Average Error: 39.7 → 0.7
Time: 7.6s
Precision: binary64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -9.502002810456572 \cdot 10^{-07} \lor \neg \left(\varepsilon \leq 8.484522352852379 \cdot 10^{-07}\right):\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(x + \varepsilon \cdot 0.5\right)\right)\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \leq -9.502002810456572 \cdot 10^{-07} \lor \neg \left(\varepsilon \leq 8.484522352852379 \cdot 10^{-07}\right):\\
\;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\

\mathbf{else}:\\
\;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(x + \varepsilon \cdot 0.5\right)\right)\\

\end{array}
(FPCore (x eps) :precision binary64 (- (cos (+ x eps)) (cos x)))
↓
(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -9.502002810456572e-07) (not (<= eps 8.484522352852379e-07)))
   (- (- (* (cos x) (cos eps)) (* (sin x) (sin eps))) (cos x))
   (* -2.0 (* (sin (/ eps 2.0)) (sin (+ x (* eps 0.5)))))))
double code(double x, double eps) {
	return cos(x + eps) - cos(x);
}
↓
double code(double x, double eps) {
	double tmp;
	if ((eps <= -9.502002810456572e-07) || !(eps <= 8.484522352852379e-07)) {
		tmp = ((cos(x) * cos(eps)) - (sin(x) * sin(eps))) - cos(x);
	} else {
		tmp = -2.0 * (sin(eps / 2.0) * sin(x + (eps * 0.5)));
	}
	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 2 regimes

  2. if eps < -9.50200281045657179e-7 or 8.4845223528523788e-7 < eps

    1. Initial program 29.9

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

    3. Applied cos-sum_binary64_5531.0

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

    if -9.50200281045657179e-7 < eps < 8.4845223528523788e-7

    1. Initial program 49.8

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

    3. Applied diff-cos_binary64_57037.9

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

    4. Simplified0.4

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

    5. Taylor expanded around inf 0.4

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

    6. Simplified0.4

      \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \color{blue}{\sin \left(x + \varepsilon \cdot 0.5\right)}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -9.502002810456572 \cdot 10^{-07} \lor \neg \left(\varepsilon \leq 8.484522352852379 \cdot 10^{-07}\right):\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(x + \varepsilon \cdot 0.5\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020354 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))