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

Average Error: 12.9 → 0.2

Time: 2.8m

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

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

↓

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

C

double f(double x, double y, double z, double a) {
        double r9993509 = x;
        double r9993510 = y;
        double r9993511 = z;
        double r9993512 = r9993510 + r9993511;
        double r9993513 = tan(r9993512);
        double r9993514 = a;
        double r9993515 = tan(r9993514);
        double r9993516 = r9993513 - r9993515;
        double r9993517 = r9993509 + r9993516;
        return r9993517;
}

↓

double f(double x, double y, double z, double a) {
        double r9993518 = y;
        double r9993519 = tan(r9993518);
        double r9993520 = z;
        double r9993521 = tan(r9993520);
        double r9993522 = r9993519 - r9993521;
        double r9993523 = r9993522 * r9993522;
        double r9993524 = r9993522 * r9993523;
        double r9993525 = r9993519 + r9993521;
        double r9993526 = r9993525 * r9993525;
        double r9993527 = r9993526 * r9993525;
        double r9993528 = r9993524 * r9993527;
        double r9993529 = cbrt(r9993528);
        double r9993530 = r9993529 / r9993522;
        double r9993531 = 1.0;
        double r9993532 = r9993521 * r9993519;
        double r9993533 = r9993531 - r9993532;
        double r9993534 = r9993530 / r9993533;
        double r9993535 = a;
        double r9993536 = tan(r9993535);
        double r9993537 = r9993534 - r9993536;
        double r9993538 = x;
        double r9993539 = r9993537 + r9993538;
        return r9993539;
}

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 flip-+ 0.2

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

6. Simplified 0.2

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

8. Applied add-cbrt-cube 0.2

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

9. Applied add-cbrt-cube 0.3

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

10. Applied cbrt-unprod 0.2

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

11. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019179 
(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))))