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

Average Error: 13.7 → 0.3

Time: 12.5s

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

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

↓

\[\mathsf{fma}\left(\frac{\tan y + \tan z}{1 - \left(\left(\tan y \cdot \left(\sqrt[3]{\tan z} \cdot \sqrt[3]{\tan z}\right)\right) \cdot \sqrt[3]{\tan z}\right) \cdot \left(\tan y \cdot \tan z\right)}, \mathsf{fma}\left(\tan y, \tan z, 1\right), \left(-\tan a\right) + x\right) + \mathsf{fma}\left(-\tan a, 1, \tan a \cdot 1\right)\]

C

double f(double x, double y, double z, double a) {
        double r152968 = x;
        double r152969 = y;
        double r152970 = z;
        double r152971 = r152969 + r152970;
        double r152972 = tan(r152971);
        double r152973 = a;
        double r152974 = tan(r152973);
        double r152975 = r152972 - r152974;
        double r152976 = r152968 + r152975;
        return r152976;
}

↓

double f(double x, double y, double z, double a) {
        double r152977 = y;
        double r152978 = tan(r152977);
        double r152979 = z;
        double r152980 = tan(r152979);
        double r152981 = r152978 + r152980;
        double r152982 = 1.0;
        double r152983 = cbrt(r152980);
        double r152984 = r152983 * r152983;
        double r152985 = r152978 * r152984;
        double r152986 = r152985 * r152983;
        double r152987 = r152978 * r152980;
        double r152988 = r152986 * r152987;
        double r152989 = r152982 - r152988;
        double r152990 = r152981 / r152989;
        double r152991 = fma(r152978, r152980, r152982);
        double r152992 = a;
        double r152993 = tan(r152992);
        double r152994 = -r152993;
        double r152995 = x;
        double r152996 = r152994 + r152995;
        double r152997 = fma(r152990, r152991, r152996);
        double r152998 = r152993 * r152982;
        double r152999 = fma(r152994, r152982, r152998);
        double r153000 = r152997 + r152999;
        return r153000;
}

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 *-un-lft-identity 0.2

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

6. Applied flip-- 0.2

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

7. Applied associate-/r/ 0.2

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

8. Applied prod-diff 0.2

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

9. Applied associate-+r+ 0.2

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

10. Simplified 0.3

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\tan y + \tan z}{1 - \left(\tan y \cdot \tan z\right) \cdot \left(\tan y \cdot \tan z\right)}, \mathsf{fma}\left(\tan y, \tan z, 1\right), \left(-\tan a\right) + x\right)} + \mathsf{fma}\left(-\tan a, 1, \tan a \cdot 1\right)\]
11. Using strategy rm

12. Applied add-cube-cbrt 0.3

    \[\leadsto \mathsf{fma}\left(\frac{\tan y + \tan z}{1 - \left(\tan y \cdot \color{blue}{\left(\left(\sqrt[3]{\tan z} \cdot \sqrt[3]{\tan z}\right) \cdot \sqrt[3]{\tan z}\right)}\right) \cdot \left(\tan y \cdot \tan z\right)}, \mathsf{fma}\left(\tan y, \tan z, 1\right), \left(-\tan a\right) + x\right) + \mathsf{fma}\left(-\tan a, 1, \tan a \cdot 1\right)\]

13. Applied associate-*r* 0.3

    \[\leadsto \mathsf{fma}\left(\frac{\tan y + \tan z}{1 - \color{blue}{\left(\left(\tan y \cdot \left(\sqrt[3]{\tan z} \cdot \sqrt[3]{\tan z}\right)\right) \cdot \sqrt[3]{\tan z}\right)} \cdot \left(\tan y \cdot \tan z\right)}, \mathsf{fma}\left(\tan y, \tan z, 1\right), \left(-\tan a\right) + x\right) + \mathsf{fma}\left(-\tan a, 1, \tan a \cdot 1\right)\]

14. Final simplification 0.3

    \[\leadsto \mathsf{fma}\left(\frac{\tan y + \tan z}{1 - \left(\left(\tan y \cdot \left(\sqrt[3]{\tan z} \cdot \sqrt[3]{\tan z}\right)\right) \cdot \sqrt[3]{\tan z}\right) \cdot \left(\tan y \cdot \tan z\right)}, \mathsf{fma}\left(\tan y, \tan z, 1\right), \left(-\tan a\right) + x\right) + \mathsf{fma}\left(-\tan a, 1, \tan a \cdot 1\right)\]

Reproduce

herbie shell --seed 2020083 +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))))