Average Error: 16.5 → 14.1
Time: 1.0m
Precision: 64
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -6.70441392016166230425642730230126840983 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}} + x}{\left(1 + a\right) + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{b}{\sqrt[3]{t}}}\\ \mathbf{elif}\;t \le 4.727405801907533555014023479091638851741 \cdot 10^{-292}:\\ \;\;\;\;\left(\sqrt[3]{\frac{z \cdot y}{t} + x} \cdot \sqrt[3]{\frac{z \cdot y}{t} + x}\right) \cdot \frac{\sqrt[3]{\frac{z \cdot y}{t} + x}}{\frac{b \cdot y}{t} + \left(1 + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}} + x}{\left(1 + a\right) + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{b}{\sqrt[3]{t}}}\\ \end{array}\]
\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
↓
\begin{array}{l}
\mathbf{if}\;t \le -6.70441392016166230425642730230126840983 \cdot 10^{-62}:\\
\;\;\;\;\frac{\frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}} + x}{\left(1 + a\right) + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{b}{\sqrt[3]{t}}}\\

\mathbf{elif}\;t \le 4.727405801907533555014023479091638851741 \cdot 10^{-292}:\\
\;\;\;\;\left(\sqrt[3]{\frac{z \cdot y}{t} + x} \cdot \sqrt[3]{\frac{z \cdot y}{t} + x}\right) \cdot \frac{\sqrt[3]{\frac{z \cdot y}{t} + x}}{\frac{b \cdot y}{t} + \left(1 + a\right)}\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r35806407 = x;
        double r35806408 = y;
        double r35806409 = z;
        double r35806410 = r35806408 * r35806409;
        double r35806411 = t;
        double r35806412 = r35806410 / r35806411;
        double r35806413 = r35806407 + r35806412;
        double r35806414 = a;
        double r35806415 = 1.0;
        double r35806416 = r35806414 + r35806415;
        double r35806417 = b;
        double r35806418 = r35806408 * r35806417;
        double r35806419 = r35806418 / r35806411;
        double r35806420 = r35806416 + r35806419;
        double r35806421 = r35806413 / r35806420;
        return r35806421;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r35806422 = t;
        double r35806423 = -6.704413920161662e-62;
        bool r35806424 = r35806422 <= r35806423;
        double r35806425 = y;
        double r35806426 = cbrt(r35806422);
        double r35806427 = r35806426 * r35806426;
        double r35806428 = r35806425 / r35806427;
        double r35806429 = z;
        double r35806430 = r35806429 / r35806426;
        double r35806431 = r35806428 * r35806430;
        double r35806432 = x;
        double r35806433 = r35806431 + r35806432;
        double r35806434 = 1.0;
        double r35806435 = a;
        double r35806436 = r35806434 + r35806435;
        double r35806437 = b;
        double r35806438 = r35806437 / r35806426;
        double r35806439 = r35806428 * r35806438;
        double r35806440 = r35806436 + r35806439;
        double r35806441 = r35806433 / r35806440;
        double r35806442 = 4.727405801907534e-292;
        bool r35806443 = r35806422 <= r35806442;
        double r35806444 = r35806429 * r35806425;
        double r35806445 = r35806444 / r35806422;
        double r35806446 = r35806445 + r35806432;
        double r35806447 = cbrt(r35806446);
        double r35806448 = r35806447 * r35806447;
        double r35806449 = r35806437 * r35806425;
        double r35806450 = r35806449 / r35806422;
        double r35806451 = r35806450 + r35806436;
        double r35806452 = r35806447 / r35806451;
        double r35806453 = r35806448 * r35806452;
        double r35806454 = r35806443 ? r35806453 : r35806441;
        double r35806455 = r35806424 ? r35806441 : r35806454;
        return r35806455;
}

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
Herbie14.1
\[\begin{array}{l} \mathbf{if}\;t \lt -1.365908536631008841640163147697088508132 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t \lt 3.036967103737245906066829435890093573122 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -6.704413920161662e-62 or 4.727405801907534e-292 < t

    1. Initial program 14.0

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

    3. Applied add-cube-cbrt14.1

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

    4. Applied times-frac12.9

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

    6. Applied add-cube-cbrt13.0

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

    7. Applied times-frac10.8

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

    if -6.704413920161662e-62 < t < 4.727405801907534e-292

    1. Initial program 26.1

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

    3. Applied *-un-lft-identity26.1

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

    4. Applied add-cube-cbrt26.6

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

    5. Applied times-frac26.6

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

  4. Final simplification14.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -6.70441392016166230425642730230126840983 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}} + x}{\left(1 + a\right) + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{b}{\sqrt[3]{t}}}\\ \mathbf{elif}\;t \le 4.727405801907533555014023479091638851741 \cdot 10^{-292}:\\ \;\;\;\;\left(\sqrt[3]{\frac{z \cdot y}{t} + x} \cdot \sqrt[3]{\frac{z \cdot y}{t} + x}\right) \cdot \frac{\sqrt[3]{\frac{z \cdot y}{t} + x}}{\frac{b \cdot y}{t} + \left(1 + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}} + x}{\left(1 + a\right) + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{b}{\sqrt[3]{t}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 
(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))))