(+ x (- (tan (+ y z)) (tan a)))

Average Error: 13.0 → 0.2

Time: 10.9s

Precision: 64

\[\left(x = 0.0 \lor 0.588414199999999998 \le x \le 505.590899999999976\right) \land \left(-1.79665800000000009 \cdot 10^{308} \le y \le -9.425585000000013 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \le y \le 1.7512240000000001 \cdot 10^{308}\right) \land \left(-1.7767070000000002 \cdot 10^{308} \le z \le -8.59979600000002 \cdot 10^{-310} \lor 3.29314499999998 \cdot 10^{-311} \le z \le 1.72515400000000009 \cdot 10^{308}\right) \land \left(-1.79665800000000009 \cdot 10^{308} \le a \le -9.425585000000013 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \le a \le 1.7512240000000001 \cdot 10^{308}\right)\]

Math

\[x + \left(\tan \left(y + z\right) - \tan a\right)\]

↓

\[x + \frac{\left(\tan y + \tan z\right) \cdot \cos a - \left(1 - \tan y \cdot \tan z\right) \cdot \sin a}{\left(1 - {\left(\tan y \cdot \tan z\right)}^{3}\right) \cdot \cos a} \cdot \left(1 \cdot 1 + \left(\left(\tan y \cdot \tan z\right) \cdot \left(\tan y \cdot \tan z\right) + 1 \cdot \left(\left(\sqrt[3]{\tan y} \cdot \sqrt[3]{\tan y}\right) \cdot \left(\sqrt[3]{\tan y} \cdot \tan z\right)\right)\right)\right)\]

C

double f(double x, double y, double z, double a) {
        double r170747 = x;
        double r170748 = y;
        double r170749 = z;
        double r170750 = r170748 + r170749;
        double r170751 = tan(r170750);
        double r170752 = a;
        double r170753 = tan(r170752);
        double r170754 = r170751 - r170753;
        double r170755 = r170747 + r170754;
        return r170755;
}

↓

double f(double x, double y, double z, double a) {
        double r170756 = x;
        double r170757 = y;
        double r170758 = tan(r170757);
        double r170759 = z;
        double r170760 = tan(r170759);
        double r170761 = r170758 + r170760;
        double r170762 = a;
        double r170763 = cos(r170762);
        double r170764 = r170761 * r170763;
        double r170765 = 1.0;
        double r170766 = r170758 * r170760;
        double r170767 = r170765 - r170766;
        double r170768 = sin(r170762);
        double r170769 = r170767 * r170768;
        double r170770 = r170764 - r170769;
        double r170771 = 3.0;
        double r170772 = pow(r170766, r170771);
        double r170773 = r170765 - r170772;
        double r170774 = r170773 * r170763;
        double r170775 = r170770 / r170774;
        double r170776 = r170765 * r170765;
        double r170777 = r170766 * r170766;
        double r170778 = cbrt(r170758);
        double r170779 = r170778 * r170778;
        double r170780 = r170778 * r170760;
        double r170781 = r170779 * r170780;
        double r170782 = r170765 * r170781;
        double r170783 = r170777 + r170782;
        double r170784 = r170776 + r170783;
        double r170785 = r170775 * r170784;
        double r170786 = r170756 + r170785;
        return r170786;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus a

Derivation

1. Initial program 13.0

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

3. Applied tan-quot 13.0

   \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\frac{\sin a}{\cos a}}\right)\]

4. Applied tan-sum 0.2

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

5. Applied frac-sub 0.2

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

7. Applied flip3-- 0.2

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

8. Applied associate-*l/ 0.2

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

9. Applied associate-/r/ 0.2

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

10. Simplified 0.2

    \[\leadsto x + \color{blue}{\frac{\left(\tan y + \tan z\right) \cdot \cos a - \left(1 - \tan y \cdot \tan z\right) \cdot \sin a}{\left(1 - {\left(\tan y \cdot \tan z\right)}^{3}\right) \cdot \cos a}} \cdot \left(1 \cdot 1 + \left(\left(\tan y \cdot \tan z\right) \cdot \left(\tan y \cdot \tan z\right) + 1 \cdot \left(\tan y \cdot \tan z\right)\right)\right)\]
11. Using strategy rm

12. Applied add-cube-cbrt 0.2

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

13. Applied associate-*l* 0.2

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

14. Final simplification 0.2

    \[\leadsto x + \frac{\left(\tan y + \tan z\right) \cdot \cos a - \left(1 - \tan y \cdot \tan z\right) \cdot \sin a}{\left(1 - {\left(\tan y \cdot \tan z\right)}^{3}\right) \cdot \cos a} \cdot \left(1 \cdot 1 + \left(\left(\tan y \cdot \tan z\right) \cdot \left(\tan y \cdot \tan z\right) + 1 \cdot \left(\left(\sqrt[3]{\tan y} \cdot \sqrt[3]{\tan y}\right) \cdot \left(\sqrt[3]{\tan y} \cdot \tan z\right)\right)\right)\right)\]

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (x y z a)
  :name "(+ x (- (tan (+ y z)) (tan a)))"
  :precision binary64
  :pre (and (or (== x 0.0) (<= 0.5884142 x 505.5909)) (or (<= -1.796658e+308 y -9.425585e-310) (<= 1.284938e-309 y 1.7512240000000001e+308)) (or (<= -1.7767070000000002e+308 z -8.599796e-310) (<= 3.293145e-311 z 1.725154e+308)) (or (<= -1.796658e+308 a -9.425585e-310) (<= 1.284938e-309 a 1.7512240000000001e+308)))
  (+ x (- (tan (+ y z)) (tan a))))