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

Average Error: 13.7 → 0.2

Time: 11.1s

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

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

↓

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

C

double f(double x, double y, double z, double a) {
        double r160767 = x;
        double r160768 = y;
        double r160769 = z;
        double r160770 = r160768 + r160769;
        double r160771 = tan(r160770);
        double r160772 = a;
        double r160773 = tan(r160772);
        double r160774 = r160771 - r160773;
        double r160775 = r160767 + r160774;
        return r160775;
}

↓

double f(double x, double y, double z, double a) {
        double r160776 = x;
        double r160777 = y;
        double r160778 = sin(r160777);
        double r160779 = z;
        double r160780 = cos(r160779);
        double r160781 = r160778 * r160780;
        double r160782 = cos(r160777);
        double r160783 = sin(r160779);
        double r160784 = r160782 * r160783;
        double r160785 = r160781 + r160784;
        double r160786 = r160782 * r160780;
        double r160787 = r160785 / r160786;
        double r160788 = 1.0;
        double r160789 = tan(r160777);
        double r160790 = tan(r160779);
        double r160791 = r160789 * r160790;
        double r160792 = r160788 - r160791;
        double r160793 = r160787 / r160792;
        double r160794 = a;
        double r160795 = tan(r160794);
        double r160796 = r160793 - r160795;
        double r160797 = r160776 + r160796;
        return r160797;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus a

Derivation

1. Initial program 13.7

   \[x + \left(\tan \left(y + z\right) - \tan a\right)\]
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. Final simplification 0.2

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

Reproduce

herbie shell --seed 2020018 
(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))))