2cos (problem 3.3.5)

Percentage Accurate: 37.6% → 99.5%

Time: 19.0s

Alternatives: 15

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 37.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 33.0%

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

   1. cos-sum 60.6%

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

   2. cancel-sign-sub-inv 60.6%

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

   3. fma-def 60.6%

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

3. Applied egg-rr 60.6%

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

4. Taylor expanded in x around inf 60.6%

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

   1. neg-mul-1 60.6%

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

   2. distribute-lft-neg-in 60.6%

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

   3. associate--l+ 87.7%

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

   4. distribute-lft-neg-in 87.7%

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

   5. distribute-rgt-neg-in 87.7%

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

   6. *-commutative 87.7%

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

   7. fma-def 87.7%

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

   8. *-rgt-identity 87.7%

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

   9. distribute-lft-out-- 87.7%

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

   10. sub-neg 87.7%

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

   11. metadata-eval 87.7%

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

   12. +-commutative 87.7%

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

6. Simplified 87.7%

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

   1. flip-+ 87.4%

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

   2. metadata-eval 87.4%

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

8. Applied egg-rr 87.4%

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

   1. 1-sub-cos 99.2%

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

10. Simplified 99.2%

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

11. Taylor expanded in eps around inf 99.2%

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

    1. metadata-eval 99.2%

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

    2. times-frac 99.2%

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

    3. *-lft-identity 99.2%

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

    4. unpow2 99.2%

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

    5. times-frac 99.2%

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

    6. hang-0p-tan 99.5%

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

13. Simplified 99.5%

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

14. Final simplification 99.5%

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

Alternative 2: 99.0% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.0%

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

   1. cos-sum 60.6%

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

   2. cancel-sign-sub-inv 60.6%

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

   3. fma-def 60.6%

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

3. Applied egg-rr 60.6%

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

4. Taylor expanded in x around inf 60.6%

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

   1. neg-mul-1 60.6%

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

   2. distribute-lft-neg-in 60.6%

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

   3. associate--l+ 87.7%

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

   4. distribute-lft-neg-in 87.7%

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

   5. distribute-rgt-neg-in 87.7%

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

   6. *-commutative 87.7%

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

   7. fma-def 87.7%

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

   8. *-rgt-identity 87.7%

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

   9. distribute-lft-out-- 87.7%

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

   10. sub-neg 87.7%

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

   11. metadata-eval 87.7%

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

   12. +-commutative 87.7%

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

6. Simplified 87.7%

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

   1. flip-+ 87.4%

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

   2. metadata-eval 87.4%

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

8. Applied egg-rr 87.4%

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

   1. 1-sub-cos 99.2%

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

10. Simplified 99.2%

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

    1. fma-udef 99.2%

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

    2. associate-*r/ 99.2%

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

    3. pow2 99.2%

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

12. Applied egg-rr 99.2%

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

    1. +-commutative 99.2%

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

    2. distribute-rgt-neg-out 99.2%

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

    3. unsub-neg 99.2%

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

    4. associate-/l* 99.1%

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

    5. associate-/r/ 99.2%

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

    6. *-commutative 99.2%

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

14. Simplified 99.2%

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

15. Final simplification 99.2%

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

Alternative 3: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\sin \varepsilon, -\sin x, \cos x \cdot \cos \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 2.1 \cdot 10^{-5}:\\ \;\;\;\;-2 \cdot \left(\left(\sin x + 0.5 \cdot \left(\varepsilon \cdot \cos x\right)\right) \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, -1 + \cos \varepsilon, \sin x \cdot \left(-\sin \varepsilon\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -4e-5)
   (- (fma (sin eps) (- (sin x)) (* (cos x) (cos eps))) (cos x))
   (if (<= eps 2.1e-5)
     (* -2.0 (* (+ (sin x) (* 0.5 (* eps (cos x)))) (sin (* eps 0.5))))
     (fma (cos x) (+ -1.0 (cos eps)) (* (sin x) (- (sin eps)))))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -4e-5) {
		tmp = fma(sin(eps), -sin(x), (cos(x) * cos(eps))) - cos(x);
	} else if (eps <= 2.1e-5) {
		tmp = -2.0 * ((sin(x) + (0.5 * (eps * cos(x)))) * sin((eps * 0.5)));
	} else {
		tmp = fma(cos(x), (-1.0 + cos(eps)), (sin(x) * -sin(eps)));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -4e-5)
		tmp = Float64(fma(sin(eps), Float64(-sin(x)), Float64(cos(x) * cos(eps))) - cos(x));
	elseif (eps <= 2.1e-5)
		tmp = Float64(-2.0 * Float64(Float64(sin(x) + Float64(0.5 * Float64(eps * cos(x)))) * sin(Float64(eps * 0.5))));
	else
		tmp = fma(cos(x), Float64(-1.0 + cos(eps)), Float64(sin(x) * Float64(-sin(eps))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -4e-5], N[(N[(N[Sin[eps], $MachinePrecision] * (-N[Sin[x], $MachinePrecision]) + N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 2.1e-5], N[(-2.0 * N[(N[(N[Sin[x], $MachinePrecision] + N[(0.5 * N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[x], $MachinePrecision] * N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * (-N[Sin[eps], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -4.00000000000000033e-5

   1. Initial program 48.6%

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

      1. cos-sum 98.8%

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

      2. sub-neg 98.8%

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

   3. Applied egg-rr 98.8%

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

      1. +-commutative 98.8%

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

      2. distribute-lft-neg-in 98.8%

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

      3. *-commutative 98.8%

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

      4. fma-def 98.9%

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

      5. *-commutative 98.9%

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

   5. Simplified 98.9%

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

   if -4.00000000000000033e-5 < eps < 2.09999999999999988e-5

   1. Initial program 20.1%

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

      1. diff-cos 43.2%

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

      2. div-inv 43.2%

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

      3. metadata-eval 43.2%

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

      4. div-inv 43.2%

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

      5. +-commutative 43.2%

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

      6. metadata-eval 43.2%

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

   3. Applied egg-rr 43.2%

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

      1. *-commutative 43.2%

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

      2. +-commutative 43.2%

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

      3. associate--l+ 98.8%

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

      4. +-inverses 98.8%

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

      5. distribute-lft-in 98.8%

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

      6. metadata-eval 98.8%

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

      7. *-commutative 98.8%

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

      8. +-commutative 98.8%

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

   5. Simplified 98.8%

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

   6. Taylor expanded in eps around 0 99.7%

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

   if 2.09999999999999988e-5 < eps

   1. Initial program 42.1%

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

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

   3. Applied egg-rr 98.7%

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

   4. Taylor expanded in x around inf 98.7%

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

      1. neg-mul-1 98.7%

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

      2. distribute-lft-neg-in 98.7%

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

      3. associate--l+ 98.8%

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

      4. distribute-lft-neg-in 98.8%

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

      5. distribute-rgt-neg-in 98.8%

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

      6. *-commutative 98.8%

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

      7. fma-def 98.8%

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

      8. *-rgt-identity 98.8%

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

      9. distribute-lft-out-- 98.8%

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

      10. sub-neg 98.8%

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

      11. metadata-eval 98.8%

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

      12. +-commutative 98.8%

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

   6. Simplified 98.8%

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

   7. Taylor expanded in x around inf 98.9%

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

      1. sub-neg 98.9%

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

      2. metadata-eval 98.9%

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

      3. +-commutative 98.9%

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

      4. mul-1-neg 98.9%

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

      5. distribute-rgt-neg-out 98.9%

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

      6. +-commutative 98.9%

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

      7. fma-def 99.0%

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

      8. +-commutative 99.0%

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

      9. distribute-rgt-neg-out 99.0%

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

      10. *-commutative 99.0%

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

      11. distribute-rgt-neg-in 99.0%

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

   9. Simplified 99.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\sin \varepsilon, -\sin x, \cos x \cdot \cos \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 2.1 \cdot 10^{-5}:\\ \;\;\;\;-2 \cdot \left(\left(\sin x + 0.5 \cdot \left(\varepsilon \cdot \cos x\right)\right) \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, -1 + \cos \varepsilon, \sin x \cdot \left(-\sin \varepsilon\right)\right)\\ \end{array} \]

Alternative 4: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -7.6 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \left(-\cos x\right) - \sin x \cdot \sin \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 2.8 \cdot 10^{-5}:\\ \;\;\;\;-2 \cdot \left(\left(\sin x + 0.5 \cdot \left(\varepsilon \cdot \cos x\right)\right) \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, -1 + \cos \varepsilon, \sin x \cdot \left(-\sin \varepsilon\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -7.6e-5)
   (fma (cos x) (cos eps) (- (- (cos x)) (* (sin x) (sin eps))))
   (if (<= eps 2.8e-5)
     (* -2.0 (* (+ (sin x) (* 0.5 (* eps (cos x)))) (sin (* eps 0.5))))
     (fma (cos x) (+ -1.0 (cos eps)) (* (sin x) (- (sin eps)))))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -7.6e-5) {
		tmp = fma(cos(x), cos(eps), (-cos(x) - (sin(x) * sin(eps))));
	} else if (eps <= 2.8e-5) {
		tmp = -2.0 * ((sin(x) + (0.5 * (eps * cos(x)))) * sin((eps * 0.5)));
	} else {
		tmp = fma(cos(x), (-1.0 + cos(eps)), (sin(x) * -sin(eps)));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -7.6e-5)
		tmp = fma(cos(x), cos(eps), Float64(Float64(-cos(x)) - Float64(sin(x) * sin(eps))));
	elseif (eps <= 2.8e-5)
		tmp = Float64(-2.0 * Float64(Float64(sin(x) + Float64(0.5 * Float64(eps * cos(x)))) * sin(Float64(eps * 0.5))));
	else
		tmp = fma(cos(x), Float64(-1.0 + cos(eps)), Float64(sin(x) * Float64(-sin(eps))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -7.6e-5], N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + N[((-N[Cos[x], $MachinePrecision]) - N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 2.8e-5], N[(-2.0 * N[(N[(N[Sin[x], $MachinePrecision] + N[(0.5 * N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[x], $MachinePrecision] * N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * (-N[Sin[eps], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -7.6000000000000004e-5

   1. Initial program 48.6%

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

      1. sub-neg 48.6%

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

      2. cos-sum 98.8%

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

      3. associate-+l- 98.8%

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

      4. fma-neg 98.9%

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

   3. Applied egg-rr 98.9%

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

   if -7.6000000000000004e-5 < eps < 2.79999999999999996e-5

   1. Initial program 20.1%

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

      1. diff-cos 43.2%

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

      2. div-inv 43.2%

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

      3. metadata-eval 43.2%

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

      4. div-inv 43.2%

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

      5. +-commutative 43.2%

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

      6. metadata-eval 43.2%

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

   3. Applied egg-rr 43.2%

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

      1. *-commutative 43.2%

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

      2. +-commutative 43.2%

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

      3. associate--l+ 98.8%

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

      4. +-inverses 98.8%

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

      5. distribute-lft-in 98.8%

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

      6. metadata-eval 98.8%

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

      7. *-commutative 98.8%

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

      8. +-commutative 98.8%

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

   5. Simplified 98.8%

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

   6. Taylor expanded in eps around 0 99.7%

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

   if 2.79999999999999996e-5 < eps

   1. Initial program 42.1%

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

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

   3. Applied egg-rr 98.7%

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

   4. Taylor expanded in x around inf 98.7%

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

      1. neg-mul-1 98.7%

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

      2. distribute-lft-neg-in 98.7%

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

      3. associate--l+ 98.8%

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

      4. distribute-lft-neg-in 98.8%

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

      5. distribute-rgt-neg-in 98.8%

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

      6. *-commutative 98.8%

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

      7. fma-def 98.8%

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

      8. *-rgt-identity 98.8%

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

      9. distribute-lft-out-- 98.8%

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

      10. sub-neg 98.8%

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

      11. metadata-eval 98.8%

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

      12. +-commutative 98.8%

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

   6. Simplified 98.8%

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

   7. Taylor expanded in x around inf 98.9%

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

      1. sub-neg 98.9%

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

      2. metadata-eval 98.9%

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

      3. +-commutative 98.9%

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

      4. mul-1-neg 98.9%

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

      5. distribute-rgt-neg-out 98.9%

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

      6. +-commutative 98.9%

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

      7. fma-def 99.0%

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

      8. +-commutative 99.0%

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

      9. distribute-rgt-neg-out 99.0%

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

      10. *-commutative 99.0%

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

      11. distribute-rgt-neg-in 99.0%

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

   9. Simplified 99.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -7.6 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \left(-\cos x\right) - \sin x \cdot \sin \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 2.8 \cdot 10^{-5}:\\ \;\;\;\;-2 \cdot \left(\left(\sin x + 0.5 \cdot \left(\varepsilon \cdot \cos x\right)\right) \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, -1 + \cos \varepsilon, \sin x \cdot \left(-\sin \varepsilon\right)\right)\\ \end{array} \]

Alternative 5: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.7 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 2.7 \cdot 10^{-5}\right):\\ \;\;\;\;\mathsf{fma}\left(\cos x, -1 + \cos \varepsilon, \sin x \cdot \left(-\sin \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(\sin x + 0.5 \cdot \left(\varepsilon \cdot \cos x\right)\right) \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -2.7e-5) (not (<= eps 2.7e-5)))
   (fma (cos x) (+ -1.0 (cos eps)) (* (sin x) (- (sin eps))))
   (* -2.0 (* (+ (sin x) (* 0.5 (* eps (cos x)))) (sin (* eps 0.5))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -2.7e-5) || !(eps <= 2.7e-5)) {
		tmp = fma(cos(x), (-1.0 + cos(eps)), (sin(x) * -sin(eps)));
	} else {
		tmp = -2.0 * ((sin(x) + (0.5 * (eps * cos(x)))) * sin((eps * 0.5)));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -2.7e-5) || !(eps <= 2.7e-5))
		tmp = fma(cos(x), Float64(-1.0 + cos(eps)), Float64(sin(x) * Float64(-sin(eps))));
	else
		tmp = Float64(-2.0 * Float64(Float64(sin(x) + Float64(0.5 * Float64(eps * cos(x)))) * sin(Float64(eps * 0.5))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -2.7e-5], N[Not[LessEqual[eps, 2.7e-5]], $MachinePrecision]], N[(N[Cos[x], $MachinePrecision] * N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * (-N[Sin[eps], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[(N[Sin[x], $MachinePrecision] + N[(0.5 * N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -2.6999999999999999e-5 or 2.6999999999999999e-5 < eps

   1. Initial program 45.6%

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

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

   3. Applied egg-rr 98.7%

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

   4. Taylor expanded in x around inf 98.7%

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

      1. neg-mul-1 98.7%

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

      2. distribute-lft-neg-in 98.7%

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

      3. associate--l+ 98.8%

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

      4. distribute-lft-neg-in 98.8%

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

      5. distribute-rgt-neg-in 98.8%

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

      6. *-commutative 98.8%

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

      7. fma-def 98.8%

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

      8. *-rgt-identity 98.8%

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

      9. distribute-lft-out-- 98.8%

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

      10. sub-neg 98.8%

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

      11. metadata-eval 98.8%

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

      12. +-commutative 98.8%

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

   6. Simplified 98.8%

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

   7. Taylor expanded in x around inf 98.8%

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

      1. sub-neg 98.8%

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

      2. metadata-eval 98.8%

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

      3. +-commutative 98.8%

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

      4. mul-1-neg 98.8%

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

      5. distribute-rgt-neg-out 98.8%

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

      6. +-commutative 98.8%

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

      7. fma-def 98.9%

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

      8. +-commutative 98.9%

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

      9. distribute-rgt-neg-out 98.9%

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

      10. *-commutative 98.9%

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

      11. distribute-rgt-neg-in 98.9%

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

   9. Simplified 98.9%

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

   if -2.6999999999999999e-5 < eps < 2.6999999999999999e-5

   1. Initial program 20.1%

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

      1. diff-cos 43.2%

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

      2. div-inv 43.2%

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

      3. metadata-eval 43.2%

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

      4. div-inv 43.2%

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

      5. +-commutative 43.2%

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

      6. metadata-eval 43.2%

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

   3. Applied egg-rr 43.2%

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

      1. *-commutative 43.2%

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

      2. +-commutative 43.2%

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

      3. associate--l+ 98.8%

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

      4. +-inverses 98.8%

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

      5. distribute-lft-in 98.8%

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

      6. metadata-eval 98.8%

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

      7. *-commutative 98.8%

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

      8. +-commutative 98.8%

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

   5. Simplified 98.8%

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

   6. Taylor expanded in eps around 0 99.7%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.7 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 2.7 \cdot 10^{-5}\right):\\ \;\;\;\;\mathsf{fma}\left(\cos x, -1 + \cos \varepsilon, \sin x \cdot \left(-\sin \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(\sin x + 0.5 \cdot \left(\varepsilon \cdot \cos x\right)\right) \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 6: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.05 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 2.75 \cdot 10^{-5}\right):\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - \sin x \cdot \sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(\sin x + 0.5 \cdot \left(\varepsilon \cdot \cos x\right)\right) \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -2.05e-5) (not (<= eps 2.75e-5)))
   (- (* (cos x) (+ -1.0 (cos eps))) (* (sin x) (sin eps)))
   (* -2.0 (* (+ (sin x) (* 0.5 (* eps (cos x)))) (sin (* eps 0.5))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -2.05e-5) || !(eps <= 2.75e-5)) {
		tmp = (cos(x) * (-1.0 + cos(eps))) - (sin(x) * sin(eps));
	} else {
		tmp = -2.0 * ((sin(x) + (0.5 * (eps * cos(x)))) * sin((eps * 0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-2.05d-5)) .or. (.not. (eps <= 2.75d-5))) then
        tmp = (cos(x) * ((-1.0d0) + cos(eps))) - (sin(x) * sin(eps))
    else
        tmp = (-2.0d0) * ((sin(x) + (0.5d0 * (eps * cos(x)))) * sin((eps * 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -2.05e-5) || !(eps <= 2.75e-5)) {
		tmp = (Math.cos(x) * (-1.0 + Math.cos(eps))) - (Math.sin(x) * Math.sin(eps));
	} else {
		tmp = -2.0 * ((Math.sin(x) + (0.5 * (eps * Math.cos(x)))) * Math.sin((eps * 0.5)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -2.05e-5) or not (eps <= 2.75e-5):
		tmp = (math.cos(x) * (-1.0 + math.cos(eps))) - (math.sin(x) * math.sin(eps))
	else:
		tmp = -2.0 * ((math.sin(x) + (0.5 * (eps * math.cos(x)))) * math.sin((eps * 0.5)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -2.05e-5) || !(eps <= 2.75e-5))
		tmp = Float64(Float64(cos(x) * Float64(-1.0 + cos(eps))) - Float64(sin(x) * sin(eps)));
	else
		tmp = Float64(-2.0 * Float64(Float64(sin(x) + Float64(0.5 * Float64(eps * cos(x)))) * sin(Float64(eps * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -2.05e-5) || ~((eps <= 2.75e-5)))
		tmp = (cos(x) * (-1.0 + cos(eps))) - (sin(x) * sin(eps));
	else
		tmp = -2.0 * ((sin(x) + (0.5 * (eps * cos(x)))) * sin((eps * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -2.05e-5], N[Not[LessEqual[eps, 2.75e-5]], $MachinePrecision]], N[(N[(N[Cos[x], $MachinePrecision] * N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[(N[Sin[x], $MachinePrecision] + N[(0.5 * N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -2.05000000000000002e-5 or 2.7500000000000001e-5 < eps

   1. Initial program 45.6%

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

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

   3. Applied egg-rr 98.7%

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

   4. Taylor expanded in x around inf 98.7%

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

      1. neg-mul-1 98.7%

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

      2. distribute-lft-neg-in 98.7%

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

      3. associate--l+ 98.8%

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

      4. distribute-lft-neg-in 98.8%

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

      5. distribute-rgt-neg-in 98.8%

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

      6. *-commutative 98.8%

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

      7. fma-def 98.8%

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

      8. *-rgt-identity 98.8%

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

      9. distribute-lft-out-- 98.8%

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

      10. sub-neg 98.8%

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

      11. metadata-eval 98.8%

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

      12. +-commutative 98.8%

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

   6. Simplified 98.8%

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

      1. fma-udef 98.8%

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

   8. Applied egg-rr 98.8%

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

   if -2.05000000000000002e-5 < eps < 2.7500000000000001e-5

   1. Initial program 20.1%

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

      1. diff-cos 43.2%

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

      2. div-inv 43.2%

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

      3. metadata-eval 43.2%

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

      4. div-inv 43.2%

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

      5. +-commutative 43.2%

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

      6. metadata-eval 43.2%

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

   3. Applied egg-rr 43.2%

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

      1. *-commutative 43.2%

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

      2. +-commutative 43.2%

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

      3. associate--l+ 98.8%

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

      4. +-inverses 98.8%

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

      5. distribute-lft-in 98.8%

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

      6. metadata-eval 98.8%

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

      7. *-commutative 98.8%

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

      8. +-commutative 98.8%

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

   5. Simplified 98.8%

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

   6. Taylor expanded in eps around 0 99.7%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.05 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 2.75 \cdot 10^{-5}\right):\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - \sin x \cdot \sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(\sin x + 0.5 \cdot \left(\varepsilon \cdot \cos x\right)\right) \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 7: 65.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(x + \varepsilon\right) - \cos x \leq -1 \cdot 10^{-12}:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- (cos (+ x eps)) (cos x)) -1e-12)
   (- (cos eps) (cos x))
   (* (sin x) (- eps))))

C

double code(double x, double eps) {
	double tmp;
	if ((cos((x + eps)) - cos(x)) <= -1e-12) {
		tmp = cos(eps) - cos(x);
	} else {
		tmp = sin(x) * -eps;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((cos((x + eps)) - cos(x)) <= (-1d-12)) then
        tmp = cos(eps) - cos(x)
    else
        tmp = sin(x) * -eps
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((Math.cos((x + eps)) - Math.cos(x)) <= -1e-12) {
		tmp = Math.cos(eps) - Math.cos(x);
	} else {
		tmp = Math.sin(x) * -eps;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (math.cos((x + eps)) - math.cos(x)) <= -1e-12:
		tmp = math.cos(eps) - math.cos(x)
	else:
		tmp = math.sin(x) * -eps
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(cos(Float64(x + eps)) - cos(x)) <= -1e-12)
		tmp = Float64(cos(eps) - cos(x));
	else
		tmp = Float64(sin(x) * Float64(-eps));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((cos((x + eps)) - cos(x)) <= -1e-12)
		tmp = cos(eps) - cos(x);
	else
		tmp = sin(x) * -eps;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(N[Cos[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], -1e-12], N[(N[Cos[eps], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * (-eps)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cos.f64 (+.f64 x eps)) (cos.f64 x)) < -9.9999999999999998e-13

   1. Initial program 72.5%

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

   2. Taylor expanded in x around 0 73.0%

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

   if -9.9999999999999998e-13 < (-.f64 (cos.f64 (+.f64 x eps)) (cos.f64 x))

   1. Initial program 16.6%

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

   2. Taylor expanded in eps around 0 54.6%

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

      1. associate-*r* 54.6%

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

      2. mul-1-neg 54.6%

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

   4. Simplified 54.6%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(x + \varepsilon\right) - \cos x \leq -1 \cdot 10^{-12}:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \end{array} \]

Alternative 8: 77.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -200 \lor \neg \left(\varepsilon \leq 165000\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) - \sin x \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -200.0) (not (<= eps 165000.0)))
   (- (cos eps) (cos x))
   (- (* -0.5 (* eps (* eps (cos x)))) (* (sin x) eps))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -200.0) || !(eps <= 165000.0)) {
		tmp = cos(eps) - cos(x);
	} else {
		tmp = (-0.5 * (eps * (eps * cos(x)))) - (sin(x) * eps);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-200.0d0)) .or. (.not. (eps <= 165000.0d0))) then
        tmp = cos(eps) - cos(x)
    else
        tmp = ((-0.5d0) * (eps * (eps * cos(x)))) - (sin(x) * eps)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -200.0) || !(eps <= 165000.0)) {
		tmp = Math.cos(eps) - Math.cos(x);
	} else {
		tmp = (-0.5 * (eps * (eps * Math.cos(x)))) - (Math.sin(x) * eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -200.0) or not (eps <= 165000.0):
		tmp = math.cos(eps) - math.cos(x)
	else:
		tmp = (-0.5 * (eps * (eps * math.cos(x)))) - (math.sin(x) * eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -200.0) || !(eps <= 165000.0))
		tmp = Float64(cos(eps) - cos(x));
	else
		tmp = Float64(Float64(-0.5 * Float64(eps * Float64(eps * cos(x)))) - Float64(sin(x) * eps));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -200.0) || ~((eps <= 165000.0)))
		tmp = cos(eps) - cos(x);
	else
		tmp = (-0.5 * (eps * (eps * cos(x)))) - (sin(x) * eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -200.0], N[Not[LessEqual[eps, 165000.0]], $MachinePrecision]], N[(N[Cos[eps], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(eps * N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -200 or 165000 < eps

   1. Initial program 47.1%

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

   2. Taylor expanded in x around 0 49.0%

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

   if -200 < eps < 165000

   1. Initial program 19.7%

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

   2. Taylor expanded in eps around 0 97.0%

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

      1. mul-1-neg 97.0%

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

      2. unsub-neg 97.0%

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

      3. unpow2 97.0%

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

      4. associate-*l* 97.0%

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

   4. Simplified 97.0%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -200 \lor \neg \left(\varepsilon \leq 165000\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) - \sin x \cdot \varepsilon\\ \end{array} \]

Alternative 9: 76.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.0%

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

   1. diff-cos 43.3%

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

   2. div-inv 43.3%

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

   3. metadata-eval 43.3%

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

   4. div-inv 43.3%

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

   5. +-commutative 43.3%

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

   6. metadata-eval 43.3%

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

3. Applied egg-rr 43.3%

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

   1. *-commutative 43.3%

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

   2. +-commutative 43.3%

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

   3. associate--l+ 72.0%

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

   4. +-inverses 72.0%

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

   5. distribute-lft-in 72.0%

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

   6. metadata-eval 72.0%

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

   7. *-commutative 72.0%

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

   8. +-commutative 72.0%

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

5. Simplified 72.0%

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

6. Taylor expanded in x around -inf 72.0%

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

7. Final simplification 72.0%

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

Alternative 10: 76.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.0%

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

   1. diff-cos 43.3%

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

   2. div-inv 43.3%

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

   3. metadata-eval 43.3%

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

   4. div-inv 43.3%

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

   5. +-commutative 43.3%

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

   6. metadata-eval 43.3%

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

3. Applied egg-rr 43.3%

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

   1. *-commutative 43.3%

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

   2. +-commutative 43.3%

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

   3. associate--l+ 72.0%

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

   4. +-inverses 72.0%

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

   5. distribute-lft-in 72.0%

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

   6. metadata-eval 72.0%

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

   7. *-commutative 72.0%

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

   8. +-commutative 72.0%

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

5. Simplified 72.0%

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

   1. add-exp-log 50.2%

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

   2. *-commutative 50.2%

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

   3. +-commutative 50.2%

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

   4. +-rgt-identity 50.2%

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

7. Applied egg-rr 50.2%

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

   1. add-exp-log 72.0%

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

   2. *-un-lft-identity 72.0%

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

   3. +-commutative 72.0%

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

   4. *-commutative 72.0%

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

9. Applied egg-rr 72.0%

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

    1. *-lft-identity 72.0%

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

    2. *-commutative 72.0%

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

11. Simplified 72.0%

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

12. Final simplification 72.0%

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

Alternative 11: 70.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\varepsilon \cdot 0.5\right)\\ \mathbf{if}\;x \leq -2.65 \cdot 10^{-51} \lor \neg \left(x \leq 6.2 \cdot 10^{-26}\right):\\ \;\;\;\;-2 \cdot \left(\sin x \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {t_0}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (* eps 0.5))))
   (if (or (<= x -2.65e-51) (not (<= x 6.2e-26)))
     (* -2.0 (* (sin x) 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 <= -2.65e-51) || !(x <= 6.2e-26)) {
		tmp = -2.0 * (sin(x) * 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 <= (-2.65d-51)) .or. (.not. (x <= 6.2d-26))) then
        tmp = (-2.0d0) * (sin(x) * 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 <= -2.65e-51) || !(x <= 6.2e-26)) {
		tmp = -2.0 * (Math.sin(x) * 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 <= -2.65e-51) or not (x <= 6.2e-26):
		tmp = -2.0 * (math.sin(x) * 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 <= -2.65e-51) || !(x <= 6.2e-26))
		tmp = Float64(-2.0 * Float64(sin(x) * 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 <= -2.65e-51) || ~((x <= 6.2e-26)))
		tmp = -2.0 * (sin(x) * 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, -2.65e-51], N[Not[LessEqual[x, 6.2e-26]], $MachinePrecision]], N[(-2.0 * N[(N[Sin[x], $MachinePrecision] * 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 < -2.64999999999999987e-51 or 6.19999999999999966e-26 < x

   1. Initial program 12.4%

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

      1. diff-cos 10.7%

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

      2. div-inv 10.7%

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

      3. metadata-eval 10.7%

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

      4. div-inv 10.7%

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

      5. +-commutative 10.7%

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

      6. metadata-eval 10.7%

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

   3. Applied egg-rr 10.7%

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

      1. *-commutative 10.7%

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

      2. +-commutative 10.7%

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

      3. associate--l+ 54.4%

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

      4. +-inverses 54.4%

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

      5. distribute-lft-in 54.4%

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

      6. metadata-eval 54.4%

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

      7. *-commutative 54.4%

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

      8. +-commutative 54.4%

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

   5. Simplified 54.4%

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

   6. Taylor expanded in eps around 0 51.4%

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

   if -2.64999999999999987e-51 < x < 6.19999999999999966e-26

   1. Initial program 65.0%

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

      1. diff-cos 94.3%

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

      2. div-inv 94.3%

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

      3. metadata-eval 94.3%

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

      4. div-inv 94.3%

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

      5. +-commutative 94.3%

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

      6. metadata-eval 94.3%

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

   3. Applied egg-rr 94.3%

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

      1. *-commutative 94.3%

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

      2. +-commutative 94.3%

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

      3. associate--l+ 99.6%

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

      4. +-inverses 99.6%

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

      5. distribute-lft-in 99.6%

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

      6. metadata-eval 99.6%

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

      7. *-commutative 99.6%

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

      8. +-commutative 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in x around 0 92.4%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.65 \cdot 10^{-51} \lor \neg \left(x \leq 6.2 \cdot 10^{-26}\right):\\ \;\;\;\;-2 \cdot \left(\sin x \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array} \]

Alternative 12: 68.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{-52} \lor \neg \left(x \leq 7.2 \cdot 10^{-27}\right):\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -4.6e-52) (not (<= x 7.2e-27)))
   (* (sin x) (- eps))
   (* -2.0 (pow (sin (* eps 0.5)) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -4.6e-52) || !(x <= 7.2e-27)) {
		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 <= (-4.6d-52)) .or. (.not. (x <= 7.2d-27))) 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 <= -4.6e-52) || !(x <= 7.2e-27)) {
		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 <= -4.6e-52) or not (x <= 7.2e-27):
		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 <= -4.6e-52) || !(x <= 7.2e-27))
		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 <= -4.6e-52) || ~((x <= 7.2e-27)))
		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, -4.6e-52], N[Not[LessEqual[x, 7.2e-27]], $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 < -4.59999999999999989e-52 or 7.1999999999999997e-27 < x

   1. Initial program 12.4%

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

   2. Taylor expanded in eps around 0 47.9%

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

      1. associate-*r* 47.9%

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

      2. mul-1-neg 47.9%

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

   4. Simplified 47.9%

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

   if -4.59999999999999989e-52 < x < 7.1999999999999997e-27

   1. Initial program 65.0%

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

      1. diff-cos 94.3%

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

      2. div-inv 94.3%

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

      3. metadata-eval 94.3%

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

      4. div-inv 94.3%

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

      5. +-commutative 94.3%

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

      6. metadata-eval 94.3%

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

   3. Applied egg-rr 94.3%

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

      1. *-commutative 94.3%

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

      2. +-commutative 94.3%

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

      3. associate--l+ 99.6%

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

      4. +-inverses 99.6%

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

      5. distribute-lft-in 99.6%

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

      6. metadata-eval 99.6%

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

      7. *-commutative 99.6%

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

      8. +-commutative 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in x around 0 92.4%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{-52} \lor \neg \left(x \leq 7.2 \cdot 10^{-27}\right):\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array} \]

Alternative 13: 65.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 + \cos \varepsilon\\ t_1 := \sin x \cdot \left(-\varepsilon\right)\\ \mathbf{if}\;\varepsilon \leq -0.00285:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 5.2 \cdot 10^{-111}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq 5.5 \cdot 10^{-36}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 4.8 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ -1.0 (cos eps))) (t_1 (* (sin x) (- eps))))
   (if (<= eps -0.00285)
     t_0
     (if (<= eps 5.2e-111)
       t_1
       (if (<= eps 5.5e-36)
         (* -0.5 (* eps eps))
         (if (<= eps 4.8e-6) t_1 t_0))))))

C

double code(double x, double eps) {
	double t_0 = -1.0 + cos(eps);
	double t_1 = sin(x) * -eps;
	double tmp;
	if (eps <= -0.00285) {
		tmp = t_0;
	} else if (eps <= 5.2e-111) {
		tmp = t_1;
	} else if (eps <= 5.5e-36) {
		tmp = -0.5 * (eps * eps);
	} else if (eps <= 4.8e-6) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (-1.0d0) + cos(eps)
    t_1 = sin(x) * -eps
    if (eps <= (-0.00285d0)) then
        tmp = t_0
    else if (eps <= 5.2d-111) then
        tmp = t_1
    else if (eps <= 5.5d-36) then
        tmp = (-0.5d0) * (eps * eps)
    else if (eps <= 4.8d-6) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = -1.0 + Math.cos(eps);
	double t_1 = Math.sin(x) * -eps;
	double tmp;
	if (eps <= -0.00285) {
		tmp = t_0;
	} else if (eps <= 5.2e-111) {
		tmp = t_1;
	} else if (eps <= 5.5e-36) {
		tmp = -0.5 * (eps * eps);
	} else if (eps <= 4.8e-6) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = -1.0 + math.cos(eps)
	t_1 = math.sin(x) * -eps
	tmp = 0
	if eps <= -0.00285:
		tmp = t_0
	elif eps <= 5.2e-111:
		tmp = t_1
	elif eps <= 5.5e-36:
		tmp = -0.5 * (eps * eps)
	elif eps <= 4.8e-6:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(-1.0 + cos(eps))
	t_1 = Float64(sin(x) * Float64(-eps))
	tmp = 0.0
	if (eps <= -0.00285)
		tmp = t_0;
	elseif (eps <= 5.2e-111)
		tmp = t_1;
	elseif (eps <= 5.5e-36)
		tmp = Float64(-0.5 * Float64(eps * eps));
	elseif (eps <= 4.8e-6)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = -1.0 + cos(eps);
	t_1 = sin(x) * -eps;
	tmp = 0.0;
	if (eps <= -0.00285)
		tmp = t_0;
	elseif (eps <= 5.2e-111)
		tmp = t_1;
	elseif (eps <= 5.5e-36)
		tmp = -0.5 * (eps * eps);
	elseif (eps <= 4.8e-6)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[x], $MachinePrecision] * (-eps)), $MachinePrecision]}, If[LessEqual[eps, -0.00285], t$95$0, If[LessEqual[eps, 5.2e-111], t$95$1, If[LessEqual[eps, 5.5e-36], N[(-0.5 * N[(eps * eps), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 4.8e-6], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.0028500000000000001 or 4.7999999999999998e-6 < eps

   1. Initial program 46.3%

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

   2. Taylor expanded in x around 0 46.4%

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

   if -0.0028500000000000001 < eps < 5.19999999999999965e-111 or 5.49999999999999984e-36 < eps < 4.7999999999999998e-6

   1. Initial program 22.2%

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

   2. Taylor expanded in eps around 0 81.4%

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

      1. associate-*r* 81.4%

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

      2. mul-1-neg 81.4%

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

   4. Simplified 81.4%

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

   if 5.19999999999999965e-111 < eps < 5.49999999999999984e-36

   1. Initial program 4.4%

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

   2. Taylor expanded in x around 0 4.4%

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

   3. Taylor expanded in eps around 0 66.8%

      \[\leadsto \color{blue}{-0.5 \cdot {\varepsilon}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 66.8%

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

   5. Simplified 66.8%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00285:\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 5.2 \cdot 10^{-111}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 5.5 \cdot 10^{-36}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 4.8 \cdot 10^{-6}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \cos \varepsilon\\ \end{array} \]

Alternative 14: 46.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00285 \lor \neg \left(\varepsilon \leq 4.3 \cdot 10^{-13}\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.00285) (not (<= eps 4.3e-13)))
   (+ -1.0 (cos eps))
   (* -0.5 (* eps eps))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.00285) || !(eps <= 4.3e-13)) {
		tmp = -1.0 + cos(eps);
	} else {
		tmp = -0.5 * (eps * eps);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-0.00285d0)) .or. (.not. (eps <= 4.3d-13))) then
        tmp = (-1.0d0) + cos(eps)
    else
        tmp = (-0.5d0) * (eps * eps)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.00285) || !(eps <= 4.3e-13)) {
		tmp = -1.0 + Math.cos(eps);
	} else {
		tmp = -0.5 * (eps * eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.00285) or not (eps <= 4.3e-13):
		tmp = -1.0 + math.cos(eps)
	else:
		tmp = -0.5 * (eps * eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.00285) || !(eps <= 4.3e-13))
		tmp = Float64(-1.0 + cos(eps));
	else
		tmp = Float64(-0.5 * Float64(eps * eps));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.00285) || ~((eps <= 4.3e-13)))
		tmp = -1.0 + cos(eps);
	else
		tmp = -0.5 * (eps * eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.00285], N[Not[LessEqual[eps, 4.3e-13]], $MachinePrecision]], N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(eps * eps), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -0.0028500000000000001 or 4.2999999999999999e-13 < eps

   1. Initial program 45.3%

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

   2. Taylor expanded in x around 0 45.5%

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

   if -0.0028500000000000001 < eps < 4.2999999999999999e-13

   1. Initial program 20.2%

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

   2. Taylor expanded in x around 0 20.3%

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

   3. Taylor expanded in eps around 0 42.2%

      \[\leadsto \color{blue}{-0.5 \cdot {\varepsilon}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 42.2%

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

   5. Simplified 42.2%

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

4. Final simplification 43.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00285 \lor \neg \left(\varepsilon \leq 4.3 \cdot 10^{-13}\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \end{array} \]

Alternative 15: 22.4% accurate, 41.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.0%

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

2. Taylor expanded in x around 0 33.1%

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

3. Taylor expanded in eps around 0 22.4%

   \[\leadsto \color{blue}{-0.5 \cdot {\varepsilon}^{2}} \]
4. Step-by-step derivation

   1. unpow2 22.4%

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

5. Simplified 22.4%

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

6. Final simplification 22.4%

   \[\leadsto -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) \]

Reproduce

herbie shell --seed 2023173 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))