Average Error: 16.5 → 12.6
Time: 20.3s
Precision: 64
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1.0\right) + \frac{y \cdot b}{t}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -9.025020228721649 \cdot 10^{+17}:\\ \;\;\;\;\frac{x + \frac{z}{t} \cdot y}{\frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{b}{\sqrt[3]{t}} + \left(1.0 + a\right)}\\ \mathbf{elif}\;t \le 0.00018452693517560944:\\ \;\;\;\;\frac{1}{\frac{\frac{b \cdot y}{t} + \left(1.0 + a\right)}{\frac{y \cdot z}{t} + x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(\frac{z}{\sqrt[3]{t}} \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{t}}\right) \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{t}}}{\frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{b}{\sqrt[3]{t}} + \left(1.0 + a\right)}\\ \end{array}\]
\frac{x + \frac{y \cdot z}{t}}{\left(a + 1.0\right) + \frac{y \cdot b}{t}}
↓
\begin{array}{l}
\mathbf{if}\;t \le -9.025020228721649 \cdot 10^{+17}:\\
\;\;\;\;\frac{x + \frac{z}{t} \cdot y}{\frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{b}{\sqrt[3]{t}} + \left(1.0 + a\right)}\\

\mathbf{elif}\;t \le 0.00018452693517560944:\\
\;\;\;\;\frac{1}{\frac{\frac{b \cdot y}{t} + \left(1.0 + a\right)}{\frac{y \cdot z}{t} + x}}\\

\mathbf{else}:\\
\;\;\;\;\frac{x + \left(\frac{z}{\sqrt[3]{t}} \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{t}}\right) \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{t}}}{\frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{b}{\sqrt[3]{t}} + \left(1.0 + a\right)}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r39106007 = x;
        double r39106008 = y;
        double r39106009 = z;
        double r39106010 = r39106008 * r39106009;
        double r39106011 = t;
        double r39106012 = r39106010 / r39106011;
        double r39106013 = r39106007 + r39106012;
        double r39106014 = a;
        double r39106015 = 1.0;
        double r39106016 = r39106014 + r39106015;
        double r39106017 = b;
        double r39106018 = r39106008 * r39106017;
        double r39106019 = r39106018 / r39106011;
        double r39106020 = r39106016 + r39106019;
        double r39106021 = r39106013 / r39106020;
        return r39106021;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r39106022 = t;
        double r39106023 = -9.025020228721649e+17;
        bool r39106024 = r39106022 <= r39106023;
        double r39106025 = x;
        double r39106026 = z;
        double r39106027 = r39106026 / r39106022;
        double r39106028 = y;
        double r39106029 = r39106027 * r39106028;
        double r39106030 = r39106025 + r39106029;
        double r39106031 = cbrt(r39106022);
        double r39106032 = r39106031 * r39106031;
        double r39106033 = r39106028 / r39106032;
        double r39106034 = b;
        double r39106035 = r39106034 / r39106031;
        double r39106036 = r39106033 * r39106035;
        double r39106037 = 1.0;
        double r39106038 = a;
        double r39106039 = r39106037 + r39106038;
        double r39106040 = r39106036 + r39106039;
        double r39106041 = r39106030 / r39106040;
        double r39106042 = 0.00018452693517560944;
        bool r39106043 = r39106022 <= r39106042;
        double r39106044 = 1.0;
        double r39106045 = r39106034 * r39106028;
        double r39106046 = r39106045 / r39106022;
        double r39106047 = r39106046 + r39106039;
        double r39106048 = r39106028 * r39106026;
        double r39106049 = r39106048 / r39106022;
        double r39106050 = r39106049 + r39106025;
        double r39106051 = r39106047 / r39106050;
        double r39106052 = r39106044 / r39106051;
        double r39106053 = r39106026 / r39106031;
        double r39106054 = cbrt(r39106028);
        double r39106055 = r39106054 / r39106031;
        double r39106056 = r39106053 * r39106055;
        double r39106057 = r39106054 * r39106054;
        double r39106058 = r39106057 / r39106031;
        double r39106059 = r39106056 * r39106058;
        double r39106060 = r39106025 + r39106059;
        double r39106061 = r39106060 / r39106040;
        double r39106062 = r39106043 ? r39106052 : r39106061;
        double r39106063 = r39106024 ? r39106041 : r39106062;
        return r39106063;
}

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
Target13.4
Herbie12.6
\[\begin{array}{l} \mathbf{if}\;t \lt -1.3659085366310088 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1.0\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t \lt 3.036967103737246 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1.0\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if t < -9.025020228721649e+17

    1. Initial program 11.9

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1.0\right) + \frac{y \cdot b}{t}}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt12.0

      \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\left(a + 1.0\right) + \frac{y \cdot b}{\color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}}}\]

    4. Applied times-frac9.0

      \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\left(a + 1.0\right) + \color{blue}{\frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{b}{\sqrt[3]{t}}}}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt9.1

      \[\leadsto \frac{x + \frac{y \cdot z}{\color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}}}{\left(a + 1.0\right) + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{b}{\sqrt[3]{t}}}\]

    7. Applied times-frac3.8

      \[\leadsto \frac{x + \color{blue}{\frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}}}{\left(a + 1.0\right) + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{b}{\sqrt[3]{t}}}\]
    8. Using strategy rm

    9. Applied *-un-lft-identity3.8

      \[\leadsto \frac{x + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}}{\color{blue}{1 \cdot \left(\left(a + 1.0\right) + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{b}{\sqrt[3]{t}}\right)}}\]

    10. Applied associate-/r*3.8

      \[\leadsto \color{blue}{\frac{\frac{x + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}}{1}}{\left(a + 1.0\right) + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{b}{\sqrt[3]{t}}}}\]

    11. Simplified3.6

      \[\leadsto \frac{\color{blue}{x + \frac{z}{t} \cdot y}}{\left(a + 1.0\right) + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{b}{\sqrt[3]{t}}}\]

    if -9.025020228721649e+17 < t < 0.00018452693517560944

    1. Initial program 21.2

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1.0\right) + \frac{y \cdot b}{t}}\]
    2. Using strategy rm

    3. Applied clear-num21.4

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(a + 1.0\right) + \frac{y \cdot b}{t}}{x + \frac{y \cdot z}{t}}}}\]

    if 0.00018452693517560944 < t

    1. Initial program 11.7

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1.0\right) + \frac{y \cdot b}{t}}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt11.8

      \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\left(a + 1.0\right) + \frac{y \cdot b}{\color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}}}\]

    4. Applied times-frac8.6

      \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\left(a + 1.0\right) + \color{blue}{\frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{b}{\sqrt[3]{t}}}}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt8.7

      \[\leadsto \frac{x + \frac{y \cdot z}{\color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}}}{\left(a + 1.0\right) + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{b}{\sqrt[3]{t}}}\]

    7. Applied times-frac4.5

      \[\leadsto \frac{x + \color{blue}{\frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}}}{\left(a + 1.0\right) + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{b}{\sqrt[3]{t}}}\]
    8. Using strategy rm

    9. Applied add-cube-cbrt4.6

      \[\leadsto \frac{x + \frac{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}}{\left(a + 1.0\right) + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{b}{\sqrt[3]{t}}}\]

    10. Applied times-frac4.6

      \[\leadsto \frac{x + \color{blue}{\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{t}} \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{t}}\right)} \cdot \frac{z}{\sqrt[3]{t}}}{\left(a + 1.0\right) + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{b}{\sqrt[3]{t}}}\]

    11. Applied associate-*l*4.3

      \[\leadsto \frac{x + \color{blue}{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{t}} \cdot \left(\frac{\sqrt[3]{y}}{\sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}\right)}}{\left(a + 1.0\right) + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{b}{\sqrt[3]{t}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification12.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -9.025020228721649 \cdot 10^{+17}:\\ \;\;\;\;\frac{x + \frac{z}{t} \cdot y}{\frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{b}{\sqrt[3]{t}} + \left(1.0 + a\right)}\\ \mathbf{elif}\;t \le 0.00018452693517560944:\\ \;\;\;\;\frac{1}{\frac{\frac{b \cdot y}{t} + \left(1.0 + a\right)}{\frac{y \cdot z}{t} + x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(\frac{z}{\sqrt[3]{t}} \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{t}}\right) \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{t}}}{\frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{b}{\sqrt[3]{t}} + \left(1.0 + a\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, B"

  :herbie-target
  (if (< t -1.3659085366310088e-271) (* 1 (* (+ x (* (/ y t) z)) (/ 1 (+ (+ a 1.0) (* (/ y t) b))))) (if (< t 3.036967103737246e-130) (/ z b) (* 1 (* (+ x (* (/ y t) z)) (/ 1 (+ (+ a 1.0) (* (/ y t) b)))))))

  (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))