Average Error: 37.1 → 0.4
Time: 25.5s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\frac{\frac{\sin \varepsilon \cdot \cos x}{\cos \varepsilon} + \frac{\sqrt{{\left(\sin x\right)}^{2}} \cdot \left(\sin \varepsilon \cdot \left|\sin x\right|\right)}{\cos x \cdot \cos \varepsilon}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\frac{\frac{\sin \varepsilon \cdot \cos x}{\cos \varepsilon} + \frac{\sqrt{{\left(\sin x\right)}^{2}} \cdot \left(\sin \varepsilon \cdot \left|\sin x\right|\right)}{\cos x \cdot \cos \varepsilon}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}
double f(double x, double eps) {
        double r83459 = x;
        double r83460 = eps;
        double r83461 = r83459 + r83460;
        double r83462 = tan(r83461);
        double r83463 = tan(r83459);
        double r83464 = r83462 - r83463;
        return r83464;
}
↓
double f(double x, double eps) {
        double r83465 = eps;
        double r83466 = sin(r83465);
        double r83467 = x;
        double r83468 = cos(r83467);
        double r83469 = r83466 * r83468;
        double r83470 = cos(r83465);
        double r83471 = r83469 / r83470;
        double r83472 = sin(r83467);
        double r83473 = 2.0;
        double r83474 = pow(r83472, r83473);
        double r83475 = sqrt(r83474);
        double r83476 = fabs(r83472);
        double r83477 = r83466 * r83476;
        double r83478 = r83475 * r83477;
        double r83479 = r83468 * r83470;
        double r83480 = r83478 / r83479;
        double r83481 = r83471 + r83480;
        double r83482 = 1.0;
        double r83483 = tan(r83467);
        double r83484 = tan(r83465);
        double r83485 = r83483 * r83484;
        double r83486 = r83482 - r83485;
        double r83487 = r83486 * r83468;
        double r83488 = r83481 / r83487;
        return r83488;
}

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
Target15.3
Herbie0.4
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 37.1

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

  3. Applied tan-quot37.1

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

  4. Applied tan-sum21.9

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

  5. Applied frac-sub21.9

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

  6. Taylor expanded around inf 0.4

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

  8. Applied add-sqr-sqrt0.4

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

  9. Applied associate-*l*0.4

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

  10. Simplified0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019326 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :precision binary64

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

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