Average Error: 10.5 → 2.1
Time: 3.9s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.06850652642206781476028426266944656188 \cdot 10^{-138} \lor \neg \left(z \le 2.010729888565966141653647925259485457877 \cdot 10^{-131}\right):\\ \;\;\;\;\frac{1}{\frac{t - a \cdot z}{x}} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y \cdot z}{\left(\sqrt[3]{t - a \cdot z} \cdot \sqrt[3]{t - a \cdot z}\right) \cdot \sqrt[3]{t - a \cdot z}}\\ \end{array}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.06850652642206781476028426266944656188 \cdot 10^{-138} \lor \neg \left(z \le 2.010729888565966141653647925259485457877 \cdot 10^{-131}\right):\\
\;\;\;\;\frac{1}{\frac{t - a \cdot z}{x}} - \frac{y}{\frac{t}{z} - a}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r727698 = x;
        double r727699 = y;
        double r727700 = z;
        double r727701 = r727699 * r727700;
        double r727702 = r727698 - r727701;
        double r727703 = t;
        double r727704 = a;
        double r727705 = r727704 * r727700;
        double r727706 = r727703 - r727705;
        double r727707 = r727702 / r727706;
        return r727707;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r727708 = z;
        double r727709 = -1.0685065264220678e-138;
        bool r727710 = r727708 <= r727709;
        double r727711 = 2.0107298885659661e-131;
        bool r727712 = r727708 <= r727711;
        double r727713 = !r727712;
        bool r727714 = r727710 || r727713;
        double r727715 = 1.0;
        double r727716 = t;
        double r727717 = a;
        double r727718 = r727717 * r727708;
        double r727719 = r727716 - r727718;
        double r727720 = x;
        double r727721 = r727719 / r727720;
        double r727722 = r727715 / r727721;
        double r727723 = y;
        double r727724 = r727716 / r727708;
        double r727725 = r727724 - r727717;
        double r727726 = r727723 / r727725;
        double r727727 = r727722 - r727726;
        double r727728 = r727720 / r727719;
        double r727729 = r727723 * r727708;
        double r727730 = cbrt(r727719);
        double r727731 = r727730 * r727730;
        double r727732 = r727731 * r727730;
        double r727733 = r727729 / r727732;
        double r727734 = r727728 - r727733;
        double r727735 = r727714 ? r727727 : r727734;
        return r727735;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original10.5
Target1.7
Herbie2.1
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958298856956410892592016 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.0685065264220678e-138 or 2.0107298885659661e-131 < z

    1. Initial program 14.5

      \[\frac{x - y \cdot z}{t - a \cdot z}\]
    2. Using strategy rm

    3. Applied div-sub14.5

      \[\leadsto \color{blue}{\frac{x}{t - a \cdot z} - \frac{y \cdot z}{t - a \cdot z}}\]
    4. Using strategy rm

    5. Applied associate-/l*9.5

      \[\leadsto \frac{x}{t - a \cdot z} - \color{blue}{\frac{y}{\frac{t - a \cdot z}{z}}}\]
    6. Using strategy rm

    7. Applied div-sub9.5

      \[\leadsto \frac{x}{t - a \cdot z} - \frac{y}{\color{blue}{\frac{t}{z} - \frac{a \cdot z}{z}}}\]

    8. Simplified2.7

      \[\leadsto \frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - \color{blue}{a}}\]
    9. Using strategy rm

    10. Applied clear-num2.8

      \[\leadsto \color{blue}{\frac{1}{\frac{t - a \cdot z}{x}}} - \frac{y}{\frac{t}{z} - a}\]

    if -1.0685065264220678e-138 < z < 2.0107298885659661e-131

    1. Initial program 0.1

      \[\frac{x - y \cdot z}{t - a \cdot z}\]
    2. Using strategy rm

    3. Applied div-sub0.1

      \[\leadsto \color{blue}{\frac{x}{t - a \cdot z} - \frac{y \cdot z}{t - a \cdot z}}\]
    4. Using strategy rm

    5. Applied add-cube-cbrt0.4

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

  4. Final simplification2.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.06850652642206781476028426266944656188 \cdot 10^{-138} \lor \neg \left(z \le 2.010729888565966141653647925259485457877 \cdot 10^{-131}\right):\\ \;\;\;\;\frac{1}{\frac{t - a \cdot z}{x}} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y \cdot z}{\left(\sqrt[3]{t - a \cdot z} \cdot \sqrt[3]{t - a \cdot z}\right) \cdot \sqrt[3]{t - a \cdot z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A"
  :precision binary64

  :herbie-target
  (if (< z -32113435955957344) (- (/ x (- t (* a z))) (/ y (- (/ t z) a))) (if (< z 3.5139522372978296e-86) (* (- x (* y z)) (/ 1 (- t (* a z)))) (- (/ x (- t (* a z))) (/ y (- (/ t z) a)))))

  (/ (- x (* y z)) (- t (* a z))))