Average Error: 37.0 → 0.4
Time: 18.5s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\frac{\frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \left(\frac{{\left(\sin x\right)}^{2}}{\cos x} + \cos x\right)}{\cos x}}{1 - \tan x \cdot \tan \varepsilon}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\frac{\frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \left(\frac{{\left(\sin x\right)}^{2}}{\cos x} + \cos x\right)}{\cos x}}{1 - \tan x \cdot \tan \varepsilon}
double f(double x, double eps) {
        double r83310 = x;
        double r83311 = eps;
        double r83312 = r83310 + r83311;
        double r83313 = tan(r83312);
        double r83314 = tan(r83310);
        double r83315 = r83313 - r83314;
        return r83315;
}
↓
double f(double x, double eps) {
        double r83316 = eps;
        double r83317 = sin(r83316);
        double r83318 = cos(r83316);
        double r83319 = r83317 / r83318;
        double r83320 = x;
        double r83321 = sin(r83320);
        double r83322 = 2.0;
        double r83323 = pow(r83321, r83322);
        double r83324 = cos(r83320);
        double r83325 = r83323 / r83324;
        double r83326 = r83325 + r83324;
        double r83327 = r83319 * r83326;
        double r83328 = r83327 / r83324;
        double r83329 = 1.0;
        double r83330 = tan(r83320);
        double r83331 = tan(r83316);
        double r83332 = r83330 * r83331;
        double r83333 = r83329 - r83332;
        double r83334 = r83328 / r83333;
        return r83334;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.0
Target15.1
Herbie0.4
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 37.0

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

  3. Applied tan-quot37.0

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

  4. Applied tan-sum21.9

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

  5. Applied frac-sub21.9

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

  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. Using strategy rm

  8. Applied associate-/l*0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2019304 
(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)))