Average Error: 13.3 → 0.2
Time: 11.8s
Precision: 64
\[\left(x = 0.0 \lor 0.588414199999999998 \le x \le 505.590899999999976\right) \land \left(-1.79665800000000009 \cdot 10^{308} \le y \le -9.425585000000013 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \le y \le 1.7512240000000001 \cdot 10^{308}\right) \land \left(-1.7767070000000002 \cdot 10^{308} \le z \le -8.59979600000002 \cdot 10^{-310} \lor 3.29314499999998 \cdot 10^{-311} \le z \le 1.72515400000000009 \cdot 10^{308}\right) \land \left(-1.79665800000000009 \cdot 10^{308} \le a \le -9.425585000000013 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \le a \le 1.7512240000000001 \cdot 10^{308}\right)\]
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[\left(x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \tan a\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\tan a}, \sqrt[3]{\tan a} \cdot \sqrt[3]{\tan a}, \sqrt[3]{\tan a} \cdot \left(\sqrt[3]{\tan a} \cdot \sqrt[3]{\tan a}\right)\right)\]
x + \left(\tan \left(y + z\right) - \tan a\right)
↓
\left(x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \tan a\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\tan a}, \sqrt[3]{\tan a} \cdot \sqrt[3]{\tan a}, \sqrt[3]{\tan a} \cdot \left(\sqrt[3]{\tan a} \cdot \sqrt[3]{\tan a}\right)\right)
double f(double x, double y, double z, double a) {
        double r131528 = x;
        double r131529 = y;
        double r131530 = z;
        double r131531 = r131529 + r131530;
        double r131532 = tan(r131531);
        double r131533 = a;
        double r131534 = tan(r131533);
        double r131535 = r131532 - r131534;
        double r131536 = r131528 + r131535;
        return r131536;
}
↓
double f(double x, double y, double z, double a) {
        double r131537 = x;
        double r131538 = y;
        double r131539 = tan(r131538);
        double r131540 = z;
        double r131541 = tan(r131540);
        double r131542 = r131539 + r131541;
        double r131543 = 1.0;
        double r131544 = r131539 * r131541;
        double r131545 = r131543 - r131544;
        double r131546 = r131542 / r131545;
        double r131547 = a;
        double r131548 = tan(r131547);
        double r131549 = r131546 - r131548;
        double r131550 = r131537 + r131549;
        double r131551 = cbrt(r131548);
        double r131552 = -r131551;
        double r131553 = r131551 * r131551;
        double r131554 = r131551 * r131553;
        double r131555 = fma(r131552, r131553, r131554);
        double r131556 = r131550 + r131555;
        return r131556;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus a

Derivation

  1. Initial program 13.3

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

  3. Applied tan-sum0.2

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

  5. Applied add-cube-cbrt0.3

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

  6. Applied add-sqr-sqrt31.7

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

  7. Applied prod-diff31.7

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

  8. Applied associate-+r+31.7

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

  9. Simplified0.2

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

  10. Final simplification0.2

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

Reproduce

herbie shell --seed 2020056 +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))))