Average Error: 10.6 → 2.0
Time: 3.7s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1226.1766839341781 \lor \neg \left(z \le 1.0351312592687676 \cdot 10^{85}\right):\\ \;\;\;\;x \cdot \frac{1}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{1}{\frac{t - a \cdot z}{y \cdot z}}\\ \end{array}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1226.1766839341781 \lor \neg \left(z \le 1.0351312592687676 \cdot 10^{85}\right):\\
\;\;\;\;x \cdot \frac{1}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r918283 = x;
        double r918284 = y;
        double r918285 = z;
        double r918286 = r918284 * r918285;
        double r918287 = r918283 - r918286;
        double r918288 = t;
        double r918289 = a;
        double r918290 = r918289 * r918285;
        double r918291 = r918288 - r918290;
        double r918292 = r918287 / r918291;
        return r918292;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r918293 = z;
        double r918294 = -1226.176683934178;
        bool r918295 = r918293 <= r918294;
        double r918296 = 1.0351312592687676e+85;
        bool r918297 = r918293 <= r918296;
        double r918298 = !r918297;
        bool r918299 = r918295 || r918298;
        double r918300 = x;
        double r918301 = 1.0;
        double r918302 = t;
        double r918303 = a;
        double r918304 = r918303 * r918293;
        double r918305 = r918302 - r918304;
        double r918306 = r918301 / r918305;
        double r918307 = r918300 * r918306;
        double r918308 = y;
        double r918309 = r918302 / r918293;
        double r918310 = r918309 - r918303;
        double r918311 = r918308 / r918310;
        double r918312 = r918307 - r918311;
        double r918313 = r918300 / r918305;
        double r918314 = r918308 * r918293;
        double r918315 = r918305 / r918314;
        double r918316 = r918301 / r918315;
        double r918317 = r918313 - r918316;
        double r918318 = r918299 ? r918312 : r918317;
        return r918318;
}

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

Original10.6
Target1.8
Herbie2.0
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1226.176683934178 or 1.0351312592687676e+85 < z

    1. Initial program 23.3

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

    3. Applied div-sub23.3

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

    5. Applied associate-/l*14.5

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

    7. Applied div-sub14.5

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

    8. Simplified3.2

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

    10. Applied div-inv3.2

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

    if -1226.176683934178 < z < 1.0351312592687676e+85

    1. Initial program 0.9

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

    3. Applied div-sub0.9

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

    5. Applied clear-num1.0

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

  4. Final simplification2.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1226.1766839341781 \lor \neg \left(z \le 1.0351312592687676 \cdot 10^{85}\right):\\ \;\;\;\;x \cdot \frac{1}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{1}{\frac{t - a \cdot z}{y \cdot z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 
(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))))