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

Average Error: 13.1 → 0.2

Time: 13.0s

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

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

↓

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

C

double f(double x, double y, double z, double a) {
        double r164806 = x;
        double r164807 = y;
        double r164808 = z;
        double r164809 = r164807 + r164808;
        double r164810 = tan(r164809);
        double r164811 = a;
        double r164812 = tan(r164811);
        double r164813 = r164810 - r164812;
        double r164814 = r164806 + r164813;
        return r164814;
}

↓

double f(double x, double y, double z, double a) {
        double r164815 = x;
        double r164816 = y;
        double r164817 = sin(r164816);
        double r164818 = z;
        double r164819 = cos(r164818);
        double r164820 = cos(r164816);
        double r164821 = sin(r164818);
        double r164822 = r164820 * r164821;
        double r164823 = fma(r164817, r164819, r164822);
        double r164824 = r164820 * r164819;
        double r164825 = r164823 / r164824;
        double r164826 = 1.0;
        double r164827 = tan(r164816);
        double r164828 = tan(r164818);
        double r164829 = r164827 * r164828;
        double r164830 = r164826 - r164829;
        double r164831 = r164825 / r164830;
        double r164832 = a;
        double r164833 = tan(r164832);
        double r164834 = r164831 - r164833;
        double r164835 = r164815 + r164834;
        return r164835;
}

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 tan-quot 0.2

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

6. Applied tan-quot 0.2

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

7. Applied frac-add 0.2

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

8. Simplified 0.2

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

9. Final simplification 0.2

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

Reproduce

herbie shell --seed 2020027 +o rules:numerics
(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))))