Average Error: 17.2 → 13.5
Time: 20.3s
Precision: 64
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.356491776467311580293233615407758052562 \cdot 10^{-8}:\\ \;\;\;\;\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{1 + \left(\frac{y}{t} \cdot b + a\right)}\\ \mathbf{elif}\;t \le 1.169708175871951089353680636282670578711 \cdot 10^{52}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{b \cdot y}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\sqrt[3]{y}}{t} \cdot z\right) \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + x}{1 + \left(\frac{y}{t} \cdot b + a\right)}\\ \end{array}\]
\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
↓
\begin{array}{l}
\mathbf{if}\;t \le -1.356491776467311580293233615407758052562 \cdot 10^{-8}:\\
\;\;\;\;\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{1 + \left(\frac{y}{t} \cdot b + a\right)}\\

\mathbf{elif}\;t \le 1.169708175871951089353680636282670578711 \cdot 10^{52}:\\
\;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{b \cdot y}{t}}\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r38190567 = x;
        double r38190568 = y;
        double r38190569 = z;
        double r38190570 = r38190568 * r38190569;
        double r38190571 = t;
        double r38190572 = r38190570 / r38190571;
        double r38190573 = r38190567 + r38190572;
        double r38190574 = a;
        double r38190575 = 1.0;
        double r38190576 = r38190574 + r38190575;
        double r38190577 = b;
        double r38190578 = r38190568 * r38190577;
        double r38190579 = r38190578 / r38190571;
        double r38190580 = r38190576 + r38190579;
        double r38190581 = r38190573 / r38190580;
        return r38190581;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r38190582 = t;
        double r38190583 = -1.3564917764673116e-08;
        bool r38190584 = r38190582 <= r38190583;
        double r38190585 = x;
        double r38190586 = y;
        double r38190587 = r38190586 / r38190582;
        double r38190588 = z;
        double r38190589 = r38190587 * r38190588;
        double r38190590 = r38190585 + r38190589;
        double r38190591 = 1.0;
        double r38190592 = 1.0;
        double r38190593 = b;
        double r38190594 = r38190587 * r38190593;
        double r38190595 = a;
        double r38190596 = r38190594 + r38190595;
        double r38190597 = r38190592 + r38190596;
        double r38190598 = r38190591 / r38190597;
        double r38190599 = r38190590 * r38190598;
        double r38190600 = 1.1697081758719511e+52;
        bool r38190601 = r38190582 <= r38190600;
        double r38190602 = r38190586 * r38190588;
        double r38190603 = r38190602 / r38190582;
        double r38190604 = r38190585 + r38190603;
        double r38190605 = r38190595 + r38190592;
        double r38190606 = r38190593 * r38190586;
        double r38190607 = r38190606 / r38190582;
        double r38190608 = r38190605 + r38190607;
        double r38190609 = r38190604 / r38190608;
        double r38190610 = cbrt(r38190586);
        double r38190611 = r38190610 / r38190582;
        double r38190612 = r38190611 * r38190588;
        double r38190613 = r38190610 * r38190610;
        double r38190614 = r38190612 * r38190613;
        double r38190615 = r38190614 + r38190585;
        double r38190616 = r38190615 / r38190597;
        double r38190617 = r38190601 ? r38190609 : r38190616;
        double r38190618 = r38190584 ? r38190599 : r38190617;
        return r38190618;
}

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

Original17.2
Target13.7
Herbie13.5
\[\begin{array}{l} \mathbf{if}\;t \lt -1.365908536631008841640163147697088508132 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t \lt 3.036967103737245906066829435890093573122 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if t < -1.3564917764673116e-08

    1. Initial program 12.0

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

    2. Simplified4.6

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

    4. Applied div-inv4.7

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

    if -1.3564917764673116e-08 < t < 1.1697081758719511e+52

    1. Initial program 21.9

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

    if 1.1697081758719511e+52 < t

    1. Initial program 11.8

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

    2. Simplified3.5

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

    4. Applied *-un-lft-identity3.5

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

    5. Applied add-cube-cbrt3.7

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

    6. Applied times-frac3.7

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

    7. Applied associate-*l*3.1

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

  4. Final simplification13.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.356491776467311580293233615407758052562 \cdot 10^{-8}:\\ \;\;\;\;\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{1 + \left(\frac{y}{t} \cdot b + a\right)}\\ \mathbf{elif}\;t \le 1.169708175871951089353680636282670578711 \cdot 10^{52}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{b \cdot y}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\sqrt[3]{y}}{t} \cdot z\right) \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + x}{1 + \left(\frac{y}{t} \cdot b + a\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(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))))