Average Error: 16.5 → 14.4
Time: 5.6s
Precision: 64
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.91683392434281339 \cdot 10^{-24} \lor \neg \left(t \le 4.10452660281528009 \cdot 10^{-61}\right):\\ \;\;\;\;\frac{x + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}}{\left(a + 1\right) + y \cdot \frac{b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}}{\sqrt[3]{\left(a + 1\right) + \frac{y \cdot b}{t}} \cdot \sqrt[3]{\left(a + 1\right) + \frac{y \cdot b}{t}}}}{\sqrt[3]{\left(a + 1\right) + \frac{y \cdot b}{t}}}\\ \end{array}\]
\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
↓
\begin{array}{l}
\mathbf{if}\;t \le -1.91683392434281339 \cdot 10^{-24} \lor \neg \left(t \le 4.10452660281528009 \cdot 10^{-61}\right):\\
\;\;\;\;\frac{x + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}}{\left(a + 1\right) + y \cdot \frac{b}{t}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{x + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}}{\sqrt[3]{\left(a + 1\right) + \frac{y \cdot b}{t}} \cdot \sqrt[3]{\left(a + 1\right) + \frac{y \cdot b}{t}}}}{\sqrt[3]{\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 r696696 = x;
        double r696697 = y;
        double r696698 = z;
        double r696699 = r696697 * r696698;
        double r696700 = t;
        double r696701 = r696699 / r696700;
        double r696702 = r696696 + r696701;
        double r696703 = a;
        double r696704 = 1.0;
        double r696705 = r696703 + r696704;
        double r696706 = b;
        double r696707 = r696697 * r696706;
        double r696708 = r696707 / r696700;
        double r696709 = r696705 + r696708;
        double r696710 = r696702 / r696709;
        return r696710;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r696711 = t;
        double r696712 = -1.9168339243428134e-24;
        bool r696713 = r696711 <= r696712;
        double r696714 = 4.10452660281528e-61;
        bool r696715 = r696711 <= r696714;
        double r696716 = !r696715;
        bool r696717 = r696713 || r696716;
        double r696718 = x;
        double r696719 = y;
        double r696720 = cbrt(r696711);
        double r696721 = r696720 * r696720;
        double r696722 = r696719 / r696721;
        double r696723 = z;
        double r696724 = r696723 / r696720;
        double r696725 = r696722 * r696724;
        double r696726 = r696718 + r696725;
        double r696727 = a;
        double r696728 = 1.0;
        double r696729 = r696727 + r696728;
        double r696730 = b;
        double r696731 = r696730 / r696711;
        double r696732 = r696719 * r696731;
        double r696733 = r696729 + r696732;
        double r696734 = r696726 / r696733;
        double r696735 = r696719 * r696730;
        double r696736 = r696735 / r696711;
        double r696737 = r696729 + r696736;
        double r696738 = cbrt(r696737);
        double r696739 = r696738 * r696738;
        double r696740 = r696726 / r696739;
        double r696741 = r696740 / r696738;
        double r696742 = r696717 ? r696734 : r696741;
        return r696742;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original16.5
Target13.2
Herbie14.4
\[\begin{array}{l} \mathbf{if}\;t \lt -1.3659085366310088 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t \lt 3.0369671037372459 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -1.9168339243428134e-24 or 4.10452660281528e-61 < t

    1. Initial program 11.5

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt11.6

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

    4. Applied times-frac8.9

      \[\leadsto \frac{x + \color{blue}{\frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
    5. Using strategy rm

    6. Applied *-un-lft-identity8.9

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

    7. Applied times-frac5.1

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

    8. Simplified5.1

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

    if -1.9168339243428134e-24 < t < 4.10452660281528e-61

    1. Initial program 23.6

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt23.9

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

    4. Applied times-frac27.3

      \[\leadsto \frac{x + \color{blue}{\frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt27.5

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

    7. Applied associate-/r*27.5

      \[\leadsto \color{blue}{\frac{\frac{x + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}}{\sqrt[3]{\left(a + 1\right) + \frac{y \cdot b}{t}} \cdot \sqrt[3]{\left(a + 1\right) + \frac{y \cdot b}{t}}}}{\sqrt[3]{\left(a + 1\right) + \frac{y \cdot b}{t}}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification14.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.91683392434281339 \cdot 10^{-24} \lor \neg \left(t \le 4.10452660281528009 \cdot 10^{-61}\right):\\ \;\;\;\;\frac{x + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}}{\left(a + 1\right) + y \cdot \frac{b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}}{\sqrt[3]{\left(a + 1\right) + \frac{y \cdot b}{t}} \cdot \sqrt[3]{\left(a + 1\right) + \frac{y \cdot b}{t}}}}{\sqrt[3]{\left(a + 1\right) + \frac{y \cdot b}{t}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 
(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))))