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

Average Error: 13.5 → 0.2

Time: 34.1s

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

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

↓

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

C

double f(double x, double y, double z, double a) {
        double r123917 = x;
        double r123918 = y;
        double r123919 = z;
        double r123920 = r123918 + r123919;
        double r123921 = tan(r123920);
        double r123922 = a;
        double r123923 = tan(r123922);
        double r123924 = r123921 - r123923;
        double r123925 = r123917 + r123924;
        return r123925;
}

↓

double f(double x, double y, double z, double a) {
        double r123926 = x;
        double r123927 = y;
        double r123928 = tan(r123927);
        double r123929 = z;
        double r123930 = tan(r123929);
        double r123931 = r123928 + r123930;
        double r123932 = 3.0;
        double r123933 = pow(r123931, r123932);
        double r123934 = cbrt(r123933);
        double r123935 = 1.0;
        double r123936 = r123928 * r123930;
        double r123937 = r123935 - r123936;
        double r123938 = r123934 / r123937;
        double r123939 = a;
        double r123940 = tan(r123939);
        double r123941 = r123938 - r123940;
        double r123942 = r123926 + r123941;
        return r123942;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus a

Derivation

1. Initial program 13.5

   \[x + \left(\tan \left(y + z\right) - \tan a\right)\]
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 add-cbrt-cube 0.2

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

6. Simplified 0.2

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

7. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019235 +o rules:numerics
(FPCore (x y z a)
  :name "(+ x (- (tan (+ y z)) (tan a)))"
  :precision binary64
  :pre (and (or (== x 0.0) (<= 0.588414199999999998 x 505.590899999999976)) (or (<= -1.79665800000000009e308 y -9.425585000000013e-310) (<= 1.284938e-309 y 1.75122399999999993e308)) (or (<= -1.776707e308 z -8.59979600000002e-310) (<= 3.29314499999998e-311 z 1.72515400000000009e308)) (or (<= -1.79665800000000009e308 a -9.425585000000013e-310) (<= 1.284938e-309 a 1.75122399999999993e308)))
  (+ x (- (tan (+ y z)) (tan a))))