Average Error: 10.1 → 2.2
Time: 24.3s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.058363476944756329672924814755261411617 \cdot 10^{-274} \lor \neg \left(z \le 1.046996143314327748479072054949128759018 \cdot 10^{-51}\right):\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t - a \cdot z}{x - y \cdot z}}\\ \end{array}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.058363476944756329672924814755261411617 \cdot 10^{-274} \lor \neg \left(z \le 1.046996143314327748479072054949128759018 \cdot 10^{-51}\right):\\
\;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\frac{t - a \cdot z}{x - y \cdot z}}\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r475987 = x;
        double r475988 = y;
        double r475989 = z;
        double r475990 = r475988 * r475989;
        double r475991 = r475987 - r475990;
        double r475992 = t;
        double r475993 = a;
        double r475994 = r475993 * r475989;
        double r475995 = r475992 - r475994;
        double r475996 = r475991 / r475995;
        return r475996;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r475997 = z;
        double r475998 = -1.0583634769447563e-274;
        bool r475999 = r475997 <= r475998;
        double r476000 = 1.0469961433143277e-51;
        bool r476001 = r475997 <= r476000;
        double r476002 = !r476001;
        bool r476003 = r475999 || r476002;
        double r476004 = x;
        double r476005 = t;
        double r476006 = a;
        double r476007 = r476006 * r475997;
        double r476008 = r476005 - r476007;
        double r476009 = r476004 / r476008;
        double r476010 = y;
        double r476011 = r476005 / r475997;
        double r476012 = r476011 - r476006;
        double r476013 = r476010 / r476012;
        double r476014 = r476009 - r476013;
        double r476015 = 1.0;
        double r476016 = r476010 * r475997;
        double r476017 = r476004 - r476016;
        double r476018 = r476008 / r476017;
        double r476019 = r476015 / r476018;
        double r476020 = r476003 ? r476014 : r476019;
        return r476020;
}

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

Original10.1
Target1.7
Herbie2.2
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958298856956410892592016 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.0583634769447563e-274 or 1.0469961433143277e-51 < z

    1. Initial program 13.2

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

    3. Applied div-sub13.2

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

    4. Simplified9.1

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

    6. Applied *-un-lft-identity9.1

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

    7. Applied associate-*l*9.1

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

    8. Simplified2.7

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

    if -1.0583634769447563e-274 < z < 1.0469961433143277e-51

    1. Initial program 0.1

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

    3. Applied clear-num0.5

      \[\leadsto \color{blue}{\frac{1}{\frac{t - a \cdot z}{x - y \cdot z}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.058363476944756329672924814755261411617 \cdot 10^{-274} \lor \neg \left(z \le 1.046996143314327748479072054949128759018 \cdot 10^{-51}\right):\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t - a \cdot z}{x - y \cdot z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(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))))