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

Average Error: 12.8 → 0.3

Time: 1.1m

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

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

↓

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

C

double f(double x, double y, double z, double a) {
        double r6787917 = x;
        double r6787918 = y;
        double r6787919 = z;
        double r6787920 = r6787918 + r6787919;
        double r6787921 = tan(r6787920);
        double r6787922 = a;
        double r6787923 = tan(r6787922);
        double r6787924 = r6787921 - r6787923;
        double r6787925 = r6787917 + r6787924;
        return r6787925;
}

↓

double f(double x, double y, double z, double a) {
        double r6787926 = y;
        double r6787927 = tan(r6787926);
        double r6787928 = z;
        double r6787929 = tan(r6787928);
        double r6787930 = r6787927 + r6787929;
        double r6787931 = 1.0;
        double r6787932 = r6787927 * r6787929;
        double r6787933 = r6787931 - r6787932;
        double r6787934 = r6787930 / r6787933;
        double r6787935 = exp(r6787934);
        double r6787936 = a;
        double r6787937 = tan(r6787936);
        double r6787938 = exp(r6787937);
        double r6787939 = r6787935 / r6787938;
        double r6787940 = x;
        double r6787941 = exp(r6787940);
        double r6787942 = r6787939 * r6787941;
        double r6787943 = log(r6787942);
        return r6787943;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus a

Derivation

1. Initial program 12.8

   \[x + \left(\tan \left(y + z\right) - \tan a\right)\]
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 add-log-exp 0.3

   \[\leadsto \color{blue}{\log \left(e^{x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \tan a\right)}\right)}\]
6. Using strategy rm

7. Applied add-log-exp 0.3

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

8. Applied add-log-exp 0.3

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

9. Applied diff-log 0.3

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

10. Applied add-log-exp 0.3

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

11. Applied sum-log 0.3

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

12. Applied rem-exp-log 0.3

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

13. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019168 
(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))))