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

Average Error: 13.3 → 0.2

Time: 1.0m

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

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

↓

\[x + \frac{\frac{\tan z + \tan y}{1 - \tan z \cdot \tan y} \cdot \frac{\tan z + \tan y}{1 - \tan z \cdot \tan y} - \tan a \cdot \tan a}{\tan a + \frac{\tan z + \tan y}{1 - \tan z \cdot \tan y}}\]

C

double f(double x, double y, double z, double a) {
        double r6593195 = x;
        double r6593196 = y;
        double r6593197 = z;
        double r6593198 = r6593196 + r6593197;
        double r6593199 = tan(r6593198);
        double r6593200 = a;
        double r6593201 = tan(r6593200);
        double r6593202 = r6593199 - r6593201;
        double r6593203 = r6593195 + r6593202;
        return r6593203;
}

↓

double f(double x, double y, double z, double a) {
        double r6593204 = x;
        double r6593205 = z;
        double r6593206 = tan(r6593205);
        double r6593207 = y;
        double r6593208 = tan(r6593207);
        double r6593209 = r6593206 + r6593208;
        double r6593210 = 1.0;
        double r6593211 = r6593206 * r6593208;
        double r6593212 = r6593210 - r6593211;
        double r6593213 = r6593209 / r6593212;
        double r6593214 = r6593213 * r6593213;
        double r6593215 = a;
        double r6593216 = tan(r6593215);
        double r6593217 = r6593216 * r6593216;
        double r6593218 = r6593214 - r6593217;
        double r6593219 = r6593216 + r6593213;
        double r6593220 = r6593218 / r6593219;
        double r6593221 = r6593204 + r6593220;
        return r6593221;
}

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 flip-- 0.2

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

6. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019200 
(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))))