Average Error: 10.1 → 2.2
Time: 23.4s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.058363476944756329672924814755261411617 \cdot 10^{-274} \lor \neg \left(z \le 1.046996143314327748479072054949128759018 \cdot 10^{-51}\right):\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t - a \cdot z}{x - y \cdot z}}\\ \end{array}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.058363476944756329672924814755261411617 \cdot 10^{-274} \lor \neg \left(z \le 1.046996143314327748479072054949128759018 \cdot 10^{-51}\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 r476299 = x;
        double r476300 = y;
        double r476301 = z;
        double r476302 = r476300 * r476301;
        double r476303 = r476299 - r476302;
        double r476304 = t;
        double r476305 = a;
        double r476306 = r476305 * r476301;
        double r476307 = r476304 - r476306;
        double r476308 = r476303 / r476307;
        return r476308;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r476309 = z;
        double r476310 = -1.0583634769447563e-274;
        bool r476311 = r476309 <= r476310;
        double r476312 = 1.0469961433143277e-51;
        bool r476313 = r476309 <= r476312;
        double r476314 = !r476313;
        bool r476315 = r476311 || r476314;
        double r476316 = x;
        double r476317 = t;
        double r476318 = a;
        double r476319 = r476318 * r476309;
        double r476320 = r476317 - r476319;
        double r476321 = r476316 / r476320;
        double r476322 = y;
        double r476323 = r476317 / r476309;
        double r476324 = r476323 - r476318;
        double r476325 = r476322 / r476324;
        double r476326 = r476321 - r476325;
        double r476327 = 1.0;
        double r476328 = r476322 * r476309;
        double r476329 = r476316 - r476328;
        double r476330 = r476320 / r476329;
        double r476331 = r476327 / r476330;
        double r476332 = r476315 ? r476326 : r476331;
        return r476332;
}

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

Original10.1
Target1.7
Herbie2.2
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958298856956410892592016 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.0583634769447563e-274 or 1.0469961433143277e-51 < z

    1. Initial program 13.2

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

    3. Applied div-sub13.2

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

    4. Simplified9.1

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

    6. Applied pow19.1

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

    7. Applied pow19.1

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

    8. Applied pow-prod-down9.1

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

    9. Simplified2.7

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

    if -1.0583634769447563e-274 < z < 1.0469961433143277e-51

    1. Initial program 0.1

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

    3. Applied clear-num0.5

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

  4. Final simplification2.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.058363476944756329672924814755261411617 \cdot 10^{-274} \lor \neg \left(z \le 1.046996143314327748479072054949128759018 \cdot 10^{-51}\right):\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t - a \cdot z}{x - y \cdot z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +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))))