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

Average Error: 13.0 → 0.2

Time: 16.8s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

\[\left(x = 0.0 \lor 0.5884141999999999983472775966220069676638 \le x \le 505.5908999999999764440872240811586380005\right) \land \left(-1.7966580000000000931214523812968299911 \cdot 10^{308} \le y \le -9.425585000000013069597555966781986720373 \cdot 10^{-310} \lor 1.284937999999999548796432976649400331091 \cdot 10^{-309} \le y \le 1.751224000000000127647232028319723370461 \cdot 10^{308}\right) \land \left(-1.776707000000000200843839711454021982841 \cdot 10^{308} \le z \le -8.599796000000016667475923823712126825539 \cdot 10^{-310} \lor 3.293144999999983071955117582595641261776 \cdot 10^{-311} \le z \le 1.725154000000000087891269878141591702413 \cdot 10^{308}\right) \land \left(-1.7966580000000000931214523812968299911 \cdot 10^{308} \le a \le -9.425585000000013069597555966781986720373 \cdot 10^{-310} \lor 1.284937999999999548796432976649400331091 \cdot 10^{-309} \le a \le 1.751224000000000127647232028319723370461 \cdot 10^{308}\right)\]

Math

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

↓

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

C

double f(double x, double y, double z, double a) {
        double r133528 = x;
        double r133529 = y;
        double r133530 = z;
        double r133531 = r133529 + r133530;
        double r133532 = tan(r133531);
        double r133533 = a;
        double r133534 = tan(r133533);
        double r133535 = r133532 - r133534;
        double r133536 = r133528 + r133535;
        return r133536;
}

↓

double f(double x, double y, double z, double a) {
        double r133537 = x;
        double r133538 = z;
        double r133539 = sin(r133538);
        double r133540 = tan(r133538);
        double r133541 = r133539 * r133540;
        double r133542 = y;
        double r133543 = cos(r133542);
        double r133544 = r133541 * r133543;
        double r133545 = cos(r133538);
        double r133546 = sin(r133542);
        double r133547 = tan(r133542);
        double r133548 = r133546 * r133547;
        double r133549 = r133545 * r133548;
        double r133550 = r133544 - r133549;
        double r133551 = r133540 - r133547;
        double r133552 = r133545 * r133543;
        double r133553 = r133551 * r133552;
        double r133554 = r133550 / r133553;
        double r133555 = 1.0;
        double r133556 = r133539 * r133547;
        double r133557 = r133556 / r133545;
        double r133558 = r133555 - r133557;
        double r133559 = r133554 / r133558;
        double r133560 = a;
        double r133561 = tan(r133560);
        double r133562 = r133559 - r133561;
        double r133563 = r133537 + r133562;
        return r133563;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus a

Derivation

1. Initial program 13.0

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

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

5. Simplified 0.2

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

7. Applied tan-quot 0.2

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

8. Applied associate-*l/ 0.2

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

10. Applied flip-+ 0.2

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

12. Applied tan-quot 0.2

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

13. Applied associate-*l/ 0.2

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

14. Applied tan-quot 0.2

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

15. Applied associate-*l/ 0.2

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

16. Applied frac-sub 0.2

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

17. Applied associate-/l/ 0.2

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

18. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019350 
(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))))