Average Error: 16.3 → 13.1
Time: 12.3s
Precision: 64
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.951712415323306753092874186654917599891 \cdot 10^{82}:\\ \;\;\;\;\frac{x + \frac{y}{t} \cdot z}{\left(a + 1\right) + y \cdot \frac{b}{t}}\\ \mathbf{elif}\;t \le 1.071734848610785363400891057909236427624 \cdot 10^{-99}:\\ \;\;\;\;\frac{x + \frac{1}{t} \cdot \left(z \cdot y\right)}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{\left(a + 1\right) + y \cdot \frac{b}{t}}\\ \end{array}\]
\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
↓
\begin{array}{l}
\mathbf{if}\;t \le -1.951712415323306753092874186654917599891 \cdot 10^{82}:\\
\;\;\;\;\frac{x + \frac{y}{t} \cdot z}{\left(a + 1\right) + y \cdot \frac{b}{t}}\\

\mathbf{elif}\;t \le 1.071734848610785363400891057909236427624 \cdot 10^{-99}:\\
\;\;\;\;\frac{x + \frac{1}{t} \cdot \left(z \cdot y\right)}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r526939 = x;
        double r526940 = y;
        double r526941 = z;
        double r526942 = r526940 * r526941;
        double r526943 = t;
        double r526944 = r526942 / r526943;
        double r526945 = r526939 + r526944;
        double r526946 = a;
        double r526947 = 1.0;
        double r526948 = r526946 + r526947;
        double r526949 = b;
        double r526950 = r526940 * r526949;
        double r526951 = r526950 / r526943;
        double r526952 = r526948 + r526951;
        double r526953 = r526945 / r526952;
        return r526953;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r526954 = t;
        double r526955 = -1.9517124153233068e+82;
        bool r526956 = r526954 <= r526955;
        double r526957 = x;
        double r526958 = y;
        double r526959 = r526958 / r526954;
        double r526960 = z;
        double r526961 = r526959 * r526960;
        double r526962 = r526957 + r526961;
        double r526963 = a;
        double r526964 = 1.0;
        double r526965 = r526963 + r526964;
        double r526966 = b;
        double r526967 = r526966 / r526954;
        double r526968 = r526958 * r526967;
        double r526969 = r526965 + r526968;
        double r526970 = r526962 / r526969;
        double r526971 = 1.0717348486107854e-99;
        bool r526972 = r526954 <= r526971;
        double r526973 = 1.0;
        double r526974 = r526973 / r526954;
        double r526975 = r526960 * r526958;
        double r526976 = r526974 * r526975;
        double r526977 = r526957 + r526976;
        double r526978 = r526958 * r526966;
        double r526979 = r526978 / r526954;
        double r526980 = r526965 + r526979;
        double r526981 = r526977 / r526980;
        double r526982 = r526954 / r526960;
        double r526983 = r526958 / r526982;
        double r526984 = r526957 + r526983;
        double r526985 = r526984 / r526969;
        double r526986 = r526972 ? r526981 : r526985;
        double r526987 = r526956 ? r526970 : r526986;
        return r526987;
}

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

Original16.3
Target13.0
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 t < -1.9517124153233068e+82

    1. Initial program 12.2

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

    3. Applied associate-/l*7.6

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

    5. Applied *-un-lft-identity7.6

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

    6. Applied times-frac2.9

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

    7. Simplified2.9

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

    9. Applied associate-/r/3.1

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

    if -1.9517124153233068e+82 < t < 1.0717348486107854e-99

    1. Initial program 21.0

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

    3. Applied associate-/l*25.3

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

    5. Applied div-inv25.3

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

    6. Applied *-un-lft-identity25.3

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

    7. Applied times-frac21.0

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

    8. Simplified21.0

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

    if 1.0717348486107854e-99 < t

    1. Initial program 12.3

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

    3. Applied associate-/l*10.7

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

    5. Applied *-un-lft-identity10.7

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

    6. Applied times-frac7.7

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

    7. Simplified7.7

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

  4. Final simplification13.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.951712415323306753092874186654917599891 \cdot 10^{82}:\\ \;\;\;\;\frac{x + \frac{y}{t} \cdot z}{\left(a + 1\right) + y \cdot \frac{b}{t}}\\ \mathbf{elif}\;t \le 1.071734848610785363400891057909236427624 \cdot 10^{-99}:\\ \;\;\;\;\frac{x + \frac{1}{t} \cdot \left(z \cdot y\right)}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{\left(a + 1\right) + y \cdot \frac{b}{t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019305 
(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 3.0369671037372459e-130) (/ z b) (* 1 (* (+ x (* (/ y t) z)) (/ 1 (+ (+ a 1) (* (/ y t) b)))))))

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