Average Error: 37.1 → 13.1
Time: 1.3m
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x}{\cos x}} - \frac{\sin x}{\cos x}\right) + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\left(\sin x \cdot \sin \varepsilon\right) \cdot \frac{1}{\cos \varepsilon}}{\cos x}}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x}{\cos x}} - \frac{\sin x}{\cos x}\right) + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\left(\sin x \cdot \sin \varepsilon\right) \cdot \frac{1}{\cos \varepsilon}}{\cos x}}
double f(double x, double eps) {
        double r3197045 = x;
        double r3197046 = eps;
        double r3197047 = r3197045 + r3197046;
        double r3197048 = tan(r3197047);
        double r3197049 = tan(r3197045);
        double r3197050 = r3197048 - r3197049;
        return r3197050;
}
↓
double f(double x, double eps) {
        double r3197051 = x;
        double r3197052 = sin(r3197051);
        double r3197053 = cos(r3197051);
        double r3197054 = r3197052 / r3197053;
        double r3197055 = 1.0;
        double r3197056 = eps;
        double r3197057 = sin(r3197056);
        double r3197058 = cos(r3197056);
        double r3197059 = r3197057 / r3197058;
        double r3197060 = r3197059 * r3197052;
        double r3197061 = r3197060 / r3197053;
        double r3197062 = r3197055 - r3197061;
        double r3197063 = r3197054 / r3197062;
        double r3197064 = r3197063 - r3197054;
        double r3197065 = r3197052 * r3197057;
        double r3197066 = r3197055 / r3197058;
        double r3197067 = r3197065 * r3197066;
        double r3197068 = r3197067 / r3197053;
        double r3197069 = r3197055 - r3197068;
        double r3197070 = r3197059 / r3197069;
        double r3197071 = r3197064 + r3197070;
        return r3197071;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.1
Target15.0
Herbie13.1
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 37.1

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

  3. Applied tan-sum22.0

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

  4. Taylor expanded around inf 22.1

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

  5. Simplified13.1

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

  7. Applied div-inv13.1

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

  8. Applied associate-*r*13.1

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

  9. Final simplification13.1

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

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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