Average Error: 39.9 → 0.7
Time: 20.0s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -4.028105222915818 \cdot 10^{-06}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 0.11407284983207572:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -4.028105222915818 \cdot 10^{-06}:\\
\;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\

\mathbf{elif}\;\varepsilon \le 0.11407284983207572:\\
\;\;\;\;-2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\

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

\end{array}
double f(double x, double eps) {
        double r1703631 = x;
        double r1703632 = eps;
        double r1703633 = r1703631 + r1703632;
        double r1703634 = cos(r1703633);
        double r1703635 = cos(r1703631);
        double r1703636 = r1703634 - r1703635;
        return r1703636;
}
↓
double f(double x, double eps) {
        double r1703637 = eps;
        double r1703638 = -4.028105222915818e-06;
        bool r1703639 = r1703637 <= r1703638;
        double r1703640 = x;
        double r1703641 = cos(r1703640);
        double r1703642 = cos(r1703637);
        double r1703643 = r1703641 * r1703642;
        double r1703644 = sin(r1703640);
        double r1703645 = sin(r1703637);
        double r1703646 = r1703644 * r1703645;
        double r1703647 = r1703643 - r1703646;
        double r1703648 = r1703647 - r1703641;
        double r1703649 = 0.11407284983207572;
        bool r1703650 = r1703637 <= r1703649;
        double r1703651 = -2.0;
        double r1703652 = 0.5;
        double r1703653 = r1703652 * r1703637;
        double r1703654 = sin(r1703653);
        double r1703655 = r1703640 + r1703637;
        double r1703656 = r1703655 + r1703640;
        double r1703657 = 2.0;
        double r1703658 = r1703656 / r1703657;
        double r1703659 = sin(r1703658);
        double r1703660 = r1703654 * r1703659;
        double r1703661 = r1703651 * r1703660;
        double r1703662 = r1703641 + r1703646;
        double r1703663 = r1703643 - r1703662;
        double r1703664 = r1703650 ? r1703661 : r1703663;
        double r1703665 = r1703639 ? r1703648 : r1703664;
        return r1703665;
}

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 < -4.028105222915818e-06

    1. Initial program 30.5

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

    3. Applied cos-sum1.0

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

    if -4.028105222915818e-06 < eps < 0.11407284983207572

    1. Initial program 49.3

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

    3. Applied diff-cos37.6

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

    4. Simplified0.5

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

    if 0.11407284983207572 < eps

    1. Initial program 29.6

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

    3. Applied cos-sum0.8

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

    4. Applied associate--l-0.8

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -4.028105222915818 \cdot 10^{-06}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 0.11407284983207572:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))