Average Error: 13.5 → 0.3
Time: 33.1s
Precision: 64
\[\left(x = 0.0 \lor 0.5884141999999999983472775966220069676638 \le x \le 505.5908999999999764440872240811586380005\right) \land \left(-1.7966580000000000931214523812968299911 \cdot 10^{308} \le y \le -9.425585000000013069597555966781986720373 \cdot 10^{-310} \lor 1.284937999999999548796432976649400331091 \cdot 10^{-309} \le y \le 1.751223999999999928063201074847742204824 \cdot 10^{308}\right) \land \left(-1.776707000000000001259808757982040817204 \cdot 10^{308} \le z \le -8.599796000000016667475923823712126825539 \cdot 10^{-310} \lor 3.293144999999983071955117582595641261776 \cdot 10^{-311} \le z \le 1.725154000000000087891269878141591702413 \cdot 10^{308}\right) \land \left(-1.7966580000000000931214523812968299911 \cdot 10^{308} \le a \le -9.425585000000013069597555966781986720373 \cdot 10^{-310} \lor 1.284937999999999548796432976649400331091 \cdot 10^{-309} \le a \le 1.751223999999999928063201074847742204824 \cdot 10^{308}\right)\]
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[\left(\mathsf{fma}\left(\mathsf{fma}\left(\tan z \cdot \tan y, \tan z \cdot \tan y, \tan z \cdot \tan y\right), \frac{\tan y + \tan z}{1 - \left(\tan z \cdot \tan y\right) \cdot \left(\left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right)\right)}, \frac{\tan y + \tan z}{1 - \left(\tan z \cdot \tan y\right) \cdot \left(\left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right)\right)}\right) - \left(\tan a - x\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\tan a}, \sqrt[3]{\tan a} \cdot \sqrt[3]{\tan a}, \left(\sqrt[3]{\tan a} \cdot \sqrt[3]{\tan a}\right) \cdot \sqrt[3]{\tan a}\right)\]
x + \left(\tan \left(y + z\right) - \tan a\right)
↓
\left(\mathsf{fma}\left(\mathsf{fma}\left(\tan z \cdot \tan y, \tan z \cdot \tan y, \tan z \cdot \tan y\right), \frac{\tan y + \tan z}{1 - \left(\tan z \cdot \tan y\right) \cdot \left(\left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right)\right)}, \frac{\tan y + \tan z}{1 - \left(\tan z \cdot \tan y\right) \cdot \left(\left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right)\right)}\right) - \left(\tan a - x\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\tan a}, \sqrt[3]{\tan a} \cdot \sqrt[3]{\tan a}, \left(\sqrt[3]{\tan a} \cdot \sqrt[3]{\tan a}\right) \cdot \sqrt[3]{\tan a}\right)
double f(double x, double y, double z, double a) {
        double r5021216 = x;
        double r5021217 = y;
        double r5021218 = z;
        double r5021219 = r5021217 + r5021218;
        double r5021220 = tan(r5021219);
        double r5021221 = a;
        double r5021222 = tan(r5021221);
        double r5021223 = r5021220 - r5021222;
        double r5021224 = r5021216 + r5021223;
        return r5021224;
}
↓
double f(double x, double y, double z, double a) {
        double r5021225 = z;
        double r5021226 = tan(r5021225);
        double r5021227 = y;
        double r5021228 = tan(r5021227);
        double r5021229 = r5021226 * r5021228;
        double r5021230 = fma(r5021229, r5021229, r5021229);
        double r5021231 = r5021228 + r5021226;
        double r5021232 = 1.0;
        double r5021233 = r5021229 * r5021229;
        double r5021234 = r5021229 * r5021233;
        double r5021235 = r5021232 - r5021234;
        double r5021236 = r5021231 / r5021235;
        double r5021237 = fma(r5021230, r5021236, r5021236);
        double r5021238 = a;
        double r5021239 = tan(r5021238);
        double r5021240 = x;
        double r5021241 = r5021239 - r5021240;
        double r5021242 = r5021237 - r5021241;
        double r5021243 = cbrt(r5021239);
        double r5021244 = -r5021243;
        double r5021245 = r5021243 * r5021243;
        double r5021246 = r5021245 * r5021243;
        double r5021247 = fma(r5021244, r5021245, r5021246);
        double r5021248 = r5021242 + r5021247;
        return r5021248;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus a

Derivation

  1. Initial program 13.5

    \[x + \left(\tan \left(y + z\right) - \tan a\right)\]
  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 flip3--0.3

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

  7. Applied associate-/r/0.3

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

  8. Applied prod-diff0.3

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

  9. Applied associate-+r+0.3

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

  10. Simplified0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2019170 +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))))