Average Error: 13.3 → 0.2
Time: 19.1s
Precision: 64
\[\left(x = 0.0 \lor 0.5884141999999999983472775966220069676638 \le x \le 505.5908999999999764440872240811586380005\right) \land \left(-1.7966580000000000931214523812968299911 \cdot 10^{308} \le y \le -9.425585000000013069597555966781986720373 \cdot 10^{-310} \lor 1.284937999999999548796432976649400331091 \cdot 10^{-309} \le y \le 1.751224000000000127647232028319723370461 \cdot 10^{308}\right) \land \left(-1.776707000000000200843839711454021982841 \cdot 10^{308} \le z \le -8.599796000000016667475923823712126825539 \cdot 10^{-310} \lor 3.293144999999983071955117582595641261776 \cdot 10^{-311} \le z \le 1.725154000000000087891269878141591702413 \cdot 10^{308}\right) \land \left(-1.7966580000000000931214523812968299911 \cdot 10^{308} \le a \le -9.425585000000013069597555966781986720373 \cdot 10^{-310} \lor 1.284937999999999548796432976649400331091 \cdot 10^{-309} \le a \le 1.751224000000000127647232028319723370461 \cdot 10^{308}\right)\]
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[\frac{\tan y + \tan z}{\mathsf{fma}\left(\tan y, -\tan z, 1\right)} + \left(x - \tan a\right)\]
x + \left(\tan \left(y + z\right) - \tan a\right)
↓
\frac{\tan y + \tan z}{\mathsf{fma}\left(\tan y, -\tan z, 1\right)} + \left(x - \tan a\right)
double f(double x, double y, double z, double a) {
        double r150192 = x;
        double r150193 = y;
        double r150194 = z;
        double r150195 = r150193 + r150194;
        double r150196 = tan(r150195);
        double r150197 = a;
        double r150198 = tan(r150197);
        double r150199 = r150196 - r150198;
        double r150200 = r150192 + r150199;
        return r150200;
}
↓
double f(double x, double y, double z, double a) {
        double r150201 = y;
        double r150202 = tan(r150201);
        double r150203 = z;
        double r150204 = tan(r150203);
        double r150205 = r150202 + r150204;
        double r150206 = -r150204;
        double r150207 = 1.0;
        double r150208 = fma(r150202, r150206, r150207);
        double r150209 = r150205 / r150208;
        double r150210 = x;
        double r150211 = a;
        double r150212 = tan(r150211);
        double r150213 = r150210 - r150212;
        double r150214 = r150209 + r150213;
        return r150214;
}

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-sum0.2

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

  5. Applied associate-+r-0.2

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

  6. Simplified0.2

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

  8. Applied associate--l+0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2019351 +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))))