sin((0.5 * (eps + (x + x))))) * -2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -1.25e-16) or not (x <= 1.6e-31):
		tmp = (math.cos(x) * (-1.0 + math.cos(eps))) - (math.sin(eps) * math.sin(x))
	else:
		tmp = (math.sin(((eps + (x - x)) * 0.5)) * math.sin((0.5 * (eps + (x + x))))) * -2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -1.25e-16) || !(x <= 1.6e-31))
		tmp = Float64(Float64(cos(x) * Float64(-1.0 + cos(eps))) - Float64(sin(eps) * sin(x)));
	else
		tmp = Float64(Float64(sin(Float64(Float64(eps + Float64(x - x)) * 0.5)) * sin(Float64(0.5 * Float64(eps + Float64(x + x))))) * -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -1.25e-16) || ~((x <= 1.6e-31)))
		tmp = (cos(x) * (-1.0 + cos(eps))) - (sin(eps) * sin(x));
	else
		tmp = (sin(((eps + (x - x)) * 0.5)) * sin((0.5 * (eps + (x + x))))) * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -1.25e-16], N[Not[LessEqual[x, 1.6e-31]], $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[Sin[N[(N[(eps + N[(x - x), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * N[(eps + N[(x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.2500000000000001e-16 or 1.60000000000000009e-31 < x

   1. Initial program 10.1%

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

      1. cos-sum 51.8%

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

      2. cancel-sign-sub-inv 51.8%

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

      3. fma-def 51.8%

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

   4. Applied egg-rr 51.8%

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

   5. Taylor expanded in x around inf 51.8%

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

      1. neg-mul-1 51.8%

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

      2. +-commutative 51.8%

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

      3. sub-neg 51.8%

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

      4. *-commutative 51.8%

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

      5. associate--r+ 51.8%

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

      6. +-commutative 51.8%

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

      7. associate--r+ 99.2%

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

      8. *-rgt-identity 99.2%

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

      9. distribute-lft-out-- 99.1%

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

      10. sub-neg 99.1%

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

      11. metadata-eval 99.1%

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

      12. +-commutative 99.1%

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

      13. *-commutative 99.1%

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

   7. Simplified 99.1%

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

   if -1.2500000000000001e-16 < x < 1.60000000000000009e-31

   1. Initial program 77.0%

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

      1. cos-sum 77.0%

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

      2. cancel-sign-sub-inv 77.0%

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

      3. fma-def 77.0%

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

   4. Applied egg-rr 77.0%

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

   5. Taylor expanded in x around inf 77.0%

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

      1. neg-mul-1 77.0%

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

      2. +-commutative 77.0%

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

      3. sub-neg 77.0%

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

      4. *-commutative 77.0%

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

      5. associate--r+ 77.0%

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

      6. +-commutative 77.0%

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

      7. associate--r+ 81.7%

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

      8. *-rgt-identity 81.7%

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

      9. distribute-lft-out-- 81.7%

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

      10. sub-neg 81.7%

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

      11. metadata-eval 81.7%

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

      12. +-commutative 81.7%

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

      13. *-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. Step-by-step derivation

      1. distribute-lft-in 81.7%

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

      2. associate--l+ 77.0%

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

      3. *-commutative 77.0%

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

      4. neg-mul-1 77.0%

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

      5. cos-sum 77.0%

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

      6. +-commutative 77.0%

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

      7. sub-neg 77.0%

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

      8. diff-cos 94.9%

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

      9. *-commutative 94.9%

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

   9. Applied egg-rr 99.6%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-16} \lor \neg \left(x \leq 1.6 \cdot 10^{-31}\right):\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\left(\sin \left(\left(\varepsilon + \left(x - x\right)\right) \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)\right) \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\varepsilon \cdot 0.5\right)\\ t_1 := \sin x \cdot \left(-2 \cdot t_0\right)\\ \mathbf{if}\;x \leq -5.2 \cdot 10^{-18}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.76 \cdot 10^{-124}:\\ \;\;\;\;-2 \cdot {t_0}^{2}\\ \mathbf{elif}\;x \leq 0.0028:\\ \;\;\;\;\left(-1 + \cos \varepsilon\right) - \sin \varepsilon \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (* eps 0.5))) (t_1 (* (sin x) (* -2.0 t_0))))
   (if (<= x -5.2e-18)
     t_1
     (if (<= x 1.76e-124)
       (* -2.0 (pow t_0 2.0))
       (if (<= x 0.0028) (- (+ -1.0 (cos eps)) (* (sin eps) x)) t_1)))))

C

double code(double x, double eps) {
	double t_0 = sin((eps * 0.5));
	double t_1 = sin(x) * (-2.0 * t_0);
	double tmp;
	if (x <= -5.2e-18) {
		tmp = t_1;
	} else if (x <= 1.76e-124) {
		tmp = -2.0 * pow(t_0, 2.0);
	} else if (x <= 0.0028) {
		tmp = (-1.0 + cos(eps)) - (sin(eps) * x);
	} else {
		tmp = t_1;
	}
	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 = sin((eps * 0.5d0))
    t_1 = sin(x) * ((-2.0d0) * t_0)
    if (x <= (-5.2d-18)) then
        tmp = t_1
    else if (x <= 1.76d-124) then
        tmp = (-2.0d0) * (t_0 ** 2.0d0)
    else if (x <= 0.0028d0) then
        tmp = ((-1.0d0) + cos(eps)) - (sin(eps) * x)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin((eps * 0.5));
	double t_1 = Math.sin(x) * (-2.0 * t_0);
	double tmp;
	if (x <= -5.2e-18) {
		tmp = t_1;
	} else if (x <= 1.76e-124) {
		tmp = -2.0 * Math.pow(t_0, 2.0);
	} else if (x <= 0.0028) {
		tmp = (-1.0 + Math.cos(eps)) - (Math.sin(eps) * x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.sin((eps * 0.5))
	t_1 = math.sin(x) * (-2.0 * t_0)
	tmp = 0
	if x <= -5.2e-18:
		tmp = t_1
	elif x <= 1.76e-124:
		tmp = -2.0 * math.pow(t_0, 2.0)
	elif x <= 0.0028:
		tmp = (-1.0 + math.cos(eps)) - (math.sin(eps) * x)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, eps)
	t_0 = sin(Float64(eps * 0.5))
	t_1 = Float64(sin(x) * Float64(-2.0 * t_0))
	tmp = 0.0
	if (x <= -5.2e-18)
		tmp = t_1;
	elseif (x <= 1.76e-124)
		tmp = Float64(-2.0 * (t_0 ^ 2.0));
	elseif (x <= 0.0028)
		tmp = Float64(Float64(-1.0 + cos(eps)) - Float64(sin(eps) * x));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = sin((eps * 0.5));
	t_1 = sin(x) * (-2.0 * t_0);
	tmp = 0.0;
	if (x <= -5.2e-18)
		tmp = t_1;
	elseif (x <= 1.76e-124)
		tmp = -2.0 * (t_0 ^ 2.0);
	elseif (x <= 0.0028)
		tmp = (-1.0 + cos(eps)) - (sin(eps) * x);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[x], $MachinePrecision] * N[(-2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.2e-18], t$95$1, If[LessEqual[x, 1.76e-124], N[(-2.0 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0028], N[(N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[eps], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.2000000000000001e-18 or 0.00279999999999999997 < x

   1. Initial program 9.1%

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

      1. diff-cos 7.5%

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

      2. div-inv 7.5%

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

      3. associate--l+ 7.5%

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

      4. metadata-eval 7.5%

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

      5. div-inv 7.5%

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

      6. +-commutative 7.5%

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

      7. associate-+l+ 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. sub-neg 7.6%

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

      8. mul-1-neg 7.6%

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

      9. +-commutative 7.6%

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

      10. associate-+r+ 57.3%

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

      11. mul-1-neg 57.3%

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

      12. sub-neg 57.3%

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

      13. +-inverses 57.3%

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

      14. remove-double-neg 57.3%

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

      15. mul-1-neg 57.3%

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

      16. sub-neg 57.3%

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

      17. neg-sub0 57.3%

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

      18. mul-1-neg 57.3%

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

      19. remove-double-neg 57.3%

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

   6. Simplified 57.3%

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

   7. Taylor expanded in eps around 0 56.5%

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

   if -5.2000000000000001e-18 < x < 1.75999999999999996e-124

   1. Initial program 79.3%

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

      1. diff-cos 99.2%

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

      2. div-inv 99.2%

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

      3. associate--l+ 99.2%

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

      4. metadata-eval 99.2%

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

      5. div-inv 99.2%

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

      6. +-commutative 99.2%

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

      7. associate-+l+ 99.2%

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

      8. metadata-eval 99.2%

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

   4. Applied egg-rr 99.2%

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

      1. associate-*r* 99.2%

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

      2. *-commutative 99.2%

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

      3. *-commutative 99.2%

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

      4. +-commutative 99.2%

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

      5. count-2 99.2%

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

      6. fma-def 99.2%

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

      7. sub-neg 99.2%

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

      8. mul-1-neg 99.2%

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

      9. +-commutative 99.2%

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

      10. associate-+r+ 99.6%

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

      11. mul-1-neg 99.6%

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

      12. sub-neg 99.6%

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

      13. +-inverses 99.6%

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

      14. remove-double-neg 99.6%

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

      15. mul-1-neg 99.6%

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

      16. sub-neg 99.6%

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

      17. neg-sub0 99.6%

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

      18. mul-1-neg 99.6%

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

      19. remove-double-neg 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in x around 0 98.5%

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

   if 1.75999999999999996e-124 < x < 0.00279999999999999997

   1. Initial program 59.2%

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

   3. Taylor expanded in x around 0 62.1%

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

      1. sub-neg 62.1%

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

      2. metadata-eval 62.1%

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

      3. +-commutative 62.1%

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

      4. associate-+r+ 91.2%

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

      5. +-commutative 91.2%

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

      6. +-commutative 91.2%

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

      7. mul-1-neg 91.2%

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

      8. unsub-neg 91.2%

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

      9. unpow2 91.2%

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

      10. associate-*l* 91.2%

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

      11. distribute-lft-out-- 91.2%

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

      12. +-commutative 91.2%

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

      13. *-commutative 91.2%

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

      14. fma-def 91.2%

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

   5. Simplified 91.2%

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

   6. Taylor expanded in x around 0 90.1%

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

      1. mul-1-neg 90.1%

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

      2. distribute-rgt-neg-out 90.1%

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

   8. Simplified 90.1%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-18}:\\ \;\;\;\;\sin x \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \mathbf{elif}\;x \leq 1.76 \cdot 10^{-124}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \mathbf{elif}\;x \leq 0.0028:\\ \;\;\;\;\left(-1 + \cos \varepsilon\right) - \sin \varepsilon \cdot x\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \varepsilon - \cos x\\ \mathbf{if}\;\varepsilon \leq -2.9 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 4.6 \cdot 10^{-17}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00014:\\ \;\;\;\;-0.5 \cdot {\varepsilon}^{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (cos eps) (cos x))))
   (if (<= eps -2.9e-5)
     t_0
     (if (<= eps 4.6e-17)
       (* (sin x) (- eps))
       (if (<= eps 0.00014) (* -0.5 (pow eps 2.0)) t_0)))))

C

double code(double x, double eps) {
	double t_0 = cos(eps) - cos(x);
	double tmp;
	if (eps <= -2.9e-5) {
		tmp = t_0;
	} else if (eps <= 4.6e-17) {
		tmp = sin(x) * -eps;
	} else if (eps <= 0.00014) {
		tmp = -0.5 * pow(eps, 2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(eps) - cos(x)
    if (eps <= (-2.9d-5)) then
        tmp = t_0
    else if (eps <= 4.6d-17) then
        tmp = sin(x) * -eps
    else if (eps <= 0.00014d0) then
        tmp = (-0.5d0) * (eps ** 2.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.cos(eps) - Math.cos(x);
	double tmp;
	if (eps <= -2.9e-5) {
		tmp = t_0;
	} else if (eps <= 4.6e-17) {
		tmp = Math.sin(x) * -eps;
	} else if (eps <= 0.00014) {
		tmp = -0.5 * Math.pow(eps, 2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.cos(eps) - math.cos(x)
	tmp = 0
	if eps <= -2.9e-5:
		tmp = t_0
	elif eps <= 4.6e-17:
		tmp = math.sin(x) * -eps
	elif eps <= 0.00014:
		tmp = -0.5 * math.pow(eps, 2.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(cos(eps) - cos(x))
	tmp = 0.0
	if (eps <= -2.9e-5)
		tmp = t_0;
	elseif (eps <= 4.6e-17)
		tmp = Float64(sin(x) * Float64(-eps));
	elseif (eps <= 0.00014)
		tmp = Float64(-0.5 * (eps ^ 2.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = cos(eps) - cos(x);
	tmp = 0.0;
	if (eps <= -2.9e-5)
		tmp = t_0;
	elseif (eps <= 4.6e-17)
		tmp = sin(x) * -eps;
	elseif (eps <= 0.00014)
		tmp = -0.5 * (eps ^ 2.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Cos[eps], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -2.9e-5], t$95$0, If[LessEqual[eps, 4.6e-17], N[(N[Sin[x], $MachinePrecision] * (-eps)), $MachinePrecision], If[LessEqual[eps, 0.00014], N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -2.9e-5 or 1.3999999999999999e-4 < eps

   1. Initial program 54.5%

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

   3. Taylor expanded in x around 0 56.9%

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

   if -2.9e-5 < eps < 4.60000000000000018e-17

   1. Initial program 25.7%

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

   3. Taylor expanded in eps around 0 86.6%

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

      1. mul-1-neg 86.6%

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

      2. *-commutative 86.6%

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

      3. distribute-rgt-neg-in 86.6%

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

   5. Simplified 86.6%

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

   if 4.60000000000000018e-17 < eps < 1.3999999999999999e-4

   1. Initial program 13.0%

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

   3. Taylor expanded in x around 0 13.0%

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

   4. Taylor expanded in eps around 0 97.5%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.9 \cdot 10^{-5}:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{elif}\;\varepsilon \leq 4.6 \cdot 10^{-17}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00014:\\ \;\;\;\;-0.5 \cdot {\varepsilon}^{2}\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\varepsilon \cdot 0.5\right)\\ \mathbf{if}\;x \leq -3 \cdot 10^{-20} \lor \neg \left(x \leq 6 \cdot 10^{-31}\right):\\ \;\;\;\;\sin x \cdot \left(-2 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {t_0}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (* eps 0.5))))
   (if (or (<= x -3e-20) (not (<= x 6e-31)))
     (* (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 <= -3e-20) || !(x <= 6e-31)) {
		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 <= (-3d-20)) .or. (.not. (x <= 6d-31))) 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 <= -3e-20) || !(x <= 6e-31)) {
		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 <= -3e-20) or not (x <= 6e-31):
		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 <= -3e-20) || !(x <= 6e-31))
		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 <= -3e-20) || ~((x <= 6e-31)))
		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, -3e-20], N[Not[LessEqual[x, 6e-31]], $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 < -3.00000000000000029e-20 or 5.99999999999999962e-31 < x

   1. Initial program 10.1%

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

      1. diff-cos 8.6%

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

      2. div-inv 8.6%

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

      3. associate--l+ 8.6%

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

      4. metadata-eval 8.6%

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

      5. div-inv 8.6%

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

      6. +-commutative 8.6%

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

      7. associate-+l+ 8.7%

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

      8. metadata-eval 8.7%

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

   4. Applied egg-rr 8.7%

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

      1. associate-*r* 8.7%

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

      2. *-commutative 8.7%

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

      3. *-commutative 8.7%

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

      4. +-commutative 8.7%

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

      5. count-2 8.7%

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

      6. fma-def 8.7%

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

      7. sub-neg 8.7%

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

      8. mul-1-neg 8.7%

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

      9. +-commutative 8.7%

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

      10. associate-+r+ 58.5%

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

      11. mul-1-neg 58.5%

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

      12. sub-neg 58.5%

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

      13. +-inverses 58.5%

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

      14. remove-double-neg 58.5%

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

      15. mul-1-neg 58.5%

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

      16. sub-neg 58.5%

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

      17. neg-sub0 58.5%

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

      18. mul-1-neg 58.5%

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

      19. remove-double-neg 58.5%

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

   6. Simplified 58.5%

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

   7. Taylor expanded in eps around 0 56.7%

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

   if -3.00000000000000029e-20 < x < 5.99999999999999962e-31

   1. Initial program 77.6%

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

      1. diff-cos 95.7%

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

      2. div-inv 95.7%

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

      3. associate--l+ 95.7%

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

      4. metadata-eval 95.7%

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

      5. div-inv 95.7%

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

      6. +-commutative 95.7%

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

      7. associate-+l+ 95.7%

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

      8. metadata-eval 95.7%

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

   4. Applied egg-rr 95.7%

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

      1. associate-*r* 95.7%

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

      2. *-commutative 95.7%

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

      3. *-commutative 95.7%

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

      4. +-commutative 95.7%

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

      5. count-2 95.7%

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

      6. fma-def 95.7%

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

      7. sub-neg 95.7%

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

      8. mul-1-neg 95.7%

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

      9. +-commutative 95.7%

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

      10. associate-+r+ 99.6%

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

      11. mul-1-neg 99.6%

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

      12. sub-neg 99.6%

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

      13. +-inverses 99.6%

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

      14. remove-double-neg 99.6%

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

      15. mul-1-neg 99.6%

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

      16. sub-neg 99.6%

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

      17. neg-sub0 99.6%

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

      18. mul-1-neg 99.6%

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

      19. remove-double-neg 99.6%

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

   6. Simplified 99.6%

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

   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 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{-20} \lor \neg \left(x \leq 6 \cdot 10^{-31}\right):\\ \;\;\;\;\sin x \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.1%

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

   1. cos-sum 63.1%

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

   2. cancel-sign-sub-inv 63.1%

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

   3. fma-def 63.1%

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

4. Applied egg-rr 63.1%

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

5. Taylor expanded in x around inf 63.1%

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

   1. neg-mul-1 63.1%

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

   2. +-commutative 63.1%

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

   3. sub-neg 63.1%

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

   4. *-commutative 63.1%

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

   5. associate--r+ 63.1%

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

   6. +-commutative 63.1%

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

   7. associate--r+ 91.3%

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

   8. *-rgt-identity 91.3%

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

   9. distribute-lft-out-- 91.3%

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

   10. sub-neg 91.3%

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

   11. metadata-eval 91.3%

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

   12. +-commutative 91.3%

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

   13. *-commutative 91.3%

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

7. Simplified 91.3%

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

   1. distribute-lft-in 91.3%

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

   2. associate--l+ 63.1%

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

   3. *-commutative 63.1%

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

   4. neg-mul-1 63.1%

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

   5. cos-sum 40.1%

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

   6. +-commutative 40.1%

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

   7. sub-neg 40.1%

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

   8. diff-cos 47.4%

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

   9. *-commutative 47.4%

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

9. Applied egg-rr 76.8%

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

10. Final simplification 76.8%

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

Alternative 11: 67.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{-17} \lor \neg \left(x \leq 4 \cdot 10^{-41}\right):\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -1.85e-17) (not (<= x 4e-41)))
   (* (sin x) (- eps))
   (* -2.0 (pow (sin (* eps 0.5)) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -1.85e-17) || !(x <= 4e-41)) {
		tmp = sin(x) * -eps;
	} 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 <= (-1.85d-17)) .or. (.not. (x <= 4d-41))) then
        tmp = sin(x) * -eps
    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 <= -1.85e-17) || !(x <= 4e-41)) {
		tmp = Math.sin(x) * -eps;
	} else {
		tmp = -2.0 * Math.pow(Math.sin((eps * 0.5)), 2.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -1.85e-17) or not (x <= 4e-41):
		tmp = math.sin(x) * -eps
	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 <= -1.85e-17) || !(x <= 4e-41))
		tmp = Float64(sin(x) * Float64(-eps));
	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 <= -1.85e-17) || ~((x <= 4e-41)))
		tmp = sin(x) * -eps;
	else
		tmp = -2.0 * (sin((eps * 0.5)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -1.85e-17], N[Not[LessEqual[x, 4e-41]], $MachinePrecision]], N[(N[Sin[x], $MachinePrecision] * (-eps)), $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 < -1.8499999999999999e-17 or 4.00000000000000002e-41 < x

   1. Initial program 10.1%

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

   3. Taylor expanded in eps around 0 52.7%

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

      1. mul-1-neg 52.7%

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

      2. *-commutative 52.7%

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

      3. distribute-rgt-neg-in 52.7%

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

   5. Simplified 52.7%

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

   if -1.8499999999999999e-17 < x < 4.00000000000000002e-41

   1. Initial program 77.6%

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

      1. diff-cos 95.7%

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

      2. div-inv 95.7%

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

      3. associate--l+ 95.7%

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

      4. metadata-eval 95.7%

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

      5. div-inv 95.7%

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

      6. +-commutative 95.7%

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

      7. associate-+l+ 95.7%

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

      8. metadata-eval 95.7%

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

   4. Applied egg-rr 95.7%

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

      1. associate-*r* 95.7%

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

      2. *-commutative 95.7%

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

      3. *-commutative 95.7%

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

      4. +-commutative 95.7%

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

      5. count-2 95.7%

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

      6. fma-def 95.7%

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

      7. sub-neg 95.7%

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

      8. mul-1-neg 95.7%

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

      9. +-commutative 95.7%

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

      10. associate-+r+ 99.6%

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

      11. mul-1-neg 99.6%

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

      12. sub-neg 99.6%

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

      13. +-inverses 99.6%

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

      14. remove-double-neg 99.6%

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

      15. mul-1-neg 99.6%

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

      16. sub-neg 99.6%

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

      17. neg-sub0 99.6%

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

      18. mul-1-neg 99.6%

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

      19. remove-double-neg 99.6%

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

   6. Simplified 99.6%

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

   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 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{-17} \lor \neg \left(x \leq 4 \cdot 10^{-41}\right):\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 66.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 + \cos \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.00078:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 3.6 \cdot 10^{-18}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 0.000135:\\ \;\;\;\;-0.5 \cdot {\varepsilon}^{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ -1.0 (cos eps))))
   (if (<= eps -0.00078)
     t_0
     (if (<= eps 3.6e-18)
       (* (sin x) (- eps))
       (if (<= eps 0.000135) (* -0.5 (pow eps 2.0)) t_0)))))

C

double code(double x, double eps) {
	double t_0 = -1.0 + cos(eps);
	double tmp;
	if (eps <= -0.00078) {
		tmp = t_0;
	} else if (eps <= 3.6e-18) {
		tmp = sin(x) * -eps;
	} else if (eps <= 0.000135) {
		tmp = -0.5 * pow(eps, 2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-1.0d0) + cos(eps)
    if (eps <= (-0.00078d0)) then
        tmp = t_0
    else if (eps <= 3.6d-18) then
        tmp = sin(x) * -eps
    else if (eps <= 0.000135d0) then
        tmp = (-0.5d0) * (eps ** 2.0d0)
    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 tmp;
	if (eps <= -0.00078) {
		tmp = t_0;
	} else if (eps <= 3.6e-18) {
		tmp = Math.sin(x) * -eps;
	} else if (eps <= 0.000135) {
		tmp = -0.5 * Math.pow(eps, 2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = -1.0 + math.cos(eps)
	tmp = 0
	if eps <= -0.00078:
		tmp = t_0
	elif eps <= 3.6e-18:
		tmp = math.sin(x) * -eps
	elif eps <= 0.000135:
		tmp = -0.5 * math.pow(eps, 2.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(-1.0 + cos(eps))
	tmp = 0.0
	if (eps <= -0.00078)
		tmp = t_0;
	elseif (eps <= 3.6e-18)
		tmp = Float64(sin(x) * Float64(-eps));
	elseif (eps <= 0.000135)
		tmp = Float64(-0.5 * (eps ^ 2.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = -1.0 + cos(eps);
	tmp = 0.0;
	if (eps <= -0.00078)
		tmp = t_0;
	elseif (eps <= 3.6e-18)
		tmp = sin(x) * -eps;
	elseif (eps <= 0.000135)
		tmp = -0.5 * (eps ^ 2.0);
	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]}, If[LessEqual[eps, -0.00078], t$95$0, If[LessEqual[eps, 3.6e-18], N[(N[Sin[x], $MachinePrecision] * (-eps)), $MachinePrecision], If[LessEqual[eps, 0.000135], N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -7.79999999999999986e-4 or 1.35000000000000002e-4 < eps

   1. Initial program 54.5%

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

   3. Taylor expanded in x around 0 54.6%

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

   if -7.79999999999999986e-4 < eps < 3.6000000000000001e-18

   1. Initial program 25.7%

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

   3. Taylor expanded in eps around 0 86.6%

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

      1. mul-1-neg 86.6%

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

      2. *-commutative 86.6%

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

      3. distribute-rgt-neg-in 86.6%

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

   5. Simplified 86.6%

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

   if 3.6000000000000001e-18 < eps < 1.35000000000000002e-4

   1. Initial program 13.0%

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

   3. Taylor expanded in x around 0 13.0%

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

   4. Taylor expanded in eps around 0 97.5%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00078:\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 3.6 \cdot 10^{-18}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 0.000135:\\ \;\;\;\;-0.5 \cdot {\varepsilon}^{2}\\ \mathbf{else}:\\ \;\;\;\;-1 + \cos \varepsilon\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 46.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00012 \lor \neg \left(\varepsilon \leq 0.000135\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot {\varepsilon}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.00012) (not (<= eps 0.000135)))
   (+ -1.0 (cos eps))
   (* -0.5 (pow eps 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.00012) || !(eps <= 0.000135)) {
		tmp = -1.0 + cos(eps);
	} else {
		tmp = -0.5 * pow(eps, 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 ((eps <= (-0.00012d0)) .or. (.not. (eps <= 0.000135d0))) then
        tmp = (-1.0d0) + cos(eps)
    else
        tmp = (-0.5d0) * (eps ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.00012) || !(eps <= 0.000135)) {
		tmp = -1.0 + Math.cos(eps);
	} else {
		tmp = -0.5 * Math.pow(eps, 2.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.00012) or not (eps <= 0.000135):
		tmp = -1.0 + math.cos(eps)
	else:
		tmp = -0.5 * math.pow(eps, 2.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.00012) || !(eps <= 0.000135))
		tmp = Float64(-1.0 + cos(eps));
	else
		tmp = Float64(-0.5 * (eps ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.00012) || ~((eps <= 0.000135)))
		tmp = -1.0 + cos(eps);
	else
		tmp = -0.5 * (eps ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.00012], N[Not[LessEqual[eps, 0.000135]], $MachinePrecision]], N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.20000000000000003e-4 or 1.35000000000000002e-4 < eps

   1. Initial program 54.5%

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

   3. Taylor expanded in x around 0 54.6%

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

   if -1.20000000000000003e-4 < eps < 1.35000000000000002e-4

   1. Initial program 25.0%

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

   3. Taylor expanded in x around 0 25.1%

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

   4. Taylor expanded in eps around 0 41.1%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00012 \lor \neg \left(\varepsilon \leq 0.000135\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot {\varepsilon}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 38.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ -1 + \cos \varepsilon \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (+ -1.0 (cos eps)))

C

double code(double x, double eps) {
	return -1.0 + cos(eps);
}

Fortran

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

Java

public static double code(double x, double eps) {
	return -1.0 + Math.cos(eps);
}

Python

def code(x, eps):
	return -1.0 + math.cos(eps)

Julia

function code(x, eps)
	return Float64(-1.0 + cos(eps))
end

MATLAB

function tmp = code(x, eps)
	tmp = -1.0 + cos(eps);
end

Wolfram

code[x_, eps_] := N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.1%

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

3. Taylor expanded in x around 0 40.2%

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

4. Final simplification 40.2%

   \[\leadsto -1 + \cos \varepsilon \]
5. Add Preprocessing

Alternative 15: 12.5% 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 40.1%

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

3. Taylor expanded in x around 0 40.2%

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

4. Taylor expanded in eps around 0 13.6%

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

5. Final simplification 13.6%

   \[\leadsto 0 \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024018 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))