Average Error: 37.6 → 0.5
Time: 11.3s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\log \left(e^{\sin x \cdot \left(\cos \varepsilon - 1\right)}\right) + \cos x \cdot \sin \varepsilon\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\log \left(e^{\sin x \cdot \left(\cos \varepsilon - 1\right)}\right) + \cos x \cdot \sin \varepsilon
double f(double x, double eps) {
        double r566 = x;
        double r567 = eps;
        double r568 = r566 + r567;
        double r569 = sin(r568);
        double r570 = sin(r566);
        double r571 = r569 - r570;
        return r571;
}
↓
double f(double x, double eps) {
        double r572 = x;
        double r573 = sin(r572);
        double r574 = eps;
        double r575 = cos(r574);
        double r576 = 1.0;
        double r577 = r575 - r576;
        double r578 = r573 * r577;
        double r579 = exp(r578);
        double r580 = log(r579);
        double r581 = cos(r572);
        double r582 = sin(r574);
        double r583 = r581 * r582;
        double r584 = r580 + r583;
        return r584;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.6
Target15.4
Herbie0.5
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 37.6

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

  3. Applied sin-sum22.1

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

  5. Applied *-un-lft-identity22.1

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

  6. Applied *-un-lft-identity22.1

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

  7. Applied distribute-lft-out--22.1

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

  8. Simplified0.4

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

  10. Applied fma-udef0.4

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

  12. Applied add-log-exp0.5

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

  13. Final simplification0.5

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

Reproduce

herbie shell --seed 2020025 +o rules:numerics
(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)))