Average Error: 37.1 → 13.1
Time: 50.7s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x}{\cos x}} - \frac{\sin x}{\cos x}\right) + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\left(\sin x \cdot \sin \varepsilon\right) \cdot \frac{1}{\cos \varepsilon}}{\cos x}}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x}{\cos x}} - \frac{\sin x}{\cos x}\right) + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\left(\sin x \cdot \sin \varepsilon\right) \cdot \frac{1}{\cos \varepsilon}}{\cos x}}
double f(double x, double eps) {
        double r2426606 = x;
        double r2426607 = eps;
        double r2426608 = r2426606 + r2426607;
        double r2426609 = tan(r2426608);
        double r2426610 = tan(r2426606);
        double r2426611 = r2426609 - r2426610;
        return r2426611;
}
↓
double f(double x, double eps) {
        double r2426612 = x;
        double r2426613 = sin(r2426612);
        double r2426614 = cos(r2426612);
        double r2426615 = r2426613 / r2426614;
        double r2426616 = 1.0;
        double r2426617 = eps;
        double r2426618 = sin(r2426617);
        double r2426619 = cos(r2426617);
        double r2426620 = r2426618 / r2426619;
        double r2426621 = r2426620 * r2426613;
        double r2426622 = r2426621 / r2426614;
        double r2426623 = r2426616 - r2426622;
        double r2426624 = r2426615 / r2426623;
        double r2426625 = r2426624 - r2426615;
        double r2426626 = r2426613 * r2426618;
        double r2426627 = r2426616 / r2426619;
        double r2426628 = r2426626 * r2426627;
        double r2426629 = r2426628 / r2426614;
        double r2426630 = r2426616 - r2426629;
        double r2426631 = r2426620 / r2426630;
        double r2426632 = r2426625 + r2426631;
        return r2426632;
}

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
Target15.0
Herbie13.1
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 37.1

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

  3. Applied tan-sum22.0

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

  4. Taylor expanded around inf 22.1

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

  5. Simplified13.1

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

  7. Applied div-inv13.1

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

  8. Applied associate-*r*13.1

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

  9. Final simplification13.1

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

Reproduce

herbie shell --seed 2019153 +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)))