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

Average Error: 12.8 → 0.2

Time: 12.9s

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

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

↓

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

C

double f(double x, double y, double z, double a) {
        double r122696 = x;
        double r122697 = y;
        double r122698 = z;
        double r122699 = r122697 + r122698;
        double r122700 = tan(r122699);
        double r122701 = a;
        double r122702 = tan(r122701);
        double r122703 = r122700 - r122702;
        double r122704 = r122696 + r122703;
        return r122704;
}

↓

double f(double x, double y, double z, double a) {
        double r122705 = x;
        double r122706 = 1.0;
        double r122707 = y;
        double r122708 = sin(r122707);
        double r122709 = z;
        double r122710 = sin(r122709);
        double r122711 = r122708 * r122710;
        double r122712 = cos(r122709);
        double r122713 = cos(r122707);
        double r122714 = r122712 * r122713;
        double r122715 = r122711 / r122714;
        double r122716 = r122706 - r122715;
        double r122717 = r122706 / r122716;
        double r122718 = r122708 / r122713;
        double r122719 = r122710 / r122712;
        double r122720 = r122718 + r122719;
        double r122721 = r122717 * r122720;
        double r122722 = a;
        double r122723 = sin(r122722);
        double r122724 = cos(r122722);
        double r122725 = r122723 / r122724;
        double r122726 = r122721 - r122725;
        double r122727 = -r122706;
        double r122728 = r122727 + r122706;
        double r122729 = r122725 * r122728;
        double r122730 = r122726 + r122729;
        double r122731 = r122705 + r122730;
        return r122731;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus a

Derivation

1. Initial program 12.8

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

3. Applied tan-sum 0.2

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

4. Taylor expanded around inf 0.2

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

6. Applied add-cube-cbrt 0.3

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

7. Applied *-un-lft-identity 0.3

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

8. Applied times-frac 0.3

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

9. Applied *-un-lft-identity 0.3

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

10. Applied times-frac 0.3

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

11. Applied distribute-lft-out 0.3

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

12. Applied prod-diff 0.3

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

13. Simplified 0.2

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

14. Simplified 0.2

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

15. Final simplification 0.2

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

Reproduce

herbie shell --seed 2020036 +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))))