Average Error: 36.5 → 0.4
Time: 5.8s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\sin x \cdot \frac{{\left(\cos \varepsilon\right)}^{3} - 1}{\cos \varepsilon \cdot \left(\cos \varepsilon + 1\right) + 1} + \cos x \cdot \sin \varepsilon\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\sin x \cdot \frac{{\left(\cos \varepsilon\right)}^{3} - 1}{\cos \varepsilon \cdot \left(\cos \varepsilon + 1\right) + 1} + \cos x \cdot \sin \varepsilon
double f(double x, double eps) {
        double r153736 = x;
        double r153737 = eps;
        double r153738 = r153736 + r153737;
        double r153739 = sin(r153738);
        double r153740 = sin(r153736);
        double r153741 = r153739 - r153740;
        return r153741;
}
↓
double f(double x, double eps) {
        double r153742 = x;
        double r153743 = sin(r153742);
        double r153744 = eps;
        double r153745 = cos(r153744);
        double r153746 = 3.0;
        double r153747 = pow(r153745, r153746);
        double r153748 = 1.0;
        double r153749 = r153747 - r153748;
        double r153750 = r153745 + r153748;
        double r153751 = r153745 * r153750;
        double r153752 = r153751 + r153748;
        double r153753 = r153749 / r153752;
        double r153754 = r153743 * r153753;
        double r153755 = cos(r153742);
        double r153756 = sin(r153744);
        double r153757 = r153755 * r153756;
        double r153758 = r153754 + r153757;
        return r153758;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.5
Target14.8
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 36.5

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

  3. Applied sin-sum21.6

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

  4. Applied associate--l+21.6

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

  5. Taylor expanded around inf 21.6

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

  6. Simplified0.4

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

  8. Applied flip3--0.4

    \[\leadsto \sin x \cdot \color{blue}{\frac{{\left(\cos \varepsilon\right)}^{3} - {1}^{3}}{\cos \varepsilon \cdot \cos \varepsilon + \left(1 \cdot 1 + \cos \varepsilon \cdot 1\right)}} + \cos x \cdot \sin \varepsilon\]

  9. Simplified0.4

    \[\leadsto \sin x \cdot \frac{\color{blue}{{\left(\cos \varepsilon\right)}^{3} - 1}}{\cos \varepsilon \cdot \cos \varepsilon + \left(1 \cdot 1 + \cos \varepsilon \cdot 1\right)} + \cos x \cdot \sin \varepsilon\]

  10. Simplified0.4

    \[\leadsto \sin x \cdot \frac{{\left(\cos \varepsilon\right)}^{3} - 1}{\color{blue}{\cos \varepsilon \cdot \left(\cos \varepsilon + 1\right) + 1}} + \cos x \cdot \sin \varepsilon\]

  11. Final simplification0.4

    \[\leadsto \sin x \cdot \frac{{\left(\cos \varepsilon\right)}^{3} - 1}{\cos \varepsilon \cdot \left(\cos \varepsilon + 1\right) + 1} + \cos x \cdot \sin \varepsilon\]

Reproduce

herbie shell --seed 2019362 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64

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

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