Average Error: 37.1 → 13.1
Time: 31.3s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\left(\frac{\frac{\sin x}{\cos x}}{1 - \sqrt[3]{\frac{\left(\left(\sin x \cdot \left(\sin x \cdot \sin x\right)\right) \cdot \sin \varepsilon\right) \cdot \left(\sin \varepsilon \cdot \sin \varepsilon\right)}{\left(\left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\cos x \cdot \cos \varepsilon\right)\right) \cdot \left(\cos x \cdot \cos \varepsilon\right)}}} - \frac{\sin x}{\cos x}\right) + \frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \sqrt[3]{\frac{\left(\left(\sin x \cdot \left(\sin x \cdot \sin x\right)\right) \cdot \sin \varepsilon\right) \cdot \left(\sin \varepsilon \cdot \sin \varepsilon\right)}{\left(\left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\cos x \cdot \cos \varepsilon\right)\right) \cdot \left(\cos x \cdot \cos \varepsilon\right)}}\right)}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\left(\frac{\frac{\sin x}{\cos x}}{1 - \sqrt[3]{\frac{\left(\left(\sin x \cdot \left(\sin x \cdot \sin x\right)\right) \cdot \sin \varepsilon\right) \cdot \left(\sin \varepsilon \cdot \sin \varepsilon\right)}{\left(\left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\cos x \cdot \cos \varepsilon\right)\right) \cdot \left(\cos x \cdot \cos \varepsilon\right)}}} - \frac{\sin x}{\cos x}\right) + \frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \sqrt[3]{\frac{\left(\left(\sin x \cdot \left(\sin x \cdot \sin x\right)\right) \cdot \sin \varepsilon\right) \cdot \left(\sin \varepsilon \cdot \sin \varepsilon\right)}{\left(\left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\cos x \cdot \cos \varepsilon\right)\right) \cdot \left(\cos x \cdot \cos \varepsilon\right)}}\right)}
double f(double x, double eps) {
        double r2082734 = x;
        double r2082735 = eps;
        double r2082736 = r2082734 + r2082735;
        double r2082737 = tan(r2082736);
        double r2082738 = tan(r2082734);
        double r2082739 = r2082737 - r2082738;
        return r2082739;
}
↓
double f(double x, double eps) {
        double r2082740 = x;
        double r2082741 = sin(r2082740);
        double r2082742 = cos(r2082740);
        double r2082743 = r2082741 / r2082742;
        double r2082744 = 1.0;
        double r2082745 = r2082741 * r2082741;
        double r2082746 = r2082741 * r2082745;
        double r2082747 = eps;
        double r2082748 = sin(r2082747);
        double r2082749 = r2082746 * r2082748;
        double r2082750 = r2082748 * r2082748;
        double r2082751 = r2082749 * r2082750;
        double r2082752 = cos(r2082747);
        double r2082753 = r2082742 * r2082752;
        double r2082754 = r2082753 * r2082753;
        double r2082755 = r2082754 * r2082753;
        double r2082756 = r2082751 / r2082755;
        double r2082757 = cbrt(r2082756);
        double r2082758 = r2082744 - r2082757;
        double r2082759 = r2082743 / r2082758;
        double r2082760 = r2082759 - r2082743;
        double r2082761 = r2082752 * r2082758;
        double r2082762 = r2082748 / r2082761;
        double r2082763 = r2082760 + r2082762;
        return r2082763;
}

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. Using strategy rm

  5. Applied add-cbrt-cube22.0

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

  7. Applied tan-quot22.0

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

  8. Applied associate-*r/22.0

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

  9. Applied tan-quot22.0

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

  10. Applied associate-*l/22.0

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

  11. Applied tan-quot22.0

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

  12. Applied tan-quot22.0

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

  13. Applied frac-times22.0

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

  14. Applied frac-times22.0

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

  15. Applied frac-times22.0

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

  16. Applied cbrt-div22.1

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

  17. Simplified22.1

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

  18. Taylor expanded around inf 33.3

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

  19. Simplified13.1

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

  20. Final simplification13.1

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

Reproduce

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

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

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