Average Error: 13.3 → 0.2
Time: 12.4s
Precision: 64
\[\left(x = 0.0 \lor 0.588414199999999998 \le x \le 505.590899999999976\right) \land \left(-1.79665800000000009 \cdot 10^{308} \le y \le -9.425585000000013 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \le y \le 1.7512240000000001 \cdot 10^{308}\right) \land \left(-1.7767070000000002 \cdot 10^{308} \le z \le -8.59979600000002 \cdot 10^{-310} \lor 3.29314499999998 \cdot 10^{-311} \le z \le 1.72515400000000009 \cdot 10^{308}\right) \land \left(-1.79665800000000009 \cdot 10^{308} \le a \le -9.425585000000013 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \le a \le 1.7512240000000001 \cdot 10^{308}\right)\]
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[x + \mathsf{fma}\left(\frac{\tan y + \tan z}{1 - {\left(\tan y \cdot \tan z\right)}^{3}}, 1 + \left(\left(\tan y \cdot \tan z\right) \cdot \left(\tan y \cdot \tan z\right) + 1 \cdot \left(\tan y \cdot \tan z\right)\right), -\tan a\right)\]
x + \left(\tan \left(y + z\right) - \tan a\right)
↓
x + \mathsf{fma}\left(\frac{\tan y + \tan z}{1 - {\left(\tan y \cdot \tan z\right)}^{3}}, 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), -\tan a\right)
double f(double x, double y, double z, double a) {
        double r124850 = x;
        double r124851 = y;
        double r124852 = z;
        double r124853 = r124851 + r124852;
        double r124854 = tan(r124853);
        double r124855 = a;
        double r124856 = tan(r124855);
        double r124857 = r124854 - r124856;
        double r124858 = r124850 + r124857;
        return r124858;
}
↓
double f(double x, double y, double z, double a) {
        double r124859 = x;
        double r124860 = y;
        double r124861 = tan(r124860);
        double r124862 = z;
        double r124863 = tan(r124862);
        double r124864 = r124861 + r124863;
        double r124865 = 1.0;
        double r124866 = r124861 * r124863;
        double r124867 = 3.0;
        double r124868 = pow(r124866, r124867);
        double r124869 = r124865 - r124868;
        double r124870 = r124864 / r124869;
        double r124871 = r124866 * r124866;
        double r124872 = r124865 * r124866;
        double r124873 = r124871 + r124872;
        double r124874 = r124865 + r124873;
        double r124875 = a;
        double r124876 = tan(r124875);
        double r124877 = -r124876;
        double r124878 = fma(r124870, r124874, r124877);
        double r124879 = r124859 + r124878;
        return r124879;
}

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 flip3--0.2

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

  6. Applied associate-/r/0.2

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

  7. Applied fma-neg0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2020089 +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))))