\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]

↓

\[\begin{array}{l} \mathbf{if}\;z \le -4.206477194529533326393607486873472590929 \cdot 10^{-59} \lor \neg \left(z \le 6.370790812859238783452018396763277563967 \cdot 10^{-53}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 \cdot \mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \end{array}\]

\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}

↓

\begin{array}{l}
\mathbf{if}\;z \le -4.206477194529533326393607486873472590929 \cdot 10^{-59} \lor \neg \left(z \le 6.370790812859238783452018396763277563967 \cdot 10^{-53}\right):\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 \cdot \mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\

\end{array}

double f(double x, double y, double z, double t, double a, double b) {
        double r839689 = x;
        double r839690 = y;
        double r839691 = z;
        double r839692 = r839690 * r839691;
        double r839693 = t;
        double r839694 = r839692 / r839693;
        double r839695 = r839689 + r839694;
        double r839696 = a;
        double r839697 = 1.0;
        double r839698 = r839696 + r839697;
        double r839699 = b;
        double r839700 = r839690 * r839699;
        double r839701 = r839700 / r839693;
        double r839702 = r839698 + r839701;
        double r839703 = r839695 / r839702;
        return r839703;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r839704 = z;
        double r839705 = -4.206477194529533e-59;
        bool r839706 = r839704 <= r839705;
        double r839707 = 6.370790812859239e-53;
        bool r839708 = r839704 <= r839707;
        double r839709 = !r839708;
        bool r839710 = r839706 || r839709;
        double r839711 = y;
        double r839712 = t;
        double r839713 = r839711 / r839712;
        double r839714 = x;
        double r839715 = fma(r839713, r839704, r839714);
        double r839716 = 1.0;
        double r839717 = b;
        double r839718 = a;
        double r839719 = 1.0;
        double r839720 = r839718 + r839719;
        double r839721 = fma(r839713, r839717, r839720);
        double r839722 = r839716 * r839721;
        double r839723 = r839715 / r839722;
        double r839724 = r839712 / r839704;
        double r839725 = r839711 / r839724;
        double r839726 = r839714 + r839725;
        double r839727 = r839711 * r839717;
        double r839728 = r839727 / r839712;
        double r839729 = r839720 + r839728;
        double r839730 = r839726 / r839729;
        double r839731 = r839710 ? r839723 : r839730;
        return r839731;
}

Original	16.8
Target	13.4
Herbie	14.0

Derivation

Split input into 2 regimes
if z < -4.206477194529533e-59 or 6.370790812859239e-53 < z
1. Initial program 22.6
  \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
2. Using strategy rm
3. Applied *-un-lft-identity22.6
  \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\color{blue}{1 \cdot \left(\left(a + 1\right) + \frac{y \cdot b}{t}\right)}}\]
4. Applied associate-/r*22.6
  \[\leadsto \color{blue}{\frac{\frac{x + \frac{y \cdot z}{t}}{1}}{\left(a + 1\right) + \frac{y \cdot b}{t}}}\]
5. Simplified20.1
  \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
6. Using strategy rm
7. Applied *-un-lft-identity20.1
  \[\leadsto \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\left(a + 1\right) + \color{blue}{1 \cdot \frac{y \cdot b}{t}}}\]
8. Applied *-un-lft-identity20.1
  \[\leadsto \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\color{blue}{1 \cdot \left(a + 1\right)} + 1 \cdot \frac{y \cdot b}{t}}\]
9. Applied distribute-lft-out20.1
  \[\leadsto \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\color{blue}{1 \cdot \left(\left(a + 1\right) + \frac{y \cdot b}{t}\right)}}\]
10. Simplified17.9
  \[\leadsto \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 \cdot \color{blue}{\mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}}\]
if -4.206477194529533e-59 < z < 6.370790812859239e-53
1. Initial program 8.9
  \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
2. Using strategy rm
3. Applied associate-/l*8.8
  \[\leadsto \frac{x + \color{blue}{\frac{y}{\frac{t}{z}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
Recombined 2 regimes into one program.
Final simplification14.0
\[\leadsto \begin{array}{l} \mathbf{if}\;z \le -4.206477194529533326393607486873472590929 \cdot 10^{-59} \lor \neg \left(z \le 6.370790812859238783452018396763277563967 \cdot 10^{-53}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 \cdot \mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +o rules:numerics
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, B"
  :precision binary64

  :herbie-target
  (if (< t -1.3659085366310088e-271) (* 1 (* (+ x (* (/ y t) z)) (/ 1 (+ (+ a 1) (* (/ y t) b))))) (if (< t 3.036967103737246e-130) (/ z b) (* 1 (* (+ x (* (/ y t) z)) (/ 1 (+ (+ a 1) (* (/ y t) b)))))))

  (/ (+ x (/ (* y z) t)) (+ (+ a 1) (/ (* y b) t))))

Error

Target

Derivation

`if z < -4.206477194529533e-59 or 6.370790812859239e-53 < z`

`if -4.206477194529533e-59 < z < 6.370790812859239e-53`

Reproduce