Average Error: 37.0 → 12.7
Time: 59.6s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin \varepsilon \cdot \frac{\sin x}{\cos x}}{\cos \varepsilon}} - \frac{\sin x}{\cos x}\right) + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin \varepsilon \cdot \frac{\sin x}{\cos x}}{\cos \varepsilon}} - \frac{\sin x}{\cos x}\right) + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}}
double f(double x, double eps) {
        double r4334160 = x;
        double r4334161 = eps;
        double r4334162 = r4334160 + r4334161;
        double r4334163 = tan(r4334162);
        double r4334164 = tan(r4334160);
        double r4334165 = r4334163 - r4334164;
        return r4334165;
}
↓
double f(double x, double eps) {
        double r4334166 = x;
        double r4334167 = sin(r4334166);
        double r4334168 = cos(r4334166);
        double r4334169 = r4334167 / r4334168;
        double r4334170 = 1.0;
        double r4334171 = eps;
        double r4334172 = sin(r4334171);
        double r4334173 = r4334172 * r4334169;
        double r4334174 = cos(r4334171);
        double r4334175 = r4334173 / r4334174;
        double r4334176 = r4334170 - r4334175;
        double r4334177 = r4334169 / r4334176;
        double r4334178 = r4334177 - r4334169;
        double r4334179 = r4334172 / r4334174;
        double r4334180 = r4334167 * r4334172;
        double r4334181 = r4334174 * r4334168;
        double r4334182 = r4334180 / r4334181;
        double r4334183 = r4334170 - r4334182;
        double r4334184 = r4334179 / r4334183;
        double r4334185 = r4334178 + r4334184;
        return r4334185;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.0
Target15.3
Herbie12.7
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 37.0

    \[\tan \left(x + \varepsilon\right) - \tan x\]
  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 21.7

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

  5. Simplified12.7

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

  7. Applied associate-*r/12.7

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

  9. Applied frac-times12.7

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

  10. Final simplification12.7

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

Reproduce

herbie shell --seed 2019129 +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)))