Average Error: 10.7 → 1.7
Time: 4.7s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -4.494157337141472818401294462701317335445 \cdot 10^{-23} \lor \neg \left(z \le 3.575354698827828217766077832854277915677 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t - a \cdot z}{x - y \cdot z}}\\ \end{array}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -4.494157337141472818401294462701317335445 \cdot 10^{-23} \lor \neg \left(z \le 3.575354698827828217766077832854277915677 \cdot 10^{-14}\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 r350276 = x;
        double r350277 = y;
        double r350278 = z;
        double r350279 = r350277 * r350278;
        double r350280 = r350276 - r350279;
        double r350281 = t;
        double r350282 = a;
        double r350283 = r350282 * r350278;
        double r350284 = r350281 - r350283;
        double r350285 = r350280 / r350284;
        return r350285;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r350286 = z;
        double r350287 = -4.494157337141473e-23;
        bool r350288 = r350286 <= r350287;
        double r350289 = 3.575354698827828e-14;
        bool r350290 = r350286 <= r350289;
        double r350291 = !r350290;
        bool r350292 = r350288 || r350291;
        double r350293 = x;
        double r350294 = t;
        double r350295 = a;
        double r350296 = r350295 * r350286;
        double r350297 = r350294 - r350296;
        double r350298 = r350293 / r350297;
        double r350299 = y;
        double r350300 = r350294 / r350286;
        double r350301 = r350300 - r350295;
        double r350302 = r350299 / r350301;
        double r350303 = r350298 - r350302;
        double r350304 = 1.0;
        double r350305 = r350299 * r350286;
        double r350306 = r350293 - r350305;
        double r350307 = r350297 / r350306;
        double r350308 = r350304 / r350307;
        double r350309 = r350292 ? r350303 : r350308;
        return r350309;
}

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

Original10.7
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 < -4.494157337141473e-23 or 3.575354698827828e-14 < z

    1. Initial program 20.3

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

    3. Applied div-sub20.3

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

    5. Applied associate-/l*12.6

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

    7. Applied div-sub12.6

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

    8. Simplified2.7

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

    if -4.494157337141473e-23 < z < 3.575354698827828e-14

    1. Initial program 0.1

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

    3. Applied clear-num0.7

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

  4. Final simplification1.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -4.494157337141472818401294462701317335445 \cdot 10^{-23} \lor \neg \left(z \le 3.575354698827828217766077832854277915677 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t - a \cdot z}{x - y \cdot z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019318 
(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))))