Average Error: 40.2 → 0.5
Time: 7.4s
Precision: binary64
\[\cos \left(x + \varepsilon\right) - \cos x \]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.007338856069565484:\\ \;\;\;\;\mathsf{fma}\left(\cos \varepsilon, \cos x, \sin x \cdot \left(-\sin \varepsilon\right)\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.0004056680490046788:\\ \;\;\;\;\mathsf{fma}\left(\cos x, 0.041666666666666664 \cdot {\varepsilon}^{4}, \mathsf{fma}\left(\varepsilon, \left(\varepsilon \cdot \cos x\right) \cdot -0.5, \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \left(-\cos x\right) - \sin x \cdot \sin \varepsilon\right)\\ \end{array} \]
(FPCore (x eps) :precision binary64 (- (cos (+ x eps)) (cos x)))
↓
(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.007338856069565484)
   (- (fma (cos eps) (cos x) (* (sin x) (- (sin eps)))) (cos x))
   (if (<= eps 0.0004056680490046788)
     (fma
      (cos x)
      (* 0.041666666666666664 (pow eps 4.0))
      (fma
       eps
       (* (* eps (cos x)) -0.5)
       (* (sin x) (- (* 0.16666666666666666 (pow eps 3.0)) eps))))
     (fma (cos x) (cos eps) (- (- (cos x)) (* (sin x) (sin eps)))))))
double code(double x, double eps) {
	return cos((x + eps)) - cos(x);
}
↓
double code(double x, double eps) {
	double tmp;
	if (eps <= -0.007338856069565484) {
		tmp = fma(cos(eps), cos(x), (sin(x) * -sin(eps))) - cos(x);
	} else if (eps <= 0.0004056680490046788) {
		tmp = fma(cos(x), (0.041666666666666664 * pow(eps, 4.0)), fma(eps, ((eps * cos(x)) * -0.5), (sin(x) * ((0.16666666666666666 * pow(eps, 3.0)) - eps))));
	} else {
		tmp = fma(cos(x), cos(eps), (-cos(x) - (sin(x) * sin(eps))));
	}
	return tmp;
}

function code(x, eps)
	return Float64(cos(Float64(x + eps)) - cos(x))
end
↓
function code(x, eps)
	tmp = 0.0
	if (eps <= -0.007338856069565484)
		tmp = Float64(fma(cos(eps), cos(x), Float64(sin(x) * Float64(-sin(eps)))) - cos(x));
	elseif (eps <= 0.0004056680490046788)
		tmp = fma(cos(x), Float64(0.041666666666666664 * (eps ^ 4.0)), fma(eps, Float64(Float64(eps * cos(x)) * -0.5), Float64(sin(x) * Float64(Float64(0.16666666666666666 * (eps ^ 3.0)) - eps))));
	else
		tmp = fma(cos(x), cos(eps), Float64(Float64(-cos(x)) - Float64(sin(x) * sin(eps))));
	end
	return tmp
end

code[x_, eps_] := N[(N[Cos[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]
↓
code[x_, eps_] := If[LessEqual[eps, -0.007338856069565484], N[(N[(N[Cos[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * (-N[Sin[eps], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.0004056680490046788], N[(N[Cos[x], $MachinePrecision] * N[(0.041666666666666664 * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision] + N[(eps * N[(N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(N[(0.16666666666666666 * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision] - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + N[((-N[Cos[x], $MachinePrecision]) - N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \leq -0.007338856069565484:\\
\;\;\;\;\mathsf{fma}\left(\cos \varepsilon, \cos x, \sin x \cdot \left(-\sin \varepsilon\right)\right) - \cos x\\

\mathbf{elif}\;\varepsilon \leq 0.0004056680490046788:\\
\;\;\;\;\mathsf{fma}\left(\cos x, 0.041666666666666664 \cdot {\varepsilon}^{4}, \mathsf{fma}\left(\varepsilon, \left(\varepsilon \cdot \cos x\right) \cdot -0.5, \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \left(-\cos x\right) - \sin x \cdot \sin \varepsilon\right)\\


\end{array}

Error

Derivation

  1. Split input into 3 regimes

  2. if eps < -0.00733885606956548395

    1. Initial program 30.3

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

    2. Applied egg-rr0.9

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

    3. Taylor expanded in x around inf 0.9

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

    4. Simplified0.8

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

    if -0.00733885606956548395 < eps < 4.0566804900467882e-4

    1. Initial program 49.7

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

    2. Taylor expanded in eps around 0 0.1

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + \left(0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + \left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right)\right)} \]

    3. Simplified0.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x, 0.041666666666666664 \cdot {\varepsilon}^{4}, \mathsf{fma}\left(\varepsilon, \left(\varepsilon \cdot \cos x\right) \cdot -0.5, \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\right)\right)} \]

    if 4.0566804900467882e-4 < eps

    1. Initial program 30.5

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

    2. Applied egg-rr0.9

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

    3. Taylor expanded in x around inf 0.9

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.007338856069565484:\\ \;\;\;\;\mathsf{fma}\left(\cos \varepsilon, \cos x, \sin x \cdot \left(-\sin \varepsilon\right)\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.0004056680490046788:\\ \;\;\;\;\mathsf{fma}\left(\cos x, 0.041666666666666664 \cdot {\varepsilon}^{4}, \mathsf{fma}\left(\varepsilon, \left(\varepsilon \cdot \cos x\right) \cdot -0.5, \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \left(-\cos x\right) - \sin x \cdot \sin \varepsilon\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022190 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))