Average Error: 37.4 → 0.5
Time: 30.6s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\frac{\frac{\cos x}{\frac{\cos \varepsilon}{\sin \varepsilon}} + \frac{\sin x}{\frac{\frac{\cos x}{\frac{\sin \varepsilon}{\cos \varepsilon}}}{\sin x}}}{\left(1 - \tan \varepsilon \cdot \tan x\right) \cdot \cos x}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\frac{\frac{\cos x}{\frac{\cos \varepsilon}{\sin \varepsilon}} + \frac{\sin x}{\frac{\frac{\cos x}{\frac{\sin \varepsilon}{\cos \varepsilon}}}{\sin x}}}{\left(1 - \tan \varepsilon \cdot \tan x\right) \cdot \cos x}
double f(double x, double eps) {
        double r3061609 = x;
        double r3061610 = eps;
        double r3061611 = r3061609 + r3061610;
        double r3061612 = tan(r3061611);
        double r3061613 = tan(r3061609);
        double r3061614 = r3061612 - r3061613;
        return r3061614;
}
↓
double f(double x, double eps) {
        double r3061615 = x;
        double r3061616 = cos(r3061615);
        double r3061617 = eps;
        double r3061618 = cos(r3061617);
        double r3061619 = sin(r3061617);
        double r3061620 = r3061618 / r3061619;
        double r3061621 = r3061616 / r3061620;
        double r3061622 = sin(r3061615);
        double r3061623 = r3061619 / r3061618;
        double r3061624 = r3061616 / r3061623;
        double r3061625 = r3061624 / r3061622;
        double r3061626 = r3061622 / r3061625;
        double r3061627 = r3061621 + r3061626;
        double r3061628 = 1.0;
        double r3061629 = tan(r3061617);
        double r3061630 = tan(r3061615);
        double r3061631 = r3061629 * r3061630;
        double r3061632 = r3061628 - r3061631;
        double r3061633 = r3061632 * r3061616;
        double r3061634 = r3061627 / r3061633;
        return r3061634;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.4
Target15.5
Herbie0.5
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 37.4

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

  3. Applied tan-sum21.8

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

  5. Applied tan-quot21.9

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

  6. Applied frac-sub22.0

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

  7. Taylor expanded around -inf 0.4

    \[\leadsto \frac{\color{blue}{\frac{{\left(\sin x\right)}^{2} \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon} + \frac{\cos x \cdot \sin \varepsilon}{\cos \varepsilon}}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\]

  8. Simplified0.5

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

  9. Final simplification0.5

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

Reproduce

herbie shell --seed 2019132 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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