Average Error: 36.3 → 0.4
Time: 31.5s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\frac{\frac{\sin \varepsilon \cdot \cos x}{\cos \varepsilon} + \frac{\sin x \cdot \sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}{\left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}\right) \cdot \cos x}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\frac{\frac{\sin \varepsilon \cdot \cos x}{\cos \varepsilon} + \frac{\sin x \cdot \sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}{\left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}\right) \cdot \cos x}
double f(double x, double eps) {
        double r4046300 = x;
        double r4046301 = eps;
        double r4046302 = r4046300 + r4046301;
        double r4046303 = tan(r4046302);
        double r4046304 = tan(r4046300);
        double r4046305 = r4046303 - r4046304;
        return r4046305;
}
↓
double f(double x, double eps) {
        double r4046306 = eps;
        double r4046307 = sin(r4046306);
        double r4046308 = x;
        double r4046309 = cos(r4046308);
        double r4046310 = r4046307 * r4046309;
        double r4046311 = cos(r4046306);
        double r4046312 = r4046310 / r4046311;
        double r4046313 = sin(r4046308);
        double r4046314 = r4046313 * r4046313;
        double r4046315 = r4046314 / r4046309;
        double r4046316 = r4046307 / r4046311;
        double r4046317 = r4046315 * r4046316;
        double r4046318 = r4046312 + r4046317;
        double r4046319 = 1.0;
        double r4046320 = r4046313 * r4046307;
        double r4046321 = r4046309 * r4046311;
        double r4046322 = r4046320 / r4046321;
        double r4046323 = r4046319 - r4046322;
        double r4046324 = r4046323 * r4046309;
        double r4046325 = r4046318 / r4046324;
        return r4046325;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.3
Target14.7
Herbie0.4
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 36.3

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

  3. Applied tan-sum21.6

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

  4. Taylor expanded around -inf 21.6

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

  6. Applied tan-quot21.7

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

  7. Applied frac-sub21.7

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

  8. Taylor expanded around inf 0.4

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

  9. Simplified0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019134 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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