Average Error: 16.6 → 12.9
Time: 12.9s
Precision: 64
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -3.591884767125234007189270192734521744478 \cdot 10^{-16}:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + \frac{1}{\frac{\frac{t}{y}}{b}}}\\ \mathbf{elif}\;t \le 3.078693643875039956031098367503131499945 \cdot 10^{59}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{\frac{1}{t}}{\frac{1}{y \cdot b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{\left(a + 1\right) + \frac{1}{\frac{\frac{t}{y}}{b}}}\\ \end{array}\]
\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
↓
\begin{array}{l}
\mathbf{if}\;t \le -3.591884767125234007189270192734521744478 \cdot 10^{-16}:\\
\;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + \frac{1}{\frac{\frac{t}{y}}{b}}}\\

\mathbf{elif}\;t \le 3.078693643875039956031098367503131499945 \cdot 10^{59}:\\
\;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{\frac{1}{t}}{\frac{1}{y \cdot b}}}\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r468778 = x;
        double r468779 = y;
        double r468780 = z;
        double r468781 = r468779 * r468780;
        double r468782 = t;
        double r468783 = r468781 / r468782;
        double r468784 = r468778 + r468783;
        double r468785 = a;
        double r468786 = 1.0;
        double r468787 = r468785 + r468786;
        double r468788 = b;
        double r468789 = r468779 * r468788;
        double r468790 = r468789 / r468782;
        double r468791 = r468787 + r468790;
        double r468792 = r468784 / r468791;
        return r468792;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r468793 = t;
        double r468794 = -3.591884767125234e-16;
        bool r468795 = r468793 <= r468794;
        double r468796 = x;
        double r468797 = y;
        double r468798 = z;
        double r468799 = r468798 / r468793;
        double r468800 = r468797 * r468799;
        double r468801 = r468796 + r468800;
        double r468802 = a;
        double r468803 = 1.0;
        double r468804 = r468802 + r468803;
        double r468805 = 1.0;
        double r468806 = r468793 / r468797;
        double r468807 = b;
        double r468808 = r468806 / r468807;
        double r468809 = r468805 / r468808;
        double r468810 = r468804 + r468809;
        double r468811 = r468801 / r468810;
        double r468812 = 3.07869364387504e+59;
        bool r468813 = r468793 <= r468812;
        double r468814 = r468797 * r468798;
        double r468815 = r468814 / r468793;
        double r468816 = r468796 + r468815;
        double r468817 = r468805 / r468793;
        double r468818 = r468797 * r468807;
        double r468819 = r468805 / r468818;
        double r468820 = r468817 / r468819;
        double r468821 = r468804 + r468820;
        double r468822 = r468816 / r468821;
        double r468823 = r468793 / r468798;
        double r468824 = r468797 / r468823;
        double r468825 = r468796 + r468824;
        double r468826 = r468825 / r468810;
        double r468827 = r468813 ? r468822 : r468826;
        double r468828 = r468795 ? r468811 : r468827;
        return r468828;
}

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

Original16.6
Target13.5
Herbie12.9
\[\begin{array}{l} \mathbf{if}\;t \lt -1.365908536631008841640163147697088508132 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t \lt 3.036967103737245906066829435890093573122 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if t < -3.591884767125234e-16

    1. Initial program 12.3

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

    3. Applied clear-num12.3

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

    5. Applied associate-/r*9.6

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

    7. Applied *-un-lft-identity9.6

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

    8. Applied times-frac5.0

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

    9. Simplified5.0

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

    if -3.591884767125234e-16 < t < 3.07869364387504e+59

    1. Initial program 20.6

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

    3. Applied clear-num20.7

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

    5. Applied div-inv20.7

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

    6. Applied associate-/r*20.7

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

    if 3.07869364387504e+59 < t

    1. Initial program 12.1

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

    3. Applied clear-num12.1

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

    5. Applied associate-/r*8.5

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

    7. Applied associate-/l*3.2

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

  4. Final simplification12.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -3.591884767125234007189270192734521744478 \cdot 10^{-16}:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + \frac{1}{\frac{\frac{t}{y}}{b}}}\\ \mathbf{elif}\;t \le 3.078693643875039956031098367503131499945 \cdot 10^{59}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{\frac{1}{t}}{\frac{1}{y \cdot b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{\left(a + 1\right) + \frac{1}{\frac{\frac{t}{y}}{b}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(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))))