2cos (problem 3.3.5)

Average Accuracy: 37.6% → 99.2%

Time: 18.9s

Precision: binary64

Cost: 39240

Math

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

↓

\[\begin{array}{l} t_0 := -1 + \cos \varepsilon\\ t_1 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0055:\\ \;\;\;\;\mathsf{fma}\left(\sin \varepsilon, -\sin x, \cos x \cdot t_0\right)\\ \mathbf{elif}\;\varepsilon \leq 0.006:\\ \;\;\;\;\cos x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5 + {\varepsilon}^{4} \cdot 0.041666666666666664\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \log \left(e^{t_0}\right) - t_1\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ -1.0 (cos eps))) (t_1 (* (sin eps) (sin x))))
   (if (<= eps -0.0055)
     (fma (sin eps) (- (sin x)) (* (cos x) t_0))
     (if (<= eps 0.006)
       (-
        (*
         (cos x)
         (+ (* (* eps eps) -0.5) (* (pow eps 4.0) 0.041666666666666664)))
        t_1)
       (- (* (cos x) (log (exp t_0))) t_1)))))

C

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

↓

double code(double x, double eps) {
	double t_0 = -1.0 + cos(eps);
	double t_1 = sin(eps) * sin(x);
	double tmp;
	if (eps <= -0.0055) {
		tmp = fma(sin(eps), -sin(x), (cos(x) * t_0));
	} else if (eps <= 0.006) {
		tmp = (cos(x) * (((eps * eps) * -0.5) + (pow(eps, 4.0) * 0.041666666666666664))) - t_1;
	} else {
		tmp = (cos(x) * log(exp(t_0))) - t_1;
	}
	return tmp;
}

Julia

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

↓

function code(x, eps)
	t_0 = Float64(-1.0 + cos(eps))
	t_1 = Float64(sin(eps) * sin(x))
	tmp = 0.0
	if (eps <= -0.0055)
		tmp = fma(sin(eps), Float64(-sin(x)), Float64(cos(x) * t_0));
	elseif (eps <= 0.006)
		tmp = Float64(Float64(cos(x) * Float64(Float64(Float64(eps * eps) * -0.5) + Float64((eps ^ 4.0) * 0.041666666666666664))) - t_1);
	else
		tmp = Float64(Float64(cos(x) * log(exp(t_0))) - t_1);
	end
	return tmp
end

Wolfram

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

↓

code[x_, eps_] := Block[{t$95$0 = N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.0055], N[(N[Sin[eps], $MachinePrecision] * (-N[Sin[x], $MachinePrecision]) + N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.006], N[(N[(N[Cos[x], $MachinePrecision] * N[(N[(N[(eps * eps), $MachinePrecision] * -0.5), $MachinePrecision] + N[(N[Power[eps, 4.0], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(N[(N[Cos[x], $MachinePrecision] * N[Log[N[Exp[t$95$0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.0054999999999999997

   1. Initial program 50.2%

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

   2. Applied egg-rr 23.3%

      \[\leadsto \color{blue}{{\left(\sqrt{\cos \left(x + \varepsilon\right)}\right)}^{2}} - \cos x \]
      Proof

      [Start] 50.2	\[ \cos \left(x + \varepsilon\right) - \cos x \]
      add-sqr-sqrt [=>] 23.3	\[ \color{blue}{\sqrt{\cos \left(x + \varepsilon\right)} \cdot \sqrt{\cos \left(x + \varepsilon\right)}} - \cos x \]
      pow2 [=>] 23.3	\[ \color{blue}{{\left(\sqrt{\cos \left(x + \varepsilon\right)}\right)}^{2}} - \cos x \]

   3. Applied egg-rr 98.7%

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

      [Start] 23.3	\[ {\left(\sqrt{\cos \left(x + \varepsilon\right)}\right)}^{2} - \cos x \]
      sub-neg [=>] 23.3	\[ \color{blue}{{\left(\sqrt{\cos \left(x + \varepsilon\right)}\right)}^{2} + \left(-\cos x\right)} \]
      unpow2 [=>] 23.3	\[ \color{blue}{\sqrt{\cos \left(x + \varepsilon\right)} \cdot \sqrt{\cos \left(x + \varepsilon\right)}} + \left(-\cos x\right) \]
      add-sqr-sqrt [<=] 50.2	\[ \color{blue}{\cos \left(x + \varepsilon\right)} + \left(-\cos x\right) \]
      cos-sum [=>] 98.7	\[ \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} + \left(-\cos x\right) \]
      cancel-sign-sub-inv [=>] 98.7	\[ \color{blue}{\left(\cos x \cdot \cos \varepsilon + \left(-\sin x\right) \cdot \sin \varepsilon\right)} + \left(-\cos x\right) \]
      associate-+l+ [=>] 98.7	\[ \color{blue}{\cos x \cdot \cos \varepsilon + \left(\left(-\sin x\right) \cdot \sin \varepsilon + \left(-\cos x\right)\right)} \]
      *-commutative [=>] 98.7	\[ \cos x \cdot \cos \varepsilon + \left(\color{blue}{\sin \varepsilon \cdot \left(-\sin x\right)} + \left(-\cos x\right)\right) \]

   4. Simplified 98.8%

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

      [Start] 98.7	\[ \cos x \cdot \cos \varepsilon + \left(\sin \varepsilon \cdot \left(-\sin x\right) + \left(-\cos x\right)\right) \]
      +-commutative [=>] 98.7	\[ \cos x \cdot \cos \varepsilon + \color{blue}{\left(\left(-\cos x\right) + \sin \varepsilon \cdot \left(-\sin x\right)\right)} \]
      *-commutative [=>] 98.7	\[ \cos x \cdot \cos \varepsilon + \left(\left(-\cos x\right) + \color{blue}{\left(-\sin x\right) \cdot \sin \varepsilon}\right) \]
      distribute-lft-neg-in [<=] 98.7	\[ \cos x \cdot \cos \varepsilon + \left(\left(-\cos x\right) + \color{blue}{\left(-\sin x \cdot \sin \varepsilon\right)}\right) \]
      unsub-neg [=>] 98.7	\[ \cos x \cdot \cos \varepsilon + \color{blue}{\left(\left(-\cos x\right) - \sin x \cdot \sin \varepsilon\right)} \]
      associate--l+ [<=] 98.7	\[ \color{blue}{\left(\cos x \cdot \cos \varepsilon + \left(-\cos x\right)\right) - \sin x \cdot \sin \varepsilon} \]
      +-commutative [<=] 98.7	\[ \color{blue}{\left(\left(-\cos x\right) + \cos x \cdot \cos \varepsilon\right)} - \sin x \cdot \sin \varepsilon \]
      *-commutative [=>] 98.7	\[ \left(\left(-\cos x\right) + \cos x \cdot \cos \varepsilon\right) - \color{blue}{\sin \varepsilon \cdot \sin x} \]
      neg-mul-1 [=>] 98.7	\[ \left(\color{blue}{-1 \cdot \cos x} + \cos x \cdot \cos \varepsilon\right) - \sin \varepsilon \cdot \sin x \]
      *-commutative [=>] 98.7	\[ \left(-1 \cdot \cos x + \color{blue}{\cos \varepsilon \cdot \cos x}\right) - \sin \varepsilon \cdot \sin x \]
      distribute-rgt-out [=>] 98.8	\[ \color{blue}{\cos x \cdot \left(-1 + \cos \varepsilon\right)} - \sin \varepsilon \cdot \sin x \]
      +-commutative [<=] 98.8	\[ \cos x \cdot \color{blue}{\left(\cos \varepsilon + -1\right)} - \sin \varepsilon \cdot \sin x \]

   5. Taylor expanded in x around inf 98.8%

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

   6. Simplified 98.8%

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

      [Start] 98.8	\[ \cos x \cdot \left(\cos \varepsilon - 1\right) - \sin x \cdot \sin \varepsilon \]
      cancel-sign-sub-inv [=>] 98.8	\[ \color{blue}{\cos x \cdot \left(\cos \varepsilon - 1\right) + \left(-\sin x\right) \cdot \sin \varepsilon} \]
      *-commutative [=>] 98.8	\[ \color{blue}{\left(\cos \varepsilon - 1\right) \cdot \cos x} + \left(-\sin x\right) \cdot \sin \varepsilon \]
      sub-neg [=>] 98.8	\[ \color{blue}{\left(\cos \varepsilon + \left(-1\right)\right)} \cdot \cos x + \left(-\sin x\right) \cdot \sin \varepsilon \]
      metadata-eval [=>] 98.8	\[ \left(\cos \varepsilon + \color{blue}{-1}\right) \cdot \cos x + \left(-\sin x\right) \cdot \sin \varepsilon \]
      *-commutative [<=] 98.8	\[ \left(\cos \varepsilon + -1\right) \cdot \cos x + \color{blue}{\sin \varepsilon \cdot \left(-\sin x\right)} \]
      +-commutative [<=] 98.8	\[ \color{blue}{\sin \varepsilon \cdot \left(-\sin x\right) + \left(\cos \varepsilon + -1\right) \cdot \cos x} \]
      fma-def [=>] 98.8	\[ \color{blue}{\mathsf{fma}\left(\sin \varepsilon, -\sin x, \left(\cos \varepsilon + -1\right) \cdot \cos x\right)} \]
      *-commutative [=>] 98.8	\[ \mathsf{fma}\left(\sin \varepsilon, -\sin x, \color{blue}{\cos x \cdot \left(\cos \varepsilon + -1\right)}\right) \]

   if -0.0054999999999999997 < eps < 0.0060000000000000001

   1. Initial program 23.1%

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

   2. Applied egg-rr 21.7%

      \[\leadsto \color{blue}{{\left(\sqrt{\cos \left(x + \varepsilon\right)}\right)}^{2}} - \cos x \]
      Proof

      [Start] 23.1	\[ \cos \left(x + \varepsilon\right) - \cos x \]
      add-sqr-sqrt [=>] 21.7	\[ \color{blue}{\sqrt{\cos \left(x + \varepsilon\right)} \cdot \sqrt{\cos \left(x + \varepsilon\right)}} - \cos x \]
      pow2 [=>] 21.7	\[ \color{blue}{{\left(\sqrt{\cos \left(x + \varepsilon\right)}\right)}^{2}} - \cos x \]

   3. Applied egg-rr 24.3%

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

      [Start] 21.7	\[ {\left(\sqrt{\cos \left(x + \varepsilon\right)}\right)}^{2} - \cos x \]
      sub-neg [=>] 21.7	\[ \color{blue}{{\left(\sqrt{\cos \left(x + \varepsilon\right)}\right)}^{2} + \left(-\cos x\right)} \]
      unpow2 [=>] 21.7	\[ \color{blue}{\sqrt{\cos \left(x + \varepsilon\right)} \cdot \sqrt{\cos \left(x + \varepsilon\right)}} + \left(-\cos x\right) \]
      add-sqr-sqrt [<=] 23.1	\[ \color{blue}{\cos \left(x + \varepsilon\right)} + \left(-\cos x\right) \]
      cos-sum [=>] 24.3	\[ \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} + \left(-\cos x\right) \]
      cancel-sign-sub-inv [=>] 24.3	\[ \color{blue}{\left(\cos x \cdot \cos \varepsilon + \left(-\sin x\right) \cdot \sin \varepsilon\right)} + \left(-\cos x\right) \]
      associate-+l+ [=>] 24.3	\[ \color{blue}{\cos x \cdot \cos \varepsilon + \left(\left(-\sin x\right) \cdot \sin \varepsilon + \left(-\cos x\right)\right)} \]
      *-commutative [=>] 24.3	\[ \cos x \cdot \cos \varepsilon + \left(\color{blue}{\sin \varepsilon \cdot \left(-\sin x\right)} + \left(-\cos x\right)\right) \]

   4. Simplified 80.8%

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

      [Start] 24.3	\[ \cos x \cdot \cos \varepsilon + \left(\sin \varepsilon \cdot \left(-\sin x\right) + \left(-\cos x\right)\right) \]
      +-commutative [=>] 24.3	\[ \cos x \cdot \cos \varepsilon + \color{blue}{\left(\left(-\cos x\right) + \sin \varepsilon \cdot \left(-\sin x\right)\right)} \]
      *-commutative [=>] 24.3	\[ \cos x \cdot \cos \varepsilon + \left(\left(-\cos x\right) + \color{blue}{\left(-\sin x\right) \cdot \sin \varepsilon}\right) \]
      distribute-lft-neg-in [<=] 24.3	\[ \cos x \cdot \cos \varepsilon + \left(\left(-\cos x\right) + \color{blue}{\left(-\sin x \cdot \sin \varepsilon\right)}\right) \]
      unsub-neg [=>] 24.3	\[ \cos x \cdot \cos \varepsilon + \color{blue}{\left(\left(-\cos x\right) - \sin x \cdot \sin \varepsilon\right)} \]
      associate--l+ [<=] 80.8	\[ \color{blue}{\left(\cos x \cdot \cos \varepsilon + \left(-\cos x\right)\right) - \sin x \cdot \sin \varepsilon} \]
      +-commutative [<=] 80.8	\[ \color{blue}{\left(\left(-\cos x\right) + \cos x \cdot \cos \varepsilon\right)} - \sin x \cdot \sin \varepsilon \]
      *-commutative [=>] 80.8	\[ \left(\left(-\cos x\right) + \cos x \cdot \cos \varepsilon\right) - \color{blue}{\sin \varepsilon \cdot \sin x} \]
      neg-mul-1 [=>] 80.8	\[ \left(\color{blue}{-1 \cdot \cos x} + \cos x \cdot \cos \varepsilon\right) - \sin \varepsilon \cdot \sin x \]
      *-commutative [=>] 80.8	\[ \left(-1 \cdot \cos x + \color{blue}{\cos \varepsilon \cdot \cos x}\right) - \sin \varepsilon \cdot \sin x \]
      distribute-rgt-out [=>] 80.8	\[ \color{blue}{\cos x \cdot \left(-1 + \cos \varepsilon\right)} - \sin \varepsilon \cdot \sin x \]
      +-commutative [<=] 80.8	\[ \cos x \cdot \color{blue}{\left(\cos \varepsilon + -1\right)} - \sin \varepsilon \cdot \sin x \]

   5. Applied egg-rr 99.8%

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

      [Start] 80.8	\[ \cos x \cdot \left(\cos \varepsilon + -1\right) - \sin \varepsilon \cdot \sin x \]
      *-commutative [=>] 80.8	\[ \color{blue}{\left(\cos \varepsilon + -1\right) \cdot \cos x} - \sin \varepsilon \cdot \sin x \]
      +-commutative [=>] 80.8	\[ \color{blue}{\left(-1 + \cos \varepsilon\right)} \cdot \cos x - \sin \varepsilon \cdot \sin x \]
      flip-+ [=>] 80.8	\[ \color{blue}{\frac{-1 \cdot -1 - \cos \varepsilon \cdot \cos \varepsilon}{-1 - \cos \varepsilon}} \cdot \cos x - \sin \varepsilon \cdot \sin x \]
      metadata-eval [=>] 80.8	\[ \frac{\color{blue}{1} - \cos \varepsilon \cdot \cos \varepsilon}{-1 - \cos \varepsilon} \cdot \cos x - \sin \varepsilon \cdot \sin x \]
      sqr-cos-a [=>] 80.9	\[ \frac{1 - \color{blue}{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \varepsilon\right)\right)}}{-1 - \cos \varepsilon} \cdot \cos x - \sin \varepsilon \cdot \sin x \]
      metadata-eval [<=] 80.9	\[ \frac{1 - \left(\color{blue}{\frac{1}{2}} + 0.5 \cdot \cos \left(2 \cdot \varepsilon\right)\right)}{-1 - \cos \varepsilon} \cdot \cos x - \sin \varepsilon \cdot \sin x \]
      associate--r+ [=>] 80.9	\[ \frac{\color{blue}{\left(1 - \frac{1}{2}\right) - 0.5 \cdot \cos \left(2 \cdot \varepsilon\right)}}{-1 - \cos \varepsilon} \cdot \cos x - \sin \varepsilon \cdot \sin x \]
      metadata-eval [=>] 80.9	\[ \frac{\left(1 - \color{blue}{0.5}\right) - 0.5 \cdot \cos \left(2 \cdot \varepsilon\right)}{-1 - \cos \varepsilon} \cdot \cos x - \sin \varepsilon \cdot \sin x \]
      metadata-eval [=>] 80.9	\[ \frac{\color{blue}{0.5} - 0.5 \cdot \cos \left(2 \cdot \varepsilon\right)}{-1 - \cos \varepsilon} \cdot \cos x - \sin \varepsilon \cdot \sin x \]
      sqr-sin-a [<=] 99.8	\[ \frac{\color{blue}{\sin \varepsilon \cdot \sin \varepsilon}}{-1 - \cos \varepsilon} \cdot \cos x - \sin \varepsilon \cdot \sin x \]
      associate-*l/ [=>] 99.8	\[ \color{blue}{\frac{\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \cos x}{-1 - \cos \varepsilon}} - \sin \varepsilon \cdot \sin x \]
      pow2 [=>] 99.8	\[ \frac{\color{blue}{{\sin \varepsilon}^{2}} \cdot \cos x}{-1 - \cos \varepsilon} - \sin \varepsilon \cdot \sin x \]

   6. Taylor expanded in eps around 0 99.8%

      \[\leadsto \color{blue}{\left(-1 \cdot \left({\varepsilon}^{4} \cdot \left(-0.16666666666666666 \cdot \cos x - -0.125 \cdot \cos x\right)\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right)} - \sin \varepsilon \cdot \sin x \]

   7. Simplified 99.8%

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

      [Start] 99.8	\[ \left(-1 \cdot \left({\varepsilon}^{4} \cdot \left(-0.16666666666666666 \cdot \cos x - -0.125 \cdot \cos x\right)\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right) - \sin \varepsilon \cdot \sin x \]
      +-commutative [=>] 99.8	\[ \color{blue}{\left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + -1 \cdot \left({\varepsilon}^{4} \cdot \left(-0.16666666666666666 \cdot \cos x - -0.125 \cdot \cos x\right)\right)\right)} - \sin \varepsilon \cdot \sin x \]
      mul-1-neg [=>] 99.8	\[ \left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + \color{blue}{\left(-{\varepsilon}^{4} \cdot \left(-0.16666666666666666 \cdot \cos x - -0.125 \cdot \cos x\right)\right)}\right) - \sin \varepsilon \cdot \sin x \]
      unsub-neg [=>] 99.8	\[ \color{blue}{\left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) - {\varepsilon}^{4} \cdot \left(-0.16666666666666666 \cdot \cos x - -0.125 \cdot \cos x\right)\right)} - \sin \varepsilon \cdot \sin x \]
      associate-*r* [=>] 99.8	\[ \left(\color{blue}{\left(-0.5 \cdot {\varepsilon}^{2}\right) \cdot \cos x} - {\varepsilon}^{4} \cdot \left(-0.16666666666666666 \cdot \cos x - -0.125 \cdot \cos x\right)\right) - \sin \varepsilon \cdot \sin x \]
      *-commutative [=>] 99.8	\[ \left(\color{blue}{\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right)} - {\varepsilon}^{4} \cdot \left(-0.16666666666666666 \cdot \cos x - -0.125 \cdot \cos x\right)\right) - \sin \varepsilon \cdot \sin x \]
      *-commutative [=>] 99.8	\[ \left(\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - \color{blue}{\left(-0.16666666666666666 \cdot \cos x - -0.125 \cdot \cos x\right) \cdot {\varepsilon}^{4}}\right) - \sin \varepsilon \cdot \sin x \]
      distribute-rgt-out-- [=>] 99.8	\[ \left(\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - \color{blue}{\left(\cos x \cdot \left(-0.16666666666666666 - -0.125\right)\right)} \cdot {\varepsilon}^{4}\right) - \sin \varepsilon \cdot \sin x \]
      metadata-eval [=>] 99.8	\[ \left(\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - \left(\cos x \cdot \color{blue}{-0.041666666666666664}\right) \cdot {\varepsilon}^{4}\right) - \sin \varepsilon \cdot \sin x \]
      associate-*l* [=>] 99.8	\[ \left(\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - \color{blue}{\cos x \cdot \left(-0.041666666666666664 \cdot {\varepsilon}^{4}\right)}\right) - \sin \varepsilon \cdot \sin x \]
      distribute-lft-out-- [=>] 99.8	\[ \color{blue}{\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2} - -0.041666666666666664 \cdot {\varepsilon}^{4}\right)} - \sin \varepsilon \cdot \sin x \]
      *-commutative [=>] 99.8	\[ \cos x \cdot \left(\color{blue}{{\varepsilon}^{2} \cdot -0.5} - -0.041666666666666664 \cdot {\varepsilon}^{4}\right) - \sin \varepsilon \cdot \sin x \]
      unpow2 [=>] 99.8	\[ \cos x \cdot \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot -0.5 - -0.041666666666666664 \cdot {\varepsilon}^{4}\right) - \sin \varepsilon \cdot \sin x \]

   if 0.0060000000000000001 < eps

   1. Initial program 53.4%

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

   2. Applied egg-rr 26.0%

      \[\leadsto \color{blue}{{\left(\sqrt{\cos \left(x + \varepsilon\right)}\right)}^{2}} - \cos x \]
      Proof

      [Start] 53.4	\[ \cos \left(x + \varepsilon\right) - \cos x \]
      add-sqr-sqrt [=>] 26.0	\[ \color{blue}{\sqrt{\cos \left(x + \varepsilon\right)} \cdot \sqrt{\cos \left(x + \varepsilon\right)}} - \cos x \]
      pow2 [=>] 26.0	\[ \color{blue}{{\left(\sqrt{\cos \left(x + \varepsilon\right)}\right)}^{2}} - \cos x \]

   3. Applied egg-rr 98.7%

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

      [Start] 26.0	\[ {\left(\sqrt{\cos \left(x + \varepsilon\right)}\right)}^{2} - \cos x \]
      sub-neg [=>] 26.0	\[ \color{blue}{{\left(\sqrt{\cos \left(x + \varepsilon\right)}\right)}^{2} + \left(-\cos x\right)} \]
      unpow2 [=>] 26.0	\[ \color{blue}{\sqrt{\cos \left(x + \varepsilon\right)} \cdot \sqrt{\cos \left(x + \varepsilon\right)}} + \left(-\cos x\right) \]
      add-sqr-sqrt [<=] 53.4	\[ \color{blue}{\cos \left(x + \varepsilon\right)} + \left(-\cos x\right) \]
      cos-sum [=>] 98.7	\[ \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} + \left(-\cos x\right) \]
      cancel-sign-sub-inv [=>] 98.7	\[ \color{blue}{\left(\cos x \cdot \cos \varepsilon + \left(-\sin x\right) \cdot \sin \varepsilon\right)} + \left(-\cos x\right) \]
      associate-+l+ [=>] 98.7	\[ \color{blue}{\cos x \cdot \cos \varepsilon + \left(\left(-\sin x\right) \cdot \sin \varepsilon + \left(-\cos x\right)\right)} \]
      *-commutative [=>] 98.7	\[ \cos x \cdot \cos \varepsilon + \left(\color{blue}{\sin \varepsilon \cdot \left(-\sin x\right)} + \left(-\cos x\right)\right) \]

   4. Simplified 98.8%

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

      [Start] 98.7	\[ \cos x \cdot \cos \varepsilon + \left(\sin \varepsilon \cdot \left(-\sin x\right) + \left(-\cos x\right)\right) \]
      +-commutative [=>] 98.7	\[ \cos x \cdot \cos \varepsilon + \color{blue}{\left(\left(-\cos x\right) + \sin \varepsilon \cdot \left(-\sin x\right)\right)} \]
      *-commutative [=>] 98.7	\[ \cos x \cdot \cos \varepsilon + \left(\left(-\cos x\right) + \color{blue}{\left(-\sin x\right) \cdot \sin \varepsilon}\right) \]
      distribute-lft-neg-in [<=] 98.7	\[ \cos x \cdot \cos \varepsilon + \left(\left(-\cos x\right) + \color{blue}{\left(-\sin x \cdot \sin \varepsilon\right)}\right) \]
      unsub-neg [=>] 98.7	\[ \cos x \cdot \cos \varepsilon + \color{blue}{\left(\left(-\cos x\right) - \sin x \cdot \sin \varepsilon\right)} \]
      associate--l+ [<=] 98.8	\[ \color{blue}{\left(\cos x \cdot \cos \varepsilon + \left(-\cos x\right)\right) - \sin x \cdot \sin \varepsilon} \]
      +-commutative [<=] 98.8	\[ \color{blue}{\left(\left(-\cos x\right) + \cos x \cdot \cos \varepsilon\right)} - \sin x \cdot \sin \varepsilon \]
      *-commutative [=>] 98.8	\[ \left(\left(-\cos x\right) + \cos x \cdot \cos \varepsilon\right) - \color{blue}{\sin \varepsilon \cdot \sin x} \]
      neg-mul-1 [=>] 98.8	\[ \left(\color{blue}{-1 \cdot \cos x} + \cos x \cdot \cos \varepsilon\right) - \sin \varepsilon \cdot \sin x \]
      *-commutative [=>] 98.8	\[ \left(-1 \cdot \cos x + \color{blue}{\cos \varepsilon \cdot \cos x}\right) - \sin \varepsilon \cdot \sin x \]
      distribute-rgt-out [=>] 98.8	\[ \color{blue}{\cos x \cdot \left(-1 + \cos \varepsilon\right)} - \sin \varepsilon \cdot \sin x \]
      +-commutative [<=] 98.8	\[ \cos x \cdot \color{blue}{\left(\cos \varepsilon + -1\right)} - \sin \varepsilon \cdot \sin x \]

   5. Applied egg-rr 98.7%

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

      [Start] 98.8	\[ \cos x \cdot \left(\cos \varepsilon + -1\right) - \sin \varepsilon \cdot \sin x \]
      add-log-exp [=>] 98.7	\[ \cos x \cdot \color{blue}{\log \left(e^{\cos \varepsilon + -1}\right)} - \sin \varepsilon \cdot \sin x \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0055:\\ \;\;\;\;\mathsf{fma}\left(\sin \varepsilon, -\sin x, \cos x \cdot \left(-1 + \cos \varepsilon\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 0.006:\\ \;\;\;\;\cos x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5 + {\varepsilon}^{4} \cdot 0.041666666666666664\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \log \left(e^{-1 + \cos \varepsilon}\right) - \sin \varepsilon \cdot \sin x\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.3%
Cost	39176

\[\begin{array}{l} t_0 := -\sin x\\ \mathbf{if}\;\varepsilon \leq -0.0055:\\ \;\;\;\;\mathsf{fma}\left(\sin \varepsilon, t_0, \cos x \cdot \left(-1 + \cos \varepsilon\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0055:\\ \;\;\;\;\cos x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5 + {\varepsilon}^{4} \cdot 0.041666666666666664\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin \varepsilon \cdot t_0\right) - \cos x\\ \end{array} \]

Alternative 2
Accuracy	99.0%
Cost	39168

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

Alternative 3
Accuracy	99.3%
Cost	32840

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0055:\\ \;\;\;\;\mathsf{fma}\left(\sin \varepsilon, -\sin x, \cos x \cdot \left(-1 + \cos \varepsilon\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0046:\\ \;\;\;\;\cos x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5 + {\varepsilon}^{4} \cdot 0.041666666666666664\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \cos x\right) - t_0\\ \end{array} \]

Alternative 4
Accuracy	99.3%
Cost	32644

\[\begin{array}{l} t_0 := \cos x \cdot \left(-1 + \cos \varepsilon\right)\\ t_1 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0055:\\ \;\;\;\;\mathsf{fma}\left(\sin \varepsilon, -\sin x, t_0\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0055:\\ \;\;\;\;\cos x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5 + {\varepsilon}^{4} \cdot 0.041666666666666664\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 - t_1\\ \end{array} \]

Alternative 5
Accuracy	99.3%
Cost	26889

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0055 \lor \neg \left(\varepsilon \leq 0.0055\right):\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5 + {\varepsilon}^{4} \cdot 0.041666666666666664\right) - t_0\\ \end{array} \]

Alternative 6
Accuracy	99.2%
Cost	26441

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.00018 \lor \neg \left(\varepsilon \leq 0.00015\right):\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\cos x \cdot -0.5\right) - t_0\\ \end{array} \]

Alternative 7
Accuracy	75.9%
Cost	13888

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

Alternative 8
Accuracy	76.6%
Cost	13769

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.02 \lor \neg \left(\varepsilon \leq 0.062\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\cos x \cdot -0.5\right) - \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 9
Accuracy	67.1%
Cost	13257

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.7 \cdot 10^{-7} \lor \neg \left(\varepsilon \leq 2.45 \cdot 10^{-5}\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \]

Alternative 10
Accuracy	66.4%
Cost	6921

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -6.2 \cdot 10^{-7} \lor \neg \left(\varepsilon \leq 0.000175\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \]

Alternative 11
Accuracy	46.8%
Cost	6857

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00018 \lor \neg \left(\varepsilon \leq 2.4 \cdot 10^{-5}\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\\ \end{array} \]

Alternative 12
Accuracy	21.4%
Cost	320

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

Alternative 13
Accuracy	12.8%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023143 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))