Average Error: 37.5 → 0.4
Time: 15.7s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\cos x \cdot \sin \varepsilon + \left(\cos \varepsilon - 1\right) \cdot \sin x\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\cos x \cdot \sin \varepsilon + \left(\cos \varepsilon - 1\right) \cdot \sin x
double f(double x, double eps) {
        double r99450 = x;
        double r99451 = eps;
        double r99452 = r99450 + r99451;
        double r99453 = sin(r99452);
        double r99454 = sin(r99450);
        double r99455 = r99453 - r99454;
        return r99455;
}
↓
double f(double x, double eps) {
        double r99456 = x;
        double r99457 = cos(r99456);
        double r99458 = eps;
        double r99459 = sin(r99458);
        double r99460 = r99457 * r99459;
        double r99461 = cos(r99458);
        double r99462 = 1.0;
        double r99463 = r99461 - r99462;
        double r99464 = sin(r99456);
        double r99465 = r99463 * r99464;
        double r99466 = r99460 + r99465;
        return r99466;
}

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
Target15.1
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 37.5

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

  3. Applied sin-sum22.0

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

  5. Applied *-un-lft-identity22.0

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

  6. Applied *-un-lft-identity22.0

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

  7. Applied distribute-lft-out--22.0

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

  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 add-cbrt-cube0.5

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

  11. Simplified0.5

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2019291 
(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)))