Average Error: 36.9 → 12.8
Time: 53.4s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\left(1 + \left(\frac{\frac{\sin \varepsilon}{\cos x} \cdot \sin x}{\cos \varepsilon} + \frac{\frac{\sin \varepsilon}{\cos x} \cdot \sin x}{\cos \varepsilon} \cdot \frac{\frac{\sin \varepsilon}{\cos x} \cdot \sin x}{\cos \varepsilon}\right)\right) \cdot \frac{\frac{\sin \varepsilon}{1 - \frac{\sqrt[3]{\left(\left(\frac{\sin \varepsilon}{\cos x} \cdot \sin x\right) \cdot \left(\frac{\sin \varepsilon}{\cos x} \cdot \sin x\right)\right) \cdot \left(\frac{\sin \varepsilon}{\cos x} \cdot \sin x\right)}}{\cos \varepsilon} \cdot \left(\frac{\frac{\sin \varepsilon}{\cos x} \cdot \sin x}{\cos \varepsilon} \cdot \frac{\frac{\sin \varepsilon}{\cos x} \cdot \sin x}{\cos \varepsilon}\right)}}{\cos \varepsilon} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\frac{\sin \varepsilon}{\cos x} \cdot \sin x}{\cos \varepsilon}} - \frac{\sin x}{\cos x}\right)\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\left(1 + \left(\frac{\frac{\sin \varepsilon}{\cos x} \cdot \sin x}{\cos \varepsilon} + \frac{\frac{\sin \varepsilon}{\cos x} \cdot \sin x}{\cos \varepsilon} \cdot \frac{\frac{\sin \varepsilon}{\cos x} \cdot \sin x}{\cos \varepsilon}\right)\right) \cdot \frac{\frac{\sin \varepsilon}{1 - \frac{\sqrt[3]{\left(\left(\frac{\sin \varepsilon}{\cos x} \cdot \sin x\right) \cdot \left(\frac{\sin \varepsilon}{\cos x} \cdot \sin x\right)\right) \cdot \left(\frac{\sin \varepsilon}{\cos x} \cdot \sin x\right)}}{\cos \varepsilon} \cdot \left(\frac{\frac{\sin \varepsilon}{\cos x} \cdot \sin x}{\cos \varepsilon} \cdot \frac{\frac{\sin \varepsilon}{\cos x} \cdot \sin x}{\cos \varepsilon}\right)}}{\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)
double f(double x, double eps) {
        double r5323250 = x;
        double r5323251 = eps;
        double r5323252 = r5323250 + r5323251;
        double r5323253 = tan(r5323252);
        double r5323254 = tan(r5323250);
        double r5323255 = r5323253 - r5323254;
        return r5323255;
}
↓
double f(double x, double eps) {
        double r5323256 = 1.0;
        double r5323257 = eps;
        double r5323258 = sin(r5323257);
        double r5323259 = x;
        double r5323260 = cos(r5323259);
        double r5323261 = r5323258 / r5323260;
        double r5323262 = sin(r5323259);
        double r5323263 = r5323261 * r5323262;
        double r5323264 = cos(r5323257);
        double r5323265 = r5323263 / r5323264;
        double r5323266 = r5323265 * r5323265;
        double r5323267 = r5323265 + r5323266;
        double r5323268 = r5323256 + r5323267;
        double r5323269 = r5323263 * r5323263;
        double r5323270 = r5323269 * r5323263;
        double r5323271 = cbrt(r5323270);
        double r5323272 = r5323271 / r5323264;
        double r5323273 = r5323272 * r5323266;
        double r5323274 = r5323256 - r5323273;
        double r5323275 = r5323258 / r5323274;
        double r5323276 = r5323275 / r5323264;
        double r5323277 = r5323268 * r5323276;
        double r5323278 = r5323262 / r5323260;
        double r5323279 = r5323256 - r5323265;
        double r5323280 = r5323278 / r5323279;
        double r5323281 = r5323280 - r5323278;
        double r5323282 = r5323277 + r5323281;
        return r5323282;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.9
Target15.5
Herbie12.8
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 36.9

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

  3. Applied tan-sum21.4

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

  4. Taylor expanded around inf 21.5

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

  5. Simplified12.7

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

  7. Applied flip3--12.8

    \[\leadsto \frac{\sin \varepsilon}{\color{blue}{\frac{{1}^{3} - {\left(\frac{\frac{\sin \varepsilon}{\cos x} \cdot \sin x}{\cos \varepsilon}\right)}^{3}}{1 \cdot 1 + \left(\frac{\frac{\sin \varepsilon}{\cos x} \cdot \sin x}{\cos \varepsilon} \cdot \frac{\frac{\sin \varepsilon}{\cos x} \cdot \sin x}{\cos \varepsilon} + 1 \cdot \frac{\frac{\sin \varepsilon}{\cos x} \cdot \sin x}{\cos \varepsilon}\right)}} \cdot \cos \varepsilon} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\frac{\sin \varepsilon}{\cos x} \cdot \sin x}{\cos \varepsilon}} - \frac{\sin x}{\cos x}\right)\]

  8. Applied associate-*l/12.8

    \[\leadsto \frac{\sin \varepsilon}{\color{blue}{\frac{\left({1}^{3} - {\left(\frac{\frac{\sin \varepsilon}{\cos x} \cdot \sin x}{\cos \varepsilon}\right)}^{3}\right) \cdot \cos \varepsilon}{1 \cdot 1 + \left(\frac{\frac{\sin \varepsilon}{\cos x} \cdot \sin x}{\cos \varepsilon} \cdot \frac{\frac{\sin \varepsilon}{\cos x} \cdot \sin x}{\cos \varepsilon} + 1 \cdot \frac{\frac{\sin \varepsilon}{\cos x} \cdot \sin x}{\cos \varepsilon}\right)}}} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\frac{\sin \varepsilon}{\cos x} \cdot \sin x}{\cos \varepsilon}} - \frac{\sin x}{\cos x}\right)\]

  9. Applied associate-/r/12.8

    \[\leadsto \color{blue}{\frac{\sin \varepsilon}{\left({1}^{3} - {\left(\frac{\frac{\sin \varepsilon}{\cos x} \cdot \sin x}{\cos \varepsilon}\right)}^{3}\right) \cdot \cos \varepsilon} \cdot \left(1 \cdot 1 + \left(\frac{\frac{\sin \varepsilon}{\cos x} \cdot \sin x}{\cos \varepsilon} \cdot \frac{\frac{\sin \varepsilon}{\cos x} \cdot \sin x}{\cos \varepsilon} + 1 \cdot \frac{\frac{\sin \varepsilon}{\cos x} \cdot \sin x}{\cos \varepsilon}\right)\right)} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\frac{\sin \varepsilon}{\cos x} \cdot \sin x}{\cos \varepsilon}} - \frac{\sin x}{\cos x}\right)\]

  10. Simplified12.8

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

  12. Applied add-cbrt-cube12.8

    \[\leadsto \frac{\frac{\sin \varepsilon}{1 - \left(\frac{\sin x \cdot \frac{\sin \varepsilon}{\cos x}}{\cos \varepsilon} \cdot \frac{\sin x \cdot \frac{\sin \varepsilon}{\cos x}}{\cos \varepsilon}\right) \cdot \frac{\color{blue}{\sqrt[3]{\left(\left(\sin x \cdot \frac{\sin \varepsilon}{\cos x}\right) \cdot \left(\sin x \cdot \frac{\sin \varepsilon}{\cos x}\right)\right) \cdot \left(\sin x \cdot \frac{\sin \varepsilon}{\cos x}\right)}}}{\cos \varepsilon}}}{\cos \varepsilon} \cdot \left(1 \cdot 1 + \left(\frac{\frac{\sin \varepsilon}{\cos x} \cdot \sin x}{\cos \varepsilon} \cdot \frac{\frac{\sin \varepsilon}{\cos x} \cdot \sin x}{\cos \varepsilon} + 1 \cdot \frac{\frac{\sin \varepsilon}{\cos x} \cdot \sin x}{\cos \varepsilon}\right)\right) + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\frac{\sin \varepsilon}{\cos x} \cdot \sin x}{\cos \varepsilon}} - \frac{\sin x}{\cos x}\right)\]

  13. Final simplification12.8

    \[\leadsto \left(1 + \left(\frac{\frac{\sin \varepsilon}{\cos x} \cdot \sin x}{\cos \varepsilon} + \frac{\frac{\sin \varepsilon}{\cos x} \cdot \sin x}{\cos \varepsilon} \cdot \frac{\frac{\sin \varepsilon}{\cos x} \cdot \sin x}{\cos \varepsilon}\right)\right) \cdot \frac{\frac{\sin \varepsilon}{1 - \frac{\sqrt[3]{\left(\left(\frac{\sin \varepsilon}{\cos x} \cdot \sin x\right) \cdot \left(\frac{\sin \varepsilon}{\cos x} \cdot \sin x\right)\right) \cdot \left(\frac{\sin \varepsilon}{\cos x} \cdot \sin x\right)}}{\cos \varepsilon} \cdot \left(\frac{\frac{\sin \varepsilon}{\cos x} \cdot \sin x}{\cos \varepsilon} \cdot \frac{\frac{\sin \varepsilon}{\cos x} \cdot \sin x}{\cos \varepsilon}\right)}}{\cos \varepsilon} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\frac{\sin \varepsilon}{\cos x} \cdot \sin x}{\cos \varepsilon}} - \frac{\sin x}{\cos x}\right)\]

Reproduce

herbie shell --seed 2019163 +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)))