Average Error: 10.2 → 2.4
Time: 26.8s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -7.433845130127019277921365656087791646311 \cdot 10^{-258}:\\ \;\;\;\;x \cdot \frac{1}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \le 6.637318488869211541788788475321548606226 \cdot 10^{-120}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \left(\frac{\sqrt[3]{y}}{\frac{\sqrt[3]{t - a \cdot z}}{\sqrt[3]{z}}} \cdot \frac{\sqrt[3]{y}}{\frac{\sqrt[3]{t - a \cdot z}}{\sqrt[3]{z}}}\right) \cdot \frac{\sqrt[3]{y}}{\frac{\sqrt[3]{t - a \cdot z}}{\sqrt[3]{z}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -7.433845130127019277921365656087791646311 \cdot 10^{-258}:\\
\;\;\;\;x \cdot \frac{1}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\

\mathbf{elif}\;z \le 6.637318488869211541788788475321548606226 \cdot 10^{-120}:\\
\;\;\;\;\frac{x}{t - a \cdot z} - \left(\frac{\sqrt[3]{y}}{\frac{\sqrt[3]{t - a \cdot z}}{\sqrt[3]{z}}} \cdot \frac{\sqrt[3]{y}}{\frac{\sqrt[3]{t - a \cdot z}}{\sqrt[3]{z}}}\right) \cdot \frac{\sqrt[3]{y}}{\frac{\sqrt[3]{t - a \cdot z}}{\sqrt[3]{z}}}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r126108848 = x;
        double r126108849 = y;
        double r126108850 = z;
        double r126108851 = r126108849 * r126108850;
        double r126108852 = r126108848 - r126108851;
        double r126108853 = t;
        double r126108854 = a;
        double r126108855 = r126108854 * r126108850;
        double r126108856 = r126108853 - r126108855;
        double r126108857 = r126108852 / r126108856;
        return r126108857;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r126108858 = z;
        double r126108859 = -7.433845130127019e-258;
        bool r126108860 = r126108858 <= r126108859;
        double r126108861 = x;
        double r126108862 = 1.0;
        double r126108863 = t;
        double r126108864 = a;
        double r126108865 = r126108864 * r126108858;
        double r126108866 = r126108863 - r126108865;
        double r126108867 = r126108862 / r126108866;
        double r126108868 = r126108861 * r126108867;
        double r126108869 = y;
        double r126108870 = r126108863 / r126108858;
        double r126108871 = r126108870 - r126108864;
        double r126108872 = r126108869 / r126108871;
        double r126108873 = r126108868 - r126108872;
        double r126108874 = 6.637318488869212e-120;
        bool r126108875 = r126108858 <= r126108874;
        double r126108876 = r126108861 / r126108866;
        double r126108877 = cbrt(r126108869);
        double r126108878 = cbrt(r126108866);
        double r126108879 = cbrt(r126108858);
        double r126108880 = r126108878 / r126108879;
        double r126108881 = r126108877 / r126108880;
        double r126108882 = r126108881 * r126108881;
        double r126108883 = r126108882 * r126108881;
        double r126108884 = r126108876 - r126108883;
        double r126108885 = r126108875 ? r126108884 : r126108873;
        double r126108886 = r126108860 ? r126108873 : r126108885;
        return r126108886;
}

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

Original10.2
Target1.8
Herbie2.4
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958298856956410892592016 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -7.433845130127019e-258 or 6.637318488869212e-120 < z

    1. Initial program 12.7

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

    3. Applied div-sub12.7

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

    5. Applied associate-/l*8.8

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

    7. Applied div-sub8.8

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

    8. Simplified2.9

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

    10. Applied div-inv2.9

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

    if -7.433845130127019e-258 < z < 6.637318488869212e-120

    1. Initial program 0.1

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

    3. Applied div-sub0.1

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

    5. Applied associate-/l*3.3

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

    7. Applied add-cube-cbrt3.5

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

    8. Applied add-cube-cbrt3.5

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

    9. Applied times-frac3.5

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

    10. Applied add-cube-cbrt3.5

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

    11. Applied times-frac1.1

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

    12. Simplified0.4

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

  4. Final simplification2.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -7.433845130127019277921365656087791646311 \cdot 10^{-258}:\\ \;\;\;\;x \cdot \frac{1}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \le 6.637318488869211541788788475321548606226 \cdot 10^{-120}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \left(\frac{\sqrt[3]{y}}{\frac{\sqrt[3]{t - a \cdot z}}{\sqrt[3]{z}}} \cdot \frac{\sqrt[3]{y}}{\frac{\sqrt[3]{t - a \cdot z}}{\sqrt[3]{z}}}\right) \cdot \frac{\sqrt[3]{y}}{\frac{\sqrt[3]{t - a \cdot z}}{\sqrt[3]{z}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019173 +o rules:numerics
(FPCore (x y z t a)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A"

  :herbie-target
  (if (< z -32113435955957344.0) (- (/ x (- t (* a z))) (/ y (- (/ t z) a))) (if (< z 3.5139522372978296e-86) (* (- x (* y z)) (/ 1.0 (- t (* a z)))) (- (/ x (- t (* a z))) (/ y (- (/ t z) a)))))

  (/ (- x (* y z)) (- t (* a z))))