Average Error: 10.8 → 1.7
Time: 27.7s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3.162433206185249868436393548434857618665 \cdot 10^{-101} \lor \neg \left(z \le 54920090.66638733446598052978515625\right):\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t - a \cdot z}{x - y \cdot z}}\\ \end{array}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -3.162433206185249868436393548434857618665 \cdot 10^{-101} \lor \neg \left(z \le 54920090.66638733446598052978515625\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 r412311 = x;
        double r412312 = y;
        double r412313 = z;
        double r412314 = r412312 * r412313;
        double r412315 = r412311 - r412314;
        double r412316 = t;
        double r412317 = a;
        double r412318 = r412317 * r412313;
        double r412319 = r412316 - r412318;
        double r412320 = r412315 / r412319;
        return r412320;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r412321 = z;
        double r412322 = -3.16243320618525e-101;
        bool r412323 = r412321 <= r412322;
        double r412324 = 54920090.666387334;
        bool r412325 = r412321 <= r412324;
        double r412326 = !r412325;
        bool r412327 = r412323 || r412326;
        double r412328 = x;
        double r412329 = t;
        double r412330 = a;
        double r412331 = r412330 * r412321;
        double r412332 = r412329 - r412331;
        double r412333 = r412328 / r412332;
        double r412334 = y;
        double r412335 = r412329 / r412321;
        double r412336 = r412335 - r412330;
        double r412337 = r412334 / r412336;
        double r412338 = r412333 - r412337;
        double r412339 = 1.0;
        double r412340 = r412334 * r412321;
        double r412341 = r412328 - r412340;
        double r412342 = r412332 / r412341;
        double r412343 = r412339 / r412342;
        double r412344 = r412327 ? r412338 : r412343;
        return r412344;
}

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

Original10.8
Target1.6
Herbie1.7
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958298856956410892592016 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -3.16243320618525e-101 or 54920090.666387334 < z

    1. Initial program 18.4

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

    3. Applied div-sub18.4

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

    4. Simplified11.9

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

    6. Applied clear-num12.0

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

    8. Applied pow112.0

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

    9. Applied pow112.0

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

    10. Applied pow-prod-down12.0

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

    11. Simplified2.5

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

    if -3.16243320618525e-101 < z < 54920090.666387334

    1. Initial program 0.1

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

    3. Applied clear-num0.6

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

  4. Final simplification1.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.162433206185249868436393548434857618665 \cdot 10^{-101} \lor \neg \left(z \le 54920090.66638733446598052978515625\right):\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t - a \cdot z}{x - y \cdot z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 +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 3.51395223729782958e-86) (* (- x (* y z)) (/ 1 (- t (* a z)))) (- (/ x (- t (* a z))) (/ y (- (/ t z) a)))))

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