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

Average Error: 13.5 → 0.2

Time: 12.2s

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

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

↓

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

C

double f(double x, double y, double z, double a) {
        double r265653 = x;
        double r265654 = y;
        double r265655 = z;
        double r265656 = r265654 + r265655;
        double r265657 = tan(r265656);
        double r265658 = a;
        double r265659 = tan(r265658);
        double r265660 = r265657 - r265659;
        double r265661 = r265653 + r265660;
        return r265661;
}

↓

double f(double x, double y, double z, double a) {
        double r265662 = x;
        double r265663 = 1.0;
        double r265664 = y;
        double r265665 = tan(r265664);
        double r265666 = z;
        double r265667 = tan(r265666);
        double r265668 = r265665 * r265667;
        double r265669 = r265663 - r265668;
        double r265670 = r265665 + r265667;
        double r265671 = r265669 / r265670;
        double r265672 = r265663 / r265671;
        double r265673 = a;
        double r265674 = tan(r265673);
        double r265675 = r265672 - r265674;
        double r265676 = r265662 + r265675;
        return r265676;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus a

Derivation

1. Initial program 13.5

   \[x + \left(\tan \left(y + z\right) - \tan a\right)\]
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 clear-num 0.2

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

6. Final simplification 0.2

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

Reproduce

herbie shell --seed 2020046 
(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))))