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

Average Error: 12.8 → 0.2

Time: 35.7s

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

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

↓

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

C

double f(double x, double y, double z, double a) {
        double r5236562 = x;
        double r5236563 = y;
        double r5236564 = z;
        double r5236565 = r5236563 + r5236564;
        double r5236566 = tan(r5236565);
        double r5236567 = a;
        double r5236568 = tan(r5236567);
        double r5236569 = r5236566 - r5236568;
        double r5236570 = r5236562 + r5236569;
        return r5236570;
}

↓

double f(double x, double y, double z, double a) {
        double r5236571 = a;
        double r5236572 = cos(r5236571);
        double r5236573 = y;
        double r5236574 = tan(r5236573);
        double r5236575 = z;
        double r5236576 = tan(r5236575);
        double r5236577 = r5236574 + r5236576;
        double r5236578 = r5236572 * r5236577;
        double r5236579 = sin(r5236571);
        double r5236580 = 1.0;
        double r5236581 = r5236576 * r5236574;
        double r5236582 = cbrt(r5236581);
        double r5236583 = r5236582 * r5236582;
        double r5236584 = r5236582 * r5236583;
        double r5236585 = r5236580 - r5236584;
        double r5236586 = r5236579 * r5236585;
        double r5236587 = r5236578 - r5236586;
        double r5236588 = r5236580 - r5236581;
        double r5236589 = r5236572 * r5236588;
        double r5236590 = r5236587 / r5236589;
        double r5236591 = x;
        double r5236592 = r5236590 + r5236591;
        return r5236592;
}

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-quot 12.8

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

4. Applied tan-sum 0.2

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

5. Applied frac-sub 0.2

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

7. Applied add-cube-cbrt 0.2

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

8. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019192 +o rules:numerics
(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))))