Average Error: 37.3 → 0.5
Time: 15.3s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -0.008247894483255535:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \mathbf{elif}\;\varepsilon \le 6.77676751873262 \cdot 10^{-09}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -0.008247894483255535:\\
\;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\

\mathbf{elif}\;\varepsilon \le 6.77676751873262 \cdot 10^{-09}:\\
\;\;\;\;2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\

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

\end{array}
double f(double x, double eps) {
        double r2382334 = x;
        double r2382335 = eps;
        double r2382336 = r2382334 + r2382335;
        double r2382337 = sin(r2382336);
        double r2382338 = sin(r2382334);
        double r2382339 = r2382337 - r2382338;
        return r2382339;
}
↓
double f(double x, double eps) {
        double r2382340 = eps;
        double r2382341 = -0.008247894483255535;
        bool r2382342 = r2382340 <= r2382341;
        double r2382343 = x;
        double r2382344 = cos(r2382343);
        double r2382345 = sin(r2382340);
        double r2382346 = r2382344 * r2382345;
        double r2382347 = sin(r2382343);
        double r2382348 = r2382346 - r2382347;
        double r2382349 = cos(r2382340);
        double r2382350 = r2382347 * r2382349;
        double r2382351 = r2382348 + r2382350;
        double r2382352 = 6.77676751873262e-09;
        bool r2382353 = r2382340 <= r2382352;
        double r2382354 = 2.0;
        double r2382355 = 0.5;
        double r2382356 = r2382355 * r2382340;
        double r2382357 = sin(r2382356);
        double r2382358 = r2382343 + r2382340;
        double r2382359 = r2382358 + r2382343;
        double r2382360 = r2382359 / r2382354;
        double r2382361 = cos(r2382360);
        double r2382362 = r2382357 * r2382361;
        double r2382363 = r2382354 * r2382362;
        double r2382364 = r2382353 ? r2382363 : r2382351;
        double r2382365 = r2382342 ? r2382351 : r2382364;
        return r2382365;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.3
Target15.1
Herbie0.5
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Split input into 2 regimes

  2. if eps < -0.008247894483255535 or 6.77676751873262e-09 < eps

    1. Initial program 30.3

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

    3. Applied sin-sum0.5

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

    4. Applied associate--l+0.5

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

    if -0.008247894483255535 < eps < 6.77676751873262e-09

    1. Initial program 44.5

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

    3. Applied diff-sin44.5

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

    4. Simplified0.5

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -0.008247894483255535:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \mathbf{elif}\;\varepsilon \le 6.77676751873262 \cdot 10^{-09}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 
(FPCore (x eps)
  :name "2sin (example 3.3)"

  :herbie-target
  (* 2 (* (cos (+ x (/ eps 2))) (sin (/ eps 2))))

  (- (sin (+ x eps)) (sin x)))