Average Error: 37.1 → 12.8
Time: 43.0s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\frac{\sin x}{\cos x}}{\frac{\cos \varepsilon}{\sin \varepsilon}}} - \frac{\sin x}{\cos x}\right) + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\frac{\sin x}{\cos x}}{\frac{\cos \varepsilon}{\sin \varepsilon}}}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\frac{\sin x}{\cos x}}{\frac{\cos \varepsilon}{\sin \varepsilon}}} - \frac{\sin x}{\cos x}\right) + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\frac{\sin x}{\cos x}}{\frac{\cos \varepsilon}{\sin \varepsilon}}}
double f(double x, double eps) {
        double r3898745 = x;
        double r3898746 = eps;
        double r3898747 = r3898745 + r3898746;
        double r3898748 = tan(r3898747);
        double r3898749 = tan(r3898745);
        double r3898750 = r3898748 - r3898749;
        return r3898750;
}
↓
double f(double x, double eps) {
        double r3898751 = x;
        double r3898752 = sin(r3898751);
        double r3898753 = cos(r3898751);
        double r3898754 = r3898752 / r3898753;
        double r3898755 = 1.0;
        double r3898756 = eps;
        double r3898757 = cos(r3898756);
        double r3898758 = sin(r3898756);
        double r3898759 = r3898757 / r3898758;
        double r3898760 = r3898754 / r3898759;
        double r3898761 = r3898755 - r3898760;
        double r3898762 = r3898754 / r3898761;
        double r3898763 = r3898762 - r3898754;
        double r3898764 = r3898758 / r3898757;
        double r3898765 = r3898764 / r3898761;
        double r3898766 = r3898763 + r3898765;
        return r3898766;
}

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

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 22.1

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

  5. Simplified22.0

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

  7. Applied div-inv22.0

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

  8. Applied div-inv22.1

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

  9. Applied times-frac22.1

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

  10. Taylor expanded around -inf 22.1

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

  11. Simplified12.8

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

  12. Final simplification12.8

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

Reproduce

herbie shell --seed 2019137 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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