Average Error: 16.6 → 13.0
Time: 4.1s
Precision: 64
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -7.57671656991631018 \cdot 10^{-43}:\\ \;\;\;\;\frac{x + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}}\\ \mathbf{elif}\;t \le 1.1851624532769905 \cdot 10^{-47}:\\ \;\;\;\;\left(x + \frac{y \cdot z}{t}\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}}\\ \end{array}\]
\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
↓
\begin{array}{l}
\mathbf{if}\;t \le -7.57671656991631018 \cdot 10^{-43}:\\
\;\;\;\;\frac{x + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}}\\

\mathbf{elif}\;t \le 1.1851624532769905 \cdot 10^{-47}:\\
\;\;\;\;\left(x + \frac{y \cdot z}{t}\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r634889 = x;
        double r634890 = y;
        double r634891 = z;
        double r634892 = r634890 * r634891;
        double r634893 = t;
        double r634894 = r634892 / r634893;
        double r634895 = r634889 + r634894;
        double r634896 = a;
        double r634897 = 1.0;
        double r634898 = r634896 + r634897;
        double r634899 = b;
        double r634900 = r634890 * r634899;
        double r634901 = r634900 / r634893;
        double r634902 = r634898 + r634901;
        double r634903 = r634895 / r634902;
        return r634903;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r634904 = t;
        double r634905 = -7.57671656991631e-43;
        bool r634906 = r634904 <= r634905;
        double r634907 = x;
        double r634908 = y;
        double r634909 = cbrt(r634904);
        double r634910 = r634909 * r634909;
        double r634911 = r634908 / r634910;
        double r634912 = z;
        double r634913 = r634912 / r634909;
        double r634914 = r634911 * r634913;
        double r634915 = r634907 + r634914;
        double r634916 = a;
        double r634917 = 1.0;
        double r634918 = r634916 + r634917;
        double r634919 = b;
        double r634920 = r634904 / r634919;
        double r634921 = r634908 / r634920;
        double r634922 = r634918 + r634921;
        double r634923 = r634915 / r634922;
        double r634924 = 1.1851624532769905e-47;
        bool r634925 = r634904 <= r634924;
        double r634926 = r634908 * r634912;
        double r634927 = r634926 / r634904;
        double r634928 = r634907 + r634927;
        double r634929 = 1.0;
        double r634930 = r634908 * r634919;
        double r634931 = r634930 / r634904;
        double r634932 = r634918 + r634931;
        double r634933 = r634929 / r634932;
        double r634934 = r634928 * r634933;
        double r634935 = r634912 / r634904;
        double r634936 = r634908 * r634935;
        double r634937 = r634907 + r634936;
        double r634938 = r634937 / r634922;
        double r634939 = r634925 ? r634934 : r634938;
        double r634940 = r634906 ? r634923 : r634939;
        return r634940;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original16.6
Target13.4
Herbie13.0
\[\begin{array}{l} \mathbf{if}\;t \lt -1.3659085366310088 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t \lt 3.0369671037372459 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if t < -7.57671656991631e-43

    1. Initial program 11.7

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
    2. Using strategy rm

    3. Applied associate-/l*9.4

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

    5. Applied add-cube-cbrt9.6

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

    6. Applied times-frac5.4

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

    if -7.57671656991631e-43 < t < 1.1851624532769905e-47

    1. Initial program 23.9

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
    2. Using strategy rm

    3. Applied div-inv23.9

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

    if 1.1851624532769905e-47 < t

    1. Initial program 11.5

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
    2. Using strategy rm

    3. Applied associate-/l*9.0

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

    5. Applied *-un-lft-identity9.0

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

    6. Applied times-frac5.4

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

    7. Simplified5.4

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

  4. Final simplification13.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -7.57671656991631018 \cdot 10^{-43}:\\ \;\;\;\;\frac{x + \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}}\\ \mathbf{elif}\;t \le 1.1851624532769905 \cdot 10^{-47}:\\ \;\;\;\;\left(x + \frac{y \cdot z}{t}\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, B"
  :precision binary64

  :herbie-target
  (if (< t -1.3659085366310088e-271) (* 1 (* (+ x (* (/ y t) z)) (/ 1 (+ (+ a 1) (* (/ y t) b))))) (if (< t 3.036967103737246e-130) (/ z b) (* 1 (* (+ x (* (/ y t) z)) (/ 1 (+ (+ a 1) (* (/ y t) b)))))))

  (/ (+ x (/ (* y z) t)) (+ (+ a 1) (/ (* y b) t))))