Average Error: 37.0 → 0.4
Time: 24.9s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\frac{\frac{\sin \varepsilon \cdot \cos x}{\cos \varepsilon} + \frac{{\left(\sin x\right)}^{2}}{\frac{\cos x \cdot \cos \varepsilon}{\sin \varepsilon}}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\frac{\frac{\sin \varepsilon \cdot \cos x}{\cos \varepsilon} + \frac{{\left(\sin x\right)}^{2}}{\frac{\cos x \cdot \cos \varepsilon}{\sin \varepsilon}}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}
double f(double x, double eps) {
        double r92964 = x;
        double r92965 = eps;
        double r92966 = r92964 + r92965;
        double r92967 = tan(r92966);
        double r92968 = tan(r92964);
        double r92969 = r92967 - r92968;
        return r92969;
}
↓
double f(double x, double eps) {
        double r92970 = eps;
        double r92971 = sin(r92970);
        double r92972 = x;
        double r92973 = cos(r92972);
        double r92974 = r92971 * r92973;
        double r92975 = cos(r92970);
        double r92976 = r92974 / r92975;
        double r92977 = sin(r92972);
        double r92978 = 2.0;
        double r92979 = pow(r92977, r92978);
        double r92980 = r92973 * r92975;
        double r92981 = r92980 / r92971;
        double r92982 = r92979 / r92981;
        double r92983 = r92976 + r92982;
        double r92984 = 1.0;
        double r92985 = tan(r92972);
        double r92986 = tan(r92970);
        double r92987 = r92985 * r92986;
        double r92988 = r92984 - r92987;
        double r92989 = r92988 * r92973;
        double r92990 = r92983 / r92989;
        return r92990;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.0
Target15.1
Herbie0.4
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 37.0

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

  3. Applied tan-quot37.0

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

  4. Applied tan-sum21.9

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

  5. Applied frac-sub21.9

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

  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 associate-/l*0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2019304 
(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)))