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

Average Error: 13.1 → 0.3

Time: 11.9s

Precision: 64

\[\left(x = 0.0 \lor 0.588414199999999998 \le x \le 505.590899999999976\right) \land \left(-1.79665800000000009 \cdot 10^{308} \le y \le -9.425585000000013 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \le y \le 1.7512240000000001 \cdot 10^{308}\right) \land \left(-1.7767070000000002 \cdot 10^{308} \le z \le -8.59979600000002 \cdot 10^{-310} \lor 3.29314499999998 \cdot 10^{-311} \le z \le 1.72515400000000009 \cdot 10^{308}\right) \land \left(-1.79665800000000009 \cdot 10^{308} \le a \le -9.425585000000013 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \le a \le 1.7512240000000001 \cdot 10^{308}\right)\]

Math

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

↓

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

C

double f(double x, double y, double z, double a) {
        double r165473 = x;
        double r165474 = y;
        double r165475 = z;
        double r165476 = r165474 + r165475;
        double r165477 = tan(r165476);
        double r165478 = a;
        double r165479 = tan(r165478);
        double r165480 = r165477 - r165479;
        double r165481 = r165473 + r165480;
        return r165481;
}

↓

double f(double x, double y, double z, double a) {
        double r165482 = x;
        double r165483 = y;
        double r165484 = tan(r165483);
        double r165485 = z;
        double r165486 = tan(r165485);
        double r165487 = r165484 + r165486;
        double r165488 = a;
        double r165489 = cos(r165488);
        double r165490 = r165487 * r165489;
        double r165491 = 1.0;
        double r165492 = r165484 * r165486;
        double r165493 = r165491 - r165492;
        double r165494 = sin(r165488);
        double r165495 = r165493 * r165494;
        double r165496 = r165490 - r165495;
        double r165497 = exp(r165492);
        double r165498 = log(r165497);
        double r165499 = r165491 - r165498;
        double r165500 = r165499 * r165489;
        double r165501 = r165496 / r165500;
        double r165502 = r165482 + r165501;
        return r165502;
}

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-quot 13.1

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

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

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

8. Final simplification 0.3

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

Reproduce

herbie shell --seed 2020049 
(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))))