Average Error: 37.1 → 12.9
Time: 31.1s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\frac{\frac{\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}}{\left(1 - \frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x}{\cos x}\right) \cdot \left(1 - \frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x}{\cos x}\right)} - \frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}}{\frac{\sin x}{\cos x} + \frac{\frac{\sin x}{\cos x}}{1 - \frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x}{\cos x}}} + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x}{\cos x}}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\frac{\frac{\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}}{\left(1 - \frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x}{\cos x}\right) \cdot \left(1 - \frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x}{\cos x}\right)} - \frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}}{\frac{\sin x}{\cos x} + \frac{\frac{\sin x}{\cos x}}{1 - \frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x}{\cos x}}} + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x}{\cos x}}
double f(double x, double eps) {
        double r5601973 = x;
        double r5601974 = eps;
        double r5601975 = r5601973 + r5601974;
        double r5601976 = tan(r5601975);
        double r5601977 = tan(r5601973);
        double r5601978 = r5601976 - r5601977;
        return r5601978;
}
↓
double f(double x, double eps) {
        double r5601979 = x;
        double r5601980 = sin(r5601979);
        double r5601981 = cos(r5601979);
        double r5601982 = r5601980 / r5601981;
        double r5601983 = r5601982 * r5601982;
        double r5601984 = 1.0;
        double r5601985 = eps;
        double r5601986 = sin(r5601985);
        double r5601987 = cos(r5601985);
        double r5601988 = r5601986 / r5601987;
        double r5601989 = r5601988 * r5601980;
        double r5601990 = r5601989 / r5601981;
        double r5601991 = r5601984 - r5601990;
        double r5601992 = r5601991 * r5601991;
        double r5601993 = r5601983 / r5601992;
        double r5601994 = r5601993 - r5601983;
        double r5601995 = r5601982 / r5601991;
        double r5601996 = r5601982 + r5601995;
        double r5601997 = r5601994 / r5601996;
        double r5601998 = r5601988 / r5601991;
        double r5601999 = r5601997 + r5601998;
        return r5601999;
}

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
Target15.1
Herbie12.9
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 37.1

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

  3. Applied tan-sum21.9

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

  4. Taylor expanded around inf 22.0

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

  5. Simplified12.9

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

  7. Applied flip--12.9

    \[\leadsto \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin x \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}{\cos x}} + \color{blue}{\frac{\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}{\cos x}} \cdot \frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}{\cos x}} - \frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}}{\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}{\cos x}} + \frac{\sin x}{\cos x}}}\]
  8. Using strategy rm

  9. Applied frac-times12.9

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

  10. Final simplification12.9

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

Reproduce

herbie shell --seed 2019171 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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