Average Error: 37.1 → 12.9
Time: 29.3s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\sqrt[3]{\left(\mathsf{fma}\left(\frac{\sin x}{\cos x}, \frac{1}{\mathsf{fma}\left(-\frac{\sin x}{\cos x}, \frac{\sin \varepsilon}{\cos \varepsilon}, 1\right)}, -\frac{\sin x}{\cos x}\right) \cdot \left(\frac{\frac{\sin x}{\cos x}}{\mathsf{fma}\left(-\frac{\sin x}{\cos x}, \frac{\sin \varepsilon}{\cos \varepsilon}, 1\right)} + \left(-\frac{\sin x}{\cos x}\right)\right)\right) \cdot \left(\frac{\frac{\sin x}{\cos x}}{\mathsf{fma}\left(-\frac{\sin x}{\cos x}, \frac{\sin \varepsilon}{\cos \varepsilon}, 1\right)} + \left(-\frac{\sin x}{\cos x}\right)\right)} + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{\mathsf{fma}\left(-\frac{\sin x}{\cos x}, \frac{\sin \varepsilon}{\cos \varepsilon}, 1\right)}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\sqrt[3]{\left(\mathsf{fma}\left(\frac{\sin x}{\cos x}, \frac{1}{\mathsf{fma}\left(-\frac{\sin x}{\cos x}, \frac{\sin \varepsilon}{\cos \varepsilon}, 1\right)}, -\frac{\sin x}{\cos x}\right) \cdot \left(\frac{\frac{\sin x}{\cos x}}{\mathsf{fma}\left(-\frac{\sin x}{\cos x}, \frac{\sin \varepsilon}{\cos \varepsilon}, 1\right)} + \left(-\frac{\sin x}{\cos x}\right)\right)\right) \cdot \left(\frac{\frac{\sin x}{\cos x}}{\mathsf{fma}\left(-\frac{\sin x}{\cos x}, \frac{\sin \varepsilon}{\cos \varepsilon}, 1\right)} + \left(-\frac{\sin x}{\cos x}\right)\right)} + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{\mathsf{fma}\left(-\frac{\sin x}{\cos x}, \frac{\sin \varepsilon}{\cos \varepsilon}, 1\right)}
double f(double x, double eps) {
        double r4298780 = x;
        double r4298781 = eps;
        double r4298782 = r4298780 + r4298781;
        double r4298783 = tan(r4298782);
        double r4298784 = tan(r4298780);
        double r4298785 = r4298783 - r4298784;
        return r4298785;
}
↓
double f(double x, double eps) {
        double r4298786 = x;
        double r4298787 = sin(r4298786);
        double r4298788 = cos(r4298786);
        double r4298789 = r4298787 / r4298788;
        double r4298790 = 1.0;
        double r4298791 = -r4298789;
        double r4298792 = eps;
        double r4298793 = sin(r4298792);
        double r4298794 = cos(r4298792);
        double r4298795 = r4298793 / r4298794;
        double r4298796 = fma(r4298791, r4298795, r4298790);
        double r4298797 = r4298790 / r4298796;
        double r4298798 = fma(r4298789, r4298797, r4298791);
        double r4298799 = r4298789 / r4298796;
        double r4298800 = r4298799 + r4298791;
        double r4298801 = r4298798 * r4298800;
        double r4298802 = r4298801 * r4298800;
        double r4298803 = cbrt(r4298802);
        double r4298804 = r4298795 / r4298796;
        double r4298805 = r4298803 + r4298804;
        return r4298805;
}

Error

Bits error versus x

Bits error versus eps

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

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

  7. Applied add-cbrt-cube12.9

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

  9. Applied div-inv12.9

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

  10. Applied fma-def12.9

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

  11. Final simplification12.9

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

Reproduce

herbie shell --seed 2019171 +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)))