Average Error: 37.2 → 0.5
Time: 29.3s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -7.240915274365558 \cdot 10^{-09}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 1.2284365724618488 \cdot 10^{-08}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -7.240915274365558 \cdot 10^{-09}:\\
\;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\

\mathbf{elif}\;\varepsilon \le 1.2284365724618488 \cdot 10^{-08}:\\
\;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\

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

\end{array}
double f(double x, double eps) {
        double r15314124 = x;
        double r15314125 = eps;
        double r15314126 = r15314124 + r15314125;
        double r15314127 = sin(r15314126);
        double r15314128 = sin(r15314124);
        double r15314129 = r15314127 - r15314128;
        return r15314129;
}
↓
double f(double x, double eps) {
        double r15314130 = eps;
        double r15314131 = -7.240915274365558e-09;
        bool r15314132 = r15314130 <= r15314131;
        double r15314133 = x;
        double r15314134 = sin(r15314133);
        double r15314135 = cos(r15314130);
        double r15314136 = r15314134 * r15314135;
        double r15314137 = cos(r15314133);
        double r15314138 = sin(r15314130);
        double r15314139 = r15314137 * r15314138;
        double r15314140 = r15314136 + r15314139;
        double r15314141 = r15314140 - r15314134;
        double r15314142 = 1.2284365724618488e-08;
        bool r15314143 = r15314130 <= r15314142;
        double r15314144 = 2.0;
        double r15314145 = r15314130 / r15314144;
        double r15314146 = sin(r15314145);
        double r15314147 = r15314133 + r15314130;
        double r15314148 = r15314147 + r15314133;
        double r15314149 = r15314148 / r15314144;
        double r15314150 = cos(r15314149);
        double r15314151 = r15314146 * r15314150;
        double r15314152 = r15314144 * r15314151;
        double r15314153 = r15314143 ? r15314152 : r15314141;
        double r15314154 = r15314132 ? r15314141 : r15314153;
        return r15314154;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.2
Target14.5
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 < -7.240915274365558e-09 or 1.2284365724618488e-08 < eps

    1. Initial program 29.6

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

    3. Applied sin-sum0.6

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

    if -7.240915274365558e-09 < eps < 1.2284365724618488e-08

    1. Initial program 44.8

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

    3. Applied diff-sin44.8

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

    4. Simplified0.3

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -7.240915274365558 \cdot 10^{-09}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 1.2284365724618488 \cdot 10^{-08}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

Reproduce

herbie shell --seed 2019125 +o rules:numerics
(FPCore (x eps)
  :name "2sin (example 3.3)"

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

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