Average Error: 16.8 → 15.4
Time: 16.1s
Precision: 64
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -2.52525423349227925938522736634805759034 \cdot 10^{-76}:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{elif}\;t \le 2.593328778428211767273946321884821130118 \cdot 10^{87}:\\ \;\;\;\;\frac{x + \left(y \cdot z\right) \cdot \frac{1}{t}}{\left(a + 1\right) + \left(y \cdot b\right) \cdot \frac{1}{t}}\\ \mathbf{elif}\;t \le 3.087220841679641963278035732560275532756 \cdot 10^{150}:\\ \;\;\;\;\left(x + \frac{y \cdot z}{t}\right) \cdot \frac{1}{\left(a + 1\right) + \frac{b}{t} \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left(a + 1\right) + \frac{b \cdot y}{t}}{\frac{z}{\frac{t}{y}} + x}}\\ \end{array}\]
\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
↓
\begin{array}{l}
\mathbf{if}\;t \le -2.52525423349227925938522736634805759034 \cdot 10^{-76}:\\
\;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\

\mathbf{elif}\;t \le 2.593328778428211767273946321884821130118 \cdot 10^{87}:\\
\;\;\;\;\frac{x + \left(y \cdot z\right) \cdot \frac{1}{t}}{\left(a + 1\right) + \left(y \cdot b\right) \cdot \frac{1}{t}}\\

\mathbf{elif}\;t \le 3.087220841679641963278035732560275532756 \cdot 10^{150}:\\
\;\;\;\;\left(x + \frac{y \cdot z}{t}\right) \cdot \frac{1}{\left(a + 1\right) + \frac{b}{t} \cdot y}\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r612687 = x;
        double r612688 = y;
        double r612689 = z;
        double r612690 = r612688 * r612689;
        double r612691 = t;
        double r612692 = r612690 / r612691;
        double r612693 = r612687 + r612692;
        double r612694 = a;
        double r612695 = 1.0;
        double r612696 = r612694 + r612695;
        double r612697 = b;
        double r612698 = r612688 * r612697;
        double r612699 = r612698 / r612691;
        double r612700 = r612696 + r612699;
        double r612701 = r612693 / r612700;
        return r612701;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r612702 = t;
        double r612703 = -2.5252542334922793e-76;
        bool r612704 = r612702 <= r612703;
        double r612705 = x;
        double r612706 = y;
        double r612707 = z;
        double r612708 = r612707 / r612702;
        double r612709 = r612706 * r612708;
        double r612710 = r612705 + r612709;
        double r612711 = a;
        double r612712 = 1.0;
        double r612713 = r612711 + r612712;
        double r612714 = b;
        double r612715 = r612706 * r612714;
        double r612716 = r612715 / r612702;
        double r612717 = r612713 + r612716;
        double r612718 = r612710 / r612717;
        double r612719 = 2.5933287784282118e+87;
        bool r612720 = r612702 <= r612719;
        double r612721 = r612706 * r612707;
        double r612722 = 1.0;
        double r612723 = r612722 / r612702;
        double r612724 = r612721 * r612723;
        double r612725 = r612705 + r612724;
        double r612726 = r612715 * r612723;
        double r612727 = r612713 + r612726;
        double r612728 = r612725 / r612727;
        double r612729 = 3.087220841679642e+150;
        bool r612730 = r612702 <= r612729;
        double r612731 = r612721 / r612702;
        double r612732 = r612705 + r612731;
        double r612733 = r612714 / r612702;
        double r612734 = r612733 * r612706;
        double r612735 = r612713 + r612734;
        double r612736 = r612722 / r612735;
        double r612737 = r612732 * r612736;
        double r612738 = r612714 * r612706;
        double r612739 = r612738 / r612702;
        double r612740 = r612713 + r612739;
        double r612741 = r612702 / r612706;
        double r612742 = r612707 / r612741;
        double r612743 = r612742 + r612705;
        double r612744 = r612740 / r612743;
        double r612745 = r612722 / r612744;
        double r612746 = r612730 ? r612737 : r612745;
        double r612747 = r612720 ? r612728 : r612746;
        double r612748 = r612704 ? r612718 : r612747;
        return r612748;
}

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

Original16.8
Target13.5
Herbie15.4
\[\begin{array}{l} \mathbf{if}\;t \lt -1.365908536631008841640163147697088508132 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t \lt 3.036967103737245906066829435890093573122 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if t < -2.5252542334922793e-76

    1. Initial program 12.2

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

    3. Applied *-un-lft-identity12.2

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

    4. Applied times-frac9.8

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

    5. Simplified9.8

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

    if -2.5252542334922793e-76 < t < 2.5933287784282118e+87

    1. Initial program 21.9

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

    3. Applied div-inv21.9

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

    5. Applied div-inv21.9

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

    if 2.5933287784282118e+87 < t < 3.087220841679642e+150

    1. Initial program 9.3

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

    3. Applied div-inv9.4

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

    5. Applied div-inv9.4

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

    6. Simplified6.9

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

    if 3.087220841679642e+150 < t

    1. Initial program 12.0

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

    3. Applied div-inv12.0

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

    5. Applied div-inv12.0

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

    7. Applied clear-num12.3

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

    8. Simplified8.5

      \[\leadsto \frac{1}{\color{blue}{\frac{\left(a + 1\right) + \frac{b \cdot y}{t}}{\frac{z}{\frac{t}{y}} + x}}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification15.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.52525423349227925938522736634805759034 \cdot 10^{-76}:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{elif}\;t \le 2.593328778428211767273946321884821130118 \cdot 10^{87}:\\ \;\;\;\;\frac{x + \left(y \cdot z\right) \cdot \frac{1}{t}}{\left(a + 1\right) + \left(y \cdot b\right) \cdot \frac{1}{t}}\\ \mathbf{elif}\;t \le 3.087220841679641963278035732560275532756 \cdot 10^{150}:\\ \;\;\;\;\left(x + \frac{y \cdot z}{t}\right) \cdot \frac{1}{\left(a + 1\right) + \frac{b}{t} \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left(a + 1\right) + \frac{b \cdot y}{t}}{\frac{z}{\frac{t}{y}} + x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 
(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))))