Average Error: 36.9 → 12.7
Time: 32.8s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x}{\cos x}} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x}{\cos x}} - \frac{\sin x}{\cos x}\right)\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x}{\cos 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)
double f(double x, double eps) {
        double r3374624 = x;
        double r3374625 = eps;
        double r3374626 = r3374624 + r3374625;
        double r3374627 = tan(r3374626);
        double r3374628 = tan(r3374624);
        double r3374629 = r3374627 - r3374628;
        return r3374629;
}
↓
double f(double x, double eps) {
        double r3374630 = eps;
        double r3374631 = sin(r3374630);
        double r3374632 = cos(r3374630);
        double r3374633 = r3374631 / r3374632;
        double r3374634 = 1.0;
        double r3374635 = x;
        double r3374636 = sin(r3374635);
        double r3374637 = r3374633 * r3374636;
        double r3374638 = cos(r3374635);
        double r3374639 = r3374637 / r3374638;
        double r3374640 = r3374634 - r3374639;
        double r3374641 = r3374633 / r3374640;
        double r3374642 = r3374636 / r3374638;
        double r3374643 = r3374642 / r3374640;
        double r3374644 = r3374643 - r3374642;
        double r3374645 = r3374641 + r3374644;
        return r3374645;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.9
Target15.5
Herbie12.7
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 36.9

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

  3. Applied tan-sum21.4

    \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
  4. Using strategy rm

  5. Applied div-inv21.4

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

  6. Applied fma-neg21.4

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

  8. Applied add-log-exp21.4

    \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \color{blue}{\log \left(e^{\frac{1}{1 - \tan x \cdot \tan \varepsilon}}\right)}, -\tan x\right)\]

  9. Taylor expanded around inf 21.5

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

  10. Simplified12.7

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

  11. Final simplification12.7

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

Reproduce

herbie shell --seed 2019163 +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)))