Average Error: 37.5 → 0.5
Time: 6.0s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\frac{\sin x \cdot \left(\log \left(e^{{\left(\cos \varepsilon\right)}^{3}}\right) - 1\right)}{\cos \varepsilon \cdot \cos \varepsilon + \left(1 \cdot 1 + \cos \varepsilon \cdot 1\right)} + \cos x \cdot \sin \varepsilon\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\frac{\sin x \cdot \left(\log \left(e^{{\left(\cos \varepsilon\right)}^{3}}\right) - 1\right)}{\cos \varepsilon \cdot \cos \varepsilon + \left(1 \cdot 1 + \cos \varepsilon \cdot 1\right)} + \cos x \cdot \sin \varepsilon
double f(double x, double eps) {
        double r96339 = x;
        double r96340 = eps;
        double r96341 = r96339 + r96340;
        double r96342 = sin(r96341);
        double r96343 = sin(r96339);
        double r96344 = r96342 - r96343;
        return r96344;
}
↓
double f(double x, double eps) {
        double r96345 = x;
        double r96346 = sin(r96345);
        double r96347 = eps;
        double r96348 = cos(r96347);
        double r96349 = 3.0;
        double r96350 = pow(r96348, r96349);
        double r96351 = exp(r96350);
        double r96352 = log(r96351);
        double r96353 = 1.0;
        double r96354 = r96352 - r96353;
        double r96355 = r96346 * r96354;
        double r96356 = r96348 * r96348;
        double r96357 = r96353 * r96353;
        double r96358 = r96348 * r96353;
        double r96359 = r96357 + r96358;
        double r96360 = r96356 + r96359;
        double r96361 = r96355 / r96360;
        double r96362 = cos(r96345);
        double r96363 = sin(r96347);
        double r96364 = r96362 * r96363;
        double r96365 = r96361 + r96364;
        return r96365;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.5
Target14.9
Herbie0.5
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 37.5

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

  3. Applied sin-sum22.5

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

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

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

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

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

  7. Applied distribute-lft-out--22.5

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

  8. Simplified0.4

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

  10. Applied flip3--0.5

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

  11. Applied associate-*r/0.5

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

  12. Simplified0.5

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

  14. Applied add-log-exp0.5

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

  15. Final simplification0.5

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

Reproduce

herbie shell --seed 2019353 
(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)))