Average Error: 10.1 → 1.8
Time: 3.3s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -2.395869675467654933412145740613551377693 \cdot 10^{-120} \lor \neg \left(z \le 8.096739799392277088151833490233000866847 \cdot 10^{-97}\right):\\ \;\;\;\;\frac{x}{t - a \cdot z} - y \cdot \frac{1}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \end{array}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -2.395869675467654933412145740613551377693 \cdot 10^{-120} \lor \neg \left(z \le 8.096739799392277088151833490233000866847 \cdot 10^{-97}\right):\\
\;\;\;\;\frac{x}{t - a \cdot z} - y \cdot \frac{1}{\frac{t}{z} - a}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r672223 = x;
        double r672224 = y;
        double r672225 = z;
        double r672226 = r672224 * r672225;
        double r672227 = r672223 - r672226;
        double r672228 = t;
        double r672229 = a;
        double r672230 = r672229 * r672225;
        double r672231 = r672228 - r672230;
        double r672232 = r672227 / r672231;
        return r672232;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r672233 = z;
        double r672234 = -2.395869675467655e-120;
        bool r672235 = r672233 <= r672234;
        double r672236 = 8.096739799392277e-97;
        bool r672237 = r672233 <= r672236;
        double r672238 = !r672237;
        bool r672239 = r672235 || r672238;
        double r672240 = x;
        double r672241 = t;
        double r672242 = a;
        double r672243 = r672242 * r672233;
        double r672244 = r672241 - r672243;
        double r672245 = r672240 / r672244;
        double r672246 = y;
        double r672247 = 1.0;
        double r672248 = r672241 / r672233;
        double r672249 = r672248 - r672242;
        double r672250 = r672247 / r672249;
        double r672251 = r672246 * r672250;
        double r672252 = r672245 - r672251;
        double r672253 = r672246 * r672233;
        double r672254 = r672240 - r672253;
        double r672255 = r672247 / r672244;
        double r672256 = r672254 * r672255;
        double r672257 = r672239 ? r672252 : r672256;
        return r672257;
}

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
Target1.6
Herbie1.8
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958298856956410892592016 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -2.395869675467655e-120 or 8.096739799392277e-97 < z

    1. Initial program 14.9

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

    3. Applied div-sub14.9

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

    5. Applied associate-/l*9.6

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

    7. Applied *-un-lft-identity9.6

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

    8. Applied *-un-lft-identity9.6

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

    9. Applied times-frac9.6

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

    10. Simplified9.6

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

    11. Simplified2.4

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

    13. Applied div-inv2.5

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

    14. Simplified2.5

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

    if -2.395869675467655e-120 < z < 8.096739799392277e-97

    1. Initial program 0.1

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

    3. Applied div-inv0.3

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

  4. Final simplification1.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.395869675467654933412145740613551377693 \cdot 10^{-120} \lor \neg \left(z \le 8.096739799392277088151833490233000866847 \cdot 10^{-97}\right):\\ \;\;\;\;\frac{x}{t - a \cdot z} - y \cdot \frac{1}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019356 
(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))))