Average Error: 39.5 → 16.6
Time: 8.3s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -9.4747810379847977 \cdot 10^{-25}:\\ \;\;\;\;\frac{\left({\left(\cos x \cdot \cos \varepsilon\right)}^{3} + 3 \cdot \left(\cos x \cdot \left({\left(\sin x\right)}^{2} \cdot \left({\left(\sin \varepsilon\right)}^{2} \cdot \cos \varepsilon\right)\right)\right)\right) - \left(\left({\left(\cos x\right)}^{3} + 3 \cdot \left({\left(\cos x\right)}^{2} \cdot \left(\sin x \cdot \left(\sin \varepsilon \cdot {\left(\cos \varepsilon\right)}^{2}\right)\right)\right)\right) + {\left(\sin x \cdot \sin \varepsilon\right)}^{3}\right)}{\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon\right) \cdot \left(\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x\right) + \cos x \cdot \cos x}\\ \mathbf{elif}\;\varepsilon \le 1.08327386020133947 \cdot 10^{-13}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^{3} - {\left(\cos x\right)}^{3}}{\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon\right) \cdot \frac{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^{2} - {\left(\cos x\right)}^{2}}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} + \cos x \cdot \cos x}\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -9.4747810379847977 \cdot 10^{-25}:\\
\;\;\;\;\frac{\left({\left(\cos x \cdot \cos \varepsilon\right)}^{3} + 3 \cdot \left(\cos x \cdot \left({\left(\sin x\right)}^{2} \cdot \left({\left(\sin \varepsilon\right)}^{2} \cdot \cos \varepsilon\right)\right)\right)\right) - \left(\left({\left(\cos x\right)}^{3} + 3 \cdot \left({\left(\cos x\right)}^{2} \cdot \left(\sin x \cdot \left(\sin \varepsilon \cdot {\left(\cos \varepsilon\right)}^{2}\right)\right)\right)\right) + {\left(\sin x \cdot \sin \varepsilon\right)}^{3}\right)}{\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon\right) \cdot \left(\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x\right) + \cos x \cdot \cos x}\\

\mathbf{elif}\;\varepsilon \le 1.08327386020133947 \cdot 10^{-13}:\\
\;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^{3} - {\left(\cos x\right)}^{3}}{\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon\right) \cdot \frac{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^{2} - {\left(\cos x\right)}^{2}}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} + \cos x \cdot \cos x}\\

\end{array}
double code(double x, double eps) {
	return ((double) (((double) cos(((double) (x + eps)))) - ((double) cos(x))));
}
↓
double code(double x, double eps) {
	double VAR;
	if ((eps <= -9.474781037984798e-25)) {
		VAR = ((double) (((double) (((double) (((double) pow(((double) (((double) cos(x)) * ((double) cos(eps)))), 3.0)) + ((double) (3.0 * ((double) (((double) cos(x)) * ((double) (((double) pow(((double) sin(x)), 2.0)) * ((double) (((double) pow(((double) sin(eps)), 2.0)) * ((double) cos(eps)))))))))))) - ((double) (((double) (((double) pow(((double) cos(x)), 3.0)) + ((double) (3.0 * ((double) (((double) pow(((double) cos(x)), 2.0)) * ((double) (((double) sin(x)) * ((double) (((double) sin(eps)) * ((double) pow(((double) cos(eps)), 2.0)))))))))))) + ((double) pow(((double) (((double) sin(x)) * ((double) sin(eps)))), 3.0)))))) / ((double) (((double) (((double) (((double) (((double) cos(eps)) * ((double) cos(x)))) - ((double) (((double) sin(x)) * ((double) sin(eps)))))) * ((double) (((double) (((double) (((double) cos(x)) * ((double) cos(eps)))) - ((double) (((double) sin(x)) * ((double) sin(eps)))))) + ((double) cos(x)))))) + ((double) (((double) cos(x)) * ((double) cos(x))))))));
	} else {
		double VAR_1;
		if ((eps <= 1.0832738602013395e-13)) {
			VAR_1 = ((double) (eps * ((double) (((double) (((double) (0.16666666666666666 * ((double) pow(x, 3.0)))) - x)) - ((double) (eps * 0.5))))));
		} else {
			VAR_1 = ((double) (((double) (((double) pow(((double) (((double) (((double) cos(x)) * ((double) cos(eps)))) - ((double) (((double) sin(x)) * ((double) sin(eps)))))), 3.0)) - ((double) pow(((double) cos(x)), 3.0)))) / ((double) (((double) (((double) (((double) (((double) cos(eps)) * ((double) cos(x)))) - ((double) (((double) sin(x)) * ((double) sin(eps)))))) * ((double) (((double) (((double) pow(((double) (((double) (((double) cos(x)) * ((double) cos(eps)))) - ((double) (((double) sin(x)) * ((double) sin(eps)))))), 2.0)) - ((double) pow(((double) cos(x)), 2.0)))) / ((double) (((double) (((double) (((double) cos(x)) * ((double) cos(eps)))) - ((double) (((double) sin(x)) * ((double) sin(eps)))))) - ((double) cos(x)))))))) + ((double) (((double) cos(x)) * ((double) cos(x))))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

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 < -9.474781037984798e-25

    1. Initial program 31.6

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

    3. Applied cos-sum3.8

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

    5. Applied flip3--4.1

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

    6. Simplified4.0

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

    7. Taylor expanded around inf 4.1

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

    8. Simplified4.1

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

    if -9.474781037984798e-25 < eps < 1.0832738602013395e-13

    1. Initial program 48.7

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

    2. Taylor expanded around 0 31.7

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \left({x}^{3} \cdot \varepsilon\right) - \left(x \cdot \varepsilon + \frac{1}{2} \cdot {\varepsilon}^{2}\right)}\]

    3. Simplified31.7

      \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)}\]

    if 1.0832738602013395e-13 < eps

    1. Initial program 31.0

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

    3. Applied cos-sum2.1

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

    5. Applied flip3--2.3

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

    6. Simplified2.3

      \[\leadsto \frac{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^{3} - {\left(\cos x\right)}^{3}}{\color{blue}{\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon\right) \cdot \left(\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x\right) + \cos x \cdot \cos x}}\]
    7. Using strategy rm

    8. Applied flip-+2.3

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

    9. Simplified2.3

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

  4. Final simplification16.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -9.4747810379847977 \cdot 10^{-25}:\\ \;\;\;\;\frac{\left({\left(\cos x \cdot \cos \varepsilon\right)}^{3} + 3 \cdot \left(\cos x \cdot \left({\left(\sin x\right)}^{2} \cdot \left({\left(\sin \varepsilon\right)}^{2} \cdot \cos \varepsilon\right)\right)\right)\right) - \left(\left({\left(\cos x\right)}^{3} + 3 \cdot \left({\left(\cos x\right)}^{2} \cdot \left(\sin x \cdot \left(\sin \varepsilon \cdot {\left(\cos \varepsilon\right)}^{2}\right)\right)\right)\right) + {\left(\sin x \cdot \sin \varepsilon\right)}^{3}\right)}{\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon\right) \cdot \left(\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x\right) + \cos x \cdot \cos x}\\ \mathbf{elif}\;\varepsilon \le 1.08327386020133947 \cdot 10^{-13}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^{3} - {\left(\cos x\right)}^{3}}{\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon\right) \cdot \frac{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^{2} - {\left(\cos x\right)}^{2}}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} + \cos x \cdot \cos x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020123 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))