Average Error: 37.2 → 0.4
Time: 9.8s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\frac{\frac{\sin \varepsilon \cdot \cos x}{\cos \varepsilon} + \frac{{\left(\sin x\right)}^{2} \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}}{\left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}\right) \cdot \cos x}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\frac{\frac{\sin \varepsilon \cdot \cos x}{\cos \varepsilon} + \frac{{\left(\sin x\right)}^{2} \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}}{\left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}\right) \cdot \cos x}
double f(double x, double eps) {
        double r136311 = x;
        double r136312 = eps;
        double r136313 = r136311 + r136312;
        double r136314 = tan(r136313);
        double r136315 = tan(r136311);
        double r136316 = r136314 - r136315;
        return r136316;
}
↓
double f(double x, double eps) {
        double r136317 = eps;
        double r136318 = sin(r136317);
        double r136319 = x;
        double r136320 = cos(r136319);
        double r136321 = r136318 * r136320;
        double r136322 = cos(r136317);
        double r136323 = r136321 / r136322;
        double r136324 = sin(r136319);
        double r136325 = 2.0;
        double r136326 = pow(r136324, r136325);
        double r136327 = r136326 * r136318;
        double r136328 = r136320 * r136322;
        double r136329 = r136327 / r136328;
        double r136330 = r136323 + r136329;
        double r136331 = 1.0;
        double r136332 = r136324 * r136318;
        double r136333 = r136332 / r136328;
        double r136334 = r136331 - r136333;
        double r136335 = r136334 * r136320;
        double r136336 = r136330 / r136335;
        return r136336;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.2
Target15.3
Herbie0.4
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 37.2

    \[\tan \left(x + \varepsilon\right) - \tan x\]
  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 21.8

    \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}}} - \tan x\]
  5. Using strategy rm

  6. Applied tan-quot21.9

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

  7. Applied frac-sub21.9

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

  8. Simplified21.9

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

  9. Taylor expanded around inf 0.4

    \[\leadsto \frac{\color{blue}{\frac{\sin \varepsilon \cdot \cos x}{\cos \varepsilon} + \frac{{\left(\sin x\right)}^{2} \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}}}{\left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}\right) \cdot \cos x}\]

  10. Final simplification0.4

    \[\leadsto \frac{\frac{\sin \varepsilon \cdot \cos x}{\cos \varepsilon} + \frac{{\left(\sin x\right)}^{2} \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}}{\left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}\right) \cdot \cos x}\]

Reproduce

herbie shell --seed 2020049 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :precision binary64

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

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