Average Error: 10.0 → 1.7
Time: 11.8s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -9.2159596611983673 \cdot 10^{-15} \lor \neg \left(z \le 3.82302836807757522 \cdot 10^{-132}\right):\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t - a \cdot z}{x - z \cdot y}}\\ \end{array}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -9.2159596611983673 \cdot 10^{-15} \lor \neg \left(z \le 3.82302836807757522 \cdot 10^{-132}\right):\\
\;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r704011 = x;
        double r704012 = y;
        double r704013 = z;
        double r704014 = r704012 * r704013;
        double r704015 = r704011 - r704014;
        double r704016 = t;
        double r704017 = a;
        double r704018 = r704017 * r704013;
        double r704019 = r704016 - r704018;
        double r704020 = r704015 / r704019;
        return r704020;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r704021 = z;
        double r704022 = -9.215959661198367e-15;
        bool r704023 = r704021 <= r704022;
        double r704024 = 3.823028368077575e-132;
        bool r704025 = r704021 <= r704024;
        double r704026 = !r704025;
        bool r704027 = r704023 || r704026;
        double r704028 = x;
        double r704029 = t;
        double r704030 = a;
        double r704031 = r704030 * r704021;
        double r704032 = r704029 - r704031;
        double r704033 = r704028 / r704032;
        double r704034 = y;
        double r704035 = r704029 / r704021;
        double r704036 = r704035 - r704030;
        double r704037 = r704034 / r704036;
        double r704038 = r704033 - r704037;
        double r704039 = 1.0;
        double r704040 = r704021 * r704034;
        double r704041 = r704028 - r704040;
        double r704042 = r704032 / r704041;
        double r704043 = r704039 / r704042;
        double r704044 = r704027 ? r704038 : r704043;
        return r704044;
}

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
Herbie1.7
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -9.215959661198367e-15 or 3.823028368077575e-132 < z

    1. Initial program 16.4

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

    3. Applied div-sub16.4

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

    4. Simplified10.5

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

    6. Applied clear-num10.6

      \[\leadsto \frac{x}{t - a \cdot z} - y \cdot \color{blue}{\frac{1}{\frac{t - a \cdot z}{z}}}\]
    7. Using strategy rm

    8. Applied *-un-lft-identity10.6

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

    9. Applied associate-*l*10.6

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

    10. Simplified2.5

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

    if -9.215959661198367e-15 < z < 3.823028368077575e-132

    1. Initial program 0.1

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

    3. Applied clear-num0.5

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

    4. Simplified0.5

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

  4. Final simplification1.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -9.2159596611983673 \cdot 10^{-15} \lor \neg \left(z \le 3.82302836807757522 \cdot 10^{-132}\right):\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t - a \cdot z}{x - z \cdot y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(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))))