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

Average Error: 12.8 → 0.3

Time: 45.5s

Precision: 64

\[\left(x = 0.0 \lor 0.5884141999999999983472775966220069676638 \le x \le 505.5908999999999764440872240811586380005\right) \land \left(-1.7966580000000000931214523812968299911 \cdot 10^{308} \le y \le -9.425585000000013069597555966781986720373 \cdot 10^{-310} \lor 1.284937999999999548796432976649400331091 \cdot 10^{-309} \le y \le 1.751223999999999928063201074847742204824 \cdot 10^{308}\right) \land \left(-1.776707000000000001259808757982040817204 \cdot 10^{308} \le z \le -8.599796000000016667475923823712126825539 \cdot 10^{-310} \lor 3.293144999999983071955117582595641261776 \cdot 10^{-311} \le z \le 1.725154000000000087891269878141591702413 \cdot 10^{308}\right) \land \left(-1.7966580000000000931214523812968299911 \cdot 10^{308} \le a \le -9.425585000000013069597555966781986720373 \cdot 10^{-310} \lor 1.284937999999999548796432976649400331091 \cdot 10^{-309} \le a \le 1.751223999999999928063201074847742204824 \cdot 10^{308}\right)\]

Math

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

↓

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

C

double f(double x, double y, double z, double a) {
        double r5813478 = x;
        double r5813479 = y;
        double r5813480 = z;
        double r5813481 = r5813479 + r5813480;
        double r5813482 = tan(r5813481);
        double r5813483 = a;
        double r5813484 = tan(r5813483);
        double r5813485 = r5813482 - r5813484;
        double r5813486 = r5813478 + r5813485;
        return r5813486;
}

↓

double f(double x, double y, double z, double a) {
        double r5813487 = y;
        double r5813488 = tan(r5813487);
        double r5813489 = z;
        double r5813490 = tan(r5813489);
        double r5813491 = r5813488 + r5813490;
        double r5813492 = 1.0;
        double r5813493 = r5813490 * r5813488;
        double r5813494 = r5813492 - r5813493;
        double r5813495 = r5813491 / r5813494;
        double r5813496 = a;
        double r5813497 = sin(r5813496);
        double r5813498 = cos(r5813496);
        double r5813499 = r5813497 / r5813498;
        double r5813500 = r5813495 - r5813499;
        double r5813501 = r5813500 * r5813500;
        double r5813502 = r5813501 * r5813500;
        double r5813503 = cbrt(r5813502);
        double r5813504 = x;
        double r5813505 = r5813503 + r5813504;
        return r5813505;
}

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-quot 12.8

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

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 add-cbrt-cube 0.3

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

8. Applied add-cbrt-cube 0.3

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

9. Applied cbrt-unprod 0.3

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

10. Applied add-cbrt-cube 0.3

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

11. Applied cbrt-undiv 0.3

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

12. Simplified 0.3

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

13. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019192 
(FPCore (x y z a)
  :name "(+ x (- (tan (+ y z)) (tan a)))"
  :pre (and (or (== x 0.0) (<= 0.5884142 x 505.5909)) (or (<= -1.796658e+308 y -9.425585e-310) (<= 1.284938e-309 y 1.751224e+308)) (or (<= -1.776707e+308 z -8.599796e-310) (<= 3.293145e-311 z 1.725154e+308)) (or (<= -1.796658e+308 a -9.425585e-310) (<= 1.284938e-309 a 1.751224e+308)))
  (+ x (- (tan (+ y z)) (tan a))))