Average Error: 37.2 → 13.2
Time: 1.1m
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\left(\frac{\sin x}{\left(1 - \frac{\sqrt[3]{\left(\left(\sin x \cdot \sin \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)}}{\cos x \cdot \cos \varepsilon}\right) \cdot \cos x} - \frac{\sin x}{\cos x}\right) + \frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}\right)}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\left(\frac{\sin x}{\left(1 - \frac{\sqrt[3]{\left(\left(\sin x \cdot \sin \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)}}{\cos x \cdot \cos \varepsilon}\right) \cdot \cos x} - \frac{\sin x}{\cos x}\right) + \frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}\right)}
double f(double x, double eps) {
        double r13664549 = x;
        double r13664550 = eps;
        double r13664551 = r13664549 + r13664550;
        double r13664552 = tan(r13664551);
        double r13664553 = tan(r13664549);
        double r13664554 = r13664552 - r13664553;
        return r13664554;
}
↓
double f(double x, double eps) {
        double r13664555 = x;
        double r13664556 = sin(r13664555);
        double r13664557 = 1.0;
        double r13664558 = eps;
        double r13664559 = sin(r13664558);
        double r13664560 = r13664556 * r13664559;
        double r13664561 = r13664560 * r13664560;
        double r13664562 = r13664561 * r13664560;
        double r13664563 = cbrt(r13664562);
        double r13664564 = cos(r13664555);
        double r13664565 = cos(r13664558);
        double r13664566 = r13664564 * r13664565;
        double r13664567 = r13664563 / r13664566;
        double r13664568 = r13664557 - r13664567;
        double r13664569 = r13664568 * r13664564;
        double r13664570 = r13664556 / r13664569;
        double r13664571 = r13664556 / r13664564;
        double r13664572 = r13664570 - r13664571;
        double r13664573 = r13664560 / r13664566;
        double r13664574 = r13664557 - r13664573;
        double r13664575 = r13664565 * r13664574;
        double r13664576 = r13664559 / r13664575;
        double r13664577 = r13664572 + r13664576;
        return r13664577;
}

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
Target14.6
Herbie13.2
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 37.2

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

  3. Applied tan-sum22.5

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

  4. Taylor expanded around -inf 22.6

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

  6. Applied associate--l+13.2

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

  8. Applied add-cbrt-cube13.2

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

  9. Final simplification13.2

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

Reproduce

herbie shell --seed 2019125 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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