Average Error: 10.1 → 2.2
Time: 27.4s
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 r444198 = x;
        double r444199 = y;
        double r444200 = z;
        double r444201 = r444199 * r444200;
        double r444202 = r444198 - r444201;
        double r444203 = t;
        double r444204 = a;
        double r444205 = r444204 * r444200;
        double r444206 = r444203 - r444205;
        double r444207 = r444202 / r444206;
        return r444207;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r444208 = z;
        double r444209 = -1.0583634769447563e-274;
        bool r444210 = r444208 <= r444209;
        double r444211 = 1.0469961433143277e-51;
        bool r444212 = r444208 <= r444211;
        double r444213 = !r444212;
        bool r444214 = r444210 || r444213;
        double r444215 = x;
        double r444216 = t;
        double r444217 = a;
        double r444218 = r444217 * r444208;
        double r444219 = r444216 - r444218;
        double r444220 = r444215 / r444219;
        double r444221 = y;
        double r444222 = r444216 / r444208;
        double r444223 = r444222 - r444217;
        double r444224 = r444221 / r444223;
        double r444225 = r444220 - r444224;
        double r444226 = 1.0;
        double r444227 = r444221 * r444208;
        double r444228 = r444215 - r444227;
        double r444229 = r444219 / r444228;
        double r444230 = r444226 / r444229;
        double r444231 = r444214 ? r444225 : r444230;
        return r444231;
}

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))))