Average Error: 37.5 → 0.5
Time: 6.4s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\left(\sin x \cdot \cos \varepsilon + \left(-\sin x\right)\right) + \cos x \cdot \sin \varepsilon\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\left(\sin x \cdot \cos \varepsilon + \left(-\sin x\right)\right) + \cos x \cdot \sin \varepsilon
double f(double x, double eps) {
        double r88355 = x;
        double r88356 = eps;
        double r88357 = r88355 + r88356;
        double r88358 = sin(r88357);
        double r88359 = sin(r88355);
        double r88360 = r88358 - r88359;
        return r88360;
}
↓
double f(double x, double eps) {
        double r88361 = x;
        double r88362 = sin(r88361);
        double r88363 = eps;
        double r88364 = cos(r88363);
        double r88365 = r88362 * r88364;
        double r88366 = -r88362;
        double r88367 = r88365 + r88366;
        double r88368 = cos(r88361);
        double r88369 = sin(r88363);
        double r88370 = r88368 * r88369;
        double r88371 = r88367 + r88370;
        return r88371;
}

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

    \[\leadsto 1 \cdot \color{blue}{\mathsf{fma}\left(\sin x, \cos \varepsilon - 1, \cos x \cdot \sin \varepsilon\right)}\]
  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 sub-neg0.4

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

  13. Applied distribute-lft-in0.5

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

  14. Simplified0.5

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

  15. Final simplification0.5

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

Reproduce

herbie shell --seed 2019353 +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)))