Average Error: 36.1 → 12.7
Time: 30.6s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\frac{\frac{\frac{\sin x}{\cos x}}{1 - \frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x}{\cos x}} \cdot \frac{\frac{\sin x}{\cos x}}{1 - \frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x}{\cos x}} - \frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}}{\frac{\sin x}{\cos x} + \frac{\frac{\sin x}{\cos x}}{1 - \frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x}{\cos x}}} + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x}{\cos x}}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\frac{\frac{\frac{\sin x}{\cos x}}{1 - \frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x}{\cos x}} \cdot \frac{\frac{\sin x}{\cos x}}{1 - \frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x}{\cos x}} - \frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}}{\frac{\sin x}{\cos x} + \frac{\frac{\sin x}{\cos x}}{1 - \frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x}{\cos x}}} + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x}{\cos x}}
double f(double x, double eps) {
        double r5087338 = x;
        double r5087339 = eps;
        double r5087340 = r5087338 + r5087339;
        double r5087341 = tan(r5087340);
        double r5087342 = tan(r5087338);
        double r5087343 = r5087341 - r5087342;
        return r5087343;
}
↓
double f(double x, double eps) {
        double r5087344 = x;
        double r5087345 = sin(r5087344);
        double r5087346 = cos(r5087344);
        double r5087347 = r5087345 / r5087346;
        double r5087348 = 1.0;
        double r5087349 = eps;
        double r5087350 = sin(r5087349);
        double r5087351 = cos(r5087349);
        double r5087352 = r5087350 / r5087351;
        double r5087353 = r5087352 * r5087345;
        double r5087354 = r5087353 / r5087346;
        double r5087355 = r5087348 - r5087354;
        double r5087356 = r5087347 / r5087355;
        double r5087357 = r5087356 * r5087356;
        double r5087358 = r5087347 * r5087347;
        double r5087359 = r5087357 - r5087358;
        double r5087360 = r5087347 + r5087356;
        double r5087361 = r5087359 / r5087360;
        double r5087362 = r5087352 / r5087355;
        double r5087363 = r5087361 + r5087362;
        return r5087363;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.1
Target14.3
Herbie12.7
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 36.1

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

  3. Applied tan-sum21.7

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

  5. Applied tan-quot21.7

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

  6. Applied associate-*r/21.7

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

  7. Taylor expanded around inf 21.8

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

  8. Simplified12.7

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

  10. Applied flip--12.7

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

  11. Final simplification12.7

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

Reproduce

herbie shell --seed 2019158 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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