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

Average Error: 13.0 → 0.3

Time: 27.9s

Precision: 64

\[\left(x = 0 \lor 0.5884142 \le x \le 505.5909\right) \land \left(-1.796658 \cdot 10^{+308} \le y \le -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \le y \le 1.751224 \cdot 10^{+308}\right) \land \left(-1.776707 \cdot 10^{+308} \le z \le -8.599796 \cdot 10^{-310} \lor 3.293145 \cdot 10^{-311} \le z \le 1.725154 \cdot 10^{+308}\right) \land \left(-1.796658 \cdot 10^{+308} \le a \le -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \le a \le 1.751224 \cdot 10^{+308}\right)\]

Math

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

↓

\[\log \left(e^{\frac{\cos a \cdot \left(\tan y + \tan z\right) - \left(1 - \tan z \cdot \tan y\right) \cdot \sin a}{\left(1 - \tan z \cdot \tan y\right) \cdot \cos a}}\right) + x\]

C

double f(double x, double y, double z, double a) {
        double r3636645 = x;
        double r3636646 = y;
        double r3636647 = z;
        double r3636648 = r3636646 + r3636647;
        double r3636649 = tan(r3636648);
        double r3636650 = a;
        double r3636651 = tan(r3636650);
        double r3636652 = r3636649 - r3636651;
        double r3636653 = r3636645 + r3636652;
        return r3636653;
}

↓

double f(double x, double y, double z, double a) {
        double r3636654 = a;
        double r3636655 = cos(r3636654);
        double r3636656 = y;
        double r3636657 = tan(r3636656);
        double r3636658 = z;
        double r3636659 = tan(r3636658);
        double r3636660 = r3636657 + r3636659;
        double r3636661 = r3636655 * r3636660;
        double r3636662 = 1.0;
        double r3636663 = r3636659 * r3636657;
        double r3636664 = r3636662 - r3636663;
        double r3636665 = sin(r3636654);
        double r3636666 = r3636664 * r3636665;
        double r3636667 = r3636661 - r3636666;
        double r3636668 = r3636664 * r3636655;
        double r3636669 = r3636667 / r3636668;
        double r3636670 = exp(r3636669);
        double r3636671 = log(r3636670);
        double r3636672 = x;
        double r3636673 = r3636671 + r3636672;
        return r3636673;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus a

Derivation

1. Initial program 13.0

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

3. Applied tan-quot 13.0

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

4. Applied tan-sum 0.2

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

5. Applied frac-sub 0.2

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

7. Applied add-log-exp 0.3

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

8. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019132 
(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))))