Average Error: 36.8 → 0.4
Time: 26.6s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x \cdot \sin x}{\cos x} + \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \cos x}{\left(1 - \tan \varepsilon \cdot \tan x\right) \cdot \cos x}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x \cdot \sin x}{\cos x} + \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \cos x}{\left(1 - \tan \varepsilon \cdot \tan x\right) \cdot \cos x}
double f(double x, double eps) {
        double r5487229 = x;
        double r5487230 = eps;
        double r5487231 = r5487229 + r5487230;
        double r5487232 = tan(r5487231);
        double r5487233 = tan(r5487229);
        double r5487234 = r5487232 - r5487233;
        return r5487234;
}
↓
double f(double x, double eps) {
        double r5487235 = eps;
        double r5487236 = sin(r5487235);
        double r5487237 = cos(r5487235);
        double r5487238 = r5487236 / r5487237;
        double r5487239 = x;
        double r5487240 = sin(r5487239);
        double r5487241 = r5487240 * r5487240;
        double r5487242 = cos(r5487239);
        double r5487243 = r5487241 / r5487242;
        double r5487244 = r5487238 * r5487243;
        double r5487245 = r5487238 * r5487242;
        double r5487246 = r5487244 + r5487245;
        double r5487247 = 1.0;
        double r5487248 = tan(r5487235);
        double r5487249 = tan(r5487239);
        double r5487250 = r5487248 * r5487249;
        double r5487251 = r5487247 - r5487250;
        double r5487252 = r5487251 * r5487242;
        double r5487253 = r5487246 / r5487252;
        return r5487253;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.8
Target15.1
Herbie0.4
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 36.8

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

  3. Applied tan-quot36.8

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

  4. Applied tan-sum21.8

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

  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

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

  7. Simplified0.4

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

  9. Applied associate-/r/0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019172 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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