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

Average Error: 13.3 → 0.2

Time: 29.0s

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

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

↓

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

C

double f(double x, double y, double z, double a) {
        double r106290 = x;
        double r106291 = y;
        double r106292 = z;
        double r106293 = r106291 + r106292;
        double r106294 = tan(r106293);
        double r106295 = a;
        double r106296 = tan(r106295);
        double r106297 = r106294 - r106296;
        double r106298 = r106290 + r106297;
        return r106298;
}

↓

double f(double x, double y, double z, double a) {
        double r106299 = x;
        double r106300 = y;
        double r106301 = tan(r106300);
        double r106302 = z;
        double r106303 = tan(r106302);
        double r106304 = r106301 + r106303;
        double r106305 = 1.0;
        double r106306 = r106301 * r106303;
        double r106307 = r106305 - r106306;
        double r106308 = a;
        double r106309 = sin(r106308);
        double r106310 = r106307 * r106309;
        double r106311 = cos(r106308);
        double r106312 = r106310 / r106311;
        double r106313 = r106304 - r106312;
        double r106314 = r106306 * r106303;
        double r106315 = r106301 * r106314;
        double r106316 = r106305 - r106315;
        double r106317 = r106313 / r106316;
        double r106318 = r106305 + r106306;
        double r106319 = r106317 * r106318;
        double r106320 = r106299 + r106319;
        return r106320;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus a

Derivation

1. Initial program 13.3

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

3. Applied tan-quot 13.3

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

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

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

8. Applied associate-*l/ 0.2

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

9. Applied associate-/r/ 0.2

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

10. Simplified 0.2

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

12. Applied associate-*l* 0.2

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

13. Simplified 0.2

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

14. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019326 
(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 1.751224e+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))))