Average Error: 37.0 → 0.4
Time: 26.3s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\left(\sin x \cdot \cos \varepsilon - \sin x\right) + \sin \varepsilon \cdot \cos x\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\left(\sin x \cdot \cos \varepsilon - \sin x\right) + \sin \varepsilon \cdot \cos x
double f(double x, double eps) {
        double r13957823 = x;
        double r13957824 = eps;
        double r13957825 = r13957823 + r13957824;
        double r13957826 = sin(r13957825);
        double r13957827 = sin(r13957823);
        double r13957828 = r13957826 - r13957827;
        return r13957828;
}
↓
double f(double x, double eps) {
        double r13957829 = x;
        double r13957830 = sin(r13957829);
        double r13957831 = eps;
        double r13957832 = cos(r13957831);
        double r13957833 = r13957830 * r13957832;
        double r13957834 = r13957833 - r13957830;
        double r13957835 = sin(r13957831);
        double r13957836 = cos(r13957829);
        double r13957837 = r13957835 * r13957836;
        double r13957838 = r13957834 + r13957837;
        return r13957838;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.0
Target15.2
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 37.0

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

  3. Applied sin-sum21.7

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

  4. Taylor expanded around -inf 21.7

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

  5. Simplified0.4

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

  7. Applied fma-udef0.4

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

  8. Final simplification0.4

    \[\leadsto \left(\sin x \cdot \cos \varepsilon - \sin x\right) + \sin \varepsilon \cdot \cos x\]

Reproduce

herbie shell --seed 2019107 +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)))