Average Error: 37.1 → 13.0
Time: 1.0m
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}\right)} + \left(\frac{1}{\cos x \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}\right)} - \frac{1}{\cos x}\right) \cdot \sin x\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}\right)} + \left(\frac{1}{\cos x \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}\right)} - \frac{1}{\cos x}\right) \cdot \sin x
double f(double x, double eps) {
        double r10359579 = x;
        double r10359580 = eps;
        double r10359581 = r10359579 + r10359580;
        double r10359582 = tan(r10359581);
        double r10359583 = tan(r10359579);
        double r10359584 = r10359582 - r10359583;
        return r10359584;
}
↓
double f(double x, double eps) {
        double r10359585 = eps;
        double r10359586 = sin(r10359585);
        double r10359587 = cos(r10359585);
        double r10359588 = 1.0;
        double r10359589 = x;
        double r10359590 = sin(r10359589);
        double r10359591 = r10359590 * r10359586;
        double r10359592 = cos(r10359589);
        double r10359593 = r10359592 * r10359587;
        double r10359594 = r10359591 / r10359593;
        double r10359595 = r10359588 - r10359594;
        double r10359596 = r10359587 * r10359595;
        double r10359597 = r10359586 / r10359596;
        double r10359598 = r10359592 * r10359595;
        double r10359599 = r10359588 / r10359598;
        double r10359600 = r10359588 / r10359592;
        double r10359601 = r10359599 - r10359600;
        double r10359602 = r10359601 * r10359590;
        double r10359603 = r10359597 + r10359602;
        return r10359603;
}

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

Derivation

  1. Initial program 37.1

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

  3. Applied tan-sum21.9

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

  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. Using strategy rm

  6. Applied associate--l+13.0

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

  8. Applied div-inv13.9

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

  9. Applied div-inv13.1

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

  10. Applied distribute-lft-out--13.0

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

  11. Final simplification13.0

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

Reproduce

herbie shell --seed 2019120 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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