Average Error: 10.0 → 1.4
Time: 13.5s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3.10185718647149596 \cdot 10^{-19} \lor \neg \left(z \le 2.3170079174536806 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{x}{t - a \cdot z} - y \cdot \frac{1}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y \cdot z}{t - z \cdot a}\\ \end{array}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -3.10185718647149596 \cdot 10^{-19} \lor \neg \left(z \le 2.3170079174536806 \cdot 10^{-12}\right):\\
\;\;\;\;\frac{x}{t - a \cdot z} - y \cdot \frac{1}{\frac{t}{z} - a}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r673699 = x;
        double r673700 = y;
        double r673701 = z;
        double r673702 = r673700 * r673701;
        double r673703 = r673699 - r673702;
        double r673704 = t;
        double r673705 = a;
        double r673706 = r673705 * r673701;
        double r673707 = r673704 - r673706;
        double r673708 = r673703 / r673707;
        return r673708;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r673709 = z;
        double r673710 = -3.101857186471496e-19;
        bool r673711 = r673709 <= r673710;
        double r673712 = 2.3170079174536806e-12;
        bool r673713 = r673709 <= r673712;
        double r673714 = !r673713;
        bool r673715 = r673711 || r673714;
        double r673716 = x;
        double r673717 = t;
        double r673718 = a;
        double r673719 = r673718 * r673709;
        double r673720 = r673717 - r673719;
        double r673721 = r673716 / r673720;
        double r673722 = y;
        double r673723 = 1.0;
        double r673724 = r673717 / r673709;
        double r673725 = r673724 - r673718;
        double r673726 = r673723 / r673725;
        double r673727 = r673722 * r673726;
        double r673728 = r673721 - r673727;
        double r673729 = r673722 * r673709;
        double r673730 = r673709 * r673718;
        double r673731 = r673717 - r673730;
        double r673732 = r673729 / r673731;
        double r673733 = r673721 - r673732;
        double r673734 = r673715 ? r673728 : r673733;
        return r673734;
}

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

Original10.0
Target1.5
Herbie1.4
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -3.101857186471496e-19 or 2.3170079174536806e-12 < z

    1. Initial program 19.4

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

    3. Applied div-sub19.4

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

    4. Simplified12.0

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

    6. Applied clear-num12.0

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

    7. Taylor expanded around 0 2.6

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

    if -3.101857186471496e-19 < z < 2.3170079174536806e-12

    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. Simplified3.1

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

    6. Applied associate-*r/0.1

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

  4. Final simplification1.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.10185718647149596 \cdot 10^{-19} \lor \neg \left(z \le 2.3170079174536806 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{x}{t - a \cdot z} - y \cdot \frac{1}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y \cdot z}{t - z \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 +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))))