Average Error: 37.4 → 12.6
Time: 1.3m
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x}{\cos x}} - \frac{\sin x}{\cos x}\right) + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sqrt[3]{\left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x\right) \cdot e^{\log \left(\left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x\right)\right)}}}{\cos x}}\]
\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{\sqrt[3]{\left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x\right) \cdot e^{\log \left(\left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x\right)\right)}}}{\cos x}}
double f(double x, double eps) {
        double r4248975 = x;
        double r4248976 = eps;
        double r4248977 = r4248975 + r4248976;
        double r4248978 = tan(r4248977);
        double r4248979 = tan(r4248975);
        double r4248980 = r4248978 - r4248979;
        return r4248980;
}
↓
double f(double x, double eps) {
        double r4248981 = x;
        double r4248982 = sin(r4248981);
        double r4248983 = cos(r4248981);
        double r4248984 = r4248982 / r4248983;
        double r4248985 = 1.0;
        double r4248986 = eps;
        double r4248987 = sin(r4248986);
        double r4248988 = cos(r4248986);
        double r4248989 = r4248987 / r4248988;
        double r4248990 = r4248989 * r4248982;
        double r4248991 = r4248990 / r4248983;
        double r4248992 = r4248985 - r4248991;
        double r4248993 = r4248984 / r4248992;
        double r4248994 = r4248993 - r4248984;
        double r4248995 = r4248990 * r4248990;
        double r4248996 = log(r4248995);
        double r4248997 = exp(r4248996);
        double r4248998 = r4248990 * r4248997;
        double r4248999 = cbrt(r4248998);
        double r4249000 = r4248999 / r4248983;
        double r4249001 = r4248985 - r4249000;
        double r4249002 = r4248989 / r4249001;
        double r4249003 = r4248994 + r4249002;
        return r4249003;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.4
Target15.9
Herbie12.6
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 37.4

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

  3. Applied tan-sum21.4

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

  4. Taylor expanded around inf 21.5

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

  5. Simplified12.6

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

  7. Applied add-cbrt-cube12.6

    \[\leadsto \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\color{blue}{\sqrt[3]{\left(\left(\sin x \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\sin x \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)\right) \cdot \left(\sin x \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}}}{\cos x}} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}{\cos x}} - \frac{\sin x}{\cos x}\right)\]
  8. Using strategy rm

  9. Applied add-exp-log12.6

    \[\leadsto \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sqrt[3]{\color{blue}{e^{\log \left(\left(\sin x \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\sin x \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)\right)}} \cdot \left(\sin x \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}}{\cos x}} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}{\cos x}} - \frac{\sin x}{\cos x}\right)\]

  10. Final simplification12.6

    \[\leadsto \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x}{\cos x}} - \frac{\sin x}{\cos x}\right) + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sqrt[3]{\left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x\right) \cdot e^{\log \left(\left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x\right)\right)}}}{\cos x}}\]

Reproduce

herbie shell --seed 2019143 +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)))