2cos (problem 3.3.5)

Percentage Accurate: 37.2% → 99.3%

Time: 18.6s

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, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0058:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - t_0\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.0042:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1 + \cos \varepsilon, \cos x, \sin x \cdot \left(-\sin \varepsilon\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (sin eps) (sin x))))
   (if (<= eps -0.0058)
     (- (- (* (cos x) (cos eps)) t_0) (cos x))
     (if (<= eps 0.0042)
       (-
        (*
         (cos x)
         (+ (* -0.5 (pow eps 2.0)) (* 0.041666666666666664 (pow eps 4.0))))
        t_0)
       (fma (+ -1.0 (cos eps)) (cos x) (* (sin x) (- (sin eps))))))))

C

double code(double x, double eps) {
	double t_0 = sin(eps) * sin(x);
	double tmp;
	if (eps <= -0.0058) {
		tmp = ((cos(x) * cos(eps)) - t_0) - cos(x);
	} else if (eps <= 0.0042) {
		tmp = (cos(x) * ((-0.5 * pow(eps, 2.0)) + (0.041666666666666664 * pow(eps, 4.0)))) - t_0;
	} else {
		tmp = fma((-1.0 + cos(eps)), cos(x), (sin(x) * -sin(eps)));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(sin(eps) * sin(x))
	tmp = 0.0
	if (eps <= -0.0058)
		tmp = Float64(Float64(Float64(cos(x) * cos(eps)) - t_0) - cos(x));
	elseif (eps <= 0.0042)
		tmp = Float64(Float64(cos(x) * Float64(Float64(-0.5 * (eps ^ 2.0)) + Float64(0.041666666666666664 * (eps ^ 4.0)))) - t_0);
	else
		tmp = fma(Float64(-1.0 + cos(eps)), cos(x), Float64(sin(x) * Float64(-sin(eps))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.0058], N[(N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.0042], N[(N[(N[Cos[x], $MachinePrecision] * N[(N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.041666666666666664 * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * (-N[Sin[eps], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.0058

   1. Initial program 59.2%

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

      1. cos-sum 98.3%

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

   3. Applied egg-rr 98.3%

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

   if -0.0058 < eps < 0.00419999999999999974

   1. Initial program 17.1%

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

      1. sub-neg 17.1%

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

      2. cos-sum 18.7%

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

      3. associate-+l- 18.7%

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

      4. fma-neg 18.7%

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

   3. Applied egg-rr 18.7%

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

   4. Taylor expanded in x around inf 18.7%

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

      1. associate--r+ 83.1%

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

      2. *-commutative 83.1%

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

      3. *-rgt-identity 83.1%

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

      4. distribute-lft-out-- 83.1%

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

      5. sub-neg 83.1%

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

      6. metadata-eval 83.1%

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

      7. +-commutative 83.1%

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

      8. *-commutative 83.1%

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

   6. Simplified 83.1%

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

   7. Taylor expanded in eps around 0 99.7%

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

   if 0.00419999999999999974 < eps

   1. Initial program 58.0%

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

      1. sub-neg 58.0%

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

      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- 98.9%

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

      4. fma-neg 99.0%

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

   3. Applied egg-rr 99.0%

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

   4. Taylor expanded in x around inf 98.9%

      \[\leadsto \color{blue}{\cos \varepsilon \cdot \cos x - \left(\cos x + \sin \varepsilon \cdot \sin x\right)} \]
   5. 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. *-commutative 98.8%

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

      3. *-rgt-identity 98.8%

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

      4. distribute-lft-out-- 98.9%

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

      5. sub-neg 98.9%

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

      6. metadata-eval 98.9%

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

      7. +-commutative 98.9%

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

      8. *-commutative 98.9%

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

   6. Simplified 98.9%

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

      1. +-commutative 98.9%

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

      2. *-commutative 98.9%

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

      3. fma-neg 99.0%

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

      4. +-commutative 99.0%

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

      5. distribute-rgt-neg-in 99.0%

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

   8. Applied egg-rr 99.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0058:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin \varepsilon \cdot \sin x\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.0042:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1 + \cos \varepsilon, \cos x, \sin x \cdot \left(-\sin \varepsilon\right)\right)\\ \end{array} \]

Alternative 2: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{{\sin \varepsilon}^{2} \cdot \cos x}{-1 - \cos \varepsilon} - \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \varepsilon \cdot \sin x\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (-
  (/ (* (pow (sin eps) 2.0) (cos x)) (- -1.0 (cos eps)))
  (expm1 (log1p (* (sin eps) (sin x))))))

C

double code(double x, double eps) {
	return ((pow(sin(eps), 2.0) * cos(x)) / (-1.0 - cos(eps))) - expm1(log1p((sin(eps) * sin(x))));
}

Java

public static double code(double x, double eps) {
	return ((Math.pow(Math.sin(eps), 2.0) * Math.cos(x)) / (-1.0 - Math.cos(eps))) - Math.expm1(Math.log1p((Math.sin(eps) * Math.sin(x))));
}

Python

def code(x, eps):
	return ((math.pow(math.sin(eps), 2.0) * math.cos(x)) / (-1.0 - math.cos(eps))) - math.expm1(math.log1p((math.sin(eps) * math.sin(x))))

Julia

function code(x, eps)
	return Float64(Float64(Float64((sin(eps) ^ 2.0) * cos(x)) / Float64(-1.0 - cos(eps))) - expm1(log1p(Float64(sin(eps) * sin(x)))))
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[Power[N[Sin[eps], $MachinePrecision], 2.0], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[(-1.0 - N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(Exp[N[Log[1 + N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.4%

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

   1. sub-neg 38.4%

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

   2. cos-sum 59.6%

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

   3. associate-+l- 59.6%

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

   4. fma-neg 59.6%

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

3. Applied egg-rr 59.6%

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

4. Taylor expanded in x around inf 59.6%

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

   1. associate--r+ 91.0%

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

   2. *-commutative 91.0%

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

   3. *-rgt-identity 91.0%

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

   4. distribute-lft-out-- 91.0%

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

   5. sub-neg 91.0%

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

   6. metadata-eval 91.0%

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

   7. +-commutative 91.0%

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

   8. *-commutative 91.0%

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

6. Simplified 91.0%

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

   1. +-commutative 91.0%

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

   2. *-commutative 91.0%

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

   3. +-commutative 91.0%

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

   4. flip-+ 90.6%

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

   5. associate-*l/ 90.6%

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

   6. metadata-eval 90.6%

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

   7. 1-sub-cos 98.8%

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

   8. pow2 98.8%

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

8. Applied egg-rr 98.8%

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

   1. expm1-log1p-u 98.8%

      \[\leadsto \frac{{\sin \varepsilon}^{2} \cdot \cos x}{-1 - \cos \varepsilon} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin x \cdot \sin \varepsilon\right)\right)} \]

   2. *-commutative 98.8%

      \[\leadsto \frac{{\sin \varepsilon}^{2} \cdot \cos x}{-1 - \cos \varepsilon} - \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sin \varepsilon \cdot \sin x}\right)\right) \]

10. Applied egg-rr 98.8%

    \[\leadsto \frac{{\sin \varepsilon}^{2} \cdot \cos x}{-1 - \cos \varepsilon} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \varepsilon \cdot \sin x\right)\right)} \]

11. Final simplification 98.8%

    \[\leadsto \frac{{\sin \varepsilon}^{2} \cdot \cos x}{-1 - \cos \varepsilon} - \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \varepsilon \cdot \sin x\right)\right) \]

Alternative 3: 99.0% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.4%

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

   1. sub-neg 38.4%

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

   2. cos-sum 59.6%

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

   3. associate-+l- 59.6%

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

   4. fma-neg 59.6%

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

3. Applied egg-rr 59.6%

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

4. Taylor expanded in x around inf 59.6%

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

   1. associate--r+ 91.0%

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

   2. *-commutative 91.0%

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

   3. *-rgt-identity 91.0%

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

   4. distribute-lft-out-- 91.0%

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

   5. sub-neg 91.0%

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

   6. metadata-eval 91.0%

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

   7. +-commutative 91.0%

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

   8. *-commutative 91.0%

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

6. Simplified 91.0%

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

   1. +-commutative 91.0%

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

   2. *-commutative 91.0%

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

   3. +-commutative 91.0%

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

   4. flip-+ 90.6%

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

   5. associate-*l/ 90.6%

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

   6. metadata-eval 90.6%

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

   7. 1-sub-cos 98.8%

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

   8. pow2 98.8%

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

8. Applied egg-rr 98.8%

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

9. Final simplification 98.8%

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

Alternative 4: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\varepsilon \cdot 0.5\right)\\ \mathbf{if}\;x \leq -4 \cdot 10^{-8} \lor \neg \left(x \leq 2.15 \cdot 10^{-6}\right):\\ \;\;\;\;\mathsf{fma}\left(-1 + \cos \varepsilon, \cos x, \sin x \cdot \left(-\sin \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-0.5 \cdot {x}^{2} + 1\right) \cdot t_0 + x \cdot \cos \left(\varepsilon \cdot 0.5\right)\right) \cdot \left(t_0 \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (* eps 0.5))))
   (if (or (<= x -4e-8) (not (<= x 2.15e-6)))
     (fma (+ -1.0 (cos eps)) (cos x) (* (sin x) (- (sin eps))))
     (*
      (+ (* (+ (* -0.5 (pow x 2.0)) 1.0) t_0) (* x (cos (* eps 0.5))))
      (* t_0 -2.0)))))

C

double code(double x, double eps) {
	double t_0 = sin((eps * 0.5));
	double tmp;
	if ((x <= -4e-8) || !(x <= 2.15e-6)) {
		tmp = fma((-1.0 + cos(eps)), cos(x), (sin(x) * -sin(eps)));
	} else {
		tmp = ((((-0.5 * pow(x, 2.0)) + 1.0) * t_0) + (x * cos((eps * 0.5)))) * (t_0 * -2.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = sin(Float64(eps * 0.5))
	tmp = 0.0
	if ((x <= -4e-8) || !(x <= 2.15e-6))
		tmp = fma(Float64(-1.0 + cos(eps)), cos(x), Float64(sin(x) * Float64(-sin(eps))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(-0.5 * (x ^ 2.0)) + 1.0) * t_0) + Float64(x * cos(Float64(eps * 0.5)))) * Float64(t_0 * -2.0));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[x, -4e-8], N[Not[LessEqual[x, 2.15e-6]], $MachinePrecision]], N[(N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * (-N[Sin[eps], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-0.5 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision] + N[(x * N[Cos[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * -2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.0000000000000001e-8 or 2.15000000000000017e-6 < x

   1. Initial program 8.5%

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

      1. sub-neg 8.5%

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

      2. cos-sum 51.4%

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

      3. associate-+l- 51.4%

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

      4. fma-neg 51.4%

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

   3. Applied egg-rr 51.4%

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

   4. Taylor expanded in x around inf 51.4%

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

      1. associate--r+ 98.6%

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

      2. *-commutative 98.6%

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

      3. *-rgt-identity 98.6%

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

      4. distribute-lft-out-- 98.6%

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

      5. sub-neg 98.6%

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

      6. metadata-eval 98.6%

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

      7. +-commutative 98.6%

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

      8. *-commutative 98.6%

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

   6. Simplified 98.6%

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

      1. +-commutative 98.6%

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

      2. *-commutative 98.6%

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

      3. fma-neg 98.7%

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

      4. +-commutative 98.7%

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

      5. distribute-rgt-neg-in 98.7%

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

   8. Applied egg-rr 98.7%

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

   if -4.0000000000000001e-8 < x < 2.15000000000000017e-6

   1. Initial program 66.5%

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

      1. diff-cos 82.3%

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

      2. div-inv 82.3%

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

      3. associate--l+ 82.3%

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

      4. metadata-eval 82.3%

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

      5. div-inv 82.3%

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

      6. +-commutative 82.3%

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

      7. associate-+l+ 82.3%

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

      8. metadata-eval 82.3%

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

   3. Applied egg-rr 82.3%

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

      1. associate-*r* 82.3%

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

      2. *-commutative 82.3%

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

      3. *-commutative 82.3%

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

      4. +-commutative 82.3%

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

      5. count-2 82.3%

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

      6. fma-def 82.3%

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

      7. sub-neg 82.3%

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

      8. mul-1-neg 82.3%

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

      9. +-commutative 82.3%

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

      10. associate-+r+ 98.7%

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

      11. mul-1-neg 98.7%

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

      12. sub-neg 98.7%

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

      13. +-inverses 98.7%

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

      14. remove-double-neg 98.7%

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

      15. mul-1-neg 98.7%

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

      16. sub-neg 98.7%

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

      17. neg-sub0 98.7%

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

      18. mul-1-neg 98.7%

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

      19. remove-double-neg 98.7%

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

   5. Simplified 98.7%

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

   6. Taylor expanded in x around 0 99.5%

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

      1. associate-+r+ 99.5%

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

      2. associate-*r* 99.5%

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

      3. distribute-rgt1-in 99.5%

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

   8. Simplified 99.5%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-8} \lor \neg \left(x \leq 2.15 \cdot 10^{-6}\right):\\ \;\;\;\;\mathsf{fma}\left(-1 + \cos \varepsilon, \cos x, \sin x \cdot \left(-\sin \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-0.5 \cdot {x}^{2} + 1\right) \cdot \sin \left(\varepsilon \cdot 0.5\right) + x \cdot \cos \left(\varepsilon \cdot 0.5\right)\right) \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot -2\right)\\ \end{array} \]

Alternative 5: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\varepsilon \cdot 0.5\right)\\ \mathbf{if}\;x \leq -2.6 \cdot 10^{-8} \lor \neg \left(x \leq 2.15 \cdot 10^{-6}\right):\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-0.5 \cdot {x}^{2} + 1\right) \cdot t_0 + x \cdot \cos \left(\varepsilon \cdot 0.5\right)\right) \cdot \left(t_0 \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (* eps 0.5))))
   (if (or (<= x -2.6e-8) (not (<= x 2.15e-6)))
     (- (* (cos x) (+ -1.0 (cos eps))) (* (sin eps) (sin x)))
     (*
      (+ (* (+ (* -0.5 (pow x 2.0)) 1.0) t_0) (* x (cos (* eps 0.5))))
      (* t_0 -2.0)))))

C

double code(double x, double eps) {
	double t_0 = sin((eps * 0.5));
	double tmp;
	if ((x <= -2.6e-8) || !(x <= 2.15e-6)) {
		tmp = (cos(x) * (-1.0 + cos(eps))) - (sin(eps) * sin(x));
	} else {
		tmp = ((((-0.5 * pow(x, 2.0)) + 1.0) * t_0) + (x * cos((eps * 0.5)))) * (t_0 * -2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((eps * 0.5d0))
    if ((x <= (-2.6d-8)) .or. (.not. (x <= 2.15d-6))) then
        tmp = (cos(x) * ((-1.0d0) + cos(eps))) - (sin(eps) * sin(x))
    else
        tmp = (((((-0.5d0) * (x ** 2.0d0)) + 1.0d0) * t_0) + (x * cos((eps * 0.5d0)))) * (t_0 * (-2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin((eps * 0.5));
	double tmp;
	if ((x <= -2.6e-8) || !(x <= 2.15e-6)) {
		tmp = (Math.cos(x) * (-1.0 + Math.cos(eps))) - (Math.sin(eps) * Math.sin(x));
	} else {
		tmp = ((((-0.5 * Math.pow(x, 2.0)) + 1.0) * t_0) + (x * Math.cos((eps * 0.5)))) * (t_0 * -2.0);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.sin((eps * 0.5))
	tmp = 0
	if (x <= -2.6e-8) or not (x <= 2.15e-6):
		tmp = (math.cos(x) * (-1.0 + math.cos(eps))) - (math.sin(eps) * math.sin(x))
	else:
		tmp = ((((-0.5 * math.pow(x, 2.0)) + 1.0) * t_0) + (x * math.cos((eps * 0.5)))) * (t_0 * -2.0)
	return tmp

Julia

function code(x, eps)
	t_0 = sin(Float64(eps * 0.5))
	tmp = 0.0
	if ((x <= -2.6e-8) || !(x <= 2.15e-6))
		tmp = Float64(Float64(cos(x) * Float64(-1.0 + cos(eps))) - Float64(sin(eps) * sin(x)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(-0.5 * (x ^ 2.0)) + 1.0) * t_0) + Float64(x * cos(Float64(eps * 0.5)))) * Float64(t_0 * -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = sin((eps * 0.5));
	tmp = 0.0;
	if ((x <= -2.6e-8) || ~((x <= 2.15e-6)))
		tmp = (cos(x) * (-1.0 + cos(eps))) - (sin(eps) * sin(x));
	else
		tmp = ((((-0.5 * (x ^ 2.0)) + 1.0) * t_0) + (x * cos((eps * 0.5)))) * (t_0 * -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[x, -2.6e-8], N[Not[LessEqual[x, 2.15e-6]], $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[(N[(-0.5 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision] + N[(x * N[Cos[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * -2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.6000000000000001e-8 or 2.15000000000000017e-6 < x

   1. Initial program 8.5%

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

      1. sub-neg 8.5%

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

      2. cos-sum 51.4%

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

      3. associate-+l- 51.4%

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

      4. fma-neg 51.4%

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

   3. Applied egg-rr 51.4%

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

   4. Taylor expanded in x around inf 51.4%

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

      1. associate--r+ 98.6%

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

      2. *-commutative 98.6%

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

      3. *-rgt-identity 98.6%

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

      4. distribute-lft-out-- 98.6%

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

      5. sub-neg 98.6%

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

      6. metadata-eval 98.6%

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

      7. +-commutative 98.6%

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

      8. *-commutative 98.6%

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

   6. Simplified 98.6%

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

   if -2.6000000000000001e-8 < x < 2.15000000000000017e-6

   1. Initial program 66.5%

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

      1. diff-cos 82.3%

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

      2. div-inv 82.3%

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

      3. associate--l+ 82.3%

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

      4. metadata-eval 82.3%

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

      5. div-inv 82.3%

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

      6. +-commutative 82.3%

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

      7. associate-+l+ 82.3%

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

      8. metadata-eval 82.3%

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

   3. Applied egg-rr 82.3%

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

      1. associate-*r* 82.3%

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

      2. *-commutative 82.3%

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

      3. *-commutative 82.3%

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

      4. +-commutative 82.3%

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

      5. count-2 82.3%

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

      6. fma-def 82.3%

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

      7. sub-neg 82.3%

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

      8. mul-1-neg 82.3%

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

      9. +-commutative 82.3%

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

      10. associate-+r+ 98.7%

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

      11. mul-1-neg 98.7%

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

      12. sub-neg 98.7%

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

      13. +-inverses 98.7%

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

      14. remove-double-neg 98.7%

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

      15. mul-1-neg 98.7%

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

      16. sub-neg 98.7%

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

      17. neg-sub0 98.7%

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

      18. mul-1-neg 98.7%

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

      19. remove-double-neg 98.7%

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

   5. Simplified 98.7%

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

   6. Taylor expanded in x around 0 99.5%

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

      1. associate-+r+ 99.5%

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

      2. associate-*r* 99.5%

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

      3. distribute-rgt1-in 99.5%

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

   8. Simplified 99.5%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-8} \lor \neg \left(x \leq 2.15 \cdot 10^{-6}\right):\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-0.5 \cdot {x}^{2} + 1\right) \cdot \sin \left(\varepsilon \cdot 0.5\right) + x \cdot \cos \left(\varepsilon \cdot 0.5\right)\right) \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot -2\right)\\ \end{array} \]

Alternative 6: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\varepsilon \cdot 0.5\right)\\ \mathbf{if}\;x \leq -2.4 \cdot 10^{-8} \lor \neg \left(x \leq 9.5 \cdot 10^{-39}\right):\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 \cdot -2\right) \cdot \left(t_0 + x \cdot \cos \left(\varepsilon \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (* eps 0.5))))
   (if (or (<= x -2.4e-8) (not (<= x 9.5e-39)))
     (- (* (cos x) (+ -1.0 (cos eps))) (* (sin eps) (sin x)))
     (* (* t_0 -2.0) (+ t_0 (* x (cos (* eps 0.5))))))))

C

double code(double x, double eps) {
	double t_0 = sin((eps * 0.5));
	double tmp;
	if ((x <= -2.4e-8) || !(x <= 9.5e-39)) {
		tmp = (cos(x) * (-1.0 + cos(eps))) - (sin(eps) * sin(x));
	} else {
		tmp = (t_0 * -2.0) * (t_0 + (x * cos((eps * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((eps * 0.5d0))
    if ((x <= (-2.4d-8)) .or. (.not. (x <= 9.5d-39))) then
        tmp = (cos(x) * ((-1.0d0) + cos(eps))) - (sin(eps) * sin(x))
    else
        tmp = (t_0 * (-2.0d0)) * (t_0 + (x * cos((eps * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin((eps * 0.5));
	double tmp;
	if ((x <= -2.4e-8) || !(x <= 9.5e-39)) {
		tmp = (Math.cos(x) * (-1.0 + Math.cos(eps))) - (Math.sin(eps) * Math.sin(x));
	} else {
		tmp = (t_0 * -2.0) * (t_0 + (x * Math.cos((eps * 0.5))));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.sin((eps * 0.5))
	tmp = 0
	if (x <= -2.4e-8) or not (x <= 9.5e-39):
		tmp = (math.cos(x) * (-1.0 + math.cos(eps))) - (math.sin(eps) * math.sin(x))
	else:
		tmp = (t_0 * -2.0) * (t_0 + (x * math.cos((eps * 0.5))))
	return tmp

Julia

function code(x, eps)
	t_0 = sin(Float64(eps * 0.5))
	tmp = 0.0
	if ((x <= -2.4e-8) || !(x <= 9.5e-39))
		tmp = Float64(Float64(cos(x) * Float64(-1.0 + cos(eps))) - Float64(sin(eps) * sin(x)));
	else
		tmp = Float64(Float64(t_0 * -2.0) * Float64(t_0 + Float64(x * cos(Float64(eps * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = sin((eps * 0.5));
	tmp = 0.0;
	if ((x <= -2.4e-8) || ~((x <= 9.5e-39)))
		tmp = (cos(x) * (-1.0 + cos(eps))) - (sin(eps) * sin(x));
	else
		tmp = (t_0 * -2.0) * (t_0 + (x * cos((eps * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[x, -2.4e-8], N[Not[LessEqual[x, 9.5e-39]], $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[(t$95$0 * -2.0), $MachinePrecision] * N[(t$95$0 + N[(x * N[Cos[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.39999999999999998e-8 or 9.4999999999999999e-39 < x

   1. Initial program 12.0%

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

      1. sub-neg 12.0%

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

      2. cos-sum 52.9%

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

      3. associate-+l- 52.9%

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

      4. fma-neg 52.9%

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

   3. Applied egg-rr 52.9%

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

   4. Taylor expanded in x around inf 52.9%

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

      1. associate--r+ 98.6%

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

      2. *-commutative 98.6%

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

      3. *-rgt-identity 98.6%

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

      4. distribute-lft-out-- 98.7%

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

      5. sub-neg 98.7%

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

      6. metadata-eval 98.7%

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

      7. +-commutative 98.7%

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

      8. *-commutative 98.7%

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

   6. Simplified 98.7%

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

   if -2.39999999999999998e-8 < x < 9.4999999999999999e-39

   1. Initial program 66.5%

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

      1. diff-cos 83.3%

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

      2. div-inv 83.3%

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

      3. associate--l+ 83.3%

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

      4. metadata-eval 83.3%

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

      5. div-inv 83.3%

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

      6. +-commutative 83.3%

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

      7. associate-+l+ 83.3%

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

      8. metadata-eval 83.3%

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

   3. Applied egg-rr 83.3%

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

      1. associate-*r* 83.3%

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

      2. *-commutative 83.3%

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

      3. *-commutative 83.3%

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

      4. +-commutative 83.3%

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

      5. count-2 83.3%

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

      6. fma-def 83.3%

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

      7. sub-neg 83.3%

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

      8. mul-1-neg 83.3%

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

      9. +-commutative 83.3%

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

      10. associate-+r+ 99.2%

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

      11. mul-1-neg 99.2%

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

      12. sub-neg 99.2%

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

      13. +-inverses 99.2%

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

      14. remove-double-neg 99.2%

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

      15. mul-1-neg 99.2%

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

      16. sub-neg 99.2%

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

      17. neg-sub0 99.2%

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

      18. mul-1-neg 99.2%

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

      19. remove-double-neg 99.2%

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

   5. Simplified 99.2%

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

   6. Taylor expanded in x around 0 99.5%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{-8} \lor \neg \left(x \leq 9.5 \cdot 10^{-39}\right):\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\left(\sin \left(\varepsilon \cdot 0.5\right) \cdot -2\right) \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) + x \cdot \cos \left(\varepsilon \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 7: 67.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\varepsilon \cdot 0.5\right)\\ \mathbf{if}\;\cos \left(\varepsilon + x\right) - \cos x \leq -0.0002:\\ \;\;\;\;-2 \cdot {t_0}^{2}\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(t_0 \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (* eps 0.5))))
   (if (<= (- (cos (+ eps x)) (cos x)) -0.0002)
     (* -2.0 (pow t_0 2.0))
     (* (sin x) (* t_0 -2.0)))))

C

double code(double x, double eps) {
	double t_0 = sin((eps * 0.5));
	double tmp;
	if ((cos((eps + x)) - cos(x)) <= -0.0002) {
		tmp = -2.0 * pow(t_0, 2.0);
	} else {
		tmp = sin(x) * (t_0 * -2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((eps * 0.5d0))
    if ((cos((eps + x)) - cos(x)) <= (-0.0002d0)) then
        tmp = (-2.0d0) * (t_0 ** 2.0d0)
    else
        tmp = sin(x) * (t_0 * (-2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin((eps * 0.5));
	double tmp;
	if ((Math.cos((eps + x)) - Math.cos(x)) <= -0.0002) {
		tmp = -2.0 * Math.pow(t_0, 2.0);
	} else {
		tmp = Math.sin(x) * (t_0 * -2.0);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.sin((eps * 0.5))
	tmp = 0
	if (math.cos((eps + x)) - math.cos(x)) <= -0.0002:
		tmp = -2.0 * math.pow(t_0, 2.0)
	else:
		tmp = math.sin(x) * (t_0 * -2.0)
	return tmp

Julia

function code(x, eps)
	t_0 = sin(Float64(eps * 0.5))
	tmp = 0.0
	if (Float64(cos(Float64(eps + x)) - cos(x)) <= -0.0002)
		tmp = Float64(-2.0 * (t_0 ^ 2.0));
	else
		tmp = Float64(sin(x) * Float64(t_0 * -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = sin((eps * 0.5));
	tmp = 0.0;
	if ((cos((eps + x)) - cos(x)) <= -0.0002)
		tmp = -2.0 * (t_0 ^ 2.0);
	else
		tmp = sin(x) * (t_0 * -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(eps + x), $MachinePrecision]], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], -0.0002], N[(-2.0 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * N[(t$95$0 * -2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cos.f64 (+.f64 x eps)) (cos.f64 x)) < -2.0000000000000001e-4

   1. Initial program 79.2%

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

      1. diff-cos 79.8%

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

      2. div-inv 79.8%

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

      3. associate--l+ 79.8%

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

      4. metadata-eval 79.8%

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

      5. div-inv 79.8%

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

      6. +-commutative 79.8%

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

      7. associate-+l+ 79.9%

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

      8. metadata-eval 79.9%

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

   3. Applied egg-rr 79.9%

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

      1. associate-*r* 79.9%

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

      2. *-commutative 79.9%

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

      3. *-commutative 79.9%

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

      4. +-commutative 79.9%

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

      5. count-2 79.9%

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

      6. fma-def 79.9%

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

      7. sub-neg 79.9%

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

      8. mul-1-neg 79.9%

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

      9. +-commutative 79.9%

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

      10. associate-+r+ 80.0%

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

      11. mul-1-neg 80.0%

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

      12. sub-neg 80.0%

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

      13. +-inverses 80.0%

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

      14. remove-double-neg 80.0%

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

      15. mul-1-neg 80.0%

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

      16. sub-neg 80.0%

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

      17. neg-sub0 80.0%

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

      18. mul-1-neg 80.0%

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

      19. remove-double-neg 80.0%

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

   5. Simplified 80.0%

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

   6. Taylor expanded in x around 0 79.9%

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

   if -2.0000000000000001e-4 < (-.f64 (cos.f64 (+.f64 x eps)) (cos.f64 x))

   1. Initial program 15.1%

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

      1. diff-cos 26.6%

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

      2. div-inv 26.6%

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

      3. associate--l+ 26.6%

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

      4. metadata-eval 26.6%

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

      5. div-inv 26.6%

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

      6. +-commutative 26.6%

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

      7. associate-+l+ 26.6%

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

      8. metadata-eval 26.6%

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

   3. Applied egg-rr 26.6%

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

      1. associate-*r* 26.6%

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

      2. *-commutative 26.6%

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

      3. *-commutative 26.6%

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

      4. +-commutative 26.6%

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

      5. count-2 26.6%

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

      6. fma-def 26.6%

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

      7. sub-neg 26.6%

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

      8. mul-1-neg 26.6%

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

      9. +-commutative 26.6%

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

      10. associate-+r+ 77.6%

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

      11. mul-1-neg 77.6%

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

      12. sub-neg 77.6%

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

      13. +-inverses 77.6%

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

      14. remove-double-neg 77.6%

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

      15. mul-1-neg 77.6%

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

      16. sub-neg 77.6%

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

      17. neg-sub0 77.6%

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

      18. mul-1-neg 77.6%

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

      19. remove-double-neg 77.6%

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

   5. Simplified 77.6%

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

   6. Taylor expanded in eps around 0 65.2%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\varepsilon + x\right) - \cos x \leq -0.0002:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot -2\right)\\ \end{array} \]

Alternative 8: 76.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 38.4%

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

   1. diff-cos 45.9%

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

   2. div-inv 45.9%

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

   3. associate--l+ 45.9%

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

   4. metadata-eval 45.9%

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

   5. div-inv 45.9%

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

   6. +-commutative 45.9%

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

   7. associate-+l+ 46.0%

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

   8. metadata-eval 46.0%

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

3. Applied egg-rr 46.0%

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

   1. associate-*r* 46.0%

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

   2. *-commutative 46.0%

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

   3. *-commutative 46.0%

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

   4. +-commutative 46.0%

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

   5. count-2 46.0%

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

   6. fma-def 46.0%

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

   7. sub-neg 46.0%

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

   8. mul-1-neg 46.0%

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

   9. +-commutative 46.0%

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

   10. associate-+r+ 78.5%

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

   11. mul-1-neg 78.5%

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

   12. sub-neg 78.5%

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

   13. +-inverses 78.5%

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

   14. remove-double-neg 78.5%

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

   15. mul-1-neg 78.5%

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

   16. sub-neg 78.5%

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

   17. neg-sub0 78.5%

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

   18. mul-1-neg 78.5%

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

   19. remove-double-neg 78.5%

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

5. Simplified 78.5%

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

6. Final simplification 78.5%

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

Alternative 9: 73.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-1 + \cos \varepsilon\right) - \sin \varepsilon \cdot x\\ t_1 := \sin \left(\varepsilon \cdot 0.5\right)\\ t_2 := \sin x \cdot \left(t_1 \cdot -2\right)\\ \mathbf{if}\;x \leq -0.0008:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-58}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-119}:\\ \;\;\;\;-2 \cdot {t_1}^{2}\\ \mathbf{elif}\;x \leq 0.00072:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (+ -1.0 (cos eps)) (* (sin eps) x)))
        (t_1 (sin (* eps 0.5)))
        (t_2 (* (sin x) (* t_1 -2.0))))
   (if (<= x -0.0008)
     t_2
     (if (<= x -5.8e-58)
       t_0
       (if (<= x 3.8e-119)
         (* -2.0 (pow t_1 2.0))
         (if (<= x 0.00072) t_0 t_2))))))

C

double code(double x, double eps) {
	double t_0 = (-1.0 + cos(eps)) - (sin(eps) * x);
	double t_1 = sin((eps * 0.5));
	double t_2 = sin(x) * (t_1 * -2.0);
	double tmp;
	if (x <= -0.0008) {
		tmp = t_2;
	} else if (x <= -5.8e-58) {
		tmp = t_0;
	} else if (x <= 3.8e-119) {
		tmp = -2.0 * pow(t_1, 2.0);
	} else if (x <= 0.00072) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = ((-1.0d0) + cos(eps)) - (sin(eps) * x)
    t_1 = sin((eps * 0.5d0))
    t_2 = sin(x) * (t_1 * (-2.0d0))
    if (x <= (-0.0008d0)) then
        tmp = t_2
    else if (x <= (-5.8d-58)) then
        tmp = t_0
    else if (x <= 3.8d-119) then
        tmp = (-2.0d0) * (t_1 ** 2.0d0)
    else if (x <= 0.00072d0) then
        tmp = t_0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = (-1.0 + Math.cos(eps)) - (Math.sin(eps) * x);
	double t_1 = Math.sin((eps * 0.5));
	double t_2 = Math.sin(x) * (t_1 * -2.0);
	double tmp;
	if (x <= -0.0008) {
		tmp = t_2;
	} else if (x <= -5.8e-58) {
		tmp = t_0;
	} else if (x <= 3.8e-119) {
		tmp = -2.0 * Math.pow(t_1, 2.0);
	} else if (x <= 0.00072) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = (-1.0 + math.cos(eps)) - (math.sin(eps) * x)
	t_1 = math.sin((eps * 0.5))
	t_2 = math.sin(x) * (t_1 * -2.0)
	tmp = 0
	if x <= -0.0008:
		tmp = t_2
	elif x <= -5.8e-58:
		tmp = t_0
	elif x <= 3.8e-119:
		tmp = -2.0 * math.pow(t_1, 2.0)
	elif x <= 0.00072:
		tmp = t_0
	else:
		tmp = t_2
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64(-1.0 + cos(eps)) - Float64(sin(eps) * x))
	t_1 = sin(Float64(eps * 0.5))
	t_2 = Float64(sin(x) * Float64(t_1 * -2.0))
	tmp = 0.0
	if (x <= -0.0008)
		tmp = t_2;
	elseif (x <= -5.8e-58)
		tmp = t_0;
	elseif (x <= 3.8e-119)
		tmp = Float64(-2.0 * (t_1 ^ 2.0));
	elseif (x <= 0.00072)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (-1.0 + cos(eps)) - (sin(eps) * x);
	t_1 = sin((eps * 0.5));
	t_2 = sin(x) * (t_1 * -2.0);
	tmp = 0.0;
	if (x <= -0.0008)
		tmp = t_2;
	elseif (x <= -5.8e-58)
		tmp = t_0;
	elseif (x <= 3.8e-119)
		tmp = -2.0 * (t_1 ^ 2.0);
	elseif (x <= 0.00072)
		tmp = t_0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[eps], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[x], $MachinePrecision] * N[(t$95$1 * -2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0008], t$95$2, If[LessEqual[x, -5.8e-58], t$95$0, If[LessEqual[x, 3.8e-119], N[(-2.0 * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.00072], t$95$0, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.00000000000000038e-4 or 7.20000000000000045e-4 < x

   1. Initial program 8.5%

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

      1. diff-cos 7.3%

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

      2. div-inv 7.3%

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

      3. associate--l+ 7.3%

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

      4. metadata-eval 7.3%

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

      5. div-inv 7.3%

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

      6. +-commutative 7.3%

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

      7. associate-+l+ 7.4%

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

      8. metadata-eval 7.4%

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

   3. Applied egg-rr 7.4%

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

      1. associate-*r* 7.4%

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

      2. *-commutative 7.4%

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

      3. *-commutative 7.4%

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

      4. +-commutative 7.4%

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

      5. count-2 7.4%

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

      6. fma-def 7.4%

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

      7. sub-neg 7.4%

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

      8. mul-1-neg 7.4%

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

      9. +-commutative 7.4%

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

      10. associate-+r+ 56.9%

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

      11. mul-1-neg 56.9%

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

      12. sub-neg 56.9%

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

      13. +-inverses 56.9%

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

      14. remove-double-neg 56.9%

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

      15. mul-1-neg 56.9%

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

      16. sub-neg 56.9%

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

      17. neg-sub0 56.9%

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

      18. mul-1-neg 56.9%

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

      19. remove-double-neg 56.9%

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

   5. Simplified 56.9%

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

   6. Taylor expanded in eps around 0 56.6%

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

   if -8.00000000000000038e-4 < x < -5.7999999999999998e-58 or 3.79999999999999975e-119 < x < 7.20000000000000045e-4

   1. Initial program 52.2%

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

   2. Taylor expanded in x around 0 54.3%

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

      1. sub-neg 54.3%

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

      2. metadata-eval 54.3%

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

      3. +-commutative 54.3%

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

      4. associate-+r+ 93.4%

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

      5. +-commutative 93.4%

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

      6. +-commutative 93.4%

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

      7. mul-1-neg 93.4%

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

      8. unsub-neg 93.4%

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

      9. unpow2 93.4%

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

      10. associate-*l* 93.4%

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

      11. distribute-lft-out-- 93.4%

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

      12. +-commutative 93.4%

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

      13. *-commutative 93.4%

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

      14. fma-def 93.4%

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

   4. Simplified 93.4%

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

   5. Taylor expanded in x around 0 93.0%

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

      1. mul-1-neg 93.0%

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

      2. distribute-rgt-neg-out 93.0%

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

   7. Simplified 93.0%

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

   if -5.7999999999999998e-58 < x < 3.79999999999999975e-119

   1. Initial program 75.7%

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

      1. diff-cos 97.5%

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

      2. div-inv 97.5%

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

      3. associate--l+ 97.5%

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

      4. metadata-eval 97.5%

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

      5. div-inv 97.5%

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

      6. +-commutative 97.5%

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

      7. associate-+l+ 97.5%

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

      8. metadata-eval 97.5%

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

   3. Applied egg-rr 97.5%

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

      1. associate-*r* 97.5%

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

      2. *-commutative 97.5%

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

      3. *-commutative 97.5%

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

      4. +-commutative 97.5%

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

      5. count-2 97.5%

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

      6. fma-def 97.5%

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

      7. sub-neg 97.5%

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

      8. mul-1-neg 97.5%

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

      9. +-commutative 97.5%

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

      10. associate-+r+ 99.4%

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

      11. mul-1-neg 99.4%

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

      12. sub-neg 99.4%

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

      13. +-inverses 99.4%

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

      14. remove-double-neg 99.4%

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

      15. mul-1-neg 99.4%

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

      16. sub-neg 99.4%

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

      17. neg-sub0 99.4%

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

      18. mul-1-neg 99.4%

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

      19. remove-double-neg 99.4%

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

   5. Simplified 99.4%

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

   6. Taylor expanded in x around 0 97.6%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0008:\\ \;\;\;\;\sin x \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot -2\right)\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-58}:\\ \;\;\;\;\left(-1 + \cos \varepsilon\right) - \sin \varepsilon \cdot x\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-119}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \mathbf{elif}\;x \leq 0.00072:\\ \;\;\;\;\left(-1 + \cos \varepsilon\right) - \sin \varepsilon \cdot x\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot -2\right)\\ \end{array} \]

Alternative 10: 76.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.4%

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

   1. sub-neg 38.4%

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

   2. cos-sum 59.6%

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

   3. associate-+l- 59.6%

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

   4. fma-neg 59.6%

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

3. Applied egg-rr 59.6%

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

4. Taylor expanded in x around inf 59.6%

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

   1. associate--r+ 91.0%

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

   2. *-commutative 91.0%

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

   3. *-rgt-identity 91.0%

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

   4. distribute-lft-out-- 91.0%

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

   5. sub-neg 91.0%

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

   6. metadata-eval 91.0%

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

   7. +-commutative 91.0%

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

   8. *-commutative 91.0%

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

6. Simplified 91.0%

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

   1. distribute-lft-in 91.0%

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

   2. associate--l+ 59.6%

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

   3. *-commutative 59.6%

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

   4. neg-mul-1 59.6%

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

   5. cos-sum 38.4%

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

   6. +-commutative 38.4%

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

   7. sub-neg 38.4%

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

   8. diff-cos 45.9%

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

   9. div-inv 45.9%

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

   10. +-commutative 45.9%

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

   11. associate--l+ 78.4%

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

   12. metadata-eval 78.4%

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

   13. div-inv 78.4%

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

   14. associate-+l+ 78.4%

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

   15. metadata-eval 78.4%

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

8. Applied egg-rr 78.4%

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

   1. *-commutative 78.4%

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

   2. associate-*l* 78.4%

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

   3. +-inverses 78.4%

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

   4. +-rgt-identity 78.4%

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

   5. *-commutative 78.4%

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

10. Simplified 78.4%

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

11. Final simplification 78.4%

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

Alternative 11: 67.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00023:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{elif}\;\varepsilon \leq 3.3 \cdot 10^{-19}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.00023)
   (- (cos eps) (cos x))
   (if (<= eps 3.3e-19)
     (* eps (- (sin x)))
     (* -2.0 (pow (sin (* eps 0.5)) 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.00023) {
		tmp = cos(eps) - cos(x);
	} else if (eps <= 3.3e-19) {
		tmp = eps * -sin(x);
	} else {
		tmp = -2.0 * pow(sin((eps * 0.5)), 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= (-0.00023d0)) then
        tmp = cos(eps) - cos(x)
    else if (eps <= 3.3d-19) then
        tmp = eps * -sin(x)
    else
        tmp = (-2.0d0) * (sin((eps * 0.5d0)) ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -0.00023) {
		tmp = Math.cos(eps) - Math.cos(x);
	} else if (eps <= 3.3e-19) {
		tmp = eps * -Math.sin(x);
	} else {
		tmp = -2.0 * Math.pow(Math.sin((eps * 0.5)), 2.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -0.00023:
		tmp = math.cos(eps) - math.cos(x)
	elif eps <= 3.3e-19:
		tmp = eps * -math.sin(x)
	else:
		tmp = -2.0 * math.pow(math.sin((eps * 0.5)), 2.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -0.00023)
		tmp = Float64(cos(eps) - cos(x));
	elseif (eps <= 3.3e-19)
		tmp = Float64(eps * Float64(-sin(x)));
	else
		tmp = Float64(-2.0 * (sin(Float64(eps * 0.5)) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -0.00023)
		tmp = cos(eps) - cos(x);
	elseif (eps <= 3.3e-19)
		tmp = eps * -sin(x);
	else
		tmp = -2.0 * (sin((eps * 0.5)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -0.00023], N[(N[Cos[eps], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 3.3e-19], N[(eps * (-N[Sin[x], $MachinePrecision])), $MachinePrecision], N[(-2.0 * N[Power[N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -2.3000000000000001e-4

   1. Initial program 59.2%

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

   2. Taylor expanded in x around 0 61.0%

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

   if -2.3000000000000001e-4 < eps < 3.2999999999999998e-19

   1. Initial program 16.9%

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

   2. Taylor expanded in eps around 0 84.0%

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

      1. mul-1-neg 84.0%

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

      2. *-commutative 84.0%

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

      3. distribute-rgt-neg-in 84.0%

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

   4. Simplified 84.0%

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

   if 3.2999999999999998e-19 < eps

   1. Initial program 56.0%

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

      1. diff-cos 58.7%

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

      2. div-inv 58.7%

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

      3. associate--l+ 58.7%

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

      4. metadata-eval 58.7%

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

      5. div-inv 58.7%

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

      6. +-commutative 58.7%

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

      7. associate-+l+ 58.6%

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

      8. metadata-eval 58.6%

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

   3. Applied egg-rr 58.6%

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

      1. associate-*r* 58.6%

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

      2. *-commutative 58.6%

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

      3. *-commutative 58.6%

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

      4. +-commutative 58.6%

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

      5. count-2 58.6%

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

      6. fma-def 58.6%

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

      7. sub-neg 58.6%

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

      8. mul-1-neg 58.6%

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

      9. +-commutative 58.6%

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

      10. associate-+r+ 61.2%

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

      11. mul-1-neg 61.2%

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

      12. sub-neg 61.2%

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

      13. +-inverses 61.2%

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

      14. remove-double-neg 61.2%

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

      15. mul-1-neg 61.2%

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

      16. sub-neg 61.2%

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

      17. neg-sub0 61.2%

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

      18. mul-1-neg 61.2%

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

      19. remove-double-neg 61.2%

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

   5. Simplified 61.2%

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

   6. Taylor expanded in x around 0 58.9%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00023:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{elif}\;\varepsilon \leq 3.3 \cdot 10^{-19}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array} \]

Alternative 12: 67.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000185 \lor \neg \left(\varepsilon \leq 0.0024\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.000185) (not (<= eps 0.0024)))
   (- (cos eps) (cos x))
   (* eps (- (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.000185) || !(eps <= 0.0024)) {
		tmp = cos(eps) - cos(x);
	} else {
		tmp = eps * -sin(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-0.000185d0)) .or. (.not. (eps <= 0.0024d0))) then
        tmp = cos(eps) - cos(x)
    else
        tmp = eps * -sin(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.000185) || !(eps <= 0.0024)) {
		tmp = Math.cos(eps) - Math.cos(x);
	} else {
		tmp = eps * -Math.sin(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.000185) or not (eps <= 0.0024):
		tmp = math.cos(eps) - math.cos(x)
	else:
		tmp = eps * -math.sin(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.000185) || !(eps <= 0.0024))
		tmp = Float64(cos(eps) - cos(x));
	else
		tmp = Float64(eps * Float64(-sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.000185) || ~((eps <= 0.0024)))
		tmp = cos(eps) - cos(x);
	else
		tmp = eps * -sin(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.000185], N[Not[LessEqual[eps, 0.0024]], $MachinePrecision]], N[(N[Cos[eps], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], N[(eps * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.85e-4 or 0.00239999999999999979 < eps

   1. Initial program 58.7%

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

   2. Taylor expanded in x around 0 60.6%

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

   if -1.85e-4 < eps < 0.00239999999999999979

   1. Initial program 17.1%

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

   2. Taylor expanded in eps around 0 82.3%

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

      1. mul-1-neg 82.3%

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

      2. *-commutative 82.3%

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

      3. distribute-rgt-neg-in 82.3%

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

   4. Simplified 82.3%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000185 \lor \neg \left(\varepsilon \leq 0.0024\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \]

Alternative 13: 66.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000185 \lor \neg \left(\varepsilon \leq 0.00185\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.000185) (not (<= eps 0.00185)))
   (+ -1.0 (cos eps))
   (* eps (- (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.000185) || !(eps <= 0.00185)) {
		tmp = -1.0 + cos(eps);
	} else {
		tmp = eps * -sin(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-0.000185d0)) .or. (.not. (eps <= 0.00185d0))) then
        tmp = (-1.0d0) + cos(eps)
    else
        tmp = eps * -sin(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.000185) || !(eps <= 0.00185)) {
		tmp = -1.0 + Math.cos(eps);
	} else {
		tmp = eps * -Math.sin(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.000185) or not (eps <= 0.00185):
		tmp = -1.0 + math.cos(eps)
	else:
		tmp = eps * -math.sin(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.000185) || !(eps <= 0.00185))
		tmp = Float64(-1.0 + cos(eps));
	else
		tmp = Float64(eps * Float64(-sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.000185) || ~((eps <= 0.00185)))
		tmp = -1.0 + cos(eps);
	else
		tmp = eps * -sin(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.000185], N[Not[LessEqual[eps, 0.00185]], $MachinePrecision]], N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision], N[(eps * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.85e-4 or 0.0018500000000000001 < eps

   1. Initial program 58.7%

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

   2. Taylor expanded in x around 0 58.6%

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

   if -1.85e-4 < eps < 0.0018500000000000001

   1. Initial program 17.1%

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

   2. Taylor expanded in eps around 0 82.3%

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

      1. mul-1-neg 82.3%

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

      2. *-commutative 82.3%

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

      3. distribute-rgt-neg-in 82.3%

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

   4. Simplified 82.3%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000185 \lor \neg \left(\varepsilon \leq 0.00185\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \]

Alternative 14: 46.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000185 \lor \neg \left(\varepsilon \leq 0.00013\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.000185) (not (<= eps 0.00013)))
   (+ -1.0 (cos eps))
   (* -0.5 (* eps eps))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.000185) || !(eps <= 0.00013)) {
		tmp = -1.0 + cos(eps);
	} else {
		tmp = -0.5 * (eps * eps);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-0.000185d0)) .or. (.not. (eps <= 0.00013d0))) then
        tmp = (-1.0d0) + cos(eps)
    else
        tmp = (-0.5d0) * (eps * eps)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.000185) || !(eps <= 0.00013)) {
		tmp = -1.0 + Math.cos(eps);
	} else {
		tmp = -0.5 * (eps * eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.000185) or not (eps <= 0.00013):
		tmp = -1.0 + math.cos(eps)
	else:
		tmp = -0.5 * (eps * eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.000185) || !(eps <= 0.00013))
		tmp = Float64(-1.0 + cos(eps));
	else
		tmp = Float64(-0.5 * Float64(eps * eps));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.000185) || ~((eps <= 0.00013)))
		tmp = -1.0 + cos(eps);
	else
		tmp = -0.5 * (eps * eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.000185], N[Not[LessEqual[eps, 0.00013]], $MachinePrecision]], N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(eps * eps), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.85e-4 or 1.29999999999999989e-4 < eps

   1. Initial program 58.9%

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

   2. Taylor expanded in x around 0 58.2%

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

   if -1.85e-4 < eps < 1.29999999999999989e-4

   1. Initial program 16.6%

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

   2. Taylor expanded in x around 0 16.7%

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

   3. Taylor expanded in eps around 0 32.5%

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

      1. unpow2 32.5%

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

   5. Applied egg-rr 32.5%

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

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000185 \lor \neg \left(\varepsilon \leq 0.00013\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \end{array} \]

Alternative 15: 21.5% accurate, 41.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.4%

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

2. Taylor expanded in x around 0 38.1%

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

3. Taylor expanded in eps around 0 17.5%

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

   1. unpow2 17.5%

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

5. Applied egg-rr 17.5%

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

6. Final simplification 17.5%

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

Alternative 16: 12.6% 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 38.4%

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

   1. add-cube-cbrt 37.9%

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

   2. pow3 37.9%

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

3. Applied egg-rr 37.9%

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

4. Taylor expanded in eps around 0 9.7%

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

   1. pow-base-1 9.7%

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

   2. *-lft-identity 9.7%

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

   3. +-inverses 9.7%

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

6. Simplified 9.7%

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

7. Final simplification 9.7%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023321 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))