Average Error: 16.3 → 13.6
Time: 15.6s
Precision: 64
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.065477972910828548290607099195452155009 \cdot 10^{-76}:\\ \;\;\;\;\frac{1}{\frac{1 + \left(b \cdot \frac{y}{t} + a\right)}{z \cdot \frac{y}{t} + x}}\\ \mathbf{elif}\;z \le 4.934506624270544260798319963136285325405 \cdot 10^{58}:\\ \;\;\;\;\frac{y \cdot \frac{z}{t} + x}{1 + \left(a + \frac{b}{t} \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \frac{y}{t} + x}{1 + \left(\left(\sqrt[3]{\frac{y}{t}} \cdot b\right) \cdot \left(\sqrt[3]{\frac{y}{t}} \cdot \sqrt[3]{\frac{y}{t}}\right) + a\right)}\\ \end{array}\]
\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.065477972910828548290607099195452155009 \cdot 10^{-76}:\\
\;\;\;\;\frac{1}{\frac{1 + \left(b \cdot \frac{y}{t} + a\right)}{z \cdot \frac{y}{t} + x}}\\

\mathbf{elif}\;z \le 4.934506624270544260798319963136285325405 \cdot 10^{58}:\\
\;\;\;\;\frac{y \cdot \frac{z}{t} + x}{1 + \left(a + \frac{b}{t} \cdot y\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{z \cdot \frac{y}{t} + x}{1 + \left(\left(\sqrt[3]{\frac{y}{t}} \cdot b\right) \cdot \left(\sqrt[3]{\frac{y}{t}} \cdot \sqrt[3]{\frac{y}{t}}\right) + a\right)}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r46272987 = x;
        double r46272988 = y;
        double r46272989 = z;
        double r46272990 = r46272988 * r46272989;
        double r46272991 = t;
        double r46272992 = r46272990 / r46272991;
        double r46272993 = r46272987 + r46272992;
        double r46272994 = a;
        double r46272995 = 1.0;
        double r46272996 = r46272994 + r46272995;
        double r46272997 = b;
        double r46272998 = r46272988 * r46272997;
        double r46272999 = r46272998 / r46272991;
        double r46273000 = r46272996 + r46272999;
        double r46273001 = r46272993 / r46273000;
        return r46273001;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r46273002 = z;
        double r46273003 = -1.0654779729108285e-76;
        bool r46273004 = r46273002 <= r46273003;
        double r46273005 = 1.0;
        double r46273006 = 1.0;
        double r46273007 = b;
        double r46273008 = y;
        double r46273009 = t;
        double r46273010 = r46273008 / r46273009;
        double r46273011 = r46273007 * r46273010;
        double r46273012 = a;
        double r46273013 = r46273011 + r46273012;
        double r46273014 = r46273006 + r46273013;
        double r46273015 = r46273002 * r46273010;
        double r46273016 = x;
        double r46273017 = r46273015 + r46273016;
        double r46273018 = r46273014 / r46273017;
        double r46273019 = r46273005 / r46273018;
        double r46273020 = 4.934506624270544e+58;
        bool r46273021 = r46273002 <= r46273020;
        double r46273022 = r46273002 / r46273009;
        double r46273023 = r46273008 * r46273022;
        double r46273024 = r46273023 + r46273016;
        double r46273025 = r46273007 / r46273009;
        double r46273026 = r46273025 * r46273008;
        double r46273027 = r46273012 + r46273026;
        double r46273028 = r46273006 + r46273027;
        double r46273029 = r46273024 / r46273028;
        double r46273030 = cbrt(r46273010);
        double r46273031 = r46273030 * r46273007;
        double r46273032 = r46273030 * r46273030;
        double r46273033 = r46273031 * r46273032;
        double r46273034 = r46273033 + r46273012;
        double r46273035 = r46273006 + r46273034;
        double r46273036 = r46273017 / r46273035;
        double r46273037 = r46273021 ? r46273029 : r46273036;
        double r46273038 = r46273004 ? r46273019 : r46273037;
        return r46273038;
}

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

Original16.3
Target12.8
Herbie13.6
\[\begin{array}{l} \mathbf{if}\;t \lt -1.365908536631008841640163147697088508132 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t \lt 3.036967103737245906066829435890093573122 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -1.0654779729108285e-76

    1. Initial program 21.8

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

    2. Simplified16.7

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

    4. Applied clear-num16.9

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

    if -1.0654779729108285e-76 < z < 4.934506624270544e+58

    1. Initial program 9.5

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

    2. Simplified11.0

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

    4. Applied div-inv11.0

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

    5. Applied associate-*l*9.6

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

    6. Simplified9.6

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

    8. Applied div-inv9.6

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

    9. Applied associate-*l*9.8

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

    10. Simplified9.8

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

    if 4.934506624270544e+58 < z

    1. Initial program 24.7

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

    2. Simplified17.8

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

    4. Applied add-cube-cbrt17.9

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

    5. Applied associate-*l*17.9

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

  4. Final simplification13.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.065477972910828548290607099195452155009 \cdot 10^{-76}:\\ \;\;\;\;\frac{1}{\frac{1 + \left(b \cdot \frac{y}{t} + a\right)}{z \cdot \frac{y}{t} + x}}\\ \mathbf{elif}\;z \le 4.934506624270544260798319963136285325405 \cdot 10^{58}:\\ \;\;\;\;\frac{y \cdot \frac{z}{t} + x}{1 + \left(a + \frac{b}{t} \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \frac{y}{t} + x}{1 + \left(\left(\sqrt[3]{\frac{y}{t}} \cdot b\right) \cdot \left(\sqrt[3]{\frac{y}{t}} \cdot \sqrt[3]{\frac{y}{t}}\right) + a\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(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) (* 1.0 (* (+ x (* (/ y t) z)) (/ 1.0 (+ (+ a 1.0) (* (/ y t) b))))) (if (< t 3.036967103737246e-130) (/ z b) (* 1.0 (* (+ x (* (/ y t) z)) (/ 1.0 (+ (+ a 1.0) (* (/ y t) b)))))))

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