Average Error: 13.3 → 0.3
Time: 32.6s
Precision: 64
\[\left(x = 0.0 \lor 0.5884141999999999983472775966220069676638 \le x \le 505.5908999999999764440872240811586380005\right) \land \left(-1.7966580000000000931214523812968299911 \cdot 10^{308} \le y \le -9.425585000000013069597555966781986720373 \cdot 10^{-310} \lor 1.284937999999999548796432976649400331091 \cdot 10^{-309} \le y \le 1.751223999999999928063201074847742204824 \cdot 10^{308}\right) \land \left(-1.776707000000000001259808757982040817204 \cdot 10^{308} \le z \le -8.599796000000016667475923823712126825539 \cdot 10^{-310} \lor 3.293144999999983071955117582595641261776 \cdot 10^{-311} \le z \le 1.725154000000000087891269878141591702413 \cdot 10^{308}\right) \land \left(-1.7966580000000000931214523812968299911 \cdot 10^{308} \le a \le -9.425585000000013069597555966781986720373 \cdot 10^{-310} \lor 1.284937999999999548796432976649400331091 \cdot 10^{-309} \le a \le 1.751223999999999928063201074847742204824 \cdot 10^{308}\right)\]
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[x + \frac{\mathsf{fma}\left(\tan z \cdot \tan y + -1, \sin a, \left(\tan y + \tan z\right) \cdot \cos a\right)}{\left(1 - \log \left(e^{\tan z \cdot \tan y}\right)\right) \cdot \cos a}\]
x + \left(\tan \left(y + z\right) - \tan a\right)
↓
x + \frac{\mathsf{fma}\left(\tan z \cdot \tan y + -1, \sin a, \left(\tan y + \tan z\right) \cdot \cos a\right)}{\left(1 - \log \left(e^{\tan z \cdot \tan y}\right)\right) \cdot \cos a}
double f(double x, double y, double z, double a) {
        double r140441 = x;
        double r140442 = y;
        double r140443 = z;
        double r140444 = r140442 + r140443;
        double r140445 = tan(r140444);
        double r140446 = a;
        double r140447 = tan(r140446);
        double r140448 = r140445 - r140447;
        double r140449 = r140441 + r140448;
        return r140449;
}
↓
double f(double x, double y, double z, double a) {
        double r140450 = x;
        double r140451 = z;
        double r140452 = tan(r140451);
        double r140453 = y;
        double r140454 = tan(r140453);
        double r140455 = r140452 * r140454;
        double r140456 = -1.0;
        double r140457 = r140455 + r140456;
        double r140458 = a;
        double r140459 = sin(r140458);
        double r140460 = r140454 + r140452;
        double r140461 = cos(r140458);
        double r140462 = r140460 * r140461;
        double r140463 = fma(r140457, r140459, r140462);
        double r140464 = 1.0;
        double r140465 = exp(r140455);
        double r140466 = log(r140465);
        double r140467 = r140464 - r140466;
        double r140468 = r140467 * r140461;
        double r140469 = r140463 / r140468;
        double r140470 = r140450 + r140469;
        return r140470;
}

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-quot13.3

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

  4. Applied tan-sum0.2

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

  5. Applied frac-sub0.2

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

  6. Simplified0.2

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

  8. Applied add-log-exp0.3

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

  9. Simplified0.3

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

  10. Final simplification0.3

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

Reproduce

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