Average Error: 37.2 → 12.9
Time: 25.0s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin x}{\cos x} \cdot \left(\sqrt[3]{\frac{\sin \varepsilon}{\cos \varepsilon}} \cdot \left(\sqrt[3]{\frac{\sin \varepsilon}{\cos \varepsilon}} \cdot \sqrt[3]{\frac{\sin \varepsilon}{\cos \varepsilon}}\right)\right)\right)} + \frac{\frac{\sin x}{1 - \frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin x}{\cos \varepsilon}} - \sin x}{\cos x}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin x}{\cos x} \cdot \left(\sqrt[3]{\frac{\sin \varepsilon}{\cos \varepsilon}} \cdot \left(\sqrt[3]{\frac{\sin \varepsilon}{\cos \varepsilon}} \cdot \sqrt[3]{\frac{\sin \varepsilon}{\cos \varepsilon}}\right)\right)\right)} + \frac{\frac{\sin x}{1 - \frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin x}{\cos \varepsilon}} - \sin x}{\cos x}
double f(double x, double eps) {
        double r67503 = x;
        double r67504 = eps;
        double r67505 = r67503 + r67504;
        double r67506 = tan(r67505);
        double r67507 = tan(r67503);
        double r67508 = r67506 - r67507;
        return r67508;
}
↓
double f(double x, double eps) {
        double r67509 = eps;
        double r67510 = sin(r67509);
        double r67511 = cos(r67509);
        double r67512 = 1.0;
        double r67513 = x;
        double r67514 = sin(r67513);
        double r67515 = cos(r67513);
        double r67516 = r67514 / r67515;
        double r67517 = r67510 / r67511;
        double r67518 = cbrt(r67517);
        double r67519 = r67518 * r67518;
        double r67520 = r67518 * r67519;
        double r67521 = r67516 * r67520;
        double r67522 = r67512 - r67521;
        double r67523 = r67511 * r67522;
        double r67524 = r67510 / r67523;
        double r67525 = r67510 / r67515;
        double r67526 = r67514 / r67511;
        double r67527 = r67525 * r67526;
        double r67528 = r67512 - r67527;
        double r67529 = r67514 / r67528;
        double r67530 = r67529 - r67514;
        double r67531 = r67530 / r67515;
        double r67532 = r67524 + r67531;
        return r67532;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.2
Target15.7
Herbie12.9
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 37.2

    \[\tan \left(x + \varepsilon\right) - \tan x\]
  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 21.7

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

  6. Simplified12.8

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

  8. Applied sub-div12.8

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

  9. Simplified12.8

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

  11. Applied add-cube-cbrt12.9

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

  12. Final simplification12.9

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

Reproduce

herbie shell --seed 2019179 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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