Average Error: 36.6 → 13.0
Time: 16.6s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \cos x + \sin x \cdot \left(1 + \left(\tan \varepsilon \cdot \tan x + -1\right)\right)}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \cos x + \sin x \cdot \left(1 + \left(\tan \varepsilon \cdot \tan x + -1\right)\right)}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}
double f(double x, double eps) {
        double r133327 = x;
        double r133328 = eps;
        double r133329 = r133327 + r133328;
        double r133330 = tan(r133329);
        double r133331 = tan(r133327);
        double r133332 = r133330 - r133331;
        return r133332;
}
↓
double f(double x, double eps) {
        double r133333 = eps;
        double r133334 = sin(r133333);
        double r133335 = cos(r133333);
        double r133336 = r133334 / r133335;
        double r133337 = x;
        double r133338 = cos(r133337);
        double r133339 = r133336 * r133338;
        double r133340 = sin(r133337);
        double r133341 = 1.0;
        double r133342 = tan(r133333);
        double r133343 = tan(r133337);
        double r133344 = r133342 * r133343;
        double r133345 = -1.0;
        double r133346 = r133344 + r133345;
        double r133347 = r133341 + r133346;
        double r133348 = r133340 * r133347;
        double r133349 = r133339 + r133348;
        double r133350 = r133343 * r133342;
        double r133351 = r133341 - r133350;
        double r133352 = r133351 * r133338;
        double r133353 = r133349 / r133352;
        return r133353;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.6
Target14.9
Herbie13.0
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 36.6

    \[\tan \left(x + \varepsilon\right) - \tan x\]
  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. Taylor expanded around inf 21.8

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

  6. Applied div-inv21.9

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

  8. Applied tan-quot21.8

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

  9. Applied frac-sub21.9

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

  10. Simplified13.0

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

  11. Final simplification13.0

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

Reproduce

herbie shell --seed 2019350 
(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)))