Average Error: 36.4 → 0.4
Time: 10.2s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\frac{\frac{\sin \varepsilon \cdot \cos x}{\cos \varepsilon} + \left({\left(\sin x\right)}^{2} \cdot \sin \varepsilon\right) \cdot \frac{1}{\cos x \cdot \cos \varepsilon}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\frac{\frac{\sin \varepsilon \cdot \cos x}{\cos \varepsilon} + \left({\left(\sin x\right)}^{2} \cdot \sin \varepsilon\right) \cdot \frac{1}{\cos x \cdot \cos \varepsilon}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}
double f(double x, double eps) {
        double r95222 = x;
        double r95223 = eps;
        double r95224 = r95222 + r95223;
        double r95225 = tan(r95224);
        double r95226 = tan(r95222);
        double r95227 = r95225 - r95226;
        return r95227;
}
↓
double f(double x, double eps) {
        double r95228 = eps;
        double r95229 = sin(r95228);
        double r95230 = x;
        double r95231 = cos(r95230);
        double r95232 = r95229 * r95231;
        double r95233 = cos(r95228);
        double r95234 = r95232 / r95233;
        double r95235 = sin(r95230);
        double r95236 = 2.0;
        double r95237 = pow(r95235, r95236);
        double r95238 = r95237 * r95229;
        double r95239 = 1.0;
        double r95240 = r95231 * r95233;
        double r95241 = r95239 / r95240;
        double r95242 = r95238 * r95241;
        double r95243 = r95234 + r95242;
        double r95244 = tan(r95230);
        double r95245 = tan(r95228);
        double r95246 = r95244 * r95245;
        double r95247 = r95239 - r95246;
        double r95248 = r95247 * r95231;
        double r95249 = r95243 / r95248;
        return r95249;
}

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

Derivation

  1. Initial program 36.4

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

  3. Applied tan-quot36.4

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

  4. Applied tan-sum21.3

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

  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 div-inv0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2020034 
(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)))