Average Error: 37.3 → 0.4
Time: 5.8s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\frac{-\sin x \cdot \left(\sin \varepsilon \cdot \sin \varepsilon\right)}{\cos \varepsilon + 1} + \cos x \cdot \sin \varepsilon\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\frac{-\sin x \cdot \left(\sin \varepsilon \cdot \sin \varepsilon\right)}{\cos \varepsilon + 1} + \cos x \cdot \sin \varepsilon
double f(double x, double eps) {
        double r92744 = x;
        double r92745 = eps;
        double r92746 = r92744 + r92745;
        double r92747 = sin(r92746);
        double r92748 = sin(r92744);
        double r92749 = r92747 - r92748;
        return r92749;
}
↓
double f(double x, double eps) {
        double r92750 = x;
        double r92751 = sin(r92750);
        double r92752 = eps;
        double r92753 = sin(r92752);
        double r92754 = r92753 * r92753;
        double r92755 = r92751 * r92754;
        double r92756 = -r92755;
        double r92757 = cos(r92752);
        double r92758 = 1.0;
        double r92759 = r92757 + r92758;
        double r92760 = r92756 / r92759;
        double r92761 = cos(r92750);
        double r92762 = r92761 * r92753;
        double r92763 = r92760 + r92762;
        return r92763;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.3
Target14.8
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 37.3

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

  3. Applied sin-sum22.3

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

  5. Applied *-un-lft-identity22.3

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

  6. Applied *-un-lft-identity22.3

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

  7. Applied distribute-lft-out--22.3

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

  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. Applied associate-*r/0.5

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

  12. Simplified0.5

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

  14. Applied sub-1-cos0.4

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

  15. Applied distribute-rgt-neg-out0.4

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

  16. Final simplification0.4

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

Reproduce

herbie shell --seed 2020039 
(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)))