Average Error: 10.7 → 1.6
Time: 26.7s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.247902517758596795481298690172105506598 \cdot 10^{-85} \lor \neg \left(z \le 1.035834224912832515763660978791449451819 \cdot 10^{-4}\right):\\ \;\;\;\;\frac{1}{\frac{t - a \cdot z}{x}} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \end{array}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.247902517758596795481298690172105506598 \cdot 10^{-85} \lor \neg \left(z \le 1.035834224912832515763660978791449451819 \cdot 10^{-4}\right):\\
\;\;\;\;\frac{1}{\frac{t - a \cdot z}{x}} - \frac{y}{\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 r388264 = x;
        double r388265 = y;
        double r388266 = z;
        double r388267 = r388265 * r388266;
        double r388268 = r388264 - r388267;
        double r388269 = t;
        double r388270 = a;
        double r388271 = r388270 * r388266;
        double r388272 = r388269 - r388271;
        double r388273 = r388268 / r388272;
        return r388273;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r388274 = z;
        double r388275 = -1.2479025177585968e-85;
        bool r388276 = r388274 <= r388275;
        double r388277 = 0.00010358342249128325;
        bool r388278 = r388274 <= r388277;
        double r388279 = !r388278;
        bool r388280 = r388276 || r388279;
        double r388281 = 1.0;
        double r388282 = t;
        double r388283 = a;
        double r388284 = r388283 * r388274;
        double r388285 = r388282 - r388284;
        double r388286 = x;
        double r388287 = r388285 / r388286;
        double r388288 = r388281 / r388287;
        double r388289 = y;
        double r388290 = r388282 / r388274;
        double r388291 = r388290 - r388283;
        double r388292 = r388289 / r388291;
        double r388293 = r388288 - r388292;
        double r388294 = r388289 * r388274;
        double r388295 = r388286 - r388294;
        double r388296 = r388281 / r388285;
        double r388297 = r388295 * r388296;
        double r388298 = r388280 ? r388293 : r388297;
        return r388298;
}

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

Derivation

  1. Split input into 2 regimes

  2. if z < -1.2479025177585968e-85 or 0.00010358342249128325 < z

    1. Initial program 18.6

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

    3. Applied div-sub18.6

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

    4. Simplified11.5

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

    6. Applied pow111.5

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

    7. Applied pow111.5

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

    8. Applied pow-prod-down11.5

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

    9. Simplified2.5

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

    11. Applied clear-num2.6

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

    if -1.2479025177585968e-85 < z < 0.00010358342249128325

    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 simplification1.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.247902517758596795481298690172105506598 \cdot 10^{-85} \lor \neg \left(z \le 1.035834224912832515763660978791449451819 \cdot 10^{-4}\right):\\ \;\;\;\;\frac{1}{\frac{t - a \cdot z}{x}} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 +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))))