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

Average Error: 12.9 → 0.2

Time: 38.1s

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

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

↓

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

C

double f(double x, double y, double z, double a) {
        double r6111510 = x;
        double r6111511 = y;
        double r6111512 = z;
        double r6111513 = r6111511 + r6111512;
        double r6111514 = tan(r6111513);
        double r6111515 = a;
        double r6111516 = tan(r6111515);
        double r6111517 = r6111514 - r6111516;
        double r6111518 = r6111510 + r6111517;
        return r6111518;
}

↓

double f(double x, double y, double z, double a) {
        double r6111519 = 1.0;
        double r6111520 = z;
        double r6111521 = tan(r6111520);
        double r6111522 = y;
        double r6111523 = tan(r6111522);
        double r6111524 = r6111521 * r6111523;
        double r6111525 = r6111519 - r6111524;
        double r6111526 = r6111519 / r6111525;
        double r6111527 = r6111523 + r6111521;
        double r6111528 = r6111526 * r6111527;
        double r6111529 = a;
        double r6111530 = tan(r6111529);
        double r6111531 = r6111528 - r6111530;
        double r6111532 = x;
        double r6111533 = r6111531 + r6111532;
        return r6111533;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus a

Derivation

1. Initial program 12.9

   \[x + \left(\tan \left(y + z\right) - \tan a\right)\]
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 div-inv 0.2

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

6. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x y z a)
  :name "(+ x (- (tan (+ y z)) (tan a)))"
  :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))))