Average Error: 37.0 → 12.6
Time: 1.1m
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}}{\cos x \cdot \cos \varepsilon} \cdot \frac{\sqrt[3]{\sin x}}{\frac{1}{\sin \varepsilon}}} - \frac{\sin x}{\cos x}\right) + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}}{\cos x \cdot \cos \varepsilon} \cdot \frac{\sqrt[3]{\sin x}}{\frac{1}{\sin \varepsilon}}} - \frac{\sin x}{\cos x}\right) + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}
double f(double x, double eps) {
        double r3622319 = x;
        double r3622320 = eps;
        double r3622321 = r3622319 + r3622320;
        double r3622322 = tan(r3622321);
        double r3622323 = tan(r3622319);
        double r3622324 = r3622322 - r3622323;
        return r3622324;
}
↓
double f(double x, double eps) {
        double r3622325 = x;
        double r3622326 = sin(r3622325);
        double r3622327 = cos(r3622325);
        double r3622328 = r3622326 / r3622327;
        double r3622329 = 1.0;
        double r3622330 = cbrt(r3622326);
        double r3622331 = r3622330 * r3622330;
        double r3622332 = eps;
        double r3622333 = cos(r3622332);
        double r3622334 = r3622327 * r3622333;
        double r3622335 = r3622331 / r3622334;
        double r3622336 = sin(r3622332);
        double r3622337 = r3622329 / r3622336;
        double r3622338 = r3622330 / r3622337;
        double r3622339 = r3622335 * r3622338;
        double r3622340 = r3622329 - r3622339;
        double r3622341 = r3622328 / r3622340;
        double r3622342 = r3622341 - r3622328;
        double r3622343 = r3622336 / r3622333;
        double r3622344 = r3622328 * r3622343;
        double r3622345 = r3622329 - r3622344;
        double r3622346 = r3622343 / r3622345;
        double r3622347 = r3622342 + r3622346;
        return r3622347;
}

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
Target15.4
Herbie12.6
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 37.0

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

  3. Applied tan-sum21.6

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

  4. Taylor expanded around -inf 21.7

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

  5. Simplified12.6

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

  6. Taylor expanded around inf 12.6

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

  7. Simplified12.6

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

  9. Applied div-inv12.6

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

  10. Applied add-cube-cbrt12.6

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

  11. Applied times-frac12.6

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

  12. Final simplification12.6

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

Reproduce

herbie shell --seed 2019133 +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)))