Average Error: 37.1 → 0.7
Time: 17.4s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1230266.91359658050350844860076904296875:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \mathbf{elif}\;\varepsilon \le 9.294677533277005823102466302784718799046 \cdot 10^{-9}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -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(\sin \left(\frac{\varepsilon}{2}\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 r7036802 = x;
        double r7036803 = eps;
        double r7036804 = r7036802 + r7036803;
        double r7036805 = sin(r7036804);
        double r7036806 = sin(r7036802);
        double r7036807 = r7036805 - r7036806;
        return r7036807;
}
↓
double f(double x, double eps) {
        double r7036808 = eps;
        double r7036809 = -1230266.9135965805;
        bool r7036810 = r7036808 <= r7036809;
        double r7036811 = x;
        double r7036812 = cos(r7036811);
        double r7036813 = sin(r7036808);
        double r7036814 = r7036812 * r7036813;
        double r7036815 = sin(r7036811);
        double r7036816 = r7036814 - r7036815;
        double r7036817 = cos(r7036808);
        double r7036818 = r7036815 * r7036817;
        double r7036819 = r7036816 + r7036818;
        double r7036820 = 9.294677533277006e-09;
        bool r7036821 = r7036808 <= r7036820;
        double r7036822 = 2.0;
        double r7036823 = r7036808 / r7036822;
        double r7036824 = sin(r7036823);
        double r7036825 = r7036811 + r7036808;
        double r7036826 = r7036825 + r7036811;
        double r7036827 = r7036826 / r7036822;
        double r7036828 = cos(r7036827);
        double r7036829 = r7036824 * r7036828;
        double r7036830 = r7036822 * r7036829;
        double r7036831 = r7036821 ? r7036830 : r7036819;
        double r7036832 = r7036810 ? r7036819 : r7036831;
        return r7036832;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.1
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

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019171 
(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)))