Average Error: 10.3 → 1.8
Time: 3.3s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -5.31566099741130341 \cdot 10^{-17} \lor \neg \left(z \le 0.014498322822577354\right):\\ \;\;\;\;\frac{1}{\sqrt[3]{t - a \cdot z} \cdot \sqrt[3]{t - a \cdot z}} \cdot \frac{x}{\sqrt[3]{t - a \cdot z}} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t - a \cdot z}{x - y \cdot z}}\\ \end{array}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -5.31566099741130341 \cdot 10^{-17} \lor \neg \left(z \le 0.014498322822577354\right):\\
\;\;\;\;\frac{1}{\sqrt[3]{t - a \cdot z} \cdot \sqrt[3]{t - a \cdot z}} \cdot \frac{x}{\sqrt[3]{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 r639492 = x;
        double r639493 = y;
        double r639494 = z;
        double r639495 = r639493 * r639494;
        double r639496 = r639492 - r639495;
        double r639497 = t;
        double r639498 = a;
        double r639499 = r639498 * r639494;
        double r639500 = r639497 - r639499;
        double r639501 = r639496 / r639500;
        return r639501;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r639502 = z;
        double r639503 = -5.3156609974113034e-17;
        bool r639504 = r639502 <= r639503;
        double r639505 = 0.014498322822577354;
        bool r639506 = r639502 <= r639505;
        double r639507 = !r639506;
        bool r639508 = r639504 || r639507;
        double r639509 = 1.0;
        double r639510 = t;
        double r639511 = a;
        double r639512 = r639511 * r639502;
        double r639513 = r639510 - r639512;
        double r639514 = cbrt(r639513);
        double r639515 = r639514 * r639514;
        double r639516 = r639509 / r639515;
        double r639517 = x;
        double r639518 = r639517 / r639514;
        double r639519 = r639516 * r639518;
        double r639520 = y;
        double r639521 = r639510 / r639502;
        double r639522 = r639521 - r639511;
        double r639523 = r639520 / r639522;
        double r639524 = r639519 - r639523;
        double r639525 = r639520 * r639502;
        double r639526 = r639517 - r639525;
        double r639527 = r639513 / r639526;
        double r639528 = r639509 / r639527;
        double r639529 = r639508 ? r639524 : r639528;
        return r639529;
}

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

Original10.3
Target1.7
Herbie1.8
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -5.3156609974113034e-17 or 0.014498322822577354 < z

    1. Initial program 19.8

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

    3. Applied div-sub19.8

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

    5. Applied associate-/l*12.3

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

    7. Applied div-sub12.3

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

    8. Simplified2.7

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

    10. Applied add-cube-cbrt3.0

      \[\leadsto \frac{x}{\color{blue}{\left(\sqrt[3]{t - a \cdot z} \cdot \sqrt[3]{t - a \cdot z}\right) \cdot \sqrt[3]{t - a \cdot z}}} - \frac{y}{\frac{t}{z} - a}\]

    11. Applied *-un-lft-identity3.0

      \[\leadsto \frac{\color{blue}{1 \cdot x}}{\left(\sqrt[3]{t - a \cdot z} \cdot \sqrt[3]{t - a \cdot z}\right) \cdot \sqrt[3]{t - a \cdot z}} - \frac{y}{\frac{t}{z} - a}\]

    12. Applied times-frac3.0

      \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{t - a \cdot z} \cdot \sqrt[3]{t - a \cdot z}} \cdot \frac{x}{\sqrt[3]{t - a \cdot z}}} - \frac{y}{\frac{t}{z} - a}\]

    if -5.3156609974113034e-17 < z < 0.014498322822577354

    1. Initial program 0.1

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

    3. Applied clear-num0.6

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

  4. Final simplification1.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -5.31566099741130341 \cdot 10^{-17} \lor \neg \left(z \le 0.014498322822577354\right):\\ \;\;\;\;\frac{1}{\sqrt[3]{t - a \cdot z} \cdot \sqrt[3]{t - a \cdot z}} \cdot \frac{x}{\sqrt[3]{t - a \cdot z}} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t - a \cdot z}{x - y \cdot z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 
(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))))