Average Error: 37.1 → 12.9
Time: 43.6s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} - \frac{\sin x}{\cos x}\right) + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\sqrt[3]{\frac{\sin x}{\cos x}} \cdot \sqrt[3]{\frac{\sin x}{\cos x}}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sqrt[3]{\frac{\sin x}{\cos x}}\right)}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\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) + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\sqrt[3]{\frac{\sin x}{\cos x}} \cdot \sqrt[3]{\frac{\sin x}{\cos x}}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sqrt[3]{\frac{\sin x}{\cos x}}\right)}
double f(double x, double eps) {
        double r2963553 = x;
        double r2963554 = eps;
        double r2963555 = r2963553 + r2963554;
        double r2963556 = tan(r2963555);
        double r2963557 = tan(r2963553);
        double r2963558 = r2963556 - r2963557;
        return r2963558;
}
↓
double f(double x, double eps) {
        double r2963559 = x;
        double r2963560 = sin(r2963559);
        double r2963561 = cos(r2963559);
        double r2963562 = r2963560 / r2963561;
        double r2963563 = 1.0;
        double r2963564 = eps;
        double r2963565 = sin(r2963564);
        double r2963566 = cos(r2963564);
        double r2963567 = r2963565 / r2963566;
        double r2963568 = r2963562 * r2963567;
        double r2963569 = r2963563 - r2963568;
        double r2963570 = r2963562 / r2963569;
        double r2963571 = r2963570 - r2963562;
        double r2963572 = cbrt(r2963562);
        double r2963573 = r2963572 * r2963572;
        double r2963574 = r2963567 * r2963572;
        double r2963575 = r2963573 * r2963574;
        double r2963576 = r2963563 - r2963575;
        double r2963577 = r2963567 / r2963576;
        double r2963578 = r2963571 + r2963577;
        return r2963578;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.1
Target15.3
Herbie12.9
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 37.1

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

  3. Applied tan-sum21.7

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

  4. Taylor expanded around inf 21.9

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

  5. Simplified12.9

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

  7. Applied add-cube-cbrt12.9

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

  8. Applied associate-*l*12.9

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

  9. Final simplification12.9

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

Reproduce

herbie shell --seed 2019141 +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)))