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

Average Error: 13.1 → 0.2

Time: 1.2m

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

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

↓

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

C

double f(double x, double y, double z, double a) {
        double r12535673 = x;
        double r12535674 = y;
        double r12535675 = z;
        double r12535676 = r12535674 + r12535675;
        double r12535677 = tan(r12535676);
        double r12535678 = a;
        double r12535679 = tan(r12535678);
        double r12535680 = r12535677 - r12535679;
        double r12535681 = r12535673 + r12535680;
        return r12535681;
}

↓

double f(double x, double y, double z, double a) {
        double r12535682 = x;
        double r12535683 = 1.0;
        double r12535684 = z;
        double r12535685 = tan(r12535684);
        double r12535686 = y;
        double r12535687 = tan(r12535686);
        double r12535688 = r12535685 * r12535687;
        double r12535689 = r12535683 - r12535688;
        double r12535690 = r12535683 / r12535689;
        double r12535691 = r12535687 + r12535685;
        double r12535692 = r12535690 * r12535691;
        double r12535693 = r12535682 + r12535692;
        double r12535694 = a;
        double r12535695 = tan(r12535694);
        double r12535696 = -r12535695;
        double r12535697 = r12535693 + r12535696;
        return r12535697;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus a

Derivation

1. Initial program 13.1

   \[x + \left(\tan \left(y + z\right) - \tan a\right)\]
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 div-inv 0.2

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

6. Applied fma-neg 0.2

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

8. Applied fma-udef 0.2

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

9. Applied associate-+r+ 0.2

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

10. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019120 +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))))