Average Error: 10.3 → 1.6
Time: 13.8s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -2.147501379453681047725513588483459387151 \cdot 10^{-37} \lor \neg \left(z \le 6.9713412473177113437097495394906217014 \cdot 10^{-63}\right):\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y \cdot z}{t - a \cdot z}\\ \end{array}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -2.147501379453681047725513588483459387151 \cdot 10^{-37} \lor \neg \left(z \le 6.9713412473177113437097495394906217014 \cdot 10^{-63}\right):\\
\;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r752621 = x;
        double r752622 = y;
        double r752623 = z;
        double r752624 = r752622 * r752623;
        double r752625 = r752621 - r752624;
        double r752626 = t;
        double r752627 = a;
        double r752628 = r752627 * r752623;
        double r752629 = r752626 - r752628;
        double r752630 = r752625 / r752629;
        return r752630;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r752631 = z;
        double r752632 = -2.147501379453681e-37;
        bool r752633 = r752631 <= r752632;
        double r752634 = 6.971341247317711e-63;
        bool r752635 = r752631 <= r752634;
        double r752636 = !r752635;
        bool r752637 = r752633 || r752636;
        double r752638 = x;
        double r752639 = t;
        double r752640 = a;
        double r752641 = r752640 * r752631;
        double r752642 = r752639 - r752641;
        double r752643 = r752638 / r752642;
        double r752644 = y;
        double r752645 = r752639 / r752631;
        double r752646 = r752645 - r752640;
        double r752647 = r752644 / r752646;
        double r752648 = r752643 - r752647;
        double r752649 = r752644 * r752631;
        double r752650 = r752638 - r752649;
        double r752651 = r752650 / r752642;
        double r752652 = r752637 ? r752648 : r752651;
        return r752652;
}

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

Original10.3
Target1.8
Herbie1.6
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958298856956410892592016 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -2.147501379453681e-37 or 6.971341247317711e-63 < z

    1. Initial program 17.8

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

    3. Applied div-sub17.8

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

    4. Simplified11.6

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

    6. Applied *-un-lft-identity11.6

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

    7. Applied associate-*l*11.6

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

    8. Simplified2.8

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

    if -2.147501379453681e-37 < z < 6.971341247317711e-63

    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. Simplified3.2

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

    6. Applied associate-*r/0.1

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

    7. Applied sub-div0.1

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

  4. Final simplification1.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.147501379453681047725513588483459387151 \cdot 10^{-37} \lor \neg \left(z \le 6.9713412473177113437097495394906217014 \cdot 10^{-63}\right):\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y \cdot z}{t - a \cdot z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 
(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))))