Average Error: 36.4 → 12.8
Time: 1.0m
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x}{\cos x}} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} - \frac{\sin x}{\cos x}\right)\]
\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{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} - \frac{\sin x}{\cos x}\right)
double f(double x, double eps) {
        double r3956255 = x;
        double r3956256 = eps;
        double r3956257 = r3956255 + r3956256;
        double r3956258 = tan(r3956257);
        double r3956259 = tan(r3956255);
        double r3956260 = r3956258 - r3956259;
        return r3956260;
}
↓
double f(double x, double eps) {
        double r3956261 = eps;
        double r3956262 = sin(r3956261);
        double r3956263 = cos(r3956261);
        double r3956264 = r3956262 / r3956263;
        double r3956265 = 1.0;
        double r3956266 = x;
        double r3956267 = sin(r3956266);
        double r3956268 = r3956264 * r3956267;
        double r3956269 = cos(r3956266);
        double r3956270 = r3956268 / r3956269;
        double r3956271 = r3956265 - r3956270;
        double r3956272 = r3956264 / r3956271;
        double r3956273 = r3956267 / r3956269;
        double r3956274 = r3956273 * r3956264;
        double r3956275 = r3956265 - r3956274;
        double r3956276 = r3956273 / r3956275;
        double r3956277 = r3956276 - r3956273;
        double r3956278 = r3956272 + r3956277;
        return r3956278;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.4
Target14.8
Herbie12.8
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 36.4

    \[\tan \left(x + \varepsilon\right) - \tan x\]
  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. Simplified12.8

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

  7. Applied associate-*l/12.8

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

  8. Final simplification12.8

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

Reproduce

herbie shell --seed 2019149 +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)))