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

Average Error: 12.9 → 0.2

Time: 38.9s

Precision: 64

\[\left(x = 0.0 \lor 0.5884141999999999983472775966220069676638 \le x \le 505.5908999999999764440872240811586380005\right) \land \left(-1.7966580000000000931214523812968299911 \cdot 10^{308} \le y \le -9.425585000000013069597555966781986720373 \cdot 10^{-310} \lor 1.284937999999999548796432976649400331091 \cdot 10^{-309} \le y \le 1.751223999999999928063201074847742204824 \cdot 10^{308}\right) \land \left(-1.776707000000000001259808757982040817204 \cdot 10^{308} \le z \le -8.599796000000016667475923823712126825539 \cdot 10^{-310} \lor 3.293144999999983071955117582595641261776 \cdot 10^{-311} \le z \le 1.725154000000000087891269878141591702413 \cdot 10^{308}\right) \land \left(-1.7966580000000000931214523812968299911 \cdot 10^{308} \le a \le -9.425585000000013069597555966781986720373 \cdot 10^{-310} \lor 1.284937999999999548796432976649400331091 \cdot 10^{-309} \le a \le 1.751223999999999928063201074847742204824 \cdot 10^{308}\right)\]

Math

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

↓

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

C

double f(double x, double y, double z, double a) {
        double r8254990 = x;
        double r8254991 = y;
        double r8254992 = z;
        double r8254993 = r8254991 + r8254992;
        double r8254994 = tan(r8254993);
        double r8254995 = a;
        double r8254996 = tan(r8254995);
        double r8254997 = r8254994 - r8254996;
        double r8254998 = r8254990 + r8254997;
        return r8254998;
}

↓

double f(double x, double y, double z, double a) {
        double r8254999 = z;
        double r8255000 = sin(r8254999);
        double r8255001 = 1.0;
        double r8255002 = y;
        double r8255003 = sin(r8255002);
        double r8255004 = cos(r8255002);
        double r8255005 = r8255003 / r8255004;
        double r8255006 = r8255000 * r8255005;
        double r8255007 = cos(r8254999);
        double r8255008 = r8255006 / r8255007;
        double r8255009 = r8255001 - r8255008;
        double r8255010 = r8255009 * r8255007;
        double r8255011 = r8255000 / r8255010;
        double r8255012 = r8255000 * r8255003;
        double r8255013 = r8255004 * r8255007;
        double r8255014 = r8255012 / r8255013;
        double r8255015 = r8255001 - r8255014;
        double r8255016 = r8255005 / r8255015;
        double r8255017 = a;
        double r8255018 = sin(r8255017);
        double r8255019 = cos(r8255017);
        double r8255020 = r8255018 / r8255019;
        double r8255021 = r8255016 - r8255020;
        double r8255022 = r8255011 + r8255021;
        double r8255023 = x;
        double r8255024 = r8255022 + r8255023;
        return r8255024;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus a

Derivation

1. Initial program 12.9

   \[x + \left(\tan \left(y + z\right) - \tan a\right)\]
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. Taylor expanded around inf 0.2

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

5. Simplified 0.2

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

6. Taylor expanded around inf 0.2

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

7. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019172 
(FPCore (x y z a)
  :name "(+ x (- (tan (+ y z)) (tan a)))"
  :pre (and (or (== x 0.0) (<= 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))))