Average Error: 36.9 → 0.4
Time: 26.1s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\frac{\frac{\sin \varepsilon \cdot \cos x}{\cos \varepsilon} + \frac{{\left(\sin x\right)}^{2} \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\frac{\frac{\sin \varepsilon \cdot \cos x}{\cos \varepsilon} + \frac{{\left(\sin x\right)}^{2} \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}
double f(double x, double eps) {
        double r79383 = x;
        double r79384 = eps;
        double r79385 = r79383 + r79384;
        double r79386 = tan(r79385);
        double r79387 = tan(r79383);
        double r79388 = r79386 - r79387;
        return r79388;
}
↓
double f(double x, double eps) {
        double r79389 = eps;
        double r79390 = sin(r79389);
        double r79391 = x;
        double r79392 = cos(r79391);
        double r79393 = r79390 * r79392;
        double r79394 = cos(r79389);
        double r79395 = r79393 / r79394;
        double r79396 = sin(r79391);
        double r79397 = 2.0;
        double r79398 = pow(r79396, r79397);
        double r79399 = r79398 * r79390;
        double r79400 = r79392 * r79394;
        double r79401 = r79399 / r79400;
        double r79402 = r79395 + r79401;
        double r79403 = 1.0;
        double r79404 = tan(r79391);
        double r79405 = tan(r79389);
        double r79406 = r79404 * r79405;
        double r79407 = r79403 - r79406;
        double r79408 = r79407 * r79392;
        double r79409 = r79402 / r79408;
        return r79409;
}

Error

Bits error versus x

Bits error versus eps

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.9
Target14.8
Herbie0.4
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 36.9

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

  3. Applied tan-quot36.9

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

  4. Applied tan-sum22.1

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

  5. Applied frac-sub22.1

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

  6. Taylor expanded around inf 0.4

    \[\leadsto \frac{\color{blue}{\frac{\sin \varepsilon \cdot \cos x}{\cos \varepsilon} + \frac{{\left(\sin x\right)}^{2} \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\]

  7. Final simplification0.4

    \[\leadsto \frac{\frac{\sin \varepsilon \cdot \cos x}{\cos \varepsilon} + \frac{{\left(\sin x\right)}^{2} \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\]

Reproduce

herbie shell --seed 2019325 
(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)))