Average Error: 39.5 → 0.5
Time: 10.5s
Precision: binary64
\[\cos \left(x + \varepsilon\right) - \cos x \]
↓
\[\begin{array}{l} t_0 := \mathsf{fma}\left(\cos \varepsilon, \cos x, \sin \varepsilon \cdot \left(-\sin x\right)\right) - \cos x\\ \mathbf{if}\;\varepsilon \leq -0.0029557493766311335:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.0020293481482823845:\\ \;\;\;\;\cos x \cdot \mathsf{fma}\left(0.041666666666666664, {\varepsilon}^{4}, \varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
(FPCore (x eps) :precision binary64 (- (cos (+ x eps)) (cos x)))
↓
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (fma (cos eps) (cos x) (* (sin eps) (- (sin x)))) (cos x))))
   (if (<= eps -0.0029557493766311335)
     t_0
     (if (<= eps 0.0020293481482823845)
       (+
        (*
         (cos x)
         (fma 0.041666666666666664 (pow eps 4.0) (* eps (* eps -0.5))))
        (* (sin x) (- (* 0.16666666666666666 (pow eps 3.0)) eps)))
       t_0))))
double code(double x, double eps) {
	return cos((x + eps)) - cos(x);
}
↓
double code(double x, double eps) {
	double t_0 = fma(cos(eps), cos(x), (sin(eps) * -sin(x))) - cos(x);
	double tmp;
	if (eps <= -0.0029557493766311335) {
		tmp = t_0;
	} else if (eps <= 0.0020293481482823845) {
		tmp = (cos(x) * fma(0.041666666666666664, pow(eps, 4.0), (eps * (eps * -0.5)))) + (sin(x) * ((0.16666666666666666 * pow(eps, 3.0)) - eps));
	} else {
		tmp = t_0;
	}
	return tmp;
}

function code(x, eps)
	return Float64(cos(Float64(x + eps)) - cos(x))
end
↓
function code(x, eps)
	t_0 = Float64(fma(cos(eps), cos(x), Float64(sin(eps) * Float64(-sin(x)))) - cos(x))
	tmp = 0.0
	if (eps <= -0.0029557493766311335)
		tmp = t_0;
	elseif (eps <= 0.0020293481482823845)
		tmp = Float64(Float64(cos(x) * fma(0.041666666666666664, (eps ^ 4.0), Float64(eps * Float64(eps * -0.5)))) + Float64(sin(x) * Float64(Float64(0.16666666666666666 * (eps ^ 3.0)) - eps)));
	else
		tmp = t_0;
	end
	return tmp
end

code[x_, eps_] := N[(N[Cos[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]
↓
code[x_, eps_] := Block[{t$95$0 = N[(N[(N[Cos[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[Sin[eps], $MachinePrecision] * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.0029557493766311335], t$95$0, If[LessEqual[eps, 0.0020293481482823845], N[(N[(N[Cos[x], $MachinePrecision] * N[(0.041666666666666664 * N[Power[eps, 4.0], $MachinePrecision] + N[(eps * N[(eps * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(N[(0.16666666666666666 * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision] - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
t_0 := \mathsf{fma}\left(\cos \varepsilon, \cos x, \sin \varepsilon \cdot \left(-\sin x\right)\right) - \cos x\\
\mathbf{if}\;\varepsilon \leq -0.0029557493766311335:\\
\;\;\;\;t_0\\

\mathbf{elif}\;\varepsilon \leq 0.0020293481482823845:\\
\;\;\;\;\cos x \cdot \mathsf{fma}\left(0.041666666666666664, {\varepsilon}^{4}, \varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\\

\mathbf{else}:\\
\;\;\;\;t_0\\


\end{array}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if eps < -0.00295574937663113354 or 0.00202934814828238446 < eps

    1. Initial program 29.9

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

    2. Applied egg-rr0.8

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

    3. Applied egg-rr0.9

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

    4. Applied egg-rr0.8

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

    if -0.00295574937663113354 < eps < 0.00202934814828238446

    1. Initial program 49.1

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

    2. Taylor expanded in eps around 0 0.1

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

    3. Simplified0.1

      \[\leadsto \color{blue}{\cos x \cdot \mathsf{fma}\left(0.041666666666666664, {\varepsilon}^{4}, \varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0029557493766311335:\\ \;\;\;\;\mathsf{fma}\left(\cos \varepsilon, \cos x, \sin \varepsilon \cdot \left(-\sin x\right)\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.0020293481482823845:\\ \;\;\;\;\cos x \cdot \mathsf{fma}\left(0.041666666666666664, {\varepsilon}^{4}, \varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos \varepsilon, \cos x, \sin \varepsilon \cdot \left(-\sin x\right)\right) - \cos x\\ \end{array} \]

Reproduce

herbie shell --seed 2022151 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))