Average Error: 16.5 → 15.2
Time: 11.9s
Precision: 64
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -6.063981578832086470410894573246159834006 \cdot 10^{-35}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{elif}\;t \le 2.89968682653951552104872010503272409208 \cdot 10^{-74}:\\ \;\;\;\;\left(\sqrt[3]{\frac{\frac{y \cdot z}{t} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}}} \cdot \sqrt[3]{\frac{\frac{y \cdot z}{t} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}}}\right) \cdot \sqrt[3]{\frac{\frac{y \cdot z}{t} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \end{array}\]
\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
↓
\begin{array}{l}
\mathbf{if}\;t \le -6.063981578832086470410894573246159834006 \cdot 10^{-35}:\\
\;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\

\mathbf{elif}\;t \le 2.89968682653951552104872010503272409208 \cdot 10^{-74}:\\
\;\;\;\;\left(\sqrt[3]{\frac{\frac{y \cdot z}{t} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}}} \cdot \sqrt[3]{\frac{\frac{y \cdot z}{t} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}}}\right) \cdot \sqrt[3]{\frac{\frac{y \cdot z}{t} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{x + y \cdot \frac{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 r480505 = x;
        double r480506 = y;
        double r480507 = z;
        double r480508 = r480506 * r480507;
        double r480509 = t;
        double r480510 = r480508 / r480509;
        double r480511 = r480505 + r480510;
        double r480512 = a;
        double r480513 = 1.0;
        double r480514 = r480512 + r480513;
        double r480515 = b;
        double r480516 = r480506 * r480515;
        double r480517 = r480516 / r480509;
        double r480518 = r480514 + r480517;
        double r480519 = r480511 / r480518;
        return r480519;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r480520 = t;
        double r480521 = -6.063981578832086e-35;
        bool r480522 = r480520 <= r480521;
        double r480523 = x;
        double r480524 = y;
        double r480525 = z;
        double r480526 = r480520 / r480525;
        double r480527 = r480524 / r480526;
        double r480528 = r480523 + r480527;
        double r480529 = a;
        double r480530 = 1.0;
        double r480531 = r480529 + r480530;
        double r480532 = b;
        double r480533 = r480524 * r480532;
        double r480534 = r480533 / r480520;
        double r480535 = r480531 + r480534;
        double r480536 = r480528 / r480535;
        double r480537 = 2.8996868265395155e-74;
        bool r480538 = r480520 <= r480537;
        double r480539 = r480524 * r480525;
        double r480540 = r480539 / r480520;
        double r480541 = r480540 + r480523;
        double r480542 = r480541 / r480535;
        double r480543 = cbrt(r480542);
        double r480544 = r480543 * r480543;
        double r480545 = r480544 * r480543;
        double r480546 = r480525 / r480520;
        double r480547 = r480524 * r480546;
        double r480548 = r480523 + r480547;
        double r480549 = r480548 / r480535;
        double r480550 = r480538 ? r480545 : r480549;
        double r480551 = r480522 ? r480536 : r480550;
        return r480551;
}

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
Target13.3
Herbie15.2
\[\begin{array}{l} \mathbf{if}\;t \lt -1.365908536631008841640163147697088508132 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t \lt 3.036967103737245906066829435890093573122 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if t < -6.063981578832086e-35

    1. Initial program 11.3

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

    3. Applied associate-/l*8.3

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

    if -6.063981578832086e-35 < t < 2.8996868265395155e-74

    1. Initial program 24.5

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

    3. Applied div-inv24.5

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

    5. Applied clear-num24.6

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

    6. Simplified24.6

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

    8. Applied add-cube-cbrt25.1

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

    9. Simplified25.1

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

    10. Simplified25.1

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

    if 2.8996868265395155e-74 < t

    1. Initial program 11.3

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

    3. Applied *-un-lft-identity11.3

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

    4. Applied times-frac9.3

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

    5. Simplified9.3

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

  4. Final simplification15.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -6.063981578832086470410894573246159834006 \cdot 10^{-35}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{elif}\;t \le 2.89968682653951552104872010503272409208 \cdot 10^{-74}:\\ \;\;\;\;\left(\sqrt[3]{\frac{\frac{y \cdot z}{t} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}}} \cdot \sqrt[3]{\frac{\frac{y \cdot z}{t} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}}}\right) \cdot \sqrt[3]{\frac{\frac{y \cdot z}{t} + x}{\left(a + 1\right) + \frac{y \cdot b}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(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 3.0369671037372459e-130) (/ z b) (* 1 (* (+ x (* (/ y t) z)) (/ 1 (+ (+ a 1) (* (/ y t) b)))))))

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