Average Error: 13.7 → 0.3
Time: 18.0s
Precision: 64
\[\left(x = 0.0 \lor 0.588414199999999998 \le x \le 505.590899999999976\right) \land \left(-1.79665800000000009 \cdot 10^{308} \le y \le -9.425585000000013 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \le y \le 1.7512240000000001 \cdot 10^{308}\right) \land \left(-1.7767070000000002 \cdot 10^{308} \le z \le -8.59979600000002 \cdot 10^{-310} \lor 3.29314499999998 \cdot 10^{-311} \le z \le 1.72515400000000009 \cdot 10^{308}\right) \land \left(-1.79665800000000009 \cdot 10^{308} \le a \le -9.425585000000013 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \le a \le 1.7512240000000001 \cdot 10^{308}\right)\]
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[\mathsf{fma}\left(\frac{\tan y + \tan z}{1 - {\left(\frac{\tan y \cdot \sin z}{\cos z}\right)}^{3}}, \mathsf{fma}\left(\frac{\tan y \cdot \sin z}{\cos z}, 1 + \frac{\tan y \cdot \sin z}{\cos z}, 1\right), \left(-\tan a\right) + x\right) + \mathsf{fma}\left(-\sqrt[3]{\tan a}, \sqrt[3]{\tan a} \cdot \sqrt[3]{\tan a}, \sqrt[3]{\tan a} \cdot \left(\sqrt[3]{\tan a} \cdot \sqrt[3]{\tan a}\right)\right)\]
x + \left(\tan \left(y + z\right) - \tan a\right)
↓
\mathsf{fma}\left(\frac{\tan y + \tan z}{1 - {\left(\frac{\tan y \cdot \sin z}{\cos z}\right)}^{3}}, \mathsf{fma}\left(\frac{\tan y \cdot \sin z}{\cos z}, 1 + \frac{\tan y \cdot \sin z}{\cos z}, 1\right), \left(-\tan a\right) + x\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 r216081 = x;
        double r216082 = y;
        double r216083 = z;
        double r216084 = r216082 + r216083;
        double r216085 = tan(r216084);
        double r216086 = a;
        double r216087 = tan(r216086);
        double r216088 = r216085 - r216087;
        double r216089 = r216081 + r216088;
        return r216089;
}
↓
double f(double x, double y, double z, double a) {
        double r216090 = y;
        double r216091 = tan(r216090);
        double r216092 = z;
        double r216093 = tan(r216092);
        double r216094 = r216091 + r216093;
        double r216095 = 1.0;
        double r216096 = sin(r216092);
        double r216097 = r216091 * r216096;
        double r216098 = cos(r216092);
        double r216099 = r216097 / r216098;
        double r216100 = 3.0;
        double r216101 = pow(r216099, r216100);
        double r216102 = r216095 - r216101;
        double r216103 = r216094 / r216102;
        double r216104 = r216095 + r216099;
        double r216105 = fma(r216099, r216104, r216095);
        double r216106 = a;
        double r216107 = tan(r216106);
        double r216108 = -r216107;
        double r216109 = x;
        double r216110 = r216108 + r216109;
        double r216111 = fma(r216103, r216105, r216110);
        double r216112 = cbrt(r216107);
        double r216113 = -r216112;
        double r216114 = r216112 * r216112;
        double r216115 = r216112 * r216114;
        double r216116 = fma(r216113, r216114, r216115);
        double r216117 = r216111 + r216116;
        return r216117;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus a

Derivation

  1. Initial program 13.7

    \[x + \left(\tan \left(y + z\right) - \tan a\right)\]
  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 tan-quot0.2

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

  6. Applied associate-*r/0.2

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

  8. Applied add-cube-cbrt0.3

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

  9. Applied flip3--0.3

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

  10. Applied associate-/r/0.3

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

  11. Applied prod-diff0.3

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

  12. Applied associate-+r+0.3

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

  13. Simplified0.3

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

  14. Final simplification0.3

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

Reproduce

herbie shell --seed 2020018 +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))))