Average Error: 39.2 → 0.8
Time: 7.5s
Precision: binary64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -4009.334003099285 \lor \neg \left(x \leq 7.492679465452055 \cdot 10^{-52}\right):\\ \;\;\;\;\frac{\left(\cos x \cdot \left(\cos \varepsilon - 1\right) - \sin x \cdot \sin \varepsilon\right) \cdot \left(\cos x \cdot \left(\cos \varepsilon + 1\right) - \sin x \cdot \sin \varepsilon\right)}{\cos x \cdot \left(\cos \varepsilon + 1\right) - \sin x \cdot \sin \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right)\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;x \leq -4009.334003099285 \lor \neg \left(x \leq 7.492679465452055 \cdot 10^{-52}\right):\\
\;\;\;\;\frac{\left(\cos x \cdot \left(\cos \varepsilon - 1\right) - \sin x \cdot \sin \varepsilon\right) \cdot \left(\cos x \cdot \left(\cos \varepsilon + 1\right) - \sin x \cdot \sin \varepsilon\right)}{\cos x \cdot \left(\cos \varepsilon + 1\right) - \sin x \cdot \sin \varepsilon}\\

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

\end{array}
(FPCore (x eps) :precision binary64 (- (cos (+ x eps)) (cos x)))
↓
(FPCore (x eps)
 :precision binary64
 (if (or (<= x -4009.334003099285) (not (<= x 7.492679465452055e-52)))
   (/
    (*
     (- (* (cos x) (- (cos eps) 1.0)) (* (sin x) (sin eps)))
     (- (* (cos x) (+ (cos eps) 1.0)) (* (sin x) (sin eps))))
    (- (* (cos x) (+ (cos eps) 1.0)) (* (sin x) (sin eps))))
   (* -2.0 (* (sin (/ eps 2.0)) (sin (/ (+ eps (+ x x)) 2.0))))))
double code(double x, double eps) {
	return cos(x + eps) - cos(x);
}
↓
double code(double x, double eps) {
	double tmp;
	if ((x <= -4009.334003099285) || !(x <= 7.492679465452055e-52)) {
		tmp = (((cos(x) * (cos(eps) - 1.0)) - (sin(x) * sin(eps))) * ((cos(x) * (cos(eps) + 1.0)) - (sin(x) * sin(eps)))) / ((cos(x) * (cos(eps) + 1.0)) - (sin(x) * sin(eps)));
	} else {
		tmp = -2.0 * (sin(eps / 2.0) * sin((eps + (x + x)) / 2.0));
	}
	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 x < -4009.3340030992849 or 7.4926794654520547e-52 < x

    1. Initial program 56.9

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

    3. Applied cos-sum_binary6429.1

      \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
    4. Using strategy rm

    5. Applied flip--_binary6429.3

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

    6. Simplified1.1

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

    7. Simplified0.9

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

    if -4009.3340030992849 < x < 7.4926794654520547e-52

    1. Initial program 19.0

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

    3. Applied diff-cos_binary646.2

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

    4. Simplified0.6

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

    6. Applied associate-+r+_binary640.6

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4009.334003099285 \lor \neg \left(x \leq 7.492679465452055 \cdot 10^{-52}\right):\\ \;\;\;\;\frac{\left(\cos x \cdot \left(\cos \varepsilon - 1\right) - \sin x \cdot \sin \varepsilon\right) \cdot \left(\cos x \cdot \left(\cos \varepsilon + 1\right) - \sin x \cdot \sin \varepsilon\right)}{\cos x \cdot \left(\cos \varepsilon + 1\right) - \sin x \cdot \sin \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020299 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))