Average Error: 16.1 → 13.1
Time: 5.9s
Precision: 64
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -7.065454724711100881523316892095087502286 \cdot 10^{-112}:\\ \;\;\;\;\frac{x + y \cdot \frac{\frac{1}{t}}{\frac{1}{z}}}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}}\\ \mathbf{elif}\;y \le 5.680955193031466208337968211151925843386 \cdot 10^{-149}:\\ \;\;\;\;\frac{x + \left(\sqrt[3]{\frac{y \cdot z}{t}} \cdot \sqrt[3]{\frac{y \cdot z}{t}}\right) \cdot \sqrt[3]{\frac{y \cdot z}{t}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}}\\ \end{array}\]
\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
↓
\begin{array}{l}
\mathbf{if}\;y \le -7.065454724711100881523316892095087502286 \cdot 10^{-112}:\\
\;\;\;\;\frac{x + y \cdot \frac{\frac{1}{t}}{\frac{1}{z}}}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}}\\

\mathbf{elif}\;y \le 5.680955193031466208337968211151925843386 \cdot 10^{-149}:\\
\;\;\;\;\frac{x + \left(\sqrt[3]{\frac{y \cdot z}{t}} \cdot \sqrt[3]{\frac{y \cdot z}{t}}\right) \cdot \sqrt[3]{\frac{y \cdot z}{t}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\

\mathbf{else}:\\
\;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{\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 r728915 = x;
        double r728916 = y;
        double r728917 = z;
        double r728918 = r728916 * r728917;
        double r728919 = t;
        double r728920 = r728918 / r728919;
        double r728921 = r728915 + r728920;
        double r728922 = a;
        double r728923 = 1.0;
        double r728924 = r728922 + r728923;
        double r728925 = b;
        double r728926 = r728916 * r728925;
        double r728927 = r728926 / r728919;
        double r728928 = r728924 + r728927;
        double r728929 = r728921 / r728928;
        return r728929;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r728930 = y;
        double r728931 = -7.065454724711101e-112;
        bool r728932 = r728930 <= r728931;
        double r728933 = x;
        double r728934 = 1.0;
        double r728935 = t;
        double r728936 = r728934 / r728935;
        double r728937 = z;
        double r728938 = r728934 / r728937;
        double r728939 = r728936 / r728938;
        double r728940 = r728930 * r728939;
        double r728941 = r728933 + r728940;
        double r728942 = a;
        double r728943 = 1.0;
        double r728944 = r728942 + r728943;
        double r728945 = b;
        double r728946 = r728935 / r728945;
        double r728947 = r728930 / r728946;
        double r728948 = r728944 + r728947;
        double r728949 = r728941 / r728948;
        double r728950 = 5.680955193031466e-149;
        bool r728951 = r728930 <= r728950;
        double r728952 = r728930 * r728937;
        double r728953 = r728952 / r728935;
        double r728954 = cbrt(r728953);
        double r728955 = r728954 * r728954;
        double r728956 = r728955 * r728954;
        double r728957 = r728933 + r728956;
        double r728958 = r728930 * r728945;
        double r728959 = r728958 / r728935;
        double r728960 = r728944 + r728959;
        double r728961 = r728957 / r728960;
        double r728962 = r728935 / r728937;
        double r728963 = r728930 / r728962;
        double r728964 = r728933 + r728963;
        double r728965 = r728964 / r728948;
        double r728966 = r728951 ? r728961 : r728965;
        double r728967 = r728932 ? r728949 : r728966;
        return r728967;
}

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

Original16.1
Target12.9
Herbie13.1
\[\begin{array}{l} \mathbf{if}\;t \lt -1.365908536631008841640163147697088508132 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t \lt 3.036967103737245906066829435890093573122 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -7.065454724711101e-112

    1. Initial program 22.8

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

    3. Applied associate-/l*20.6

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

    5. Applied associate-/l*18.2

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

    7. Applied div-inv18.3

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

    9. Applied div-inv18.3

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

    10. Applied associate-/r*18.3

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

    if -7.065454724711101e-112 < y < 5.680955193031466e-149

    1. Initial program 1.7

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

    3. Applied add-cube-cbrt1.9

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

    if 5.680955193031466e-149 < y

    1. Initial program 21.5

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

    3. Applied associate-/l*19.5

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

    5. Applied associate-/l*17.4

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

  4. Final simplification13.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -7.065454724711100881523316892095087502286 \cdot 10^{-112}:\\ \;\;\;\;\frac{x + y \cdot \frac{\frac{1}{t}}{\frac{1}{z}}}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}}\\ \mathbf{elif}\;y \le 5.680955193031466208337968211151925843386 \cdot 10^{-149}:\\ \;\;\;\;\frac{x + \left(\sqrt[3]{\frac{y \cdot z}{t}} \cdot \sqrt[3]{\frac{y \cdot z}{t}}\right) \cdot \sqrt[3]{\frac{y \cdot z}{t}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(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))))