Average Error: 37.1 → 0.6
Time: 14.3s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\left(\mathsf{fma}\left(\frac{\sin x}{{\left(\cos \varepsilon\right)}^{2}}, \frac{{\left(\sin \varepsilon\right)}^{2}}{\left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right) \cdot \cos x}, \left(\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}} + 1\right) \cdot \frac{\sin \varepsilon}{\left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right) \cdot \cos \varepsilon}\right) + \left(\frac{\sin x}{\left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right) \cdot \cos x} - \frac{\sin x}{\cos x}\right)\right) + \tan x \cdot 0\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\left(\mathsf{fma}\left(\frac{\sin x}{{\left(\cos \varepsilon\right)}^{2}}, \frac{{\left(\sin \varepsilon\right)}^{2}}{\left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right) \cdot \cos x}, \left(\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}} + 1\right) \cdot \frac{\sin \varepsilon}{\left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right) \cdot \cos \varepsilon}\right) + \left(\frac{\sin x}{\left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right) \cdot \cos x} - \frac{\sin x}{\cos x}\right)\right) + \tan x \cdot 0
double f(double x, double eps) {
        double r132240 = x;
        double r132241 = eps;
        double r132242 = r132240 + r132241;
        double r132243 = tan(r132242);
        double r132244 = tan(r132240);
        double r132245 = r132243 - r132244;
        return r132245;
}
↓
double f(double x, double eps) {
        double r132246 = x;
        double r132247 = sin(r132246);
        double r132248 = eps;
        double r132249 = cos(r132248);
        double r132250 = 2.0;
        double r132251 = pow(r132249, r132250);
        double r132252 = r132247 / r132251;
        double r132253 = sin(r132248);
        double r132254 = pow(r132253, r132250);
        double r132255 = 1.0;
        double r132256 = pow(r132247, r132250);
        double r132257 = r132256 * r132254;
        double r132258 = cos(r132246);
        double r132259 = pow(r132258, r132250);
        double r132260 = r132259 * r132251;
        double r132261 = r132257 / r132260;
        double r132262 = r132255 - r132261;
        double r132263 = r132262 * r132258;
        double r132264 = r132254 / r132263;
        double r132265 = r132256 / r132259;
        double r132266 = r132265 + r132255;
        double r132267 = r132262 * r132249;
        double r132268 = r132253 / r132267;
        double r132269 = r132266 * r132268;
        double r132270 = fma(r132252, r132264, r132269);
        double r132271 = r132247 / r132263;
        double r132272 = r132247 / r132258;
        double r132273 = r132271 - r132272;
        double r132274 = r132270 + r132273;
        double r132275 = tan(r132246);
        double r132276 = 0.0;
        double r132277 = r132275 * r132276;
        double r132278 = r132274 + r132277;
        return r132278;
}

Error

Bits error versus x

Bits error versus eps

Target

Original37.1
Target15.1
Herbie0.6
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 37.1

    \[\tan \left(x + \varepsilon\right) - \tan x\]
  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 add-cube-cbrt22.4

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

  6. Applied flip--22.4

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

  7. Applied associate-/r/22.4

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

  8. Applied prod-diff22.5

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

  9. Simplified22.3

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

  10. Simplified21.9

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

  11. Taylor expanded around inf 22.1

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

  12. Simplified0.6

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

  13. Final simplification0.6

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

Reproduce

herbie shell --seed 2020042 +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)))