Average Error: 36.8 → 12.9
Time: 43.6s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\log \left(e^{\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} - \frac{\sin x}{\cos x}}\right) + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\log \left(e^{\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} - \frac{\sin x}{\cos x}}\right) + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}}
double f(double x, double eps) {
        double r3426285 = x;
        double r3426286 = eps;
        double r3426287 = r3426285 + r3426286;
        double r3426288 = tan(r3426287);
        double r3426289 = tan(r3426285);
        double r3426290 = r3426288 - r3426289;
        return r3426290;
}
↓
double f(double x, double eps) {
        double r3426291 = x;
        double r3426292 = sin(r3426291);
        double r3426293 = cos(r3426291);
        double r3426294 = r3426292 / r3426293;
        double r3426295 = 1.0;
        double r3426296 = eps;
        double r3426297 = sin(r3426296);
        double r3426298 = cos(r3426296);
        double r3426299 = r3426297 / r3426298;
        double r3426300 = r3426299 * r3426294;
        double r3426301 = r3426295 - r3426300;
        double r3426302 = r3426294 / r3426301;
        double r3426303 = r3426302 - r3426294;
        double r3426304 = exp(r3426303);
        double r3426305 = log(r3426304);
        double r3426306 = r3426299 / r3426301;
        double r3426307 = r3426305 + r3426306;
        return r3426307;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.8
Target14.7
Herbie12.9
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 36.8

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

  3. Applied tan-sum22.1

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

  5. Applied tan-quot22.1

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

  6. Applied associate-*l/22.1

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

  7. Taylor expanded around inf 22.2

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

  8. Simplified12.9

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

  10. Applied add-log-exp21.8

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

  11. Applied add-log-exp12.9

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

  12. Applied diff-log12.9

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

  13. Simplified12.9

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

  14. Final simplification12.9

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

Reproduce

herbie shell --seed 2019146 +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)))