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

Average Error: 12.9 → 0.2

Time: 36.2s

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 r5766036 = x;
        double r5766037 = y;
        double r5766038 = z;
        double r5766039 = r5766037 + r5766038;
        double r5766040 = tan(r5766039);
        double r5766041 = a;
        double r5766042 = tan(r5766041);
        double r5766043 = r5766040 - r5766042;
        double r5766044 = r5766036 + r5766043;
        return r5766044;
}

↓

double f(double x, double y, double z, double a) {
        double r5766045 = 1.0;
        double r5766046 = z;
        double r5766047 = tan(r5766046);
        double r5766048 = y;
        double r5766049 = tan(r5766048);
        double r5766050 = r5766047 * r5766049;
        double r5766051 = r5766045 - r5766050;
        double r5766052 = r5766045 / r5766051;
        double r5766053 = r5766049 + r5766047;
        double r5766054 = r5766052 * r5766053;
        double r5766055 = a;
        double r5766056 = tan(r5766055);
        double r5766057 = r5766054 - r5766056;
        double r5766058 = x;
        double r5766059 = r5766057 + r5766058;
        return r5766059;
}

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))))