Average Error: 36.9 → 0.4
Time: 19.3s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\cos x \cdot \sin \varepsilon + \frac{-\sin x}{\frac{1 + \cos \varepsilon}{{\left(\sin \varepsilon\right)}^{2}}}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\cos x \cdot \sin \varepsilon + \frac{-\sin x}{\frac{1 + \cos \varepsilon}{{\left(\sin \varepsilon\right)}^{2}}}
double f(double x, double eps) {
        double r110852 = x;
        double r110853 = eps;
        double r110854 = r110852 + r110853;
        double r110855 = sin(r110854);
        double r110856 = sin(r110852);
        double r110857 = r110855 - r110856;
        return r110857;
}
↓
double f(double x, double eps) {
        double r110858 = x;
        double r110859 = cos(r110858);
        double r110860 = eps;
        double r110861 = sin(r110860);
        double r110862 = r110859 * r110861;
        double r110863 = sin(r110858);
        double r110864 = -r110863;
        double r110865 = 1.0;
        double r110866 = cos(r110860);
        double r110867 = r110865 + r110866;
        double r110868 = 2.0;
        double r110869 = pow(r110861, r110868);
        double r110870 = r110867 / r110869;
        double r110871 = r110864 / r110870;
        double r110872 = r110862 + r110871;
        return r110872;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.9
Target15.2
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 36.9

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

  2. Simplified36.9

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

  4. Applied sin-sum21.7

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

  5. Applied associate--l+0.4

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

  7. Applied *-un-lft-identity0.4

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

  8. Applied distribute-rgt-out--0.4

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

  10. Applied flip--0.5

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

  11. Simplified0.4

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

  12. Simplified0.4

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

  13. Taylor expanded around inf 0.4

    \[\leadsto \sin \varepsilon \cdot \cos x + \color{blue}{-1 \cdot \frac{\sin x \cdot {\left(\sin \varepsilon\right)}^{2}}{\cos \varepsilon + 1}}\]

  14. Simplified0.4

    \[\leadsto \sin \varepsilon \cdot \cos x + \color{blue}{\frac{-\sin x}{\frac{1 + \cos \varepsilon}{{\left(\sin \varepsilon\right)}^{2}}}}\]

  15. Final simplification0.4

    \[\leadsto \cos x \cdot \sin \varepsilon + \frac{-\sin x}{\frac{1 + \cos \varepsilon}{{\left(\sin \varepsilon\right)}^{2}}}\]

Reproduce

herbie shell --seed 2019194 
(FPCore (x eps)
  :name "2sin (example 3.3)"

  :herbie-target
  (* 2.0 (* (cos (+ x (/ eps 2.0))) (sin (/ eps 2.0))))

  (- (sin (+ x eps)) (sin x)))