Average Error: 37.0 → 0.4
Time: 12.4s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\sin x \cdot \frac{{\left(\cos \varepsilon\right)}^{3} - {1}^{3}}{\cos \varepsilon \cdot \left(\cos \varepsilon + 1\right) + 1} + \cos x \cdot \sin \varepsilon\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\sin x \cdot \frac{{\left(\cos \varepsilon\right)}^{3} - {1}^{3}}{\cos \varepsilon \cdot \left(\cos \varepsilon + 1\right) + 1} + \cos x \cdot \sin \varepsilon
double f(double x, double eps) {
        double r90604 = x;
        double r90605 = eps;
        double r90606 = r90604 + r90605;
        double r90607 = sin(r90606);
        double r90608 = sin(r90604);
        double r90609 = r90607 - r90608;
        return r90609;
}
↓
double f(double x, double eps) {
        double r90610 = x;
        double r90611 = sin(r90610);
        double r90612 = eps;
        double r90613 = cos(r90612);
        double r90614 = 3.0;
        double r90615 = pow(r90613, r90614);
        double r90616 = 1.0;
        double r90617 = pow(r90616, r90614);
        double r90618 = r90615 - r90617;
        double r90619 = r90613 + r90616;
        double r90620 = r90613 * r90619;
        double r90621 = r90620 + r90616;
        double r90622 = r90618 / r90621;
        double r90623 = r90611 * r90622;
        double r90624 = cos(r90610);
        double r90625 = sin(r90612);
        double r90626 = r90624 * r90625;
        double r90627 = r90623 + r90626;
        return r90627;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.0
Target15.0
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 37.0

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

  3. Applied sin-sum21.9

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

  5. Applied *-un-lft-identity21.9

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

  6. Applied *-un-lft-identity21.9

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

  7. Applied distribute-lft-out--21.9

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

  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--0.4

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

  11. Simplified0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2019304 
(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)))