Average Error: 36.9 → 0.4
Time: 9.6s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\frac{\frac{\sin \varepsilon \cdot \cos x}{\cos \varepsilon} + \frac{1}{\frac{\cos x \cdot \cos \varepsilon}{{\left(\sin x\right)}^{2} \cdot \sin \varepsilon}}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\frac{\frac{\sin \varepsilon \cdot \cos x}{\cos \varepsilon} + \frac{1}{\frac{\cos x \cdot \cos \varepsilon}{{\left(\sin x\right)}^{2} \cdot \sin \varepsilon}}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}
double f(double x, double eps) {
        double r120189 = x;
        double r120190 = eps;
        double r120191 = r120189 + r120190;
        double r120192 = tan(r120191);
        double r120193 = tan(r120189);
        double r120194 = r120192 - r120193;
        return r120194;
}
↓
double f(double x, double eps) {
        double r120195 = eps;
        double r120196 = sin(r120195);
        double r120197 = x;
        double r120198 = cos(r120197);
        double r120199 = r120196 * r120198;
        double r120200 = cos(r120195);
        double r120201 = r120199 / r120200;
        double r120202 = 1.0;
        double r120203 = r120198 * r120200;
        double r120204 = sin(r120197);
        double r120205 = 2.0;
        double r120206 = pow(r120204, r120205);
        double r120207 = r120206 * r120196;
        double r120208 = r120203 / r120207;
        double r120209 = r120202 / r120208;
        double r120210 = r120201 + r120209;
        double r120211 = tan(r120197);
        double r120212 = tan(r120195);
        double r120213 = r120211 * r120212;
        double r120214 = r120202 - r120213;
        double r120215 = r120214 * r120198;
        double r120216 = r120210 / r120215;
        return r120216;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.9
Target15.6
Herbie0.4
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 36.9

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

  3. Applied tan-quot36.9

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

  4. Applied tan-sum21.4

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

  5. Applied frac-sub21.4

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

  6. Taylor expanded around inf 0.4

    \[\leadsto \frac{\color{blue}{\frac{\sin \varepsilon \cdot \cos x}{\cos \varepsilon} + \frac{{\left(\sin x\right)}^{2} \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\]
  7. Using strategy rm

  8. Applied clear-num0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2020033 
(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)))