2cos (problem 3.3.5)

Percentage Accurate: 37.5% → 99.2%

Time: 20.4s

Alternatives: 12

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: 37.5% 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: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00015:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.000125:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos \varepsilon + -1, \cos x, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.00015)
   (- (* (cos x) (cos eps)) (fma (sin eps) (sin x) (cos x)))
   (if (<= eps 0.000125)
     (- (* (cos x) (* -0.5 (pow eps 2.0))) (* (sin eps) (sin x)))
     (fma (+ (cos eps) -1.0) (cos x) (* (sin eps) (- (sin x)))))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.00015) {
		tmp = (cos(x) * cos(eps)) - fma(sin(eps), sin(x), cos(x));
	} else if (eps <= 0.000125) {
		tmp = (cos(x) * (-0.5 * pow(eps, 2.0))) - (sin(eps) * sin(x));
	} else {
		tmp = fma((cos(eps) + -1.0), cos(x), (sin(eps) * -sin(x)));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -0.00015)
		tmp = Float64(Float64(cos(x) * cos(eps)) - fma(sin(eps), sin(x), cos(x)));
	elseif (eps <= 0.000125)
		tmp = Float64(Float64(cos(x) * Float64(-0.5 * (eps ^ 2.0))) - Float64(sin(eps) * sin(x)));
	else
		tmp = fma(Float64(cos(eps) + -1.0), cos(x), Float64(sin(eps) * Float64(-sin(x))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -0.00015], N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision] + N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.000125], N[(N[(N[Cos[x], $MachinePrecision] * N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[Sin[eps], $MachinePrecision] * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -1.49999999999999987e-4

   1. Initial program 48.9%

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

      1. sub-neg 48.9%

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

      2. cos-sum 98.7%

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

      3. cancel-sign-sub-inv 98.7%

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

      4. associate-+l+ 98.7%

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

      5. *-commutative 98.7%

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

   3. Applied egg-rr 98.7%

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

      1. +-commutative 98.7%

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

      2. distribute-rgt-neg-out 98.7%

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

      3. *-commutative 98.7%

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

      4. unsub-neg 98.7%

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

      5. associate-+r- 98.6%

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

      6. *-commutative 98.6%

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

      7. neg-mul-1 98.6%

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

      8. distribute-rgt-out 98.7%

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

      9. *-commutative 98.7%

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

   5. Simplified 98.7%

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

      1. sub-neg 98.7%

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

      2. +-commutative 98.7%

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

      3. distribute-rgt-in 98.6%

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

      4. *-commutative 98.6%

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

      5. mul-1-neg 98.6%

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

      6. associate-+r+ 98.7%

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

      7. +-commutative 98.7%

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

      8. sub-neg 98.7%

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

      9. *-commutative 98.7%

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

      10. sub-neg 98.7%

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

      11. *-commutative 98.7%

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

      12. associate--l- 98.7%

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

      13. sub-neg 98.7%

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

      14. *-commutative 98.7%

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

   7. Applied egg-rr 98.8%

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

   if -1.49999999999999987e-4 < eps < 1.25e-4

   1. Initial program 24.4%

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

      1. sub-neg 24.4%

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

      2. cos-sum 25.4%

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

      3. cancel-sign-sub-inv 25.4%

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

      4. associate-+l+ 25.4%

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

      5. *-commutative 25.4%

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

   3. Applied egg-rr 25.4%

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

      1. +-commutative 25.4%

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

      2. distribute-rgt-neg-out 25.4%

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

      3. *-commutative 25.4%

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

      4. unsub-neg 25.4%

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

      5. associate-+r- 77.6%

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

      6. *-commutative 77.6%

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

      7. neg-mul-1 77.6%

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

      8. distribute-rgt-out 77.6%

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

      9. *-commutative 77.6%

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

   5. Simplified 77.6%

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

   6. Taylor expanded in eps around 0 99.8%

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

   if 1.25e-4 < eps

   1. Initial program 55.1%

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

      1. sub-neg 55.1%

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

      2. cos-sum 98.9%

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

      3. cancel-sign-sub-inv 98.9%

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

      4. associate-+l+ 98.8%

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

      5. *-commutative 98.8%

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

   3. Applied egg-rr 98.8%

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

      1. +-commutative 98.8%

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

      2. distribute-rgt-neg-out 98.8%

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

      3. *-commutative 98.8%

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

      4. unsub-neg 98.8%

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

      5. associate-+r- 98.9%

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

      6. *-commutative 98.9%

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

      7. neg-mul-1 98.9%

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

      8. distribute-rgt-out 99.1%

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

      9. *-commutative 99.1%

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

   5. Simplified 99.1%

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

      1. *-commutative 99.1%

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

      2. fma-neg 99.1%

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

      3. *-commutative 99.1%

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

      4. distribute-lft-neg-in 99.1%

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

      5. *-commutative 99.1%

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

   7. Applied egg-rr 99.1%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00015:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.000125:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos \varepsilon + -1, \cos x, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \end{array} \]

Alternative 2: 99.0% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.6%

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

   1. sub-neg 38.6%

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

   2. cos-sum 63.3%

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

   3. cancel-sign-sub-inv 63.3%

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

   4. associate-+l+ 63.2%

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

   5. *-commutative 63.2%

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

3. Applied egg-rr 63.2%

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

   1. +-commutative 63.2%

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

   2. distribute-rgt-neg-out 63.2%

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

   3. *-commutative 63.2%

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

   4. unsub-neg 63.2%

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

   5. associate-+r- 88.5%

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

   6. *-commutative 88.5%

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

   7. neg-mul-1 88.5%

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

   8. distribute-rgt-out 88.6%

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

   9. *-commutative 88.6%

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

5. Simplified 88.6%

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

   1. *-commutative 88.6%

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

   2. metadata-eval 88.6%

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

   3. sub-neg 88.6%

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

   4. flip-- 88.1%

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

   5. metadata-eval 88.1%

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

   6. metadata-eval 88.1%

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

   7. associate-*l/ 88.1%

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

   8. metadata-eval 88.1%

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

   9. sub-1-cos 99.1%

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

   10. pow2 99.1%

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

7. Applied egg-rr 99.1%

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

8. Final simplification 99.1%

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

Alternative 3: 99.0% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.6%

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

   1. sub-neg 38.6%

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

   2. cos-sum 63.3%

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

   3. cancel-sign-sub-inv 63.3%

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

   4. associate-+l+ 63.2%

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

   5. *-commutative 63.2%

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

3. Applied egg-rr 63.2%

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

   1. +-commutative 63.2%

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

   2. distribute-rgt-neg-out 63.2%

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

   3. *-commutative 63.2%

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

   4. unsub-neg 63.2%

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

   5. associate-+r- 88.5%

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

   6. *-commutative 88.5%

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

   7. neg-mul-1 88.5%

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

   8. distribute-rgt-out 88.6%

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

   9. *-commutative 88.6%

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

5. Simplified 88.6%

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

   1. metadata-eval 88.6%

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

   2. sub-neg 88.6%

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

   3. flip-- 88.1%

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

   4. metadata-eval 88.1%

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

   5. metadata-eval 88.1%

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

   6. frac-2neg 88.1%

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

   7. metadata-eval 88.1%

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

   8. sub-1-cos 99.0%

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

   9. pow2 99.0%

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

7. Applied egg-rr 99.0%

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

   1. remove-double-neg 99.0%

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

   2. neg-sub0 99.0%

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

   3. +-commutative 99.0%

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

   4. associate--r+ 99.0%

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

   5. metadata-eval 99.0%

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

9. Simplified 99.0%

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

10. Final simplification 99.0%

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

Alternative 4: 99.2% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 51.9%

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

      1. sub-neg 51.9%

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

      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. cancel-sign-sub-inv 98.8%

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

      4. associate-+l+ 98.8%

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

      5. *-commutative 98.8%

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

   3. Applied egg-rr 98.8%

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

      1. +-commutative 98.8%

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

      2. distribute-rgt-neg-out 98.8%

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

      3. *-commutative 98.8%

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

      4. unsub-neg 98.8%

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

      5. associate-+r- 98.8%

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

      6. *-commutative 98.8%

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

      7. neg-mul-1 98.8%

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

      8. distribute-rgt-out 98.9%

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

      9. *-commutative 98.9%

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

   5. Simplified 98.9%

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

      1. *-commutative 98.9%

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

      2. fma-neg 98.9%

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

      3. *-commutative 98.9%

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

      4. distribute-lft-neg-in 98.9%

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

      5. *-commutative 98.9%

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

   7. Applied egg-rr 98.9%

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

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

   1. Initial program 24.4%

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

      1. sub-neg 24.4%

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

      2. cos-sum 25.4%

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

      3. cancel-sign-sub-inv 25.4%

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

      4. associate-+l+ 25.4%

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

      5. *-commutative 25.4%

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

   3. Applied egg-rr 25.4%

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

      1. +-commutative 25.4%

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

      2. distribute-rgt-neg-out 25.4%

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

      3. *-commutative 25.4%

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

      4. unsub-neg 25.4%

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

      5. associate-+r- 77.6%

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

      6. *-commutative 77.6%

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

      7. neg-mul-1 77.6%

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

      8. distribute-rgt-out 77.6%

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

      9. *-commutative 77.6%

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

   5. Simplified 77.6%

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

   6. Taylor expanded in eps around 0 99.8%

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

4. Final simplification 99.4%

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

Alternative 5: 99.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(eps) * sin(x)
    if ((eps <= (-0.000125d0)) .or. (.not. (eps <= 0.000125d0))) then
        tmp = (cos(x) * (cos(eps) + (-1.0d0))) - t_0
    else
        tmp = (cos(x) * ((-0.5d0) * (eps ** 2.0d0))) - t_0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 51.9%

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

      1. sub-neg 51.9%

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

      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. cancel-sign-sub-inv 98.8%

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

      4. associate-+l+ 98.8%

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

      5. *-commutative 98.8%

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

   3. Applied egg-rr 98.8%

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

      1. +-commutative 98.8%

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

      2. distribute-rgt-neg-out 98.8%

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

      3. *-commutative 98.8%

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

      4. unsub-neg 98.8%

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

      5. associate-+r- 98.8%

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

      6. *-commutative 98.8%

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

      7. neg-mul-1 98.8%

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

      8. distribute-rgt-out 98.9%

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

      9. *-commutative 98.9%

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

   5. Simplified 98.9%

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

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

   1. Initial program 24.4%

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

      1. sub-neg 24.4%

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

      2. cos-sum 25.4%

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

      3. cancel-sign-sub-inv 25.4%

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

      4. associate-+l+ 25.4%

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

      5. *-commutative 25.4%

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

   3. Applied egg-rr 25.4%

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

      1. +-commutative 25.4%

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

      2. distribute-rgt-neg-out 25.4%

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

      3. *-commutative 25.4%

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

      4. unsub-neg 25.4%

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

      5. associate-+r- 77.6%

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

      6. *-commutative 77.6%

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

      7. neg-mul-1 77.6%

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

      8. distribute-rgt-out 77.6%

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

      9. *-commutative 77.6%

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

   5. Simplified 77.6%

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

   6. Taylor expanded in eps around 0 99.8%

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

4. Final simplification 99.3%

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

Alternative 6: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 4.8 \cdot 10^{-5}\right):\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - \varepsilon \cdot \sin x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -3e-5) (not (<= eps 4.8e-5)))
   (- (* (cos x) (+ (cos eps) -1.0)) (* (sin eps) (sin x)))
   (- (* (cos x) (* -0.5 (pow eps 2.0))) (* eps (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -3e-5) || !(eps <= 4.8e-5)) {
		tmp = (cos(x) * (cos(eps) + -1.0)) - (sin(eps) * sin(x));
	} else {
		tmp = (cos(x) * (-0.5 * pow(eps, 2.0))) - (eps * sin(x));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-3d-5)) .or. (.not. (eps <= 4.8d-5))) then
        tmp = (cos(x) * (cos(eps) + (-1.0d0))) - (sin(eps) * sin(x))
    else
        tmp = (cos(x) * ((-0.5d0) * (eps ** 2.0d0))) - (eps * sin(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -3e-5) || !(eps <= 4.8e-5)) {
		tmp = (Math.cos(x) * (Math.cos(eps) + -1.0)) - (Math.sin(eps) * Math.sin(x));
	} else {
		tmp = (Math.cos(x) * (-0.5 * Math.pow(eps, 2.0))) - (eps * Math.sin(x));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -3e-5) or not (eps <= 4.8e-5):
		tmp = (math.cos(x) * (math.cos(eps) + -1.0)) - (math.sin(eps) * math.sin(x))
	else:
		tmp = (math.cos(x) * (-0.5 * math.pow(eps, 2.0))) - (eps * math.sin(x))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -3e-5) || !(eps <= 4.8e-5))
		tmp = Float64(Float64(cos(x) * Float64(cos(eps) + -1.0)) - Float64(sin(eps) * sin(x)));
	else
		tmp = Float64(Float64(cos(x) * Float64(-0.5 * (eps ^ 2.0))) - Float64(eps * sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -3e-5) || ~((eps <= 4.8e-5)))
		tmp = (cos(x) * (cos(eps) + -1.0)) - (sin(eps) * sin(x));
	else
		tmp = (cos(x) * (-0.5 * (eps ^ 2.0))) - (eps * sin(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -3e-5], N[Not[LessEqual[eps, 4.8e-5]], $MachinePrecision]], N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[x], $MachinePrecision] * N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -3.00000000000000008e-5 or 4.8000000000000001e-5 < eps

   1. Initial program 51.9%

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

      1. sub-neg 51.9%

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

      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. cancel-sign-sub-inv 98.8%

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

      4. associate-+l+ 98.8%

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

      5. *-commutative 98.8%

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

   3. Applied egg-rr 98.8%

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

      1. +-commutative 98.8%

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

      2. distribute-rgt-neg-out 98.8%

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

      3. *-commutative 98.8%

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

      4. unsub-neg 98.8%

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

      5. associate-+r- 98.8%

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

      6. *-commutative 98.8%

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

      7. neg-mul-1 98.8%

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

      8. distribute-rgt-out 98.9%

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

      9. *-commutative 98.9%

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

   5. Simplified 98.9%

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

   if -3.00000000000000008e-5 < eps < 4.8000000000000001e-5

   1. Initial program 24.4%

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

   2. Taylor expanded in eps around 0 99.7%

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

      1. +-commutative 99.7%

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

      2. mul-1-neg 99.7%

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

      3. unsub-neg 99.7%

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

      4. associate-*r* 99.7%

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

      5. *-commutative 99.7%

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

   4. Simplified 99.7%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 4.8 \cdot 10^{-5}\right):\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 7: 75.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.6%

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

   1. diff-cos 48.8%

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

   2. div-inv 48.8%

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

   3. +-commutative 48.8%

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

   4. associate--l+ 75.3%

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

   5. *-un-lft-identity 75.3%

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

   6. *-un-lft-identity 75.3%

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

   7. distribute-rgt-out-- 75.3%

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

   8. metadata-eval 75.3%

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

   9. metadata-eval 75.3%

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

   10. div-inv 75.3%

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

   11. +-commutative 75.3%

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

   12. associate-+l+ 75.3%

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

   13. count-2 75.3%

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

   14. *-commutative 75.3%

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

   15. metadata-eval 75.3%

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

3. Applied egg-rr 75.3%

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

   1. associate-*r* 75.3%

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

   2. *-commutative 75.3%

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

   3. *-commutative 75.3%

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

   4. *-commutative 75.3%

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

   5. mul0-rgt 75.3%

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

5. Simplified 75.3%

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

6. Final simplification 75.3%

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

Alternative 8: 67.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -6.8 \cdot 10^{-6} \lor \neg \left(\varepsilon \leq 4.3 \cdot 10^{-6}\right):\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -6.8e-6) (not (<= eps 4.3e-6)))
   (* (cos x) (+ (cos eps) -1.0))
   (* eps (- (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -6.8e-6) || !(eps <= 4.3e-6)) {
		tmp = cos(x) * (cos(eps) + -1.0);
	} else {
		tmp = eps * -sin(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-6.8d-6)) .or. (.not. (eps <= 4.3d-6))) then
        tmp = cos(x) * (cos(eps) + (-1.0d0))
    else
        tmp = eps * -sin(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -6.8e-6) || !(eps <= 4.3e-6)) {
		tmp = Math.cos(x) * (Math.cos(eps) + -1.0);
	} else {
		tmp = eps * -Math.sin(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -6.8e-6) or not (eps <= 4.3e-6):
		tmp = math.cos(x) * (math.cos(eps) + -1.0)
	else:
		tmp = eps * -math.sin(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -6.8e-6) || !(eps <= 4.3e-6))
		tmp = Float64(cos(x) * Float64(cos(eps) + -1.0));
	else
		tmp = Float64(eps * Float64(-sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -6.8e-6) || ~((eps <= 4.3e-6)))
		tmp = cos(x) * (cos(eps) + -1.0);
	else
		tmp = eps * -sin(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -6.8e-6], N[Not[LessEqual[eps, 4.3e-6]], $MachinePrecision]], N[(N[Cos[x], $MachinePrecision] * N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(eps * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -6.80000000000000012e-6 or 4.30000000000000033e-6 < eps

   1. Initial program 51.9%

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

      1. sub-neg 51.9%

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

      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. cancel-sign-sub-inv 98.8%

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

      4. associate-+l+ 98.8%

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

      5. *-commutative 98.8%

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

   3. Applied egg-rr 98.8%

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

      1. +-commutative 98.8%

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

      2. distribute-rgt-neg-out 98.8%

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

      3. *-commutative 98.8%

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

      4. unsub-neg 98.8%

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

      5. associate-+r- 98.8%

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

      6. *-commutative 98.8%

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

      7. neg-mul-1 98.8%

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

      8. distribute-rgt-out 98.9%

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

      9. *-commutative 98.9%

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

   5. Simplified 98.9%

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

      1. add-cube-cbrt 98.4%

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

      2. pow3 98.4%

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

   7. Applied egg-rr 98.4%

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

      1. rem-cube-cbrt 98.9%

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

      2. *-commutative 98.9%

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

      3. sin-mult 54.3%

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

      4. sub-neg 54.3%

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

      5. add-sqr-sqrt 17.9%

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

      6. sqrt-unprod 28.0%

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

      7. sqr-neg 28.0%

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

      8. sqrt-unprod 18.8%

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

      9. add-sqr-sqrt 53.6%

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

      10. +-commutative 53.6%

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

      11. +-commutative 53.6%

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

   9. Applied egg-rr 53.6%

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

       1. +-inverses 53.6%

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

       2. metadata-eval 53.6%

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

   11. Simplified 53.6%

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

   if -6.80000000000000012e-6 < eps < 4.30000000000000033e-6

   1. Initial program 24.4%

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

   2. Taylor expanded in eps around 0 77.5%

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

      1. associate-*r* 77.5%

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

      2. mul-1-neg 77.5%

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

   4. Simplified 77.5%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -6.8 \cdot 10^{-6} \lor \neg \left(\varepsilon \leq 4.3 \cdot 10^{-6}\right):\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \]

Alternative 9: 67.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -9.5 \cdot 10^{-6} \lor \neg \left(\varepsilon \leq 2.9 \cdot 10^{-7}\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -9.5e-6) (not (<= eps 2.9e-7)))
   (- (cos eps) (cos x))
   (* eps (- (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -9.5e-6) || !(eps <= 2.9e-7)) {
		tmp = cos(eps) - cos(x);
	} else {
		tmp = eps * -sin(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-9.5d-6)) .or. (.not. (eps <= 2.9d-7))) then
        tmp = cos(eps) - cos(x)
    else
        tmp = eps * -sin(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -9.5e-6) || !(eps <= 2.9e-7)) {
		tmp = Math.cos(eps) - Math.cos(x);
	} else {
		tmp = eps * -Math.sin(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -9.5e-6) or not (eps <= 2.9e-7):
		tmp = math.cos(eps) - math.cos(x)
	else:
		tmp = eps * -math.sin(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -9.5e-6) || !(eps <= 2.9e-7))
		tmp = Float64(cos(eps) - cos(x));
	else
		tmp = Float64(eps * Float64(-sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -9.5e-6) || ~((eps <= 2.9e-7)))
		tmp = cos(eps) - cos(x);
	else
		tmp = eps * -sin(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -9.5e-6], N[Not[LessEqual[eps, 2.9e-7]], $MachinePrecision]], N[(N[Cos[eps], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], N[(eps * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -9.5000000000000005e-6 or 2.8999999999999998e-7 < eps

   1. Initial program 51.9%

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

   2. Taylor expanded in x around 0 53.3%

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

   if -9.5000000000000005e-6 < eps < 2.8999999999999998e-7

   1. Initial program 24.4%

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

   2. Taylor expanded in eps around 0 77.5%

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

      1. associate-*r* 77.5%

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

      2. mul-1-neg 77.5%

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

   4. Simplified 77.5%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -9.5 \cdot 10^{-6} \lor \neg \left(\varepsilon \leq 2.9 \cdot 10^{-7}\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \]

Alternative 10: 66.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000125 \lor \neg \left(\varepsilon \leq 1.95 \cdot 10^{-5}\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.000125) (not (<= eps 1.95e-5)))
   (+ (cos eps) -1.0)
   (* eps (- (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.000125) || !(eps <= 1.95e-5)) {
		tmp = cos(eps) + -1.0;
	} else {
		tmp = eps * -sin(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-0.000125d0)) .or. (.not. (eps <= 1.95d-5))) then
        tmp = cos(eps) + (-1.0d0)
    else
        tmp = eps * -sin(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.000125) || !(eps <= 1.95e-5)) {
		tmp = Math.cos(eps) + -1.0;
	} else {
		tmp = eps * -Math.sin(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.000125) or not (eps <= 1.95e-5):
		tmp = math.cos(eps) + -1.0
	else:
		tmp = eps * -math.sin(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.000125) || !(eps <= 1.95e-5))
		tmp = Float64(cos(eps) + -1.0);
	else
		tmp = Float64(eps * Float64(-sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.000125) || ~((eps <= 1.95e-5)))
		tmp = cos(eps) + -1.0;
	else
		tmp = eps * -sin(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.000125], N[Not[LessEqual[eps, 1.95e-5]], $MachinePrecision]], N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision], N[(eps * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.25e-4 or 1.95e-5 < eps

   1. Initial program 51.9%

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

   2. Taylor expanded in x around 0 52.3%

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

   if -1.25e-4 < eps < 1.95e-5

   1. Initial program 24.4%

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

   2. Taylor expanded in eps around 0 77.5%

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

      1. associate-*r* 77.5%

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

      2. mul-1-neg 77.5%

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

   4. Simplified 77.5%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000125 \lor \neg \left(\varepsilon \leq 1.95 \cdot 10^{-5}\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \]

Alternative 11: 43.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -7.5 \cdot 10^{-11} \lor \neg \left(\varepsilon \leq 1.02 \cdot 10^{-8}\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-\varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -7.5e-11) (not (<= eps 1.02e-8)))
   (+ (cos eps) -1.0)
   (* x (- eps))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -7.5e-11) || !(eps <= 1.02e-8)) {
		tmp = cos(eps) + -1.0;
	} else {
		tmp = 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 <= (-7.5d-11)) .or. (.not. (eps <= 1.02d-8))) then
        tmp = cos(eps) + (-1.0d0)
    else
        tmp = x * -eps
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -7.5e-11) || !(eps <= 1.02e-8)) {
		tmp = Math.cos(eps) + -1.0;
	} else {
		tmp = x * -eps;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -7.5e-11) or not (eps <= 1.02e-8):
		tmp = math.cos(eps) + -1.0
	else:
		tmp = x * -eps
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -7.5e-11) || !(eps <= 1.02e-8))
		tmp = Float64(cos(eps) + -1.0);
	else
		tmp = Float64(x * Float64(-eps));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -7.5e-11) || ~((eps <= 1.02e-8)))
		tmp = cos(eps) + -1.0;
	else
		tmp = x * -eps;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -7.5e-11], N[Not[LessEqual[eps, 1.02e-8]], $MachinePrecision]], N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision], N[(x * (-eps)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -7.5e-11 or 1.02000000000000003e-8 < eps

   1. Initial program 51.5%

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

   2. Taylor expanded in x around 0 52.0%

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

   if -7.5e-11 < eps < 1.02000000000000003e-8

   1. Initial program 24.5%

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

   2. Taylor expanded in eps around 0 77.7%

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

      1. associate-*r* 77.7%

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

      2. mul-1-neg 77.7%

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

   4. Simplified 77.7%

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

   5. Taylor expanded in x around 0 35.2%

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

      1. associate-*r* 35.2%

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

      2. mul-1-neg 35.2%

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

   7. Simplified 35.2%

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

4. Final simplification 43.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -7.5 \cdot 10^{-11} \lor \neg \left(\varepsilon \leq 1.02 \cdot 10^{-8}\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-\varepsilon\right)\\ \end{array} \]

Alternative 12: 17.7% accurate, 51.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.6%

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

2. Taylor expanded in eps around 0 39.2%

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

   1. associate-*r* 39.2%

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

   2. mul-1-neg 39.2%

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

4. Simplified 39.2%

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

5. Taylor expanded in x around 0 18.5%

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

   1. associate-*r* 18.5%

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

   2. mul-1-neg 18.5%

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

7. Simplified 18.5%

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

8. Final simplification 18.5%

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

Reproduce

herbie shell --seed 2023305 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))