Average Error: 39.6 → 0.5
Time: 9.9s
Precision: binary64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0033612638349102826:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0022378297832547816:\\ \;\;\;\;\sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right) + \cos x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4} - 0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \leq -0.0033612638349102826:\\
\;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\

\mathbf{elif}\;\varepsilon \leq 0.0022378297832547816:\\
\;\;\;\;\sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right) + \cos x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4} - 0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\

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

\end{array}
(FPCore (x eps) :precision binary64 (- (cos (+ x eps)) (cos x)))
↓
(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.0033612638349102826)
   (- (* (cos x) (cos eps)) (+ (cos x) (* (sin x) (sin eps))))
   (if (<= eps 0.0022378297832547816)
     (+
      (* (sin x) (- (* 0.16666666666666666 (pow eps 3.0)) eps))
      (*
       (cos x)
       (- (* 0.041666666666666664 (pow eps 4.0)) (* 0.5 (* eps eps)))))
     (- (- (* (cos x) (cos eps)) (* (sin x) (sin eps))) (cos x)))))
double code(double x, double eps) {
	return cos(x + eps) - cos(x);
}
↓
double code(double x, double eps) {
	double tmp;
	if (eps <= -0.0033612638349102826) {
		tmp = (cos(x) * cos(eps)) - (cos(x) + (sin(x) * sin(eps)));
	} else if (eps <= 0.0022378297832547816) {
		tmp = (sin(x) * ((0.16666666666666666 * pow(eps, 3.0)) - eps)) + (cos(x) * ((0.041666666666666664 * pow(eps, 4.0)) - (0.5 * (eps * eps))));
	} else {
		tmp = ((cos(x) * cos(eps)) - (sin(x) * sin(eps))) - cos(x);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if eps < -0.00336126383491028259

    1. Initial program 30.2

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    2. Using strategy rm

    3. Applied cos-sum_binary64_2120.9

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

    4. Applied associate--l-_binary64_160.9

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

    if -0.00336126383491028259 < eps < 0.0022378297832547816

    1. Initial program 48.9

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

    2. Taylor expanded around 0 0.1

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

    3. Simplified0.1

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

    if 0.0022378297832547816 < eps

    1. Initial program 30.7

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    2. Using strategy rm

    3. Applied cos-sum_binary64_2120.8

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

    4. Simplified0.8

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

    5. Simplified0.8

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0033612638349102826:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0022378297832547816:\\ \;\;\;\;\sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right) + \cos x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4} - 0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

Reproduce

herbie shell --seed 2021015 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))