Average Error: 39.4 → 0.7
Time: 7.0s
Precision: binary64
\[\cos \left(x + \varepsilon\right) - \cos x \]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.718011094065103 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 0.07710856464853962\right):\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(\sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \log e\right)\\ \end{array} \]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \leq -3.718011094065103 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 0.07710856464853962\right):\\
\;\;\;\;\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) - \cos x\\

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


\end{array}

(FPCore (x eps) :precision binary64 (- (cos (+ x eps)) (cos x)))
↓
(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -3.718011094065103e-5) (not (<= eps 0.07710856464853962)))
   (- (- (* (cos eps) (cos x)) (* (sin eps) (sin x))) (cos x))
   (* -2.0 (* (* (sin (/ (+ eps (+ x x)) 2.0)) (sin (/ eps 2.0))) (log E)))))
double code(double x, double eps) {
	return cos(x + eps) - cos(x);
}
↓
double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.718011094065103e-5) || !(eps <= 0.07710856464853962)) {
		tmp = ((cos(eps) * cos(x)) - (sin(eps) * sin(x))) - cos(x);
	} else {
		tmp = -2.0 * ((sin((eps + (x + x)) / 2.0) * sin(eps / 2.0)) * log((double) M_E));
	}
	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 < -3.7180110940651027e-5 or 0.0771085646485396153 < eps

    1. Initial program 30.0

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

    3. Applied cos-sum_binary640.8

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

    4. Simplified0.8

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

    5. Simplified0.8

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

    if -3.7180110940651027e-5 < eps < 0.0771085646485396153

    1. Initial program 49.0

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

    3. Applied diff-cos_binary6437.4

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

    4. Simplified0.6

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

    6. Applied add-log-exp_binary6418.3

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

    7. Simplified18.3

      \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \log \color{blue}{\left(e^{\sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)}\right)}\right) \]
    8. Using strategy rm

    9. Applied *-un-lft-identity_binary6418.3

      \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \log \left(e^{\color{blue}{1 \cdot \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)}}\right)\right) \]

    10. Applied exp-prod_binary6418.4

      \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \log \color{blue}{\left({\left(e^{1}\right)}^{\sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)}\right)}\right) \]

    11. Applied log-pow_binary640.6

      \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \color{blue}{\left(\sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right) \cdot \log \left(e^{1}\right)\right)}\right) \]

    12. Applied associate-*r*_binary640.6

      \[\leadsto -2 \cdot \color{blue}{\left(\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right) \cdot \log \left(e^{1}\right)\right)} \]

    13. Simplified0.6

      \[\leadsto -2 \cdot \left(\color{blue}{\left(\sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)} \cdot \log \left(e^{1}\right)\right) \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.718011094065103 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 0.07710856464853962\right):\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(\sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \log e\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021204 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))