Average Error: 16.4 → 12.9
Time: 8.7s
Precision: 64
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -453438913939.10449 \lor \neg \left(t \le 1.599714736526996 \cdot 10^{-7}\right):\\ \;\;\;\;\left(x + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(\sqrt[3]{\frac{y \cdot z}{t}} \cdot \sqrt[3]{\frac{y \cdot z}{t}}\right) \cdot \sqrt[3]{\frac{y \cdot z}{t}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \end{array}\]
\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
↓
\begin{array}{l}
\mathbf{if}\;t \le -453438913939.10449 \lor \neg \left(t \le 1.599714736526996 \cdot 10^{-7}\right):\\
\;\;\;\;\left(x + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{x + \left(\sqrt[3]{\frac{y \cdot z}{t}} \cdot \sqrt[3]{\frac{y \cdot z}{t}}\right) \cdot \sqrt[3]{\frac{y \cdot z}{t}}}{\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 r843716 = x;
        double r843717 = y;
        double r843718 = z;
        double r843719 = r843717 * r843718;
        double r843720 = t;
        double r843721 = r843719 / r843720;
        double r843722 = r843716 + r843721;
        double r843723 = a;
        double r843724 = 1.0;
        double r843725 = r843723 + r843724;
        double r843726 = b;
        double r843727 = r843717 * r843726;
        double r843728 = r843727 / r843720;
        double r843729 = r843725 + r843728;
        double r843730 = r843722 / r843729;
        return r843730;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r843731 = t;
        double r843732 = -453438913939.1045;
        bool r843733 = r843731 <= r843732;
        double r843734 = 1.599714736526996e-07;
        bool r843735 = r843731 <= r843734;
        double r843736 = !r843735;
        bool r843737 = r843733 || r843736;
        double r843738 = x;
        double r843739 = y;
        double r843740 = cbrt(r843731);
        double r843741 = r843740 * r843740;
        double r843742 = r843739 / r843741;
        double r843743 = z;
        double r843744 = r843743 / r843740;
        double r843745 = r843742 * r843744;
        double r843746 = r843738 + r843745;
        double r843747 = 1.0;
        double r843748 = a;
        double r843749 = 1.0;
        double r843750 = r843748 + r843749;
        double r843751 = b;
        double r843752 = r843731 / r843751;
        double r843753 = r843739 / r843752;
        double r843754 = r843750 + r843753;
        double r843755 = r843747 / r843754;
        double r843756 = r843746 * r843755;
        double r843757 = r843739 * r843743;
        double r843758 = r843757 / r843731;
        double r843759 = cbrt(r843758);
        double r843760 = r843759 * r843759;
        double r843761 = r843760 * r843759;
        double r843762 = r843738 + r843761;
        double r843763 = r843739 * r843751;
        double r843764 = r843763 / r843731;
        double r843765 = r843750 + r843764;
        double r843766 = r843762 / r843765;
        double r843767 = r843737 ? r843756 : r843766;
        return r843767;
}

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

Original16.4
Target13.2
Herbie12.9
\[\begin{array}{l} \mathbf{if}\;t \lt -1.3659085366310088 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t \lt 3.0369671037372459 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -453438913939.1045 or 1.599714736526996e-07 < t

    1. Initial program 11.2

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

    3. Applied add-cube-cbrt11.3

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

    4. Applied times-frac7.9

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

    6. Applied associate-/l*4.1

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

    8. Applied div-inv4.2

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

    if -453438913939.1045 < t < 1.599714736526996e-07

    1. Initial program 21.9

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

    3. Applied add-cube-cbrt22.1

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

  4. Final simplification12.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -453438913939.10449 \lor \neg \left(t \le 1.599714736526996 \cdot 10^{-7}\right):\\ \;\;\;\;\left(x + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(\sqrt[3]{\frac{y \cdot z}{t}} \cdot \sqrt[3]{\frac{y \cdot z}{t}}\right) \cdot \sqrt[3]{\frac{y \cdot z}{t}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 +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))))