Average Error: 37.4 → 13.0
Time: 54.1s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\log \left(e^{\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} - \frac{\sin x}{\cos x}}\right) + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \log \left(e^{\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}\right)}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\log \left(e^{\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} - \frac{\sin x}{\cos x}}\right) + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \log \left(e^{\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}\right)}
double f(double x, double eps) {
        double r3904266 = x;
        double r3904267 = eps;
        double r3904268 = r3904266 + r3904267;
        double r3904269 = tan(r3904268);
        double r3904270 = tan(r3904266);
        double r3904271 = r3904269 - r3904270;
        return r3904271;
}
↓
double f(double x, double eps) {
        double r3904272 = x;
        double r3904273 = sin(r3904272);
        double r3904274 = cos(r3904272);
        double r3904275 = r3904273 / r3904274;
        double r3904276 = 1.0;
        double r3904277 = eps;
        double r3904278 = sin(r3904277);
        double r3904279 = cos(r3904277);
        double r3904280 = r3904278 / r3904279;
        double r3904281 = r3904275 * r3904280;
        double r3904282 = r3904276 - r3904281;
        double r3904283 = r3904275 / r3904282;
        double r3904284 = r3904283 - r3904275;
        double r3904285 = exp(r3904284);
        double r3904286 = log(r3904285);
        double r3904287 = exp(r3904281);
        double r3904288 = log(r3904287);
        double r3904289 = r3904276 - r3904288;
        double r3904290 = r3904280 / r3904289;
        double r3904291 = r3904286 + r3904290;
        return r3904291;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.4
Target15.5
Herbie13.0
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 37.4

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

  3. Applied tan-sum21.8

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

  4. Taylor expanded around -inf 22.0

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

  5. Simplified12.9

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

  7. Applied add-log-exp12.9

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

  9. Applied add-log-exp13.0

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

  10. Final simplification13.0

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

Reproduce

herbie shell --seed 2019132 +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)))