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

Average Error: 13.1 → 0.2

Time: 15.4s

Precision: 64

\[\left(x = 0.0 \lor 0.5884141999999999983472775966220069676638 \le x \le 505.5908999999999764440872240811586380005\right) \land \left(-1.7966580000000000931214523812968299911 \cdot 10^{308} \le y \le -9.425585000000013069597555966781986720373 \cdot 10^{-310} \lor 1.284937999999999548796432976649400331091 \cdot 10^{-309} \le y \le 1.751224000000000127647232028319723370461 \cdot 10^{308}\right) \land \left(-1.776707000000000200843839711454021982841 \cdot 10^{308} \le z \le -8.599796000000016667475923823712126825539 \cdot 10^{-310} \lor 3.293144999999983071955117582595641261776 \cdot 10^{-311} \le z \le 1.725154000000000087891269878141591702413 \cdot 10^{308}\right) \land \left(-1.7966580000000000931214523812968299911 \cdot 10^{308} \le a \le -9.425585000000013069597555966781986720373 \cdot 10^{-310} \lor 1.284937999999999548796432976649400331091 \cdot 10^{-309} \le a \le 1.751224000000000127647232028319723370461 \cdot 10^{308}\right)\]

Math

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

↓

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

C

double f(double x, double y, double z, double a) {
        double r165295 = x;
        double r165296 = y;
        double r165297 = z;
        double r165298 = r165296 + r165297;
        double r165299 = tan(r165298);
        double r165300 = a;
        double r165301 = tan(r165300);
        double r165302 = r165299 - r165301;
        double r165303 = r165295 + r165302;
        return r165303;
}

↓

double f(double x, double y, double z, double a) {
        double r165304 = x;
        double r165305 = y;
        double r165306 = tan(r165305);
        double r165307 = z;
        double r165308 = tan(r165307);
        double r165309 = r165306 + r165308;
        double r165310 = 1.0;
        double r165311 = fma(r165306, r165308, r165310);
        double r165312 = r165309 * r165311;
        double r165313 = r165306 * r165308;
        double r165314 = r165313 * r165313;
        double r165315 = r165314 * r165314;
        double r165316 = r165310 - r165315;
        double r165317 = r165312 / r165316;
        double r165318 = r165310 + r165314;
        double r165319 = a;
        double r165320 = tan(r165319);
        double r165321 = -r165320;
        double r165322 = fma(r165317, r165318, r165321);
        double r165323 = fma(r165321, r165310, r165320);
        double r165324 = r165322 + r165323;
        double r165325 = r165304 + r165324;
        return r165325;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus a

Derivation

1. Initial program 13.1

   \[x + \left(\tan \left(y + z\right) - \tan a\right)\]
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 add-cube-cbrt 0.3

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

6. Applied flip-- 0.4

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

7. Applied associate-/r/ 0.4

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

8. Applied prod-diff 0.4

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

9. Simplified 0.2

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

10. Simplified 0.2

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

12. Applied flip-- 0.2

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

13. Applied associate-/r/ 0.2

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

14. Applied fma-neg 0.2

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

15. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019356 +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))))