Average Error: 36.8 → 13.0
Time: 26.7s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x}{\cos x}} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\log \left(e^{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x}\right)}{\cos x}} - \frac{\sin x}{\cos x}\right)\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x}{\cos x}} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\log \left(e^{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x}\right)}{\cos x}} - \frac{\sin x}{\cos x}\right)
double f(double x, double eps) {
        double r6996260 = x;
        double r6996261 = eps;
        double r6996262 = r6996260 + r6996261;
        double r6996263 = tan(r6996262);
        double r6996264 = tan(r6996260);
        double r6996265 = r6996263 - r6996264;
        return r6996265;
}
↓
double f(double x, double eps) {
        double r6996266 = eps;
        double r6996267 = sin(r6996266);
        double r6996268 = cos(r6996266);
        double r6996269 = r6996267 / r6996268;
        double r6996270 = 1.0;
        double r6996271 = x;
        double r6996272 = sin(r6996271);
        double r6996273 = r6996269 * r6996272;
        double r6996274 = cos(r6996271);
        double r6996275 = r6996273 / r6996274;
        double r6996276 = r6996270 - r6996275;
        double r6996277 = r6996269 / r6996276;
        double r6996278 = r6996272 / r6996274;
        double r6996279 = exp(r6996273);
        double r6996280 = log(r6996279);
        double r6996281 = r6996280 / r6996274;
        double r6996282 = r6996270 - r6996281;
        double r6996283 = r6996278 / r6996282;
        double r6996284 = r6996283 - r6996278;
        double r6996285 = r6996277 + r6996284;
        return r6996285;
}

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
Herbie13.0
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 36.8

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

  3. Applied tan-sum21.7

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

  4. Taylor expanded around inf 21.9

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

  5. Simplified12.9

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

  7. Applied add-log-exp13.0

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

  8. Final simplification13.0

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

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)))