Average Error: 37.1 → 13.1
Time: 1.4m
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\frac{\frac{\sin \varepsilon}{1 - \frac{\frac{\sin x}{\cos x} \cdot \sin \varepsilon}{\cos \varepsilon}}}{\cos \varepsilon} - \left(\sin x - \frac{\sin x}{1 - \frac{\frac{\sin x}{\cos x} \cdot \sin \varepsilon}{\cos \varepsilon}}\right) \cdot \frac{1}{\cos x}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\frac{\frac{\sin \varepsilon}{1 - \frac{\frac{\sin x}{\cos x} \cdot \sin \varepsilon}{\cos \varepsilon}}}{\cos \varepsilon} - \left(\sin x - \frac{\sin x}{1 - \frac{\frac{\sin x}{\cos x} \cdot \sin \varepsilon}{\cos \varepsilon}}\right) \cdot \frac{1}{\cos x}
double f(double x, double eps) {
        double r5241406 = x;
        double r5241407 = eps;
        double r5241408 = r5241406 + r5241407;
        double r5241409 = tan(r5241408);
        double r5241410 = tan(r5241406);
        double r5241411 = r5241409 - r5241410;
        return r5241411;
}
↓
double f(double x, double eps) {
        double r5241412 = eps;
        double r5241413 = sin(r5241412);
        double r5241414 = 1.0;
        double r5241415 = x;
        double r5241416 = sin(r5241415);
        double r5241417 = cos(r5241415);
        double r5241418 = r5241416 / r5241417;
        double r5241419 = r5241418 * r5241413;
        double r5241420 = cos(r5241412);
        double r5241421 = r5241419 / r5241420;
        double r5241422 = r5241414 - r5241421;
        double r5241423 = r5241413 / r5241422;
        double r5241424 = r5241423 / r5241420;
        double r5241425 = r5241416 / r5241422;
        double r5241426 = r5241416 - r5241425;
        double r5241427 = r5241414 / r5241417;
        double r5241428 = r5241426 * r5241427;
        double r5241429 = r5241424 - r5241428;
        return r5241429;
}

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-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

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

  5. Simplified13.1

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

  7. Applied div-inv13.9

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

  8. Applied div-inv13.1

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

  9. Applied distribute-rgt-out--13.1

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

  10. Final simplification13.1

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

Reproduce

herbie shell --seed 2019200 +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)))