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

Average Error: 13.2 → 0.3

Time: 17.8s

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

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

↓

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

C

double f(double x, double y, double z, double a) {
        double r219328 = x;
        double r219329 = y;
        double r219330 = z;
        double r219331 = r219329 + r219330;
        double r219332 = tan(r219331);
        double r219333 = a;
        double r219334 = tan(r219333);
        double r219335 = r219332 - r219334;
        double r219336 = r219328 + r219335;
        return r219336;
}

↓

double f(double x, double y, double z, double a) {
        double r219337 = y;
        double r219338 = tan(r219337);
        double r219339 = z;
        double r219340 = tan(r219339);
        double r219341 = r219338 + r219340;
        double r219342 = 1.0;
        double r219343 = r219338 * r219340;
        double r219344 = 3.0;
        double r219345 = pow(r219343, r219344);
        double r219346 = pow(r219345, r219344);
        double r219347 = cbrt(r219346);
        double r219348 = r219342 - r219347;
        double r219349 = r219341 / r219348;
        double r219350 = fma(r219338, r219340, r219342);
        double r219351 = fma(r219350, r219343, r219342);
        double r219352 = a;
        double r219353 = tan(r219352);
        double r219354 = -r219353;
        double r219355 = x;
        double r219356 = r219354 + r219355;
        double r219357 = fma(r219349, r219351, r219356);
        double r219358 = r219353 * r219342;
        double r219359 = fma(r219354, r219342, r219358);
        double r219360 = r219357 + r219359;
        return r219360;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus a

Derivation

1. Initial program 13.2

   \[x + \left(\tan \left(y + z\right) - \tan a\right)\]
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 *-un-lft-identity 0.2

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

6. Applied flip3-- 0.2

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

7. Applied associate-/r/ 0.2

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

8. Applied prod-diff 0.2

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

9. Applied associate-+r+ 0.2

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

10. Simplified 0.3

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

12. Applied add-cbrt-cube 0.3

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

13. Simplified 0.3

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

14. Final simplification 0.3

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

Reproduce

herbie shell --seed 2020057 +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))))