Average Error: 37.4 → 0.6
Time: 25.5s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\mathsf{fma}\left(\frac{{\left(\sin x\right)}^{2}}{\mathsf{fma}\left({\left(\frac{\sin x}{\cos x}\right)}^{3}, -{\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3}, 1\right) \cdot \cos \varepsilon}, \frac{\sin \varepsilon}{{\left(\cos x\right)}^{2}}, \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{\mathsf{fma}\left({\left(\frac{\sin x}{\cos x}\right)}^{3}, -{\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3}, 1\right)} + \left(\mathsf{fma}\left(\frac{\frac{{\left(\sin \varepsilon\right)}^{2}}{{\left(\cos \varepsilon\right)}^{2}}}{\mathsf{fma}\left({\left(\frac{\sin x}{\cos x}\right)}^{3}, -{\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3}, 1\right)}, \frac{\sin x}{\cos x} + {\left(\frac{\sin x}{\cos x}\right)}^{3}, \mathsf{fma}\left(\frac{{\left(\sin x\right)}^{2}}{\mathsf{fma}\left({\left(\frac{\sin x}{\cos x}\right)}^{3}, -{\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3}, 1\right)}, \frac{{\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3}}{{\left(\cos x\right)}^{2}}, \frac{\frac{\sin x}{\mathsf{fma}\left({\left(\frac{\sin x}{\cos x}\right)}^{3}, -{\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3}, 1\right)}}{\cos x}\right)\right) - \frac{\sin x}{\cos x}\right)\right)\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\mathsf{fma}\left(\frac{{\left(\sin x\right)}^{2}}{\mathsf{fma}\left({\left(\frac{\sin x}{\cos x}\right)}^{3}, -{\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3}, 1\right) \cdot \cos \varepsilon}, \frac{\sin \varepsilon}{{\left(\cos x\right)}^{2}}, \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{\mathsf{fma}\left({\left(\frac{\sin x}{\cos x}\right)}^{3}, -{\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3}, 1\right)} + \left(\mathsf{fma}\left(\frac{\frac{{\left(\sin \varepsilon\right)}^{2}}{{\left(\cos \varepsilon\right)}^{2}}}{\mathsf{fma}\left({\left(\frac{\sin x}{\cos x}\right)}^{3}, -{\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3}, 1\right)}, \frac{\sin x}{\cos x} + {\left(\frac{\sin x}{\cos x}\right)}^{3}, \mathsf{fma}\left(\frac{{\left(\sin x\right)}^{2}}{\mathsf{fma}\left({\left(\frac{\sin x}{\cos x}\right)}^{3}, -{\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3}, 1\right)}, \frac{{\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3}}{{\left(\cos x\right)}^{2}}, \frac{\frac{\sin x}{\mathsf{fma}\left({\left(\frac{\sin x}{\cos x}\right)}^{3}, -{\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3}, 1\right)}}{\cos x}\right)\right) - \frac{\sin x}{\cos x}\right)\right)
double f(double x, double eps) {
        double r93300 = x;
        double r93301 = eps;
        double r93302 = r93300 + r93301;
        double r93303 = tan(r93302);
        double r93304 = tan(r93300);
        double r93305 = r93303 - r93304;
        return r93305;
}
↓
double f(double x, double eps) {
        double r93306 = x;
        double r93307 = sin(r93306);
        double r93308 = 2.0;
        double r93309 = pow(r93307, r93308);
        double r93310 = cos(r93306);
        double r93311 = r93307 / r93310;
        double r93312 = 3.0;
        double r93313 = pow(r93311, r93312);
        double r93314 = eps;
        double r93315 = sin(r93314);
        double r93316 = cos(r93314);
        double r93317 = r93315 / r93316;
        double r93318 = pow(r93317, r93312);
        double r93319 = -r93318;
        double r93320 = 1.0;
        double r93321 = fma(r93313, r93319, r93320);
        double r93322 = r93321 * r93316;
        double r93323 = r93309 / r93322;
        double r93324 = pow(r93310, r93308);
        double r93325 = r93315 / r93324;
        double r93326 = r93317 / r93321;
        double r93327 = pow(r93315, r93308);
        double r93328 = pow(r93316, r93308);
        double r93329 = r93327 / r93328;
        double r93330 = r93329 / r93321;
        double r93331 = r93311 + r93313;
        double r93332 = r93309 / r93321;
        double r93333 = r93318 / r93324;
        double r93334 = r93307 / r93321;
        double r93335 = r93334 / r93310;
        double r93336 = fma(r93332, r93333, r93335);
        double r93337 = fma(r93330, r93331, r93336);
        double r93338 = r93337 - r93311;
        double r93339 = r93326 + r93338;
        double r93340 = fma(r93323, r93325, r93339);
        return r93340;
}

Error

Bits error versus x

Bits error versus eps

Target

Original37.4
Target15.4
Herbie0.6
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 37.4

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

  3. Applied tan-sum21.9

    \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
  4. Using strategy rm

  5. Applied tan-quot21.9

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

  6. Applied associate-*r/21.9

    \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}}} - \tan x\]
  7. Using strategy rm

  8. Applied add-cube-cbrt22.4

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

  9. Applied flip3--22.4

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

  10. Applied associate-/r/22.4

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

  11. Applied prod-diff22.5

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

  12. Simplified22.3

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

  13. Simplified21.9

    \[\leadsto \mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 - {\left(\frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}\right)}^{3}}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{\tan x}{\cos \varepsilon}, \sin \varepsilon, 1\right), \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}, 1\right), -\tan x\right) + \color{blue}{0}\]

  14. Taylor expanded around -inf 22.0

    \[\leadsto \color{blue}{\left(\left(\frac{{\left(\sin x\right)}^{2} \cdot \sin \varepsilon}{\cos \varepsilon \cdot \left(\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot {\left(\cos x\right)}^{2}\right)} + \left(\frac{\sin \varepsilon}{\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot \cos \varepsilon} + \left(\frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{3}}{\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot \left({\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{3}\right)} + \left(\frac{\sin x}{\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot \cos x} + \left(\frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos \varepsilon\right)}^{2} \cdot \left(\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot {\left(\cos x\right)}^{3}\right)} + \frac{\sin x \cdot {\left(\sin \varepsilon\right)}^{2}}{\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot \left(\cos x \cdot {\left(\cos \varepsilon\right)}^{2}\right)}\right)\right)\right)\right)\right) - \frac{\sin x}{\cos x}\right)} + 0\]

  15. Simplified0.6

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(\sin x\right)}^{2}}{\mathsf{fma}\left({\left(\frac{\sin x}{\cos x}\right)}^{3}, -{\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3}, 1\right) \cdot \cos \varepsilon}, \frac{\sin \varepsilon}{{\left(\cos x\right)}^{2}}, \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{\mathsf{fma}\left({\left(\frac{\sin x}{\cos x}\right)}^{3}, -{\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3}, 1\right)} + \left(\mathsf{fma}\left(\frac{\frac{{\left(\sin \varepsilon\right)}^{2}}{{\left(\cos \varepsilon\right)}^{2}}}{\mathsf{fma}\left({\left(\frac{\sin x}{\cos x}\right)}^{3}, -{\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3}, 1\right)}, \frac{\sin x}{\cos x} + {\left(\frac{\sin x}{\cos x}\right)}^{3}, \mathsf{fma}\left(\frac{{\left(\sin x\right)}^{2}}{\mathsf{fma}\left({\left(\frac{\sin x}{\cos x}\right)}^{3}, -{\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3}, 1\right)}, \frac{{\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3}}{{\left(\cos x\right)}^{2}}, \frac{\frac{\sin x}{\mathsf{fma}\left({\left(\frac{\sin x}{\cos x}\right)}^{3}, -{\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3}, 1\right)}}{\cos x}\right)\right) - \frac{\sin x}{\cos x}\right)\right)} + 0\]

  16. Final simplification0.6

    \[\leadsto \mathsf{fma}\left(\frac{{\left(\sin x\right)}^{2}}{\mathsf{fma}\left({\left(\frac{\sin x}{\cos x}\right)}^{3}, -{\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3}, 1\right) \cdot \cos \varepsilon}, \frac{\sin \varepsilon}{{\left(\cos x\right)}^{2}}, \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{\mathsf{fma}\left({\left(\frac{\sin x}{\cos x}\right)}^{3}, -{\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3}, 1\right)} + \left(\mathsf{fma}\left(\frac{\frac{{\left(\sin \varepsilon\right)}^{2}}{{\left(\cos \varepsilon\right)}^{2}}}{\mathsf{fma}\left({\left(\frac{\sin x}{\cos x}\right)}^{3}, -{\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3}, 1\right)}, \frac{\sin x}{\cos x} + {\left(\frac{\sin x}{\cos x}\right)}^{3}, \mathsf{fma}\left(\frac{{\left(\sin x\right)}^{2}}{\mathsf{fma}\left({\left(\frac{\sin x}{\cos x}\right)}^{3}, -{\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3}, 1\right)}, \frac{{\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3}}{{\left(\cos x\right)}^{2}}, \frac{\frac{\sin x}{\mathsf{fma}\left({\left(\frac{\sin x}{\cos x}\right)}^{3}, -{\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3}, 1\right)}}{\cos x}\right)\right) - \frac{\sin x}{\cos x}\right)\right)\]

Reproduce

herbie shell --seed 2019306 +o rules:numerics
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :precision binary64

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

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