Average Error: 37.1 → 0.7
Time: 23.1s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1230266.91359658050350844860076904296875:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \mathbf{elif}\;\varepsilon \le 9.294677533277005823102466302784718799046 \cdot 10^{-9}:\\ \;\;\;\;2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -1230266.91359658050350844860076904296875:\\
\;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\

\mathbf{elif}\;\varepsilon \le 9.294677533277005823102466302784718799046 \cdot 10^{-9}:\\
\;\;\;\;2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\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 r5273290 = x;
        double r5273291 = eps;
        double r5273292 = r5273290 + r5273291;
        double r5273293 = sin(r5273292);
        double r5273294 = sin(r5273290);
        double r5273295 = r5273293 - r5273294;
        return r5273295;
}
↓
double f(double x, double eps) {
        double r5273296 = eps;
        double r5273297 = -1230266.9135965805;
        bool r5273298 = r5273296 <= r5273297;
        double r5273299 = x;
        double r5273300 = cos(r5273299);
        double r5273301 = sin(r5273296);
        double r5273302 = r5273300 * r5273301;
        double r5273303 = sin(r5273299);
        double r5273304 = r5273302 - r5273303;
        double r5273305 = cos(r5273296);
        double r5273306 = r5273303 * r5273305;
        double r5273307 = r5273304 + r5273306;
        double r5273308 = 9.294677533277006e-09;
        bool r5273309 = r5273296 <= r5273308;
        double r5273310 = 2.0;
        double r5273311 = fma(r5273310, r5273299, r5273296);
        double r5273312 = r5273311 / r5273310;
        double r5273313 = cos(r5273312);
        double r5273314 = 0.5;
        double r5273315 = r5273314 * r5273296;
        double r5273316 = sin(r5273315);
        double r5273317 = r5273313 * r5273316;
        double r5273318 = r5273310 * r5273317;
        double r5273319 = r5273309 ? r5273318 : r5273307;
        double r5273320 = r5273298 ? r5273307 : r5273319;
        return r5273320;
}

Error

Bits error versus x

Bits error versus eps

Target

Original37.1
Target15.0
Herbie0.7
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Split input into 2 regimes

  2. if eps < -1230266.9135965805 or 9.294677533277006e-09 < eps

    1. Initial program 29.9

      \[\sin \left(x + \varepsilon\right) - \sin x\]
    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 -1230266.9135965805 < eps < 9.294677533277006e-09

    1. Initial program 44.4

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

    3. Applied diff-sin44.4

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

    4. Simplified0.9

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1230266.91359658050350844860076904296875:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \mathbf{elif}\;\varepsilon \le 9.294677533277005823102466302784718799046 \cdot 10^{-9}:\\ \;\;\;\;2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (x eps)
  :name "2sin (example 3.3)"

  :herbie-target
  (* 2.0 (* (cos (+ x (/ eps 2.0))) (sin (/ eps 2.0))))

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