Average Error: 36.4 → 13.1
Time: 17.0s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\left(\frac{\sin x}{\left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}\right) \cdot \cos x} - \frac{\sin x}{\cos x}\right) + \frac{\frac{\sin \varepsilon}{1 - \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}}}{\cos \varepsilon}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\left(\frac{\sin x}{\left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}\right) \cdot \cos x} - \frac{\sin x}{\cos x}\right) + \frac{\frac{\sin \varepsilon}{1 - \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}}}{\cos \varepsilon}
double f(double x, double eps) {
        double r131267 = x;
        double r131268 = eps;
        double r131269 = r131267 + r131268;
        double r131270 = tan(r131269);
        double r131271 = tan(r131267);
        double r131272 = r131270 - r131271;
        return r131272;
}
↓
double f(double x, double eps) {
        double r131273 = x;
        double r131274 = sin(r131273);
        double r131275 = 1.0;
        double r131276 = eps;
        double r131277 = sin(r131276);
        double r131278 = r131274 * r131277;
        double r131279 = cos(r131273);
        double r131280 = cos(r131276);
        double r131281 = r131279 * r131280;
        double r131282 = r131278 / r131281;
        double r131283 = r131275 - r131282;
        double r131284 = r131283 * r131279;
        double r131285 = r131274 / r131284;
        double r131286 = r131274 / r131279;
        double r131287 = r131285 - r131286;
        double r131288 = r131277 / r131283;
        double r131289 = r131288 / r131280;
        double r131290 = r131287 + r131289;
        return r131290;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.4
Target14.6
Herbie13.1
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 36.4

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

  3. Applied tan-sum21.8

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

  4. Taylor expanded around inf 21.9

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

  6. Applied add-cube-cbrt21.9

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

  7. Applied associate-*r*21.9

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

  8. Taylor expanded around inf 21.9

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

  9. Simplified13.1

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

  10. Final simplification13.1

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

Reproduce

herbie shell --seed 2019351 
(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)))