Average Error: 37.2 → 0.4
Time: 25.7s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[2 \cdot \left(\left(\cos x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right) - \log \left(e^{\sin x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
2 \cdot \left(\left(\cos x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right) - \log \left(e^{\sin x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)
double f(double x, double eps) {
        double r8334704 = x;
        double r8334705 = eps;
        double r8334706 = r8334704 + r8334705;
        double r8334707 = sin(r8334706);
        double r8334708 = sin(r8334704);
        double r8334709 = r8334707 - r8334708;
        return r8334709;
}
↓
double f(double x, double eps) {
        double r8334710 = 2.0;
        double r8334711 = x;
        double r8334712 = cos(r8334711);
        double r8334713 = eps;
        double r8334714 = 0.5;
        double r8334715 = r8334713 * r8334714;
        double r8334716 = cos(r8334715);
        double r8334717 = r8334712 * r8334716;
        double r8334718 = sin(r8334711);
        double r8334719 = sin(r8334715);
        double r8334720 = r8334718 * r8334719;
        double r8334721 = exp(r8334720);
        double r8334722 = log(r8334721);
        double r8334723 = r8334717 - r8334722;
        double r8334724 = r8334713 / r8334710;
        double r8334725 = sin(r8334724);
        double r8334726 = r8334723 * r8334725;
        double r8334727 = r8334710 * r8334726;
        return r8334727;
}

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
Target15.1
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 37.2

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

  3. Applied diff-sin37.6

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

  4. Simplified15.1

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

  5. Taylor expanded around inf 15.1

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

  6. Simplified15.1

    \[\leadsto 2 \cdot \left(\color{blue}{\cos \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right)} \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]
  7. Using strategy rm

  8. Applied fma-udef15.1

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

  9. Applied cos-sum0.3

    \[\leadsto 2 \cdot \left(\color{blue}{\left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x - \sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right)} \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]
  10. Using strategy rm

  11. Applied add-log-exp0.4

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

  12. Final simplification0.4

    \[\leadsto 2 \cdot \left(\left(\cos x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right) - \log \left(e^{\sin x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Reproduce

herbie shell --seed 2019120 +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)))