Average Error: 13.3 → 0.2
Time: 1.4m
Precision: 64
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[\left((\left(\frac{\tan y + \tan z}{1 - \left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right)}\right) \cdot \left(\tan z \cdot \tan y\right) + \left(\frac{\tan y + \tan z}{1 - \frac{\left(\sin z \cdot \sin y\right) \cdot \left(\sin z \cdot \sin y\right)}{\left(\cos z \cdot \cos y\right) \cdot \left(\cos z \cdot \cos y\right)}}\right))_* - \tan a\right) + x\]
double f(double x, double y, double z, double a) {
        double r12787649 = x;
        double r12787650 = y;
        double r12787651 = z;
        double r12787652 = r12787650 + r12787651;
        double r12787653 = tan(r12787652);
        double r12787654 = a;
        double r12787655 = tan(r12787654);
        double r12787656 = r12787653 - r12787655;
        double r12787657 = r12787649 + r12787656;
        return r12787657;
}
↓
double f(double x, double y, double z, double a) {
        double r12787658 = y;
        double r12787659 = tan(r12787658);
        double r12787660 = z;
        double r12787661 = tan(r12787660);
        double r12787662 = r12787659 + r12787661;
        double r12787663 = 1.0;
        double r12787664 = r12787661 * r12787659;
        double r12787665 = r12787664 * r12787664;
        double r12787666 = r12787663 - r12787665;
        double r12787667 = r12787662 / r12787666;
        double r12787668 = sin(r12787660);
        double r12787669 = sin(r12787658);
        double r12787670 = r12787668 * r12787669;
        double r12787671 = r12787670 * r12787670;
        double r12787672 = cos(r12787660);
        double r12787673 = cos(r12787658);
        double r12787674 = r12787672 * r12787673;
        double r12787675 = r12787674 * r12787674;
        double r12787676 = r12787671 / r12787675;
        double r12787677 = r12787663 - r12787676;
        double r12787678 = r12787662 / r12787677;
        double r12787679 = fma(r12787667, r12787664, r12787678);
        double r12787680 = a;
        double r12787681 = tan(r12787680);
        double r12787682 = r12787679 - r12787681;
        double r12787683 = x;
        double r12787684 = r12787682 + r12787683;
        return r12787684;
}


x + \left(\tan \left(y + z\right) - \tan a\right)
↓
\left((\left(\frac{\tan y + \tan z}{1 - \left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right)}\right) \cdot \left(\tan z \cdot \tan y\right) + \left(\frac{\tan y + \tan z}{1 - \frac{\left(\sin z \cdot \sin y\right) \cdot \left(\sin z \cdot \sin y\right)}{\left(\cos z \cdot \cos y\right) \cdot \left(\cos z \cdot \cos y\right)}}\right))_* - \tan a\right) + x

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 flip--0.3

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

  7. Applied associate-/r/0.3

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

  8. Applied prod-diff0.3

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

  9. Simplified0.2

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

  10. Simplified0.2

    \[\leadsto x + \left(\left((\left(\frac{\tan y + \tan z}{1 - \left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right)}\right) \cdot \left(\tan z \cdot \tan y\right) + \left(\frac{\tan y + \tan z}{1 - \left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right)}\right))_* - \tan a\right) + \color{blue}{0}\right)\]
  11. Using strategy rm

  12. Applied tan-quot0.2

    \[\leadsto x + \left(\left((\left(\frac{\tan y + \tan z}{1 - \left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right)}\right) \cdot \left(\tan z \cdot \tan y\right) + \left(\frac{\tan y + \tan z}{1 - \left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \color{blue}{\frac{\sin y}{\cos y}}\right)}\right))_* - \tan a\right) + 0\right)\]

  13. Applied tan-quot0.2

    \[\leadsto x + \left(\left((\left(\frac{\tan y + \tan z}{1 - \left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right)}\right) \cdot \left(\tan z \cdot \tan y\right) + \left(\frac{\tan y + \tan z}{1 - \left(\tan z \cdot \tan y\right) \cdot \left(\color{blue}{\frac{\sin z}{\cos z}} \cdot \frac{\sin y}{\cos y}\right)}\right))_* - \tan a\right) + 0\right)\]

  14. Applied frac-times0.2

    \[\leadsto x + \left(\left((\left(\frac{\tan y + \tan z}{1 - \left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right)}\right) \cdot \left(\tan z \cdot \tan y\right) + \left(\frac{\tan y + \tan z}{1 - \left(\tan z \cdot \tan y\right) \cdot \color{blue}{\frac{\sin z \cdot \sin y}{\cos z \cdot \cos y}}}\right))_* - \tan a\right) + 0\right)\]

  15. Applied tan-quot0.2

    \[\leadsto x + \left(\left((\left(\frac{\tan y + \tan z}{1 - \left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right)}\right) \cdot \left(\tan z \cdot \tan y\right) + \left(\frac{\tan y + \tan z}{1 - \left(\tan z \cdot \color{blue}{\frac{\sin y}{\cos y}}\right) \cdot \frac{\sin z \cdot \sin y}{\cos z \cdot \cos y}}\right))_* - \tan a\right) + 0\right)\]

  16. Applied tan-quot0.2

    \[\leadsto x + \left(\left((\left(\frac{\tan y + \tan z}{1 - \left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right)}\right) \cdot \left(\tan z \cdot \tan y\right) + \left(\frac{\tan y + \tan z}{1 - \left(\color{blue}{\frac{\sin z}{\cos z}} \cdot \frac{\sin y}{\cos y}\right) \cdot \frac{\sin z \cdot \sin y}{\cos z \cdot \cos y}}\right))_* - \tan a\right) + 0\right)\]

  17. Applied frac-times0.2

    \[\leadsto x + \left(\left((\left(\frac{\tan y + \tan z}{1 - \left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right)}\right) \cdot \left(\tan z \cdot \tan y\right) + \left(\frac{\tan y + \tan z}{1 - \color{blue}{\frac{\sin z \cdot \sin y}{\cos z \cdot \cos y}} \cdot \frac{\sin z \cdot \sin y}{\cos z \cdot \cos y}}\right))_* - \tan a\right) + 0\right)\]

  18. Applied frac-times0.2

    \[\leadsto x + \left(\left((\left(\frac{\tan y + \tan z}{1 - \left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right)}\right) \cdot \left(\tan z \cdot \tan y\right) + \left(\frac{\tan y + \tan z}{1 - \color{blue}{\frac{\left(\sin z \cdot \sin y\right) \cdot \left(\sin z \cdot \sin y\right)}{\left(\cos z \cdot \cos y\right) \cdot \left(\cos z \cdot \cos y\right)}}}\right))_* - \tan a\right) + 0\right)\]

  19. Final simplification0.2

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

Reproduce

herbie shell --seed 2019101 +o rules:numerics
(FPCore (x y z a)
  :name "(+ x (- (tan (+ y z)) (tan a)))"
  :pre (and (or (== x 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))))