Average Error: 10.3 → 6.5
Time: 12.6s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -4.06401190083563283 \cdot 10^{38} \lor \neg \left(z \le 2.50952591182416396 \cdot 10^{-42}\right):\\ \;\;\;\;\frac{x}{t - a \cdot z} - y \cdot \frac{1}{\frac{t - a \cdot z}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y \cdot z}{t - a \cdot z}\\ \end{array}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -4.06401190083563283 \cdot 10^{38} \lor \neg \left(z \le 2.50952591182416396 \cdot 10^{-42}\right):\\
\;\;\;\;\frac{x}{t - a \cdot z} - y \cdot \frac{1}{\frac{t - a \cdot z}{z}}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r926419 = x;
        double r926420 = y;
        double r926421 = z;
        double r926422 = r926420 * r926421;
        double r926423 = r926419 - r926422;
        double r926424 = t;
        double r926425 = a;
        double r926426 = r926425 * r926421;
        double r926427 = r926424 - r926426;
        double r926428 = r926423 / r926427;
        return r926428;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r926429 = z;
        double r926430 = -4.064011900835633e+38;
        bool r926431 = r926429 <= r926430;
        double r926432 = 2.509525911824164e-42;
        bool r926433 = r926429 <= r926432;
        double r926434 = !r926433;
        bool r926435 = r926431 || r926434;
        double r926436 = x;
        double r926437 = t;
        double r926438 = a;
        double r926439 = r926438 * r926429;
        double r926440 = r926437 - r926439;
        double r926441 = r926436 / r926440;
        double r926442 = y;
        double r926443 = 1.0;
        double r926444 = r926440 / r926429;
        double r926445 = r926443 / r926444;
        double r926446 = r926442 * r926445;
        double r926447 = r926441 - r926446;
        double r926448 = r926442 * r926429;
        double r926449 = r926448 / r926440;
        double r926450 = r926441 - r926449;
        double r926451 = r926435 ? r926447 : r926450;
        return r926451;
}

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

Original10.3
Target1.7
Herbie6.5
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -4.064011900835633e+38 or 2.509525911824164e-42 < z

    1. Initial program 20.2

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

    3. Applied div-sub20.2

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

    4. Simplified12.7

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

    6. Applied clear-num12.8

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

    if -4.064011900835633e+38 < z < 2.509525911824164e-42

    1. Initial program 0.3

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

    3. Applied div-sub0.3

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

    4. Simplified2.9

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

    6. Applied associate-*r/0.3

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

  4. Final simplification6.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -4.06401190083563283 \cdot 10^{38} \lor \neg \left(z \le 2.50952591182416396 \cdot 10^{-42}\right):\\ \;\;\;\;\frac{x}{t - a \cdot z} - y \cdot \frac{1}{\frac{t - a \cdot z}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y \cdot z}{t - a \cdot z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 +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))))