Average Error: 13.4 → 0.2
Time: 48.2s
Precision: 64
\[\left(x = 0 \lor 0.5884142 \le x \le 505.5909\right) \land \left(-1.796658 \cdot 10^{+308} \le y \le -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \le y \le 1.751224 \cdot 10^{+308}\right) \land \left(-1.776707 \cdot 10^{+308} \le z \le -8.599796 \cdot 10^{-310} \lor 3.293145 \cdot 10^{-311} \le z \le 1.725154 \cdot 10^{+308}\right) \land \left(-1.796658 \cdot 10^{+308} \le a \le -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \le a \le 1.751224 \cdot 10^{+308}\right)\]
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[\mathsf{fma}\left(\frac{\tan y + \tan z}{1 - {\left(\frac{\tan z \cdot \sin y}{\cos y}\right)}^{3}}, \left(\frac{\tan z \cdot \sin y}{\cos y} \cdot \frac{\tan z \cdot \sin y}{\cos y} + \frac{\tan z \cdot \sin y}{\cos y}\right) + 1, -\tan a\right) + x\]
x + \left(\tan \left(y + z\right) - \tan a\right)
↓
\mathsf{fma}\left(\frac{\tan y + \tan z}{1 - {\left(\frac{\tan z \cdot \sin y}{\cos y}\right)}^{3}}, \left(\frac{\tan z \cdot \sin y}{\cos y} \cdot \frac{\tan z \cdot \sin y}{\cos y} + \frac{\tan z \cdot \sin y}{\cos y}\right) + 1, -\tan a\right) + x
double f(double x, double y, double z, double a) {
        double r3815277 = x;
        double r3815278 = y;
        double r3815279 = z;
        double r3815280 = r3815278 + r3815279;
        double r3815281 = tan(r3815280);
        double r3815282 = a;
        double r3815283 = tan(r3815282);
        double r3815284 = r3815281 - r3815283;
        double r3815285 = r3815277 + r3815284;
        return r3815285;
}
↓
double f(double x, double y, double z, double a) {
        double r3815286 = y;
        double r3815287 = tan(r3815286);
        double r3815288 = z;
        double r3815289 = tan(r3815288);
        double r3815290 = r3815287 + r3815289;
        double r3815291 = 1.0;
        double r3815292 = sin(r3815286);
        double r3815293 = r3815289 * r3815292;
        double r3815294 = cos(r3815286);
        double r3815295 = r3815293 / r3815294;
        double r3815296 = 3.0;
        double r3815297 = pow(r3815295, r3815296);
        double r3815298 = r3815291 - r3815297;
        double r3815299 = r3815290 / r3815298;
        double r3815300 = r3815295 * r3815295;
        double r3815301 = r3815300 + r3815295;
        double r3815302 = r3815301 + r3815291;
        double r3815303 = a;
        double r3815304 = tan(r3815303);
        double r3815305 = -r3815304;
        double r3815306 = fma(r3815299, r3815302, r3815305);
        double r3815307 = x;
        double r3815308 = r3815306 + r3815307;
        return r3815308;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus a

Derivation

  1. Initial program 13.4

    \[x + \left(\tan \left(y + z\right) - \tan a\right)\]
  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 tan-quot0.2

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

  6. Applied associate-*l/0.2

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

  8. Applied flip3--0.2

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

  9. Applied associate-/r/0.2

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

  10. Applied fma-neg0.2

    \[\leadsto x + \color{blue}{\mathsf{fma}\left(\frac{\tan y + \tan z}{{1}^{3} - {\left(\frac{\sin y \cdot \tan z}{\cos y}\right)}^{3}}, 1 \cdot 1 + \left(\frac{\sin y \cdot \tan z}{\cos y} \cdot \frac{\sin y \cdot \tan z}{\cos y} + 1 \cdot \frac{\sin y \cdot \tan z}{\cos y}\right), -\tan a\right)}\]

  11. Final simplification0.2

    \[\leadsto \mathsf{fma}\left(\frac{\tan y + \tan z}{1 - {\left(\frac{\tan z \cdot \sin y}{\cos y}\right)}^{3}}, \left(\frac{\tan z \cdot \sin y}{\cos y} \cdot \frac{\tan z \cdot \sin y}{\cos y} + \frac{\tan z \cdot \sin y}{\cos y}\right) + 1, -\tan a\right) + x\]

Reproduce

herbie shell --seed 2019149 +o rules:numerics
(FPCore (x y z a)
  :name "(+ x (- (tan (+ y z)) (tan a)))"
  :pre (and (or (== x 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))))