Average Error: 37.3 → 13.0
Time: 47.8s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x}{\cos x}} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x}{\cos x}} - \frac{\sin x}{\cos x}\right)\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x}{\cos x}} + \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)
double f(double x, double eps) {
        double r2108335 = x;
        double r2108336 = eps;
        double r2108337 = r2108335 + r2108336;
        double r2108338 = tan(r2108337);
        double r2108339 = tan(r2108335);
        double r2108340 = r2108338 - r2108339;
        return r2108340;
}
↓
double f(double x, double eps) {
        double r2108341 = eps;
        double r2108342 = sin(r2108341);
        double r2108343 = cos(r2108341);
        double r2108344 = r2108342 / r2108343;
        double r2108345 = 1.0;
        double r2108346 = x;
        double r2108347 = sin(r2108346);
        double r2108348 = r2108344 * r2108347;
        double r2108349 = cos(r2108346);
        double r2108350 = r2108348 / r2108349;
        double r2108351 = r2108345 - r2108350;
        double r2108352 = r2108344 / r2108351;
        double r2108353 = r2108347 / r2108349;
        double r2108354 = r2108353 / r2108351;
        double r2108355 = r2108354 - r2108353;
        double r2108356 = r2108352 + r2108355;
        return r2108356;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.3
Target15.2
Herbie13.0
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 37.3

    \[\tan \left(x + \varepsilon\right) - \tan x\]
  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. Taylor expanded around -inf 22.0

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

  5. Simplified13.0

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

  6. Final simplification13.0

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

Reproduce

herbie shell --seed 2019151 +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)))