Average Error: 37.0 → 13.4
Time: 44.8s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin \varepsilon \cdot \frac{\sin x}{\cos x}}{\cos \varepsilon}} - \frac{\sin x}{\cos x}\right) + \frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\frac{\sin \varepsilon}{\cos x} \cdot \sin x}{\cos \varepsilon}\right)}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin \varepsilon \cdot \frac{\sin x}{\cos x}}{\cos \varepsilon}} - \frac{\sin x}{\cos x}\right) + \frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\frac{\sin \varepsilon}{\cos x} \cdot \sin x}{\cos \varepsilon}\right)}
double f(double x, double eps) {
        double r4635235 = x;
        double r4635236 = eps;
        double r4635237 = r4635235 + r4635236;
        double r4635238 = tan(r4635237);
        double r4635239 = tan(r4635235);
        double r4635240 = r4635238 - r4635239;
        return r4635240;
}
↓
double f(double x, double eps) {
        double r4635241 = x;
        double r4635242 = sin(r4635241);
        double r4635243 = cos(r4635241);
        double r4635244 = r4635242 / r4635243;
        double r4635245 = 1.0;
        double r4635246 = eps;
        double r4635247 = sin(r4635246);
        double r4635248 = r4635247 * r4635244;
        double r4635249 = cos(r4635246);
        double r4635250 = r4635248 / r4635249;
        double r4635251 = r4635245 - r4635250;
        double r4635252 = r4635244 / r4635251;
        double r4635253 = r4635252 - r4635244;
        double r4635254 = r4635247 / r4635243;
        double r4635255 = r4635254 * r4635242;
        double r4635256 = r4635255 / r4635249;
        double r4635257 = r4635245 - r4635256;
        double r4635258 = r4635249 * r4635257;
        double r4635259 = r4635247 / r4635258;
        double r4635260 = r4635253 + r4635259;
        return r4635260;
}

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
Target14.6
Herbie13.4
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 37.0

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

  3. Applied tan-sum22.5

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

  4. Taylor expanded around inf 22.6

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

  5. Simplified13.4

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

  7. Applied div-inv13.4

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

  8. Applied associate-*l*13.4

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

  9. Simplified13.4

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

  10. Final simplification13.4

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

Reproduce

herbie shell --seed 2019164 +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)))