Average Error: 36.9 → 12.8
Time: 52.4s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\left(1 + \left(\frac{\frac{\sin \varepsilon}{\cos x} \cdot \sin x}{\cos \varepsilon} + \frac{\frac{\sin \varepsilon}{\cos x} \cdot \sin x}{\cos \varepsilon} \cdot \frac{\frac{\sin \varepsilon}{\cos x} \cdot \sin x}{\cos \varepsilon}\right)\right) \cdot \frac{\frac{\sin \varepsilon}{1 - \frac{\sqrt[3]{\left(\left(\frac{\sin \varepsilon}{\cos x} \cdot \sin x\right) \cdot \left(\frac{\sin \varepsilon}{\cos x} \cdot \sin x\right)\right) \cdot \left(\frac{\sin \varepsilon}{\cos x} \cdot \sin x\right)}}{\cos \varepsilon} \cdot \left(\frac{\frac{\sin \varepsilon}{\cos x} \cdot \sin x}{\cos \varepsilon} \cdot \frac{\frac{\sin \varepsilon}{\cos x} \cdot \sin x}{\cos \varepsilon}\right)}}{\cos \varepsilon} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\frac{\sin \varepsilon}{\cos x} \cdot \sin x}{\cos \varepsilon}} - \frac{\sin x}{\cos x}\right)\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\left(1 + \left(\frac{\frac{\sin \varepsilon}{\cos x} \cdot \sin x}{\cos \varepsilon} + \frac{\frac{\sin \varepsilon}{\cos x} \cdot \sin x}{\cos \varepsilon} \cdot \frac{\frac{\sin \varepsilon}{\cos x} \cdot \sin x}{\cos \varepsilon}\right)\right) \cdot \frac{\frac{\sin \varepsilon}{1 - \frac{\sqrt[3]{\left(\left(\frac{\sin \varepsilon}{\cos x} \cdot \sin x\right) \cdot \left(\frac{\sin \varepsilon}{\cos x} \cdot \sin x\right)\right) \cdot \left(\frac{\sin \varepsilon}{\cos x} \cdot \sin x\right)}}{\cos \varepsilon} \cdot \left(\frac{\frac{\sin \varepsilon}{\cos x} \cdot \sin x}{\cos \varepsilon} \cdot \frac{\frac{\sin \varepsilon}{\cos x} \cdot \sin x}{\cos \varepsilon}\right)}}{\cos \varepsilon} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\frac{\sin \varepsilon}{\cos x} \cdot \sin x}{\cos \varepsilon}} - \frac{\sin x}{\cos x}\right)
double f(double x, double eps) {
        double r6025901 = x;
        double r6025902 = eps;
        double r6025903 = r6025901 + r6025902;
        double r6025904 = tan(r6025903);
        double r6025905 = tan(r6025901);
        double r6025906 = r6025904 - r6025905;
        return r6025906;
}
↓
double f(double x, double eps) {
        double r6025907 = 1.0;
        double r6025908 = eps;
        double r6025909 = sin(r6025908);
        double r6025910 = x;
        double r6025911 = cos(r6025910);
        double r6025912 = r6025909 / r6025911;
        double r6025913 = sin(r6025910);
        double r6025914 = r6025912 * r6025913;
        double r6025915 = cos(r6025908);
        double r6025916 = r6025914 / r6025915;
        double r6025917 = r6025916 * r6025916;
        double r6025918 = r6025916 + r6025917;
        double r6025919 = r6025907 + r6025918;
        double r6025920 = r6025914 * r6025914;
        double r6025921 = r6025920 * r6025914;
        double r6025922 = cbrt(r6025921);
        double r6025923 = r6025922 / r6025915;
        double r6025924 = r6025923 * r6025917;
        double r6025925 = r6025907 - r6025924;
        double r6025926 = r6025909 / r6025925;
        double r6025927 = r6025926 / r6025915;
        double r6025928 = r6025919 * r6025927;
        double r6025929 = r6025913 / r6025911;
        double r6025930 = r6025907 - r6025916;
        double r6025931 = r6025929 / r6025930;
        double r6025932 = r6025931 - r6025929;
        double r6025933 = r6025928 + r6025932;
        return r6025933;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.9
Target15.5
Herbie12.8
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 36.9

    \[\tan \left(x + \varepsilon\right) - \tan x\]
  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. Simplified12.7

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

  7. Applied flip3--12.8

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

  8. Applied associate-*l/12.8

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

  9. Applied associate-/r/12.8

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

  10. Simplified12.8

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

  12. Applied add-cbrt-cube12.8

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

  13. Final simplification12.8

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

Reproduce

herbie shell --seed 2019163 +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)))