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

Average Error: 13.4 → 0.2

Time: 12.8s

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

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

↓

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

C

double f(double x, double y, double z, double a) {
        double r155678 = x;
        double r155679 = y;
        double r155680 = z;
        double r155681 = r155679 + r155680;
        double r155682 = tan(r155681);
        double r155683 = a;
        double r155684 = tan(r155683);
        double r155685 = r155682 - r155684;
        double r155686 = r155678 + r155685;
        return r155686;
}

↓

double f(double x, double y, double z, double a) {
        double r155687 = x;
        double r155688 = y;
        double r155689 = tan(r155688);
        double r155690 = z;
        double r155691 = tan(r155690);
        double r155692 = r155689 + r155691;
        double r155693 = 1.0;
        double r155694 = r155689 * r155691;
        double r155695 = r155693 - r155694;
        double r155696 = r155692 / r155695;
        double r155697 = r155696 / r155695;
        double r155698 = r155692 * r155697;
        double r155699 = a;
        double r155700 = tan(r155699);
        double r155701 = r155700 * r155700;
        double r155702 = r155698 - r155701;
        double r155703 = r155693 * r155694;
        double r155704 = r155693 - r155703;
        double r155705 = r155692 / r155704;
        double r155706 = r155705 + r155700;
        double r155707 = r155702 / r155706;
        double r155708 = r155687 + r155707;
        return r155708;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus a

Derivation

1. Initial program 13.4

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

5. Applied flip-- 0.2

   \[\leadsto x + \color{blue}{\frac{\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} \cdot \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \tan a \cdot \tan a}{\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} + \tan a}}\]
6. Using strategy rm

7. Applied *-un-lft-identity 0.2

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

8. Applied add-sqr-sqrt 31.9

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

9. Applied times-frac 31.9

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

10. Applied *-un-lft-identity 31.9

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

11. Applied add-sqr-sqrt 32.0

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

12. Applied times-frac 32.0

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

13. Applied swap-sqr 32.0

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

14. Simplified 31.9

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

15. Simplified 0.2

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

17. Applied *-un-lft-identity 0.2

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

18. Final simplification 0.2

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

Reproduce

herbie shell --seed 2020047 
(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))))