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

Average Error: 13.1 → 0.2

Time: 11.4s

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

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

↓

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

C

double f(double x, double y, double z, double a) {
        double r221362 = x;
        double r221363 = y;
        double r221364 = z;
        double r221365 = r221363 + r221364;
        double r221366 = tan(r221365);
        double r221367 = a;
        double r221368 = tan(r221367);
        double r221369 = r221366 - r221368;
        double r221370 = r221362 + r221369;
        return r221370;
}

↓

double f(double x, double y, double z, double a) {
        double r221371 = x;
        double r221372 = y;
        double r221373 = tan(r221372);
        double r221374 = r221373 * r221373;
        double r221375 = z;
        double r221376 = tan(r221375);
        double r221377 = r221376 * r221376;
        double r221378 = r221374 - r221377;
        double r221379 = a;
        double r221380 = cos(r221379);
        double r221381 = r221378 * r221380;
        double r221382 = r221373 - r221376;
        double r221383 = r221381 / r221382;
        double r221384 = 1.0;
        double r221385 = r221373 * r221376;
        double r221386 = r221384 - r221385;
        double r221387 = sin(r221379);
        double r221388 = r221386 * r221387;
        double r221389 = r221383 - r221388;
        double r221390 = r221386 * r221380;
        double r221391 = r221389 / r221390;
        double r221392 = r221371 + r221391;
        return r221392;
}

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-quot 13.1

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

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 flip-+ 0.2

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

8. Applied associate-*l/ 0.2

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

9. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019354 +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))))