Average Error: 36.3 → 0.3
Time: 17.3s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[2 \cdot \left(\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right) \cdot \left(-\sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) + \sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \left(\cos x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right)\right)\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
2 \cdot \left(\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right) \cdot \left(-\sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) + \sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \left(\cos x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right)\right)\right)
double f(double x, double eps) {
        double r3469895 = x;
        double r3469896 = eps;
        double r3469897 = r3469895 + r3469896;
        double r3469898 = sin(r3469897);
        double r3469899 = sin(r3469895);
        double r3469900 = r3469898 - r3469899;
        return r3469900;
}
↓
double f(double x, double eps) {
        double r3469901 = 2.0;
        double r3469902 = 0.5;
        double r3469903 = eps;
        double r3469904 = r3469902 * r3469903;
        double r3469905 = sin(r3469904);
        double r3469906 = x;
        double r3469907 = sin(r3469906);
        double r3469908 = r3469905 * r3469907;
        double r3469909 = -r3469905;
        double r3469910 = r3469908 * r3469909;
        double r3469911 = cos(r3469906);
        double r3469912 = cos(r3469904);
        double r3469913 = r3469911 * r3469912;
        double r3469914 = r3469905 * r3469913;
        double r3469915 = r3469910 + r3469914;
        double r3469916 = r3469901 * r3469915;
        return r3469916;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.3
Target15.2
Herbie0.3
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 36.3

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

  3. Applied diff-sin36.7

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

  4. Simplified15.3

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

  5. Taylor expanded around inf 15.2

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

  6. Simplified15.2

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

  8. Applied cos-sum0.3

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

  10. Applied sub-neg0.3

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

  11. Applied distribute-lft-in0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2019165 
(FPCore (x eps)
  :name "2sin (example 3.3)"

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

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