Average Error: 10.7 → 1.7
Time: 3.3s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3.7911710689350596 \cdot 10^{47}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{1}{\frac{\frac{t}{z} - a}{y}}\\ \mathbf{elif}\;z \le 99120762.3127832562:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{1}{t - a \cdot z} \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\sqrt[3]{t - a \cdot z} \cdot \sqrt[3]{t - a \cdot z}}}{\sqrt[3]{t - a \cdot z}} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -3.7911710689350596 \cdot 10^{47}:\\
\;\;\;\;\frac{x}{t - a \cdot z} - \frac{1}{\frac{\frac{t}{z} - a}{y}}\\

\mathbf{elif}\;z \le 99120762.3127832562:\\
\;\;\;\;\frac{x}{t - a \cdot z} - \frac{1}{t - a \cdot z} \cdot \left(y \cdot z\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{x}{\sqrt[3]{t - a \cdot z} \cdot \sqrt[3]{t - a \cdot z}}}{\sqrt[3]{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 r775877 = x;
        double r775878 = y;
        double r775879 = z;
        double r775880 = r775878 * r775879;
        double r775881 = r775877 - r775880;
        double r775882 = t;
        double r775883 = a;
        double r775884 = r775883 * r775879;
        double r775885 = r775882 - r775884;
        double r775886 = r775881 / r775885;
        return r775886;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r775887 = z;
        double r775888 = -3.79117106893506e+47;
        bool r775889 = r775887 <= r775888;
        double r775890 = x;
        double r775891 = t;
        double r775892 = a;
        double r775893 = r775892 * r775887;
        double r775894 = r775891 - r775893;
        double r775895 = r775890 / r775894;
        double r775896 = 1.0;
        double r775897 = r775891 / r775887;
        double r775898 = r775897 - r775892;
        double r775899 = y;
        double r775900 = r775898 / r775899;
        double r775901 = r775896 / r775900;
        double r775902 = r775895 - r775901;
        double r775903 = 99120762.31278326;
        bool r775904 = r775887 <= r775903;
        double r775905 = r775896 / r775894;
        double r775906 = r775899 * r775887;
        double r775907 = r775905 * r775906;
        double r775908 = r775895 - r775907;
        double r775909 = cbrt(r775894);
        double r775910 = r775909 * r775909;
        double r775911 = r775890 / r775910;
        double r775912 = r775911 / r775909;
        double r775913 = r775899 / r775898;
        double r775914 = r775912 - r775913;
        double r775915 = r775904 ? r775908 : r775914;
        double r775916 = r775889 ? r775902 : r775915;
        return r775916;
}

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

Derivation

  1. Split input into 3 regimes

  2. if z < -3.79117106893506e+47

    1. Initial program 25.3

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

    3. Applied div-sub25.3

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

    5. Applied associate-/l*16.7

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

    7. Applied div-sub16.7

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

    8. Simplified3.3

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

    10. Applied clear-num3.7

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

    if -3.79117106893506e+47 < z < 99120762.31278326

    1. Initial program 0.3

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

    3. Applied div-sub0.3

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

    5. Applied associate-/l*2.9

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

    7. Applied div-inv2.9

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

    8. Applied *-un-lft-identity2.9

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

    9. Applied times-frac0.4

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

    10. Simplified0.3

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

    if 99120762.31278326 < z

    1. Initial program 21.5

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

    3. Applied div-sub21.5

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

    5. Applied associate-/l*13.4

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

    7. Applied div-sub13.4

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

    8. Simplified2.9

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

    10. Applied add-cube-cbrt3.1

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

    11. Applied associate-/r*3.1

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

  4. Final simplification1.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.7911710689350596 \cdot 10^{47}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{1}{\frac{\frac{t}{z} - a}{y}}\\ \mathbf{elif}\;z \le 99120762.3127832562:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{1}{t - a \cdot z} \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\sqrt[3]{t - a \cdot z} \cdot \sqrt[3]{t - a \cdot z}}}{\sqrt[3]{t - a \cdot z}} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 
(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))))