Average Error: 36.9 → 13.0
Time: 23.4s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\left(\frac{\sin x}{\left(1 - \frac{\frac{\sin x}{\cos x} \cdot \sin \varepsilon}{\cos \varepsilon}\right) \cdot \cos x} - \frac{\sin x}{\cos x}\right) + \frac{\frac{\sin \varepsilon}{1 - \frac{\frac{\log \left(e^{\sin \varepsilon \cdot \sin x}\right)}{\cos x}}{\cos \varepsilon}}}{\cos \varepsilon}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\left(\frac{\sin x}{\left(1 - \frac{\frac{\sin x}{\cos x} \cdot \sin \varepsilon}{\cos \varepsilon}\right) \cdot \cos x} - \frac{\sin x}{\cos x}\right) + \frac{\frac{\sin \varepsilon}{1 - \frac{\frac{\log \left(e^{\sin \varepsilon \cdot \sin x}\right)}{\cos x}}{\cos \varepsilon}}}{\cos \varepsilon}
double f(double x, double eps) {
        double r109187 = x;
        double r109188 = eps;
        double r109189 = r109187 + r109188;
        double r109190 = tan(r109189);
        double r109191 = tan(r109187);
        double r109192 = r109190 - r109191;
        return r109192;
}
↓
double f(double x, double eps) {
        double r109193 = x;
        double r109194 = sin(r109193);
        double r109195 = 1.0;
        double r109196 = cos(r109193);
        double r109197 = r109194 / r109196;
        double r109198 = eps;
        double r109199 = sin(r109198);
        double r109200 = r109197 * r109199;
        double r109201 = cos(r109198);
        double r109202 = r109200 / r109201;
        double r109203 = r109195 - r109202;
        double r109204 = r109203 * r109196;
        double r109205 = r109194 / r109204;
        double r109206 = r109205 - r109197;
        double r109207 = r109199 * r109194;
        double r109208 = exp(r109207);
        double r109209 = log(r109208);
        double r109210 = r109209 / r109196;
        double r109211 = r109210 / r109201;
        double r109212 = r109195 - r109211;
        double r109213 = r109199 / r109212;
        double r109214 = r109213 / r109201;
        double r109215 = r109206 + r109214;
        return r109215;
}

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

Derivation

  1. Initial program 36.9

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

  3. Applied tan-sum21.5

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

  4. Simplified21.5

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

  5. Taylor expanded around inf 21.6

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

  6. Simplified13.0

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

  8. Applied *-un-lft-identity13.0

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

  9. Applied times-frac13.0

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

  10. Simplified13.0

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

  12. Applied add-log-exp13.0

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

  13. Simplified13.0

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

  14. Final simplification13.0

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

Reproduce

herbie shell --seed 2019174 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

  :herbie-target
  (/ (sin eps) (* (cos x) (cos (+ x eps))))

  (- (tan (+ x eps)) (tan x)))