Average Error: 36.3 → 12.5
Time: 26.9s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\frac{\frac{\frac{\sin x}{\cos x}}{1 - \frac{\frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon}}{\cos x}} \cdot \frac{\frac{\sin x}{\cos x}}{1 - \frac{\frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon}}{\cos x}} - \frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}}{\frac{\sin x}{\cos x} + \frac{\frac{\sin x}{\cos x}}{1 - \frac{\frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon}}{\cos x}}} + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon}}{\cos x}}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\frac{\frac{\frac{\sin x}{\cos x}}{1 - \frac{\frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon}}{\cos x}} \cdot \frac{\frac{\sin x}{\cos x}}{1 - \frac{\frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon}}{\cos x}} - \frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}}{\frac{\sin x}{\cos x} + \frac{\frac{\sin x}{\cos x}}{1 - \frac{\frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon}}{\cos x}}} + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon}}{\cos x}}
double f(double x, double eps) {
        double r3658280 = x;
        double r3658281 = eps;
        double r3658282 = r3658280 + r3658281;
        double r3658283 = tan(r3658282);
        double r3658284 = tan(r3658280);
        double r3658285 = r3658283 - r3658284;
        return r3658285;
}
↓
double f(double x, double eps) {
        double r3658286 = x;
        double r3658287 = sin(r3658286);
        double r3658288 = cos(r3658286);
        double r3658289 = r3658287 / r3658288;
        double r3658290 = 1.0;
        double r3658291 = eps;
        double r3658292 = sin(r3658291);
        double r3658293 = r3658287 * r3658292;
        double r3658294 = cos(r3658291);
        double r3658295 = r3658293 / r3658294;
        double r3658296 = r3658295 / r3658288;
        double r3658297 = r3658290 - r3658296;
        double r3658298 = r3658289 / r3658297;
        double r3658299 = r3658298 * r3658298;
        double r3658300 = r3658289 * r3658289;
        double r3658301 = r3658299 - r3658300;
        double r3658302 = r3658289 + r3658298;
        double r3658303 = r3658301 / r3658302;
        double r3658304 = r3658292 / r3658294;
        double r3658305 = r3658304 / r3658297;
        double r3658306 = r3658303 + r3658305;
        return r3658306;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.3
Target15.3
Herbie12.5
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 36.3

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

  3. Applied tan-sum21.0

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

  4. Taylor expanded around inf 21.1

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

  5. Simplified12.5

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

  7. Applied flip--12.5

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

  8. Final simplification12.5

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

Reproduce

herbie shell --seed 2019165 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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