Average Error: 37.1 → 0.5
Time: 6.3s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\sin x \cdot \frac{\cos \varepsilon \cdot \cos \varepsilon - 1}{\cos \varepsilon + 1} + \cos x \cdot \sin \varepsilon\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\sin x \cdot \frac{\cos \varepsilon \cdot \cos \varepsilon - 1}{\cos \varepsilon + 1} + \cos x \cdot \sin \varepsilon
double f(double x, double eps) {
        double r126624 = x;
        double r126625 = eps;
        double r126626 = r126624 + r126625;
        double r126627 = sin(r126626);
        double r126628 = sin(r126624);
        double r126629 = r126627 - r126628;
        return r126629;
}
↓
double f(double x, double eps) {
        double r126630 = x;
        double r126631 = sin(r126630);
        double r126632 = eps;
        double r126633 = cos(r126632);
        double r126634 = r126633 * r126633;
        double r126635 = 1.0;
        double r126636 = r126634 - r126635;
        double r126637 = r126633 + r126635;
        double r126638 = r126636 / r126637;
        double r126639 = r126631 * r126638;
        double r126640 = cos(r126630);
        double r126641 = sin(r126632);
        double r126642 = r126640 * r126641;
        double r126643 = r126639 + r126642;
        return r126643;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.1
Target15.0
Herbie0.5
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 37.1

    \[\sin \left(x + \varepsilon\right) - \sin x\]
  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 flip--0.5

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

  11. Simplified0.5

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

  12. Final simplification0.5

    \[\leadsto \sin x \cdot \frac{\cos \varepsilon \cdot \cos \varepsilon - 1}{\cos \varepsilon + 1} + \cos x \cdot \sin \varepsilon\]

Reproduce

herbie shell --seed 2020062 
(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)))