Average Error: 13.2 → 0.2
Time: 16.2s
Precision: 64
\[\left(x = 0.0 \lor 0.5884141999999999983472775966220069676638 \le x \le 505.5908999999999764440872240811586380005\right) \land \left(-1.7966580000000000931214523812968299911 \cdot 10^{308} \le y \le -9.425585000000013069597555966781986720373 \cdot 10^{-310} \lor 1.284937999999999548796432976649400331091 \cdot 10^{-309} \le y \le 1.751223999999999928063201074847742204824 \cdot 10^{308}\right) \land \left(-1.776707000000000001259808757982040817204 \cdot 10^{308} \le z \le -8.599796000000016667475923823712126825539 \cdot 10^{-310} \lor 3.293144999999983071955117582595641261776 \cdot 10^{-311} \le z \le 1.725154000000000087891269878141591702413 \cdot 10^{308}\right) \land \left(-1.7966580000000000931214523812968299911 \cdot 10^{308} \le a \le -9.425585000000013069597555966781986720373 \cdot 10^{-310} \lor 1.284937999999999548796432976649400331091 \cdot 10^{-309} \le a \le 1.751223999999999928063201074847742204824 \cdot 10^{308}\right)\]
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[x + \frac{\left(\tan y + \tan z\right) \cdot \cos a - \left(1 - \frac{\sin y \cdot \sin z}{\cos z \cdot \cos y}\right) \cdot \sin a}{\left(1 - \frac{\sin y \cdot \tan z}{\cos y}\right) \cdot \cos a}\]
x + \left(\tan \left(y + z\right) - \tan a\right)
↓
x + \frac{\left(\tan y + \tan z\right) \cdot \cos a - \left(1 - \frac{\sin y \cdot \sin z}{\cos z \cdot \cos y}\right) \cdot \sin a}{\left(1 - \frac{\sin y \cdot \tan z}{\cos y}\right) \cdot \cos a}
double f(double x, double y, double z, double a) {
        double r210527 = x;
        double r210528 = y;
        double r210529 = z;
        double r210530 = r210528 + r210529;
        double r210531 = tan(r210530);
        double r210532 = a;
        double r210533 = tan(r210532);
        double r210534 = r210531 - r210533;
        double r210535 = r210527 + r210534;
        return r210535;
}
↓
double f(double x, double y, double z, double a) {
        double r210536 = x;
        double r210537 = y;
        double r210538 = tan(r210537);
        double r210539 = z;
        double r210540 = tan(r210539);
        double r210541 = r210538 + r210540;
        double r210542 = a;
        double r210543 = cos(r210542);
        double r210544 = r210541 * r210543;
        double r210545 = 1.0;
        double r210546 = sin(r210537);
        double r210547 = sin(r210539);
        double r210548 = r210546 * r210547;
        double r210549 = cos(r210539);
        double r210550 = cos(r210537);
        double r210551 = r210549 * r210550;
        double r210552 = r210548 / r210551;
        double r210553 = r210545 - r210552;
        double r210554 = sin(r210542);
        double r210555 = r210553 * r210554;
        double r210556 = r210544 - r210555;
        double r210557 = r210546 * r210540;
        double r210558 = r210557 / r210550;
        double r210559 = r210545 - r210558;
        double r210560 = r210559 * r210543;
        double r210561 = r210556 / r210560;
        double r210562 = r210536 + r210561;
        return r210562;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 13.2

    \[x + \left(\tan \left(y + z\right) - \tan a\right)\]
  2. Using strategy rm

  3. Applied tan-quot13.2

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

  4. Applied tan-sum0.2

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

  5. Applied frac-sub0.2

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

  7. Applied tan-quot0.2

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

  8. Applied associate-*l/0.2

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

  9. Taylor expanded around inf 0.2

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

  10. Final simplification0.2

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

Reproduce

herbie shell --seed 2019322 
(FPCore (x y z a)
  :name "(+ x (- (tan (+ y z)) (tan a)))"
  :precision binary64
  :pre (and (or (== x 0.0) (<= 0.588414199999999998 x 505.590899999999976)) (or (<= -1.79665800000000009e308 y -9.425585000000013e-310) (<= 1.284938e-309 y 1.75122399999999993e308)) (or (<= -1.776707e308 z -8.59979600000002e-310) (<= 3.29314499999998e-311 z 1.72515400000000009e308)) (or (<= -1.79665800000000009e308 a -9.425585000000013e-310) (<= 1.284938e-309 a 1.75122399999999993e308)))
  (+ x (- (tan (+ y z)) (tan a))))