Average Error: 10.7 → 7.6
Time: 11.4s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - a \cdot z} \le -2.388206450000805047309333686608094243085 \cdot 10^{257}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - z \cdot \frac{y}{t - a \cdot z}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - a \cdot z} \le 4.407308869629805887214722361549895283734 \cdot 10^{165}:\\ \;\;\;\;\frac{1}{\frac{t - a \cdot z}{x - y \cdot z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{z}{\frac{t - a \cdot z}{y}}\\ \end{array}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x - y \cdot z}{t - a \cdot z} \le -2.388206450000805047309333686608094243085 \cdot 10^{257}:\\
\;\;\;\;\frac{x}{t - a \cdot z} - z \cdot \frac{y}{t - a \cdot z}\\

\mathbf{elif}\;\frac{x - y \cdot z}{t - a \cdot z} \le 4.407308869629805887214722361549895283734 \cdot 10^{165}:\\
\;\;\;\;\frac{1}{\frac{t - a \cdot z}{x - y \cdot z}}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r1732440 = x;
        double r1732441 = y;
        double r1732442 = z;
        double r1732443 = r1732441 * r1732442;
        double r1732444 = r1732440 - r1732443;
        double r1732445 = t;
        double r1732446 = a;
        double r1732447 = r1732446 * r1732442;
        double r1732448 = r1732445 - r1732447;
        double r1732449 = r1732444 / r1732448;
        return r1732449;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r1732450 = x;
        double r1732451 = y;
        double r1732452 = z;
        double r1732453 = r1732451 * r1732452;
        double r1732454 = r1732450 - r1732453;
        double r1732455 = t;
        double r1732456 = a;
        double r1732457 = r1732456 * r1732452;
        double r1732458 = r1732455 - r1732457;
        double r1732459 = r1732454 / r1732458;
        double r1732460 = -2.388206450000805e+257;
        bool r1732461 = r1732459 <= r1732460;
        double r1732462 = r1732450 / r1732458;
        double r1732463 = r1732451 / r1732458;
        double r1732464 = r1732452 * r1732463;
        double r1732465 = r1732462 - r1732464;
        double r1732466 = 4.407308869629806e+165;
        bool r1732467 = r1732459 <= r1732466;
        double r1732468 = 1.0;
        double r1732469 = r1732458 / r1732454;
        double r1732470 = r1732468 / r1732469;
        double r1732471 = r1732458 / r1732451;
        double r1732472 = r1732452 / r1732471;
        double r1732473 = r1732462 - r1732472;
        double r1732474 = r1732467 ? r1732470 : r1732473;
        double r1732475 = r1732461 ? r1732465 : r1732474;
        return r1732475;
}

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
Target1.8
Herbie7.6
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958298856956410892592016 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (- x (* y z)) (- t (* a z))) < -2.388206450000805e+257

    1. Initial program 42.7

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

    3. Applied div-sub42.7

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

    4. Simplified42.7

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

    6. Applied *-un-lft-identity42.7

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

    7. Applied times-frac5.5

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

    8. Simplified5.5

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

    if -2.388206450000805e+257 < (/ (- x (* y z)) (- t (* a z))) < 4.407308869629806e+165

    1. Initial program 4.7

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

    3. Applied clear-num5.1

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

    if 4.407308869629806e+165 < (/ (- x (* y z)) (- t (* a z)))

    1. Initial program 35.7

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

    3. Applied div-sub35.7

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

    4. Simplified35.7

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

    6. Applied associate-/l*23.3

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

  4. Final simplification7.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - a \cdot z} \le -2.388206450000805047309333686608094243085 \cdot 10^{257}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - z \cdot \frac{y}{t - a \cdot z}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - a \cdot z} \le 4.407308869629805887214722361549895283734 \cdot 10^{165}:\\ \;\;\;\;\frac{1}{\frac{t - a \cdot z}{x - y \cdot z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{z}{\frac{t - a \cdot z}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 
(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))))