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

Average Error: 13.3 → 0.3

Time: 26.3s

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

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

↓

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

C

double f(double x, double y, double z, double a) {
        double r106992 = x;
        double r106993 = y;
        double r106994 = z;
        double r106995 = r106993 + r106994;
        double r106996 = tan(r106995);
        double r106997 = a;
        double r106998 = tan(r106997);
        double r106999 = r106996 - r106998;
        double r107000 = r106992 + r106999;
        return r107000;
}

↓

double f(double x, double y, double z, double a) {
        double r107001 = x;
        double r107002 = y;
        double r107003 = tan(r107002);
        double r107004 = z;
        double r107005 = tan(r107004);
        double r107006 = r107003 + r107005;
        double r107007 = 1.0;
        double r107008 = cbrt(r107005);
        double r107009 = r107008 * r107008;
        double r107010 = r107003 * r107009;
        double r107011 = r107010 * r107008;
        double r107012 = r107007 - r107011;
        double r107013 = r107006 / r107012;
        double r107014 = a;
        double r107015 = tan(r107014);
        double r107016 = r107013 - r107015;
        double r107017 = r107001 + r107016;
        return r107017;
}

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-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-cube-cbrt 0.3

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

6. Applied associate-*r* 0.3

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

7. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019347 +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))))