2cos (problem 3.3.5)

Average Accuracy: 37.6% → 99.2%

Time: 21.2s

Precision: binary64

Cost: 45448

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0038:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin x \cdot \left(-\sin \varepsilon\right)\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.00305:\\ \;\;\;\;\left(\cos x \cdot \left(\varepsilon \cdot 0.5 + -0.020833333333333332 \cdot {\varepsilon}^{3}\right) + \sin x \cdot \left(-0.125 \cdot \left(\varepsilon \cdot \varepsilon\right) + 1\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, -\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.0038)
   (- (fma (cos x) (cos eps) (* (sin x) (- (sin eps)))) (cos x))
   (if (<= eps 0.00305)
     (*
      (+
       (* (cos x) (+ (* eps 0.5) (* -0.020833333333333332 (pow eps 3.0))))
       (* (sin x) (+ (* -0.125 (* eps eps)) 1.0)))
      (* -2.0 (sin (* eps 0.5))))
     (fma (cos x) (cos eps) (- (fma (sin x) (sin eps) (cos x)))))))

C

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

↓

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.0038) {
		tmp = fma(cos(x), cos(eps), (sin(x) * -sin(eps))) - cos(x);
	} else if (eps <= 0.00305) {
		tmp = ((cos(x) * ((eps * 0.5) + (-0.020833333333333332 * pow(eps, 3.0)))) + (sin(x) * ((-0.125 * (eps * eps)) + 1.0))) * (-2.0 * sin((eps * 0.5)));
	} else {
		tmp = fma(cos(x), cos(eps), -fma(sin(x), sin(eps), cos(x)));
	}
	return tmp;
}

Julia

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

↓

function code(x, eps)
	tmp = 0.0
	if (eps <= -0.0038)
		tmp = Float64(fma(cos(x), cos(eps), Float64(sin(x) * Float64(-sin(eps)))) - cos(x));
	elseif (eps <= 0.00305)
		tmp = Float64(Float64(Float64(cos(x) * Float64(Float64(eps * 0.5) + Float64(-0.020833333333333332 * (eps ^ 3.0)))) + Float64(sin(x) * Float64(Float64(-0.125 * Float64(eps * eps)) + 1.0))) * Float64(-2.0 * sin(Float64(eps * 0.5))));
	else
		tmp = fma(cos(x), cos(eps), Float64(-fma(sin(x), sin(eps), cos(x))));
	end
	return tmp
end

Wolfram

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

↓

code[x_, eps_] := If[LessEqual[eps, -0.0038], N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * (-N[Sin[eps], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.00305], N[(N[(N[(N[Cos[x], $MachinePrecision] * N[(N[(eps * 0.5), $MachinePrecision] + N[(-0.020833333333333332 * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(N[(-0.125 * N[(eps * eps), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-2.0 * N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + (-N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision] + N[Cos[x], $MachinePrecision]), $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.00379999999999999999

   1. Initial program 51.9%

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

   2. Applied egg-rr 98.7%

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

      [Start] 51.9	\[ \cos \left(x + \varepsilon\right) - \cos x \]
      cos-sum [=>] 98.7	\[ \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x \]
      cancel-sign-sub-inv [=>] 98.7	\[ \color{blue}{\left(\cos x \cdot \cos \varepsilon + \left(-\sin x\right) \cdot \sin \varepsilon\right)} - \cos x \]
      fma-def [=>] 98.7	\[ \color{blue}{\mathsf{fma}\left(\cos x, \cos \varepsilon, \left(-\sin x\right) \cdot \sin \varepsilon\right)} - \cos x \]

   if -0.00379999999999999999 < eps < 0.00305000000000000019

   1. Initial program 22.6%

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

   2. Applied egg-rr 22.6%

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

      [Start] 22.6	\[ \cos \left(x + \varepsilon\right) - \cos x \]
      add-cube-cbrt [=>] 22.6	\[ \color{blue}{\left(\sqrt[3]{\cos \left(x + \varepsilon\right) - \cos x} \cdot \sqrt[3]{\cos \left(x + \varepsilon\right) - \cos x}\right) \cdot \sqrt[3]{\cos \left(x + \varepsilon\right) - \cos x}} \]
      pow3 [=>] 22.6	\[ \color{blue}{{\left(\sqrt[3]{\cos \left(x + \varepsilon\right) - \cos x}\right)}^{3}} \]

   3. Applied egg-rr 41.1%

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

      [Start] 22.6	\[ {\left(\sqrt[3]{\cos \left(x + \varepsilon\right) - \cos x}\right)}^{3} \]
      rem-cube-cbrt [=>] 22.6	\[ \color{blue}{\cos \left(x + \varepsilon\right) - \cos x} \]
      diff-cos [=>] 41.1	\[ \color{blue}{-2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)} \]
      div-inv [=>] 41.1	\[ -2 \cdot \left(\sin \color{blue}{\left(\left(\left(x + \varepsilon\right) - x\right) \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]
      associate--l+ [=>] 41.1	\[ -2 \cdot \left(\sin \left(\color{blue}{\left(x + \left(\varepsilon - x\right)\right)} \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]
      metadata-eval [=>] 41.1	\[ -2 \cdot \left(\sin \left(\left(x + \left(\varepsilon - x\right)\right) \cdot \color{blue}{0.5}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]
      div-inv [=>] 41.1	\[ -2 \cdot \left(\sin \left(\left(x + \left(\varepsilon - x\right)\right) \cdot 0.5\right) \cdot \sin \color{blue}{\left(\left(\left(x + \varepsilon\right) + x\right) \cdot \frac{1}{2}\right)}\right) \]
      +-commutative [=>] 41.1	\[ -2 \cdot \left(\sin \left(\left(x + \left(\varepsilon - x\right)\right) \cdot 0.5\right) \cdot \sin \left(\color{blue}{\left(x + \left(x + \varepsilon\right)\right)} \cdot \frac{1}{2}\right)\right) \]
      metadata-eval [=>] 41.1	\[ -2 \cdot \left(\sin \left(\left(x + \left(\varepsilon - x\right)\right) \cdot 0.5\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot \color{blue}{0.5}\right)\right) \]

   4. Simplified 99.0%

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

      [Start] 41.1	\[ -2 \cdot \left(\sin \left(\left(x + \left(\varepsilon - x\right)\right) \cdot 0.5\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]
      associate-*r* [=>] 41.1	\[ \color{blue}{\left(-2 \cdot \sin \left(\left(x + \left(\varepsilon - x\right)\right) \cdot 0.5\right)\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)} \]
      *-commutative [=>] 41.1	\[ \color{blue}{\sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right) \cdot \left(-2 \cdot \sin \left(\left(x + \left(\varepsilon - x\right)\right) \cdot 0.5\right)\right)} \]
      *-commutative [=>] 41.1	\[ \sin \color{blue}{\left(0.5 \cdot \left(x + \left(x + \varepsilon\right)\right)\right)} \cdot \left(-2 \cdot \sin \left(\left(x + \left(\varepsilon - x\right)\right) \cdot 0.5\right)\right) \]
      associate-+r+ [=>] 41.1	\[ \sin \left(0.5 \cdot \color{blue}{\left(\left(x + x\right) + \varepsilon\right)}\right) \cdot \left(-2 \cdot \sin \left(\left(x + \left(\varepsilon - x\right)\right) \cdot 0.5\right)\right) \]
      +-commutative [=>] 41.1	\[ \sin \left(0.5 \cdot \color{blue}{\left(\varepsilon + \left(x + x\right)\right)}\right) \cdot \left(-2 \cdot \sin \left(\left(x + \left(\varepsilon - x\right)\right) \cdot 0.5\right)\right) \]
      sub-neg [=>] 41.1	\[ \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(\left(x + \color{blue}{\left(\varepsilon + \left(-x\right)\right)}\right) \cdot 0.5\right)\right) \]
      mul-1-neg [<=] 41.1	\[ \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(\left(x + \left(\varepsilon + \color{blue}{-1 \cdot x}\right)\right) \cdot 0.5\right)\right) \]
      +-commutative [=>] 41.1	\[ \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(\left(x + \color{blue}{\left(-1 \cdot x + \varepsilon\right)}\right) \cdot 0.5\right)\right) \]
      associate-+r+ [=>] 99.0	\[ \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(\color{blue}{\left(\left(x + -1 \cdot x\right) + \varepsilon\right)} \cdot 0.5\right)\right) \]
      mul-1-neg [=>] 99.0	\[ \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(\left(\left(x + \color{blue}{\left(-x\right)}\right) + \varepsilon\right) \cdot 0.5\right)\right) \]
      sub-neg [<=] 99.0	\[ \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(\left(\color{blue}{\left(x - x\right)} + \varepsilon\right) \cdot 0.5\right)\right) \]
      +-inverses [=>] 99.0	\[ \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(\left(\color{blue}{0} + \varepsilon\right) \cdot 0.5\right)\right) \]
      remove-double-neg [<=] 99.0	\[ \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(\left(0 + \color{blue}{\left(-\left(-\varepsilon\right)\right)}\right) \cdot 0.5\right)\right) \]
      mul-1-neg [<=] 99.0	\[ \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(\left(0 + \left(-\color{blue}{-1 \cdot \varepsilon}\right)\right) \cdot 0.5\right)\right) \]
      sub-neg [<=] 99.0	\[ \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(\color{blue}{\left(0 - -1 \cdot \varepsilon\right)} \cdot 0.5\right)\right) \]
      neg-sub0 [<=] 99.0	\[ \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(\color{blue}{\left(--1 \cdot \varepsilon\right)} \cdot 0.5\right)\right) \]
      mul-1-neg [=>] 99.0	\[ \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(\left(-\color{blue}{\left(-\varepsilon\right)}\right) \cdot 0.5\right)\right) \]
      remove-double-neg [=>] 99.0	\[ \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(\color{blue}{\varepsilon} \cdot 0.5\right)\right) \]

   5. Taylor expanded in eps around 0 99.8%

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

   6. Simplified 99.8%

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

      [Start] 99.8	\[ \left(-0.125 \cdot \left({\varepsilon}^{2} \cdot \sin x\right) + \left(-0.020833333333333332 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + \left(0.5 \cdot \left(\varepsilon \cdot \cos x\right) + \sin x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]
      +-commutative [=>] 99.8	\[ \color{blue}{\left(\left(-0.020833333333333332 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + \left(0.5 \cdot \left(\varepsilon \cdot \cos x\right) + \sin x\right)\right) + -0.125 \cdot \left({\varepsilon}^{2} \cdot \sin x\right)\right)} \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]
      associate-+r+ [=>] 99.8	\[ \left(\color{blue}{\left(\left(-0.020833333333333332 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + 0.5 \cdot \left(\varepsilon \cdot \cos x\right)\right) + \sin x\right)} + -0.125 \cdot \left({\varepsilon}^{2} \cdot \sin x\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]
      associate-+l+ [=>] 99.8	\[ \color{blue}{\left(\left(-0.020833333333333332 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + 0.5 \cdot \left(\varepsilon \cdot \cos x\right)\right) + \left(\sin x + -0.125 \cdot \left({\varepsilon}^{2} \cdot \sin x\right)\right)\right)} \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]
      associate-*r* [=>] 99.8	\[ \left(\left(\color{blue}{\left(-0.020833333333333332 \cdot {\varepsilon}^{3}\right) \cdot \cos x} + 0.5 \cdot \left(\varepsilon \cdot \cos x\right)\right) + \left(\sin x + -0.125 \cdot \left({\varepsilon}^{2} \cdot \sin x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]
      associate-*r* [=>] 99.8	\[ \left(\left(\left(-0.020833333333333332 \cdot {\varepsilon}^{3}\right) \cdot \cos x + \color{blue}{\left(0.5 \cdot \varepsilon\right) \cdot \cos x}\right) + \left(\sin x + -0.125 \cdot \left({\varepsilon}^{2} \cdot \sin x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]
      distribute-rgt-out [=>] 99.8	\[ \left(\color{blue}{\cos x \cdot \left(-0.020833333333333332 \cdot {\varepsilon}^{3} + 0.5 \cdot \varepsilon\right)} + \left(\sin x + -0.125 \cdot \left({\varepsilon}^{2} \cdot \sin x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]
      +-commutative [=>] 99.8	\[ \left(\cos x \cdot \color{blue}{\left(0.5 \cdot \varepsilon + -0.020833333333333332 \cdot {\varepsilon}^{3}\right)} + \left(\sin x + -0.125 \cdot \left({\varepsilon}^{2} \cdot \sin x\right)\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]
      associate-*r* [=>] 99.8	\[ \left(\cos x \cdot \left(0.5 \cdot \varepsilon + -0.020833333333333332 \cdot {\varepsilon}^{3}\right) + \left(\sin x + \color{blue}{\left(-0.125 \cdot {\varepsilon}^{2}\right) \cdot \sin x}\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]
      distribute-rgt1-in [=>] 99.8	\[ \left(\cos x \cdot \left(0.5 \cdot \varepsilon + -0.020833333333333332 \cdot {\varepsilon}^{3}\right) + \color{blue}{\left(-0.125 \cdot {\varepsilon}^{2} + 1\right) \cdot \sin x}\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]
      unpow2 [=>] 99.8	\[ \left(\cos x \cdot \left(0.5 \cdot \varepsilon + -0.020833333333333332 \cdot {\varepsilon}^{3}\right) + \left(-0.125 \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} + 1\right) \cdot \sin x\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]

   if 0.00305000000000000019 < eps

   1. Initial program 51.9%

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

   2. Applied egg-rr 98.7%

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

      [Start] 51.9	\[ \cos \left(x + \varepsilon\right) - \cos x \]
      cos-sum [=>] 98.7	\[ \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x \]
      associate--l- [=>] 98.6	\[ \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)} \]
      fma-neg [=>] 98.7	\[ \color{blue}{\mathsf{fma}\left(\cos x, \cos \varepsilon, -\left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)} \]
      fma-def [=>] 98.7	\[ \mathsf{fma}\left(\cos x, \cos \varepsilon, -\color{blue}{\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0038:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin x \cdot \left(-\sin \varepsilon\right)\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.00305:\\ \;\;\;\;\left(\cos x \cdot \left(\varepsilon \cdot 0.5 + -0.020833333333333332 \cdot {\varepsilon}^{3}\right) + \sin x \cdot \left(-0.125 \cdot \left(\varepsilon \cdot \varepsilon\right) + 1\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, -\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.2%
Cost	39176

\[\begin{array}{l} t_0 := \sin x \cdot \left(-\sin \varepsilon\right)\\ \mathbf{if}\;\varepsilon \leq -0.0031:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, t_0\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.00335:\\ \;\;\;\;\left(\cos x \cdot \left(\varepsilon \cdot 0.5 + -0.020833333333333332 \cdot {\varepsilon}^{3}\right) + \sin x \cdot \left(-0.125 \cdot \left(\varepsilon \cdot \varepsilon\right) + 1\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, t_0 - \cos x\right)\\ \end{array} \]

Alternative 2
Accuracy	99.2%
Cost	39112

\[\begin{array}{l} t_0 := \cos x \cdot \cos \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.0038:\\ \;\;\;\;\left(t_0 - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.0035:\\ \;\;\;\;\left(\cos x \cdot \left(\varepsilon \cdot 0.5 + -0.020833333333333332 \cdot {\varepsilon}^{3}\right) + \sin x \cdot \left(-0.125 \cdot \left(\varepsilon \cdot \varepsilon\right) + 1\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 - \mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\\ \end{array} \]

Alternative 3
Accuracy	99.2%
Cost	39112

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00315:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin x \cdot \left(-\sin \varepsilon\right)\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.0035:\\ \;\;\;\;\left(\cos x \cdot \left(\varepsilon \cdot 0.5 + -0.020833333333333332 \cdot {\varepsilon}^{3}\right) + \sin x \cdot \left(-0.125 \cdot \left(\varepsilon \cdot \varepsilon\right) + 1\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\\ \end{array} \]

Alternative 4
Accuracy	99.2%
Cost	32841

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0034 \lor \neg \left(\varepsilon \leq 0.0034\right):\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \left(\varepsilon \cdot 0.5 + -0.020833333333333332 \cdot {\varepsilon}^{3}\right) + \sin x \cdot \left(-0.125 \cdot \left(\varepsilon \cdot \varepsilon\right) + 1\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 5
Accuracy	99.2%
Cost	32841

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0038 \lor \neg \left(\varepsilon \leq 0.0038\right):\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \left(\varepsilon \cdot 0.5 + -0.020833333333333332 \cdot {\varepsilon}^{3}\right) + \sin x \cdot \left(-0.125 \cdot \left(\varepsilon \cdot \varepsilon\right) + 1\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 6
Accuracy	76.0%
Cost	19652

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.8 \cdot 10^{+31}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\varepsilon \cdot 0.5\right) \cdot \left(-2 \cdot \sin \left(0.5 \cdot \left(x + \left(\varepsilon + x\right)\right)\right)\right)\\ \end{array} \]

Alternative 7
Accuracy	76.2%
Cost	13769

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.011 \lor \neg \left(\varepsilon \leq 0.0062\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\cos x \cdot \left(\varepsilon \cdot -0.5\right)\right) - \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 8
Accuracy	75.6%
Cost	13632

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

Alternative 9
Accuracy	76.0%
Cost	13257

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.006 \lor \neg \left(\varepsilon \leq 0.0063\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5 - \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 10
Accuracy	75.3%
Cost	7241

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0032 \lor \neg \left(\varepsilon \leq 0.0049\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5 - \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 11
Accuracy	66.3%
Cost	6921

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.6 \cdot 10^{-7} \lor \neg \left(\varepsilon \leq 0.0042\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \]

Alternative 12
Accuracy	46.5%
Cost	6857

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00011 \lor \neg \left(\varepsilon \leq 0.000165\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \end{array} \]

Alternative 13
Accuracy	20.7%
Cost	320

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

Alternative 14
Accuracy	20.7%
Cost	320

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

Alternative 15
Accuracy	12.5%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023147 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))