Average Error: 37.0 → 12.7
Time: 28.7s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{\mathsf{fma}\left(\frac{\sin x}{\cos x}, -\frac{\sin \varepsilon}{\cos \varepsilon}, 1\right)} + \left(\mathsf{fma}\left(\frac{-1}{\cos x}, \sin x, \frac{\sin x}{\cos x}\right) + \mathsf{fma}\left(\frac{\frac{\sin x}{\sqrt[3]{\mathsf{fma}\left(\frac{\sin x}{\cos x}, -\frac{\sin \varepsilon}{\cos \varepsilon}, 1\right)}}}{\sqrt[3]{\mathsf{fma}\left(\frac{\sin x}{\cos x}, -\frac{\sin \varepsilon}{\cos \varepsilon}, 1\right)}}, \frac{1}{\sqrt[3]{\mathsf{fma}\left(\frac{\sin x}{\cos x}, -\frac{\sin \varepsilon}{\cos \varepsilon}, 1\right)} \cdot \cos x}, -\frac{\sin x}{\cos x}\right)\right)\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{\mathsf{fma}\left(\frac{\sin x}{\cos x}, -\frac{\sin \varepsilon}{\cos \varepsilon}, 1\right)} + \left(\mathsf{fma}\left(\frac{-1}{\cos x}, \sin x, \frac{\sin x}{\cos x}\right) + \mathsf{fma}\left(\frac{\frac{\sin x}{\sqrt[3]{\mathsf{fma}\left(\frac{\sin x}{\cos x}, -\frac{\sin \varepsilon}{\cos \varepsilon}, 1\right)}}}{\sqrt[3]{\mathsf{fma}\left(\frac{\sin x}{\cos x}, -\frac{\sin \varepsilon}{\cos \varepsilon}, 1\right)}}, \frac{1}{\sqrt[3]{\mathsf{fma}\left(\frac{\sin x}{\cos x}, -\frac{\sin \varepsilon}{\cos \varepsilon}, 1\right)} \cdot \cos x}, -\frac{\sin x}{\cos x}\right)\right)
double f(double x, double eps) {
        double r126686 = x;
        double r126687 = eps;
        double r126688 = r126686 + r126687;
        double r126689 = tan(r126688);
        double r126690 = tan(r126686);
        double r126691 = r126689 - r126690;
        return r126691;
}
↓
double f(double x, double eps) {
        double r126692 = eps;
        double r126693 = sin(r126692);
        double r126694 = cos(r126692);
        double r126695 = r126693 / r126694;
        double r126696 = x;
        double r126697 = sin(r126696);
        double r126698 = cos(r126696);
        double r126699 = r126697 / r126698;
        double r126700 = -r126695;
        double r126701 = 1.0;
        double r126702 = fma(r126699, r126700, r126701);
        double r126703 = r126695 / r126702;
        double r126704 = -1.0;
        double r126705 = r126704 / r126698;
        double r126706 = fma(r126705, r126697, r126699);
        double r126707 = cbrt(r126702);
        double r126708 = r126697 / r126707;
        double r126709 = r126708 / r126707;
        double r126710 = r126707 * r126698;
        double r126711 = r126701 / r126710;
        double r126712 = -r126699;
        double r126713 = fma(r126709, r126711, r126712);
        double r126714 = r126706 + r126713;
        double r126715 = r126703 + r126714;
        return r126715;
}

Error

Bits error versus x

Bits error versus eps

Target

Original37.0
Target15.2
Herbie12.7
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 37.0

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

  3. Applied tan-sum21.6

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

  4. Simplified21.6

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

  5. Simplified21.6

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

  6. Taylor expanded around inf 21.7

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

  7. Simplified12.5

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

  9. Applied add-cube-cbrt20.9

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

  10. Applied add-cube-cbrt20.9

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

  11. Applied div-inv20.9

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

  12. Applied times-frac20.9

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

  13. Applied prod-diff22.3

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

  14. Simplified22.2

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

  15. Simplified12.7

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

  16. Final simplification12.7

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

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

  (- (tan (+ x eps)) (tan x)))