2cos (problem 3.3.5)

Percentage Accurate: 37.2% → 99.3%

Time: 17.5s

Alternatives: 16

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: 37.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.004:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin \varepsilon \cdot \left(-\sin x\right)\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.004:\\ \;\;\;\;\left(\sin x \cdot \left(-0.125 \cdot \left(\varepsilon \cdot \varepsilon\right) + 1\right) + \cos x \cdot \left(\varepsilon \cdot 0.5 + -0.020833333333333332 \cdot {\varepsilon}^{3}\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \cos x\right) - \sin \varepsilon \cdot \sin x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.004)
   (- (fma (cos x) (cos eps) (* (sin eps) (- (sin x)))) (cos x))
   (if (<= eps 0.004)
     (*
      (+
       (* (sin x) (+ (* -0.125 (* eps eps)) 1.0))
       (* (cos x) (+ (* eps 0.5) (* -0.020833333333333332 (pow eps 3.0)))))
      (* -2.0 (sin (* eps 0.5))))
     (- (- (* (cos x) (cos eps)) (cos x)) (* (sin eps) (sin x))))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.004) {
		tmp = fma(cos(x), cos(eps), (sin(eps) * -sin(x))) - cos(x);
	} else if (eps <= 0.004) {
		tmp = ((sin(x) * ((-0.125 * (eps * eps)) + 1.0)) + (cos(x) * ((eps * 0.5) + (-0.020833333333333332 * pow(eps, 3.0))))) * (-2.0 * sin((eps * 0.5)));
	} else {
		tmp = ((cos(x) * cos(eps)) - cos(x)) - (sin(eps) * sin(x));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -0.004)
		tmp = Float64(fma(cos(x), cos(eps), Float64(sin(eps) * Float64(-sin(x)))) - cos(x));
	elseif (eps <= 0.004)
		tmp = Float64(Float64(Float64(sin(x) * Float64(Float64(-0.125 * Float64(eps * eps)) + 1.0)) + Float64(cos(x) * Float64(Float64(eps * 0.5) + Float64(-0.020833333333333332 * (eps ^ 3.0))))) * Float64(-2.0 * sin(Float64(eps * 0.5))));
	else
		tmp = Float64(Float64(Float64(cos(x) * cos(eps)) - cos(x)) - Float64(sin(eps) * sin(x)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -0.004], N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + N[(N[Sin[eps], $MachinePrecision] * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.004], N[(N[(N[(N[Sin[x], $MachinePrecision] * N[(N[(-0.125 * N[(eps * eps), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[(eps * 0.5), $MachinePrecision] + N[(-0.020833333333333332 * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-2.0 * N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.0040000000000000001

   1. Initial program 46.5%

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

      1. cos-sum 98.5%

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

      2. cancel-sign-sub-inv 98.5%

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

      3. fma-def 98.6%

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

   3. Applied egg-rr 98.6%

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

   if -0.0040000000000000001 < eps < 0.0040000000000000001

   1. Initial program 21.4%

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

      1. diff-cos 41.4%

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

      2. div-inv 41.4%

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

      3. associate--l+ 41.4%

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

      4. metadata-eval 41.4%

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

      5. div-inv 41.4%

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

      6. +-commutative 41.4%

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

      7. associate-+l+ 41.4%

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

      8. metadata-eval 41.4%

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

   3. Applied egg-rr 41.4%

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

      1. associate-*r* 41.4%

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

      2. *-commutative 41.4%

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

      3. *-commutative 41.4%

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

      4. +-commutative 41.4%

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

      5. count-2 41.4%

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

      6. fma-def 41.4%

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

      7. sub-neg 41.4%

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

      8. mul-1-neg 41.4%

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

      9. +-commutative 41.4%

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

      10. associate-+r+ 99.0%

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

      11. mul-1-neg 99.0%

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

      12. sub-neg 99.0%

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

      13. +-inverses 99.0%

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

      14. remove-double-neg 99.0%

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

      15. mul-1-neg 99.0%

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

      16. sub-neg 99.0%

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

      17. neg-sub0 99.0%

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

      18. mul-1-neg 99.0%

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

      19. remove-double-neg 99.0%

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

   5. Simplified 99.0%

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

   6. Taylor expanded in eps around 0 99.8%

      \[\leadsto \color{blue}{\left(\sin x + \left(-0.125 \cdot \left({\varepsilon}^{2} \cdot \sin x\right) + \left(-0.020833333333333332 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + 0.5 \cdot \left(\varepsilon \cdot \cos x\right)\right)\right)\right)} \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]
   7. Step-by-step derivation

      1. associate-+r+ 99.8%

         \[\leadsto \color{blue}{\left(\left(\sin x + -0.125 \cdot \left({\varepsilon}^{2} \cdot \sin x\right)\right) + \left(-0.020833333333333332 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + 0.5 \cdot \left(\varepsilon \cdot \cos x\right)\right)\right)} \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]

      2. associate-*r* 99.8%

         \[\leadsto \left(\left(\sin x + \color{blue}{\left(-0.125 \cdot {\varepsilon}^{2}\right) \cdot \sin x}\right) + \left(-0.020833333333333332 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + 0.5 \cdot \left(\varepsilon \cdot \cos x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]

      3. distribute-rgt1-in 99.8%

         \[\leadsto \left(\color{blue}{\left(-0.125 \cdot {\varepsilon}^{2} + 1\right) \cdot \sin x} + \left(-0.020833333333333332 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + 0.5 \cdot \left(\varepsilon \cdot \cos x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]

      4. +-commutative 99.8%

         \[\leadsto \left(\left(-0.125 \cdot {\varepsilon}^{2} + 1\right) \cdot \sin x + \color{blue}{\left(0.5 \cdot \left(\varepsilon \cdot \cos x\right) + -0.020833333333333332 \cdot \left({\varepsilon}^{3} \cdot \cos x\right)\right)}\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]

      5. associate-*r* 99.8%

         \[\leadsto \left(\left(-0.125 \cdot {\varepsilon}^{2} + 1\right) \cdot \sin x + \left(\color{blue}{\left(0.5 \cdot \varepsilon\right) \cdot \cos x} + -0.020833333333333332 \cdot \left({\varepsilon}^{3} \cdot \cos x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]

      6. associate-*r* 99.8%

         \[\leadsto \left(\left(-0.125 \cdot {\varepsilon}^{2} + 1\right) \cdot \sin x + \left(\left(0.5 \cdot \varepsilon\right) \cdot \cos x + \color{blue}{\left(-0.020833333333333332 \cdot {\varepsilon}^{3}\right) \cdot \cos x}\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]

      7. distribute-rgt-out 99.8%

         \[\leadsto \left(\left(-0.125 \cdot {\varepsilon}^{2} + 1\right) \cdot \sin x + \color{blue}{\cos x \cdot \left(0.5 \cdot \varepsilon + -0.020833333333333332 \cdot {\varepsilon}^{3}\right)}\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]

   8. Simplified 99.8%

      \[\leadsto \color{blue}{\left(\left(-0.125 \cdot {\varepsilon}^{2} + 1\right) \cdot \sin x + \cos x \cdot \left(0.5 \cdot \varepsilon + -0.020833333333333332 \cdot {\varepsilon}^{3}\right)\right)} \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]
   9. Step-by-step derivation

      1. unpow2 99.8%

         \[\leadsto \left(\left(-0.125 \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} + 1\right) \cdot \sin x + \cos x \cdot \left(0.5 \cdot \varepsilon + -0.020833333333333332 \cdot {\varepsilon}^{3}\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]

   10. Applied egg-rr 99.8%

       \[\leadsto \left(\left(-0.125 \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} + 1\right) \cdot \sin x + \cos x \cdot \left(0.5 \cdot \varepsilon + -0.020833333333333332 \cdot {\varepsilon}^{3}\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]

   if 0.0040000000000000001 < eps

   1. Initial program 56.2%

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

      1. sub-neg 56.2%

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

      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. 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%

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

      1. fma-neg 98.8%

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

      2. *-commutative 98.8%

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

      3. *-commutative 98.8%

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

      4. fma-neg 98.9%

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

      5. remove-double-neg 98.9%

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

   5. Simplified 98.9%

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

   6. Taylor expanded in eps around inf 98.8%

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

      1. associate--r+ 99.1%

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

      2. *-lft-identity 99.1%

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

      3. distribute-rgt-out-- 98.9%

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

      4. sub-neg 98.9%

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

      5. metadata-eval 98.9%

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

      6. +-commutative 98.9%

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

      7. *-commutative 98.9%

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

   8. Simplified 98.9%

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

      1. distribute-lft-in 99.1%

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

   10. Applied egg-rr 99.1%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.004:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin \varepsilon \cdot \left(-\sin x\right)\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.004:\\ \;\;\;\;\left(\sin x \cdot \left(-0.125 \cdot \left(\varepsilon \cdot \varepsilon\right) + 1\right) + \cos x \cdot \left(\varepsilon \cdot 0.5 + -0.020833333333333332 \cdot {\varepsilon}^{3}\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \cos x\right) - \sin \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 2: 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 36.8%

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

   1. sub-neg 36.8%

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

   2. cos-sum 61.5%

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

   3. associate-+l- 61.5%

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

   4. fma-neg 61.5%

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

3. Applied egg-rr 61.5%

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

   1. fma-neg 61.5%

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

   2. *-commutative 61.5%

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

   3. *-commutative 61.5%

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

   4. fma-neg 61.5%

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

   5. remove-double-neg 61.5%

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

5. Simplified 61.5%

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

6. Taylor expanded in eps around inf 61.5%

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

   1. associate--r+ 89.4%

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

   2. *-lft-identity 89.4%

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

   3. distribute-rgt-out-- 89.4%

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

   4. sub-neg 89.4%

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

   5. metadata-eval 89.4%

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

   6. +-commutative 89.4%

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

   7. *-commutative 89.4%

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

8. Simplified 89.4%

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

   1. flip-+ 89.0%

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

   2. metadata-eval 89.0%

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

   3. 1-sub-cos 98.9%

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

   4. pow2 98.9%

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

10. Applied egg-rr 98.9%

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

11. Final simplification 98.9%

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

Alternative 3: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0039:\\ \;\;\;\;\mathsf{fma}\left(-1 + \cos \varepsilon, \cos x, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0039:\\ \;\;\;\;\left(\sin x \cdot \left(-0.125 \cdot \left(\varepsilon \cdot \varepsilon\right) + 1\right) + \cos x \cdot \left(\varepsilon \cdot 0.5 + -0.020833333333333332 \cdot {\varepsilon}^{3}\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \cos x\right) - \sin \varepsilon \cdot \sin x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.0039)
   (fma (+ -1.0 (cos eps)) (cos x) (* (sin eps) (- (sin x))))
   (if (<= eps 0.0039)
     (*
      (+
       (* (sin x) (+ (* -0.125 (* eps eps)) 1.0))
       (* (cos x) (+ (* eps 0.5) (* -0.020833333333333332 (pow eps 3.0)))))
      (* -2.0 (sin (* eps 0.5))))
     (- (- (* (cos x) (cos eps)) (cos x)) (* (sin eps) (sin x))))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.0039) {
		tmp = fma((-1.0 + cos(eps)), cos(x), (sin(eps) * -sin(x)));
	} else if (eps <= 0.0039) {
		tmp = ((sin(x) * ((-0.125 * (eps * eps)) + 1.0)) + (cos(x) * ((eps * 0.5) + (-0.020833333333333332 * pow(eps, 3.0))))) * (-2.0 * sin((eps * 0.5)));
	} else {
		tmp = ((cos(x) * cos(eps)) - cos(x)) - (sin(eps) * sin(x));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -0.0039)
		tmp = fma(Float64(-1.0 + cos(eps)), cos(x), Float64(sin(eps) * Float64(-sin(x))));
	elseif (eps <= 0.0039)
		tmp = Float64(Float64(Float64(sin(x) * Float64(Float64(-0.125 * Float64(eps * eps)) + 1.0)) + Float64(cos(x) * Float64(Float64(eps * 0.5) + Float64(-0.020833333333333332 * (eps ^ 3.0))))) * Float64(-2.0 * sin(Float64(eps * 0.5))));
	else
		tmp = Float64(Float64(Float64(cos(x) * cos(eps)) - cos(x)) - Float64(sin(eps) * sin(x)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -0.0039], N[(N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[Sin[eps], $MachinePrecision] * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.0039], N[(N[(N[(N[Sin[x], $MachinePrecision] * N[(N[(-0.125 * N[(eps * eps), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[(eps * 0.5), $MachinePrecision] + N[(-0.020833333333333332 * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-2.0 * N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.0038999999999999998

   1. Initial program 46.5%

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

      1. sub-neg 46.5%

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

      2. cos-sum 98.5%

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

      3. associate-+l- 98.4%

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

      4. fma-neg 98.4%

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

   3. Applied egg-rr 98.4%

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

      1. fma-neg 98.4%

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

      2. *-commutative 98.4%

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

      3. *-commutative 98.4%

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

      4. fma-neg 98.5%

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

      5. remove-double-neg 98.5%

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

   5. Simplified 98.5%

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

   6. Taylor expanded in eps around inf 98.4%

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

      1. associate--r+ 98.4%

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

      2. *-lft-identity 98.4%

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

      3. distribute-rgt-out-- 98.5%

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

      4. sub-neg 98.5%

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

      5. metadata-eval 98.5%

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

      6. +-commutative 98.5%

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

      7. *-commutative 98.5%

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

   8. Simplified 98.5%

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

      1. cancel-sign-sub-inv 98.5%

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

      2. *-commutative 98.5%

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

      3. fma-def 98.5%

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

   10. Applied egg-rr 98.5%

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

   if -0.0038999999999999998 < eps < 0.0038999999999999998

   1. Initial program 21.4%

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

      1. diff-cos 41.4%

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

      2. div-inv 41.4%

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

      3. associate--l+ 41.4%

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

      4. metadata-eval 41.4%

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

      5. div-inv 41.4%

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

      6. +-commutative 41.4%

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

      7. associate-+l+ 41.4%

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

      8. metadata-eval 41.4%

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

   3. Applied egg-rr 41.4%

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

      1. associate-*r* 41.4%

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

      2. *-commutative 41.4%

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

      3. *-commutative 41.4%

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

      4. +-commutative 41.4%

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

      5. count-2 41.4%

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

      6. fma-def 41.4%

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

      7. sub-neg 41.4%

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

      8. mul-1-neg 41.4%

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

      9. +-commutative 41.4%

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

      10. associate-+r+ 99.0%

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

      11. mul-1-neg 99.0%

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

      12. sub-neg 99.0%

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

      13. +-inverses 99.0%

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

      14. remove-double-neg 99.0%

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

      15. mul-1-neg 99.0%

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

      16. sub-neg 99.0%

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

      17. neg-sub0 99.0%

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

      18. mul-1-neg 99.0%

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

      19. remove-double-neg 99.0%

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

   5. Simplified 99.0%

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

   6. Taylor expanded in eps around 0 99.8%

      \[\leadsto \color{blue}{\left(\sin x + \left(-0.125 \cdot \left({\varepsilon}^{2} \cdot \sin x\right) + \left(-0.020833333333333332 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + 0.5 \cdot \left(\varepsilon \cdot \cos x\right)\right)\right)\right)} \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]
   7. Step-by-step derivation

      1. associate-+r+ 99.8%

         \[\leadsto \color{blue}{\left(\left(\sin x + -0.125 \cdot \left({\varepsilon}^{2} \cdot \sin x\right)\right) + \left(-0.020833333333333332 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + 0.5 \cdot \left(\varepsilon \cdot \cos x\right)\right)\right)} \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]

      2. associate-*r* 99.8%

         \[\leadsto \left(\left(\sin x + \color{blue}{\left(-0.125 \cdot {\varepsilon}^{2}\right) \cdot \sin x}\right) + \left(-0.020833333333333332 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + 0.5 \cdot \left(\varepsilon \cdot \cos x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]

      3. distribute-rgt1-in 99.8%

         \[\leadsto \left(\color{blue}{\left(-0.125 \cdot {\varepsilon}^{2} + 1\right) \cdot \sin x} + \left(-0.020833333333333332 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + 0.5 \cdot \left(\varepsilon \cdot \cos x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]

      4. +-commutative 99.8%

         \[\leadsto \left(\left(-0.125 \cdot {\varepsilon}^{2} + 1\right) \cdot \sin x + \color{blue}{\left(0.5 \cdot \left(\varepsilon \cdot \cos x\right) + -0.020833333333333332 \cdot \left({\varepsilon}^{3} \cdot \cos x\right)\right)}\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]

      5. associate-*r* 99.8%

         \[\leadsto \left(\left(-0.125 \cdot {\varepsilon}^{2} + 1\right) \cdot \sin x + \left(\color{blue}{\left(0.5 \cdot \varepsilon\right) \cdot \cos x} + -0.020833333333333332 \cdot \left({\varepsilon}^{3} \cdot \cos x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]

      6. associate-*r* 99.8%

         \[\leadsto \left(\left(-0.125 \cdot {\varepsilon}^{2} + 1\right) \cdot \sin x + \left(\left(0.5 \cdot \varepsilon\right) \cdot \cos x + \color{blue}{\left(-0.020833333333333332 \cdot {\varepsilon}^{3}\right) \cdot \cos x}\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]

      7. distribute-rgt-out 99.8%

         \[\leadsto \left(\left(-0.125 \cdot {\varepsilon}^{2} + 1\right) \cdot \sin x + \color{blue}{\cos x \cdot \left(0.5 \cdot \varepsilon + -0.020833333333333332 \cdot {\varepsilon}^{3}\right)}\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]

   8. Simplified 99.8%

      \[\leadsto \color{blue}{\left(\left(-0.125 \cdot {\varepsilon}^{2} + 1\right) \cdot \sin x + \cos x \cdot \left(0.5 \cdot \varepsilon + -0.020833333333333332 \cdot {\varepsilon}^{3}\right)\right)} \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]
   9. Step-by-step derivation

      1. unpow2 99.8%

         \[\leadsto \left(\left(-0.125 \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} + 1\right) \cdot \sin x + \cos x \cdot \left(0.5 \cdot \varepsilon + -0.020833333333333332 \cdot {\varepsilon}^{3}\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]

   10. Applied egg-rr 99.8%

       \[\leadsto \left(\left(-0.125 \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} + 1\right) \cdot \sin x + \cos x \cdot \left(0.5 \cdot \varepsilon + -0.020833333333333332 \cdot {\varepsilon}^{3}\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]

   if 0.0038999999999999998 < eps

   1. Initial program 56.2%

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

      1. sub-neg 56.2%

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

      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. 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%

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

      1. fma-neg 98.8%

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

      2. *-commutative 98.8%

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

      3. *-commutative 98.8%

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

      4. fma-neg 98.9%

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

      5. remove-double-neg 98.9%

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

   5. Simplified 98.9%

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

   6. Taylor expanded in eps around inf 98.8%

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

      1. associate--r+ 99.1%

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

      2. *-lft-identity 99.1%

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

      3. distribute-rgt-out-- 98.9%

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

      4. sub-neg 98.9%

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

      5. metadata-eval 98.9%

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

      6. +-commutative 98.9%

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

      7. *-commutative 98.9%

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

   8. Simplified 98.9%

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

      1. distribute-lft-in 99.1%

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

   10. Applied egg-rr 99.1%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0039:\\ \;\;\;\;\mathsf{fma}\left(-1 + \cos \varepsilon, \cos x, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0039:\\ \;\;\;\;\left(\sin x \cdot \left(-0.125 \cdot \left(\varepsilon \cdot \varepsilon\right) + 1\right) + \cos x \cdot \left(\varepsilon \cdot 0.5 + -0.020833333333333332 \cdot {\varepsilon}^{3}\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \cos x\right) - \sin \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 4: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0035 \lor \neg \left(\varepsilon \leq 0.0039\right):\\ \;\;\;\;\mathsf{fma}\left(-1 + \cos \varepsilon, \cos x, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \left(-0.125 \cdot \left(\varepsilon \cdot \varepsilon\right) + 1\right) + \cos x \cdot \left(\varepsilon \cdot 0.5 + -0.020833333333333332 \cdot {\varepsilon}^{3}\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.0035) (not (<= eps 0.0039)))
   (fma (+ -1.0 (cos eps)) (cos x) (* (sin eps) (- (sin x))))
   (*
    (+
     (* (sin x) (+ (* -0.125 (* eps eps)) 1.0))
     (* (cos x) (+ (* eps 0.5) (* -0.020833333333333332 (pow eps 3.0)))))
    (* -2.0 (sin (* eps 0.5))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.0035) || !(eps <= 0.0039)) {
		tmp = fma((-1.0 + cos(eps)), cos(x), (sin(eps) * -sin(x)));
	} else {
		tmp = ((sin(x) * ((-0.125 * (eps * eps)) + 1.0)) + (cos(x) * ((eps * 0.5) + (-0.020833333333333332 * pow(eps, 3.0))))) * (-2.0 * sin((eps * 0.5)));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.0035) || !(eps <= 0.0039))
		tmp = fma(Float64(-1.0 + cos(eps)), cos(x), Float64(sin(eps) * Float64(-sin(x))));
	else
		tmp = Float64(Float64(Float64(sin(x) * Float64(Float64(-0.125 * Float64(eps * eps)) + 1.0)) + Float64(cos(x) * Float64(Float64(eps * 0.5) + Float64(-0.020833333333333332 * (eps ^ 3.0))))) * Float64(-2.0 * sin(Float64(eps * 0.5))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.0035], N[Not[LessEqual[eps, 0.0039]], $MachinePrecision]], N[(N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[Sin[eps], $MachinePrecision] * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sin[x], $MachinePrecision] * N[(N[(-0.125 * N[(eps * eps), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[(eps * 0.5), $MachinePrecision] + N[(-0.020833333333333332 * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-2.0 * N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -0.00350000000000000007 or 0.0038999999999999998 < eps

   1. Initial program 51.4%

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

      1. sub-neg 51.4%

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

      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. associate-+l- 98.6%

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

      4. fma-neg 98.7%

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

   3. Applied egg-rr 98.7%

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

      1. fma-neg 98.6%

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

      2. *-commutative 98.6%

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

      3. *-commutative 98.6%

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

      4. fma-neg 98.7%

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

      5. remove-double-neg 98.7%

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

   5. Simplified 98.7%

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

   6. Taylor expanded in eps around inf 98.6%

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

      1. associate--r+ 98.8%

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

      2. *-lft-identity 98.8%

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

      3. distribute-rgt-out-- 98.7%

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

      4. sub-neg 98.7%

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

      5. metadata-eval 98.7%

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

      6. +-commutative 98.7%

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

      7. *-commutative 98.7%

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

   8. Simplified 98.7%

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

      1. cancel-sign-sub-inv 98.7%

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

      2. *-commutative 98.7%

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

      3. fma-def 98.8%

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

   10. Applied egg-rr 98.8%

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

   if -0.00350000000000000007 < eps < 0.0038999999999999998

   1. Initial program 21.4%

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

      1. diff-cos 41.4%

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

      2. div-inv 41.4%

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

      3. associate--l+ 41.4%

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

      4. metadata-eval 41.4%

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

      5. div-inv 41.4%

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

      6. +-commutative 41.4%

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

      7. associate-+l+ 41.4%

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

      8. metadata-eval 41.4%

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

   3. Applied egg-rr 41.4%

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

      1. associate-*r* 41.4%

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

      2. *-commutative 41.4%

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

      3. *-commutative 41.4%

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

      4. +-commutative 41.4%

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

      5. count-2 41.4%

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

      6. fma-def 41.4%

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

      7. sub-neg 41.4%

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

      8. mul-1-neg 41.4%

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

      9. +-commutative 41.4%

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

      10. associate-+r+ 99.0%

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

      11. mul-1-neg 99.0%

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

      12. sub-neg 99.0%

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

      13. +-inverses 99.0%

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

      14. remove-double-neg 99.0%

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

      15. mul-1-neg 99.0%

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

      16. sub-neg 99.0%

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

      17. neg-sub0 99.0%

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

      18. mul-1-neg 99.0%

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

      19. remove-double-neg 99.0%

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

   5. Simplified 99.0%

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

   6. Taylor expanded in eps around 0 99.8%

      \[\leadsto \color{blue}{\left(\sin x + \left(-0.125 \cdot \left({\varepsilon}^{2} \cdot \sin x\right) + \left(-0.020833333333333332 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + 0.5 \cdot \left(\varepsilon \cdot \cos x\right)\right)\right)\right)} \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]
   7. Step-by-step derivation

      1. associate-+r+ 99.8%

         \[\leadsto \color{blue}{\left(\left(\sin x + -0.125 \cdot \left({\varepsilon}^{2} \cdot \sin x\right)\right) + \left(-0.020833333333333332 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + 0.5 \cdot \left(\varepsilon \cdot \cos x\right)\right)\right)} \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]

      2. associate-*r* 99.8%

         \[\leadsto \left(\left(\sin x + \color{blue}{\left(-0.125 \cdot {\varepsilon}^{2}\right) \cdot \sin x}\right) + \left(-0.020833333333333332 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + 0.5 \cdot \left(\varepsilon \cdot \cos x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]

      3. distribute-rgt1-in 99.8%

         \[\leadsto \left(\color{blue}{\left(-0.125 \cdot {\varepsilon}^{2} + 1\right) \cdot \sin x} + \left(-0.020833333333333332 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + 0.5 \cdot \left(\varepsilon \cdot \cos x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]

      4. +-commutative 99.8%

         \[\leadsto \left(\left(-0.125 \cdot {\varepsilon}^{2} + 1\right) \cdot \sin x + \color{blue}{\left(0.5 \cdot \left(\varepsilon \cdot \cos x\right) + -0.020833333333333332 \cdot \left({\varepsilon}^{3} \cdot \cos x\right)\right)}\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]

      5. associate-*r* 99.8%

         \[\leadsto \left(\left(-0.125 \cdot {\varepsilon}^{2} + 1\right) \cdot \sin x + \left(\color{blue}{\left(0.5 \cdot \varepsilon\right) \cdot \cos x} + -0.020833333333333332 \cdot \left({\varepsilon}^{3} \cdot \cos x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]

      6. associate-*r* 99.8%

         \[\leadsto \left(\left(-0.125 \cdot {\varepsilon}^{2} + 1\right) \cdot \sin x + \left(\left(0.5 \cdot \varepsilon\right) \cdot \cos x + \color{blue}{\left(-0.020833333333333332 \cdot {\varepsilon}^{3}\right) \cdot \cos x}\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]

      7. distribute-rgt-out 99.8%

         \[\leadsto \left(\left(-0.125 \cdot {\varepsilon}^{2} + 1\right) \cdot \sin x + \color{blue}{\cos x \cdot \left(0.5 \cdot \varepsilon + -0.020833333333333332 \cdot {\varepsilon}^{3}\right)}\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]

   8. Simplified 99.8%

      \[\leadsto \color{blue}{\left(\left(-0.125 \cdot {\varepsilon}^{2} + 1\right) \cdot \sin x + \cos x \cdot \left(0.5 \cdot \varepsilon + -0.020833333333333332 \cdot {\varepsilon}^{3}\right)\right)} \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]
   9. Step-by-step derivation

      1. unpow2 99.8%

         \[\leadsto \left(\left(-0.125 \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} + 1\right) \cdot \sin x + \cos x \cdot \left(0.5 \cdot \varepsilon + -0.020833333333333332 \cdot {\varepsilon}^{3}\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]

   10. Applied egg-rr 99.8%

       \[\leadsto \left(\left(-0.125 \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} + 1\right) \cdot \sin x + \cos x \cdot \left(0.5 \cdot \varepsilon + -0.020833333333333332 \cdot {\varepsilon}^{3}\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0035 \lor \neg \left(\varepsilon \leq 0.0039\right):\\ \;\;\;\;\mathsf{fma}\left(-1 + \cos \varepsilon, \cos x, \sin \varepsilon \cdot \left(-\sin x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \left(-0.125 \cdot \left(\varepsilon \cdot \varepsilon\right) + 1\right) + \cos x \cdot \left(\varepsilon \cdot 0.5 + -0.020833333333333332 \cdot {\varepsilon}^{3}\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 5: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0035 \lor \neg \left(\varepsilon \leq 0.0035\right):\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \left(-0.125 \cdot \left(\varepsilon \cdot \varepsilon\right) + 1\right) + \cos x \cdot \left(\varepsilon \cdot 0.5 + -0.020833333333333332 \cdot {\varepsilon}^{3}\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.0035) (not (<= eps 0.0035)))
   (- (* (cos x) (+ -1.0 (cos eps))) (* (sin eps) (sin x)))
   (*
    (+
     (* (sin x) (+ (* -0.125 (* eps eps)) 1.0))
     (* (cos x) (+ (* eps 0.5) (* -0.020833333333333332 (pow eps 3.0)))))
    (* -2.0 (sin (* eps 0.5))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.0035) || !(eps <= 0.0035)) {
		tmp = (cos(x) * (-1.0 + cos(eps))) - (sin(eps) * sin(x));
	} else {
		tmp = ((sin(x) * ((-0.125 * (eps * eps)) + 1.0)) + (cos(x) * ((eps * 0.5) + (-0.020833333333333332 * pow(eps, 3.0))))) * (-2.0 * sin((eps * 0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-0.0035d0)) .or. (.not. (eps <= 0.0035d0))) then
        tmp = (cos(x) * ((-1.0d0) + cos(eps))) - (sin(eps) * sin(x))
    else
        tmp = ((sin(x) * (((-0.125d0) * (eps * eps)) + 1.0d0)) + (cos(x) * ((eps * 0.5d0) + ((-0.020833333333333332d0) * (eps ** 3.0d0))))) * ((-2.0d0) * sin((eps * 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.0035) || !(eps <= 0.0035)) {
		tmp = (Math.cos(x) * (-1.0 + Math.cos(eps))) - (Math.sin(eps) * Math.sin(x));
	} else {
		tmp = ((Math.sin(x) * ((-0.125 * (eps * eps)) + 1.0)) + (Math.cos(x) * ((eps * 0.5) + (-0.020833333333333332 * Math.pow(eps, 3.0))))) * (-2.0 * Math.sin((eps * 0.5)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.0035) or not (eps <= 0.0035):
		tmp = (math.cos(x) * (-1.0 + math.cos(eps))) - (math.sin(eps) * math.sin(x))
	else:
		tmp = ((math.sin(x) * ((-0.125 * (eps * eps)) + 1.0)) + (math.cos(x) * ((eps * 0.5) + (-0.020833333333333332 * math.pow(eps, 3.0))))) * (-2.0 * math.sin((eps * 0.5)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.0035) || !(eps <= 0.0035))
		tmp = Float64(Float64(cos(x) * Float64(-1.0 + cos(eps))) - Float64(sin(eps) * sin(x)));
	else
		tmp = Float64(Float64(Float64(sin(x) * Float64(Float64(-0.125 * Float64(eps * eps)) + 1.0)) + Float64(cos(x) * Float64(Float64(eps * 0.5) + Float64(-0.020833333333333332 * (eps ^ 3.0))))) * Float64(-2.0 * sin(Float64(eps * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.0035) || ~((eps <= 0.0035)))
		tmp = (cos(x) * (-1.0 + cos(eps))) - (sin(eps) * sin(x));
	else
		tmp = ((sin(x) * ((-0.125 * (eps * eps)) + 1.0)) + (cos(x) * ((eps * 0.5) + (-0.020833333333333332 * (eps ^ 3.0))))) * (-2.0 * sin((eps * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.0035], N[Not[LessEqual[eps, 0.0035]], $MachinePrecision]], N[(N[(N[Cos[x], $MachinePrecision] * N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sin[x], $MachinePrecision] * N[(N[(-0.125 * N[(eps * eps), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[(eps * 0.5), $MachinePrecision] + N[(-0.020833333333333332 * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-2.0 * N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -0.00350000000000000007 or 0.00350000000000000007 < eps

   1. Initial program 51.4%

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

      1. sub-neg 51.4%

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

      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. associate-+l- 98.6%

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

      4. fma-neg 98.7%

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

   3. Applied egg-rr 98.7%

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

      1. fma-neg 98.6%

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

      2. *-commutative 98.6%

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

      3. *-commutative 98.6%

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

      4. fma-neg 98.7%

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

      5. remove-double-neg 98.7%

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

   5. Simplified 98.7%

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

   6. Taylor expanded in eps around inf 98.6%

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

      1. associate--r+ 98.8%

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

      2. *-lft-identity 98.8%

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

      3. distribute-rgt-out-- 98.7%

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

      4. sub-neg 98.7%

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

      5. metadata-eval 98.7%

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

      6. +-commutative 98.7%

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

      7. *-commutative 98.7%

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

   8. Simplified 98.7%

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

   if -0.00350000000000000007 < eps < 0.00350000000000000007

   1. Initial program 21.4%

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

      1. diff-cos 41.4%

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

      2. div-inv 41.4%

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

      3. associate--l+ 41.4%

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

      4. metadata-eval 41.4%

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

      5. div-inv 41.4%

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

      6. +-commutative 41.4%

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

      7. associate-+l+ 41.4%

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

      8. metadata-eval 41.4%

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

   3. Applied egg-rr 41.4%

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

      1. associate-*r* 41.4%

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

      2. *-commutative 41.4%

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

      3. *-commutative 41.4%

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

      4. +-commutative 41.4%

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

      5. count-2 41.4%

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

      6. fma-def 41.4%

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

      7. sub-neg 41.4%

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

      8. mul-1-neg 41.4%

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

      9. +-commutative 41.4%

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

      10. associate-+r+ 99.0%

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

      11. mul-1-neg 99.0%

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

      12. sub-neg 99.0%

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

      13. +-inverses 99.0%

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

      14. remove-double-neg 99.0%

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

      15. mul-1-neg 99.0%

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

      16. sub-neg 99.0%

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

      17. neg-sub0 99.0%

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

      18. mul-1-neg 99.0%

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

      19. remove-double-neg 99.0%

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

   5. Simplified 99.0%

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

   6. Taylor expanded in eps around 0 99.8%

      \[\leadsto \color{blue}{\left(\sin x + \left(-0.125 \cdot \left({\varepsilon}^{2} \cdot \sin x\right) + \left(-0.020833333333333332 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + 0.5 \cdot \left(\varepsilon \cdot \cos x\right)\right)\right)\right)} \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]
   7. Step-by-step derivation

      1. associate-+r+ 99.8%

         \[\leadsto \color{blue}{\left(\left(\sin x + -0.125 \cdot \left({\varepsilon}^{2} \cdot \sin x\right)\right) + \left(-0.020833333333333332 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + 0.5 \cdot \left(\varepsilon \cdot \cos x\right)\right)\right)} \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]

      2. associate-*r* 99.8%

         \[\leadsto \left(\left(\sin x + \color{blue}{\left(-0.125 \cdot {\varepsilon}^{2}\right) \cdot \sin x}\right) + \left(-0.020833333333333332 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + 0.5 \cdot \left(\varepsilon \cdot \cos x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]

      3. distribute-rgt1-in 99.8%

         \[\leadsto \left(\color{blue}{\left(-0.125 \cdot {\varepsilon}^{2} + 1\right) \cdot \sin x} + \left(-0.020833333333333332 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + 0.5 \cdot \left(\varepsilon \cdot \cos x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]

      4. +-commutative 99.8%

         \[\leadsto \left(\left(-0.125 \cdot {\varepsilon}^{2} + 1\right) \cdot \sin x + \color{blue}{\left(0.5 \cdot \left(\varepsilon \cdot \cos x\right) + -0.020833333333333332 \cdot \left({\varepsilon}^{3} \cdot \cos x\right)\right)}\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]

      5. associate-*r* 99.8%

         \[\leadsto \left(\left(-0.125 \cdot {\varepsilon}^{2} + 1\right) \cdot \sin x + \left(\color{blue}{\left(0.5 \cdot \varepsilon\right) \cdot \cos x} + -0.020833333333333332 \cdot \left({\varepsilon}^{3} \cdot \cos x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]

      6. associate-*r* 99.8%

         \[\leadsto \left(\left(-0.125 \cdot {\varepsilon}^{2} + 1\right) \cdot \sin x + \left(\left(0.5 \cdot \varepsilon\right) \cdot \cos x + \color{blue}{\left(-0.020833333333333332 \cdot {\varepsilon}^{3}\right) \cdot \cos x}\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]

      7. distribute-rgt-out 99.8%

         \[\leadsto \left(\left(-0.125 \cdot {\varepsilon}^{2} + 1\right) \cdot \sin x + \color{blue}{\cos x \cdot \left(0.5 \cdot \varepsilon + -0.020833333333333332 \cdot {\varepsilon}^{3}\right)}\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]

   8. Simplified 99.8%

      \[\leadsto \color{blue}{\left(\left(-0.125 \cdot {\varepsilon}^{2} + 1\right) \cdot \sin x + \cos x \cdot \left(0.5 \cdot \varepsilon + -0.020833333333333332 \cdot {\varepsilon}^{3}\right)\right)} \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]
   9. Step-by-step derivation

      1. unpow2 99.8%

         \[\leadsto \left(\left(-0.125 \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} + 1\right) \cdot \sin x + \cos x \cdot \left(0.5 \cdot \varepsilon + -0.020833333333333332 \cdot {\varepsilon}^{3}\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]

   10. Applied egg-rr 99.8%

       \[\leadsto \left(\left(-0.125 \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} + 1\right) \cdot \sin x + \cos x \cdot \left(0.5 \cdot \varepsilon + -0.020833333333333332 \cdot {\varepsilon}^{3}\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0035 \lor \neg \left(\varepsilon \leq 0.0035\right):\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \left(-0.125 \cdot \left(\varepsilon \cdot \varepsilon\right) + 1\right) + \cos x \cdot \left(\varepsilon \cdot 0.5 + -0.020833333333333332 \cdot {\varepsilon}^{3}\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 6: 98.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.4 \lor \neg \left(x \leq 1.3 \cdot 10^{-28}\right):\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \sin \left(\frac{\varepsilon + x \cdot 2}{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -3.4) (not (<= x 1.3e-28)))
   (- (* (cos x) (+ -1.0 (cos eps))) (* (sin eps) (sin x)))
   (* (* -2.0 (sin (/ eps 2.0))) (sin (/ (+ eps (* x 2.0)) 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -3.4) || !(x <= 1.3e-28)) {
		tmp = (cos(x) * (-1.0 + cos(eps))) - (sin(eps) * sin(x));
	} else {
		tmp = (-2.0 * sin((eps / 2.0))) * sin(((eps + (x * 2.0)) / 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 <= (-3.4d0)) .or. (.not. (x <= 1.3d-28))) then
        tmp = (cos(x) * ((-1.0d0) + cos(eps))) - (sin(eps) * sin(x))
    else
        tmp = ((-2.0d0) * sin((eps / 2.0d0))) * sin(((eps + (x * 2.0d0)) / 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -3.4) || !(x <= 1.3e-28)) {
		tmp = (Math.cos(x) * (-1.0 + Math.cos(eps))) - (Math.sin(eps) * Math.sin(x));
	} else {
		tmp = (-2.0 * Math.sin((eps / 2.0))) * Math.sin(((eps + (x * 2.0)) / 2.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -3.4) or not (x <= 1.3e-28):
		tmp = (math.cos(x) * (-1.0 + math.cos(eps))) - (math.sin(eps) * math.sin(x))
	else:
		tmp = (-2.0 * math.sin((eps / 2.0))) * math.sin(((eps + (x * 2.0)) / 2.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -3.4) || !(x <= 1.3e-28))
		tmp = Float64(Float64(cos(x) * Float64(-1.0 + cos(eps))) - Float64(sin(eps) * sin(x)));
	else
		tmp = Float64(Float64(-2.0 * sin(Float64(eps / 2.0))) * sin(Float64(Float64(eps + Float64(x * 2.0)) / 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -3.4) || ~((x <= 1.3e-28)))
		tmp = (cos(x) * (-1.0 + cos(eps))) - (sin(eps) * sin(x));
	else
		tmp = (-2.0 * sin((eps / 2.0))) * sin(((eps + (x * 2.0)) / 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -3.4], N[Not[LessEqual[x, 1.3e-28]], $MachinePrecision]], N[(N[(N[Cos[x], $MachinePrecision] * N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[Sin[N[(eps / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(eps + N[(x * 2.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.39999999999999991 or 1.3e-28 < x

   1. Initial program 8.8%

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

      1. sub-neg 8.8%

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

      2. cos-sum 56.8%

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

      3. associate-+l- 56.7%

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

      4. fma-neg 56.8%

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

   3. Applied egg-rr 56.8%

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

      1. fma-neg 56.7%

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

      2. *-commutative 56.7%

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

      3. *-commutative 56.7%

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

      4. fma-neg 56.8%

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

      5. remove-double-neg 56.8%

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

   5. Simplified 56.8%

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

   6. Taylor expanded in eps around inf 56.7%

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

      1. associate--r+ 98.7%

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

      2. *-lft-identity 98.7%

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

      3. distribute-rgt-out-- 98.7%

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

      4. sub-neg 98.7%

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

      5. metadata-eval 98.7%

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

      6. +-commutative 98.7%

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

      7. *-commutative 98.7%

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

   8. Simplified 98.7%

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

   if -3.39999999999999991 < x < 1.3e-28

   1. Initial program 66.5%

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

      1. add-log-exp 66.4%

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

   3. Applied egg-rr 66.4%

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

      1. diff-cos 66.4%

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

      2. add-log-exp 87.0%

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

      3. +-commutative 87.0%

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

      4. +-commutative 87.0%

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

   5. Applied egg-rr 87.0%

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

      1. associate-*r* 87.0%

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

      2. associate--l+ 99.6%

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

      3. +-inverses 99.6%

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

      4. associate-+l+ 99.6%

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

      5. count-2 99.6%

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

   7. Simplified 99.6%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4 \lor \neg \left(x \leq 1.3 \cdot 10^{-28}\right):\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \sin \left(\frac{\varepsilon + x \cdot 2}{2}\right)\\ \end{array} \]

Alternative 7: 76.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.8%

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

   1. diff-cos 46.3%

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

   2. *-commutative 46.3%

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

   3. div-inv 46.3%

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

   4. associate--l+ 46.3%

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

   5. metadata-eval 46.3%

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

   6. div-inv 46.3%

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

   7. +-commutative 46.3%

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

   8. associate-+l+ 46.5%

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

   9. metadata-eval 46.5%

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

3. Applied egg-rr 46.5%

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

4. Taylor expanded in x around 0 75.9%

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

5. Final simplification 75.9%

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

Alternative 8: 76.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.8%

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

   1. add-log-exp 36.7%

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

3. Applied egg-rr 36.7%

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

   1. diff-cos 36.3%

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

   2. add-log-exp 46.3%

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

   3. +-commutative 46.3%

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

   4. +-commutative 46.3%

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

5. Applied egg-rr 46.3%

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

   1. associate-*r* 46.3%

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

   2. associate--l+ 75.8%

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

   3. +-inverses 75.8%

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

   4. associate-+l+ 75.9%

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

   5. count-2 75.9%

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

7. Simplified 75.9%

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

8. Final simplification 75.9%

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

Alternative 9: 66.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -6.2 \cdot 10^{-58} \lor \neg \left(\varepsilon \leq 1.7 \cdot 10^{-23}\right):\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -6.2e-58) (not (<= eps 1.7e-23)))
   (* -2.0 (pow (sin (* eps 0.5)) 2.0))
   (* eps (- (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -6.2e-58) || !(eps <= 1.7e-23)) {
		tmp = -2.0 * pow(sin((eps * 0.5)), 2.0);
	} 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 <= (-6.2d-58)) .or. (.not. (eps <= 1.7d-23))) then
        tmp = (-2.0d0) * (sin((eps * 0.5d0)) ** 2.0d0)
    else
        tmp = eps * -sin(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -6.2e-58) || !(eps <= 1.7e-23)) {
		tmp = -2.0 * Math.pow(Math.sin((eps * 0.5)), 2.0);
	} else {
		tmp = eps * -Math.sin(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -6.2e-58) or not (eps <= 1.7e-23):
		tmp = -2.0 * math.pow(math.sin((eps * 0.5)), 2.0)
	else:
		tmp = eps * -math.sin(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -6.2e-58) || !(eps <= 1.7e-23))
		tmp = Float64(-2.0 * (sin(Float64(eps * 0.5)) ^ 2.0));
	else
		tmp = Float64(eps * Float64(-sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -6.2e-58) || ~((eps <= 1.7e-23)))
		tmp = -2.0 * (sin((eps * 0.5)) ^ 2.0);
	else
		tmp = eps * -sin(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -6.2e-58], N[Not[LessEqual[eps, 1.7e-23]], $MachinePrecision]], N[(-2.0 * N[Power[N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(eps * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -6.1999999999999998e-58 or 1.7e-23 < eps

   1. Initial program 46.4%

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

      1. diff-cos 54.9%

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

      2. div-inv 54.9%

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

      3. associate--l+ 54.9%

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

      4. metadata-eval 54.9%

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

      5. div-inv 54.9%

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

      6. +-commutative 54.9%

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

      7. associate-+l+ 55.3%

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

      8. metadata-eval 55.3%

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

   3. Applied egg-rr 55.3%

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

      1. associate-*r* 55.3%

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

      2. *-commutative 55.3%

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

      3. *-commutative 55.3%

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

      4. +-commutative 55.3%

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

      5. count-2 55.3%

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

      6. fma-def 55.3%

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

      7. sub-neg 55.3%

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

      8. mul-1-neg 55.3%

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

      9. +-commutative 55.3%

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

      10. associate-+r+ 58.7%

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

      11. mul-1-neg 58.7%

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

      12. sub-neg 58.7%

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

      13. +-inverses 58.7%

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

      14. remove-double-neg 58.7%

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

      15. mul-1-neg 58.7%

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

      16. sub-neg 58.7%

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

      17. neg-sub0 58.7%

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

      18. mul-1-neg 58.7%

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

      19. remove-double-neg 58.7%

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

   5. Simplified 58.7%

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

   6. Taylor expanded in x around 0 57.1%

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

   if -6.1999999999999998e-58 < eps < 1.7e-23

   1. Initial program 23.4%

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

   2. Taylor expanded in eps around 0 89.1%

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

      1. mul-1-neg 89.1%

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

      2. *-commutative 89.1%

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

      3. distribute-rgt-neg-in 89.1%

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

   4. Simplified 89.1%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -6.2 \cdot 10^{-58} \lor \neg \left(\varepsilon \leq 1.7 \cdot 10^{-23}\right):\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \]

Alternative 10: 67.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \varepsilon - \cos x\\ \mathbf{if}\;\varepsilon \leq -12.2:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -6.2 \cdot 10^{-58}:\\ \;\;\;\;-0.5 \cdot {\varepsilon}^{2}\\ \mathbf{elif}\;\varepsilon \leq 1.65 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (cos eps) (cos x))))
   (if (<= eps -12.2)
     t_0
     (if (<= eps -6.2e-58)
       (* -0.5 (pow eps 2.0))
       (if (<= eps 1.65e-7) (* eps (- (sin x))) t_0)))))

C

double code(double x, double eps) {
	double t_0 = cos(eps) - cos(x);
	double tmp;
	if (eps <= -12.2) {
		tmp = t_0;
	} else if (eps <= -6.2e-58) {
		tmp = -0.5 * pow(eps, 2.0);
	} else if (eps <= 1.65e-7) {
		tmp = eps * -sin(x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(eps) - cos(x)
    if (eps <= (-12.2d0)) then
        tmp = t_0
    else if (eps <= (-6.2d-58)) then
        tmp = (-0.5d0) * (eps ** 2.0d0)
    else if (eps <= 1.65d-7) then
        tmp = eps * -sin(x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.cos(eps) - Math.cos(x);
	double tmp;
	if (eps <= -12.2) {
		tmp = t_0;
	} else if (eps <= -6.2e-58) {
		tmp = -0.5 * Math.pow(eps, 2.0);
	} else if (eps <= 1.65e-7) {
		tmp = eps * -Math.sin(x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.cos(eps) - math.cos(x)
	tmp = 0
	if eps <= -12.2:
		tmp = t_0
	elif eps <= -6.2e-58:
		tmp = -0.5 * math.pow(eps, 2.0)
	elif eps <= 1.65e-7:
		tmp = eps * -math.sin(x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(cos(eps) - cos(x))
	tmp = 0.0
	if (eps <= -12.2)
		tmp = t_0;
	elseif (eps <= -6.2e-58)
		tmp = Float64(-0.5 * (eps ^ 2.0));
	elseif (eps <= 1.65e-7)
		tmp = Float64(eps * Float64(-sin(x)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = cos(eps) - cos(x);
	tmp = 0.0;
	if (eps <= -12.2)
		tmp = t_0;
	elseif (eps <= -6.2e-58)
		tmp = -0.5 * (eps ^ 2.0);
	elseif (eps <= 1.65e-7)
		tmp = eps * -sin(x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Cos[eps], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -12.2], t$95$0, If[LessEqual[eps, -6.2e-58], N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 1.65e-7], N[(eps * (-N[Sin[x], $MachinePrecision])), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -12.199999999999999 or 1.6500000000000001e-7 < eps

   1. Initial program 51.6%

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

   2. Taylor expanded in x around 0 55.1%

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

   if -12.199999999999999 < eps < -6.1999999999999998e-58

   1. Initial program 6.5%

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

   2. Taylor expanded in x around 0 8.1%

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

   3. Taylor expanded in eps around 0 70.3%

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

   if -6.1999999999999998e-58 < eps < 1.6500000000000001e-7

   1. Initial program 22.8%

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

   2. Taylor expanded in eps around 0 86.8%

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

      1. mul-1-neg 86.8%

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

      2. *-commutative 86.8%

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

      3. distribute-rgt-neg-in 86.8%

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

   4. Simplified 86.8%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -12.2:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{elif}\;\varepsilon \leq -6.2 \cdot 10^{-58}:\\ \;\;\;\;-0.5 \cdot {\varepsilon}^{2}\\ \mathbf{elif}\;\varepsilon \leq 1.65 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \end{array} \]

Alternative 11: 66.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 + \cos \varepsilon\\ \mathbf{if}\;\varepsilon \leq -1.18 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -1.15 \cdot 10^{-58}:\\ \;\;\;\;-0.5 \cdot {\varepsilon}^{2}\\ \mathbf{elif}\;\varepsilon \leq 2.9 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ -1.0 (cos eps))))
   (if (<= eps -1.18e-5)
     t_0
     (if (<= eps -1.15e-58)
       (* -0.5 (pow eps 2.0))
       (if (<= eps 2.9e-7) (* eps (- (sin x))) t_0)))))

C

double code(double x, double eps) {
	double t_0 = -1.0 + cos(eps);
	double tmp;
	if (eps <= -1.18e-5) {
		tmp = t_0;
	} else if (eps <= -1.15e-58) {
		tmp = -0.5 * pow(eps, 2.0);
	} else if (eps <= 2.9e-7) {
		tmp = eps * -sin(x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-1.0d0) + cos(eps)
    if (eps <= (-1.18d-5)) then
        tmp = t_0
    else if (eps <= (-1.15d-58)) then
        tmp = (-0.5d0) * (eps ** 2.0d0)
    else if (eps <= 2.9d-7) then
        tmp = eps * -sin(x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = -1.0 + Math.cos(eps);
	double tmp;
	if (eps <= -1.18e-5) {
		tmp = t_0;
	} else if (eps <= -1.15e-58) {
		tmp = -0.5 * Math.pow(eps, 2.0);
	} else if (eps <= 2.9e-7) {
		tmp = eps * -Math.sin(x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = -1.0 + math.cos(eps)
	tmp = 0
	if eps <= -1.18e-5:
		tmp = t_0
	elif eps <= -1.15e-58:
		tmp = -0.5 * math.pow(eps, 2.0)
	elif eps <= 2.9e-7:
		tmp = eps * -math.sin(x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(-1.0 + cos(eps))
	tmp = 0.0
	if (eps <= -1.18e-5)
		tmp = t_0;
	elseif (eps <= -1.15e-58)
		tmp = Float64(-0.5 * (eps ^ 2.0));
	elseif (eps <= 2.9e-7)
		tmp = Float64(eps * Float64(-sin(x)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = -1.0 + cos(eps);
	tmp = 0.0;
	if (eps <= -1.18e-5)
		tmp = t_0;
	elseif (eps <= -1.15e-58)
		tmp = -0.5 * (eps ^ 2.0);
	elseif (eps <= 2.9e-7)
		tmp = eps * -sin(x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -1.18e-5], t$95$0, If[LessEqual[eps, -1.15e-58], N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 2.9e-7], N[(eps * (-N[Sin[x], $MachinePrecision])), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -1.18000000000000005e-5 or 2.8999999999999998e-7 < eps

   1. Initial program 50.9%

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

   2. Taylor expanded in x around 0 52.8%

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

   if -1.18000000000000005e-5 < eps < -1.1499999999999999e-58

   1. Initial program 7.1%

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

   2. Taylor expanded in x around 0 7.1%

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

   3. Taylor expanded in eps around 0 79.9%

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

   if -1.1499999999999999e-58 < eps < 2.8999999999999998e-7

   1. Initial program 22.8%

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

   2. Taylor expanded in eps around 0 86.8%

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

      1. mul-1-neg 86.8%

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

      2. *-commutative 86.8%

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

      3. distribute-rgt-neg-in 86.8%

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

   4. Simplified 86.8%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.18 \cdot 10^{-5}:\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{elif}\;\varepsilon \leq -1.15 \cdot 10^{-58}:\\ \;\;\;\;-0.5 \cdot {\varepsilon}^{2}\\ \mathbf{elif}\;\varepsilon \leq 2.9 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \cos \varepsilon\\ \end{array} \]

Alternative 12: 46.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.18 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 0.00015\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot {\varepsilon}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -1.18e-5) (not (<= eps 0.00015)))
   (+ -1.0 (cos eps))
   (* -0.5 (pow eps 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.18e-5) || !(eps <= 0.00015)) {
		tmp = -1.0 + cos(eps);
	} else {
		tmp = -0.5 * pow(eps, 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-1.18d-5)) .or. (.not. (eps <= 0.00015d0))) then
        tmp = (-1.0d0) + cos(eps)
    else
        tmp = (-0.5d0) * (eps ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.18e-5) || !(eps <= 0.00015)) {
		tmp = -1.0 + Math.cos(eps);
	} else {
		tmp = -0.5 * Math.pow(eps, 2.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -1.18e-5) or not (eps <= 0.00015):
		tmp = -1.0 + math.cos(eps)
	else:
		tmp = -0.5 * math.pow(eps, 2.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -1.18e-5) || !(eps <= 0.00015))
		tmp = Float64(-1.0 + cos(eps));
	else
		tmp = Float64(-0.5 * (eps ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -1.18e-5) || ~((eps <= 0.00015)))
		tmp = -1.0 + cos(eps);
	else
		tmp = -0.5 * (eps ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -1.18e-5], N[Not[LessEqual[eps, 0.00015]], $MachinePrecision]], N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.18000000000000005e-5 or 1.49999999999999987e-4 < eps

   1. Initial program 51.0%

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

   2. Taylor expanded in x around 0 53.0%

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

   if -1.18000000000000005e-5 < eps < 1.49999999999999987e-4

   1. Initial program 21.6%

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

   2. Taylor expanded in x around 0 21.6%

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

   3. Taylor expanded in eps around 0 40.9%

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

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.18 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 0.00015\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot {\varepsilon}^{2}\\ \end{array} \]

Alternative 13: 27.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.9 \lor \neg \left(\varepsilon \leq 1.65\right):\\ \;\;\;\;-1 - \cos x\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -1.9) (not (<= eps 1.65)))
   (- -1.0 (cos x))
   (* (* eps eps) -0.5)))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.9) || !(eps <= 1.65)) {
		tmp = -1.0 - cos(x);
	} else {
		tmp = (eps * 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 <= (-1.9d0)) .or. (.not. (eps <= 1.65d0))) then
        tmp = (-1.0d0) - cos(x)
    else
        tmp = (eps * eps) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.9) || !(eps <= 1.65)) {
		tmp = -1.0 - Math.cos(x);
	} else {
		tmp = (eps * eps) * -0.5;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -1.9) or not (eps <= 1.65):
		tmp = -1.0 - math.cos(x)
	else:
		tmp = (eps * eps) * -0.5
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -1.9) || !(eps <= 1.65))
		tmp = Float64(-1.0 - cos(x));
	else
		tmp = Float64(Float64(eps * eps) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -1.9) || ~((eps <= 1.65)))
		tmp = -1.0 - cos(x);
	else
		tmp = (eps * eps) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -1.9], N[Not[LessEqual[eps, 1.65]], $MachinePrecision]], N[(-1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision], N[(N[(eps * eps), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.8999999999999999 or 1.6499999999999999 < eps

   1. Initial program 51.8%

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

      1. add-cube-cbrt 51.8%

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

      2. pow3 51.8%

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

   3. Applied egg-rr 51.8%

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

   4. Taylor expanded in x around 0 25.5%

      \[\leadsto {\color{blue}{\left({\cos \varepsilon}^{0.3333333333333333}\right)}}^{3} - \cos x \]
   5. Step-by-step derivation

      1. unpow1/3 54.6%

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

   6. Simplified 54.6%

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

   7. Taylor expanded in eps around 0 6.7%

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

   9. Simplified 15.9%

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

   if -1.8999999999999999 < eps < 1.6499999999999999

   1. Initial program 21.3%

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

   2. Taylor expanded in x around 0 21.4%

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

   3. Taylor expanded in eps around 0 40.4%

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

      1. unpow2 99.4%

         \[\leadsto \left(\left(-0.125 \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} + 1\right) \cdot \sin x + \cos x \cdot \left(0.5 \cdot \varepsilon + -0.020833333333333332 \cdot {\varepsilon}^{3}\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]

   5. Applied egg-rr 40.4%

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

4. Final simplification 28.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.9 \lor \neg \left(\varepsilon \leq 1.65\right):\\ \;\;\;\;-1 - \cos x\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\\ \end{array} \]

Alternative 14: 46.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.18 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 0.00015\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -1.18e-5) (not (<= eps 0.00015)))
   (+ -1.0 (cos eps))
   (* (* eps eps) -0.5)))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.18e-5) || !(eps <= 0.00015)) {
		tmp = -1.0 + cos(eps);
	} else {
		tmp = (eps * 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 <= (-1.18d-5)) .or. (.not. (eps <= 0.00015d0))) then
        tmp = (-1.0d0) + cos(eps)
    else
        tmp = (eps * eps) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.18e-5) || !(eps <= 0.00015)) {
		tmp = -1.0 + Math.cos(eps);
	} else {
		tmp = (eps * eps) * -0.5;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -1.18e-5) or not (eps <= 0.00015):
		tmp = -1.0 + math.cos(eps)
	else:
		tmp = (eps * eps) * -0.5
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -1.18e-5) || !(eps <= 0.00015))
		tmp = Float64(-1.0 + cos(eps));
	else
		tmp = Float64(Float64(eps * eps) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -1.18e-5) || ~((eps <= 0.00015)))
		tmp = -1.0 + cos(eps);
	else
		tmp = (eps * eps) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -1.18e-5], N[Not[LessEqual[eps, 0.00015]], $MachinePrecision]], N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision], N[(N[(eps * eps), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.18000000000000005e-5 or 1.49999999999999987e-4 < eps

   1. Initial program 51.0%

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

   2. Taylor expanded in x around 0 53.0%

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

   if -1.18000000000000005e-5 < eps < 1.49999999999999987e-4

   1. Initial program 21.6%

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

   2. Taylor expanded in x around 0 21.6%

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

   3. Taylor expanded in eps around 0 40.9%

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

      1. unpow2 99.8%

         \[\leadsto \left(\left(-0.125 \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} + 1\right) \cdot \sin x + \cos x \cdot \left(0.5 \cdot \varepsilon + -0.020833333333333332 \cdot {\varepsilon}^{3}\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]

   5. Applied egg-rr 40.9%

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

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.18 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 0.00015\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\\ \end{array} \]

Alternative 15: 22.1% accurate, 41.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.8%

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

2. Taylor expanded in x around 0 37.8%

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

3. Taylor expanded in eps around 0 21.6%

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

   1. unpow2 49.9%

      \[\leadsto \left(\left(-0.125 \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} + 1\right) \cdot \sin x + \cos x \cdot \left(0.5 \cdot \varepsilon + -0.020833333333333332 \cdot {\varepsilon}^{3}\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]

5. Applied egg-rr 21.6%

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

6. Final simplification 21.6%

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

Alternative 16: 13.4% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

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

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 36.8%

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

   1. add-cube-cbrt 36.5%

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

   2. pow3 36.6%

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

3. Applied egg-rr 36.6%

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

4. Taylor expanded in eps around 0 11.6%

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

   1. pow-base-1 11.6%

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

   2. *-lft-identity 11.6%

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

   3. +-inverses 11.6%

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

6. Simplified 11.6%

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

7. Final simplification 11.6%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023311 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))