Average Error: 13.2 → 0.2
Time: 12.3s
Precision: 64
\[\left(x = 0.0 \lor 0.5884141999999999983472775966220069676638 \le x \le 505.5908999999999764440872240811586380005\right) \land \left(-1.7966580000000000931214523812968299911 \cdot 10^{308} \le y \le -9.425585000000013069597555966781986720373 \cdot 10^{-310} \lor 1.284937999999999548796432976649400331091 \cdot 10^{-309} \le y \le 1.751224000000000127647232028319723370461 \cdot 10^{308}\right) \land \left(-1.776707000000000200843839711454021982841 \cdot 10^{308} \le z \le -8.599796000000016667475923823712126825539 \cdot 10^{-310} \lor 3.293144999999983071955117582595641261776 \cdot 10^{-311} \le z \le 1.725154000000000087891269878141591702413 \cdot 10^{308}\right) \land \left(-1.7966580000000000931214523812968299911 \cdot 10^{308} \le a \le -9.425585000000013069597555966781986720373 \cdot 10^{-310} \lor 1.284937999999999548796432976649400331091 \cdot 10^{-309} \le a \le 1.751224000000000127647232028319723370461 \cdot 10^{308}\right)\]
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[x + \frac{\mathsf{fma}\left(\tan y + \tan z, \cos a, -\left(1 - \tan y \cdot \tan z\right) \cdot \sin a\right)}{\left(1 - \tan y \cdot \tan z\right) \cdot \cos a}\]
x + \left(\tan \left(y + z\right) - \tan a\right)
↓
x + \frac{\mathsf{fma}\left(\tan y + \tan z, \cos a, -\left(1 - \tan y \cdot \tan z\right) \cdot \sin a\right)}{\left(1 - \tan y \cdot \tan z\right) \cdot \cos a}
double f(double x, double y, double z, double a) {
        double r184238 = x;
        double r184239 = y;
        double r184240 = z;
        double r184241 = r184239 + r184240;
        double r184242 = tan(r184241);
        double r184243 = a;
        double r184244 = tan(r184243);
        double r184245 = r184242 - r184244;
        double r184246 = r184238 + r184245;
        return r184246;
}
↓
double f(double x, double y, double z, double a) {
        double r184247 = x;
        double r184248 = y;
        double r184249 = tan(r184248);
        double r184250 = z;
        double r184251 = tan(r184250);
        double r184252 = r184249 + r184251;
        double r184253 = a;
        double r184254 = cos(r184253);
        double r184255 = 1.0;
        double r184256 = r184249 * r184251;
        double r184257 = r184255 - r184256;
        double r184258 = sin(r184253);
        double r184259 = r184257 * r184258;
        double r184260 = -r184259;
        double r184261 = fma(r184252, r184254, r184260);
        double r184262 = r184257 * r184254;
        double r184263 = r184261 / r184262;
        double r184264 = r184247 + r184263;
        return r184264;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus a

Derivation

  1. Initial program 13.2

    \[x + \left(\tan \left(y + z\right) - \tan a\right)\]
  2. Using strategy rm

  3. Applied tan-quot13.2

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

  4. Applied tan-sum0.2

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

  5. Applied frac-sub0.2

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

  7. Applied fma-neg0.2

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

  8. Final simplification0.2

    \[\leadsto x + \frac{\mathsf{fma}\left(\tan y + \tan z, \cos a, -\left(1 - \tan y \cdot \tan z\right) \cdot \sin a\right)}{\left(1 - \tan y \cdot \tan z\right) \cdot \cos a}\]

Reproduce

herbie shell --seed 2020001 +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))))