Average Error: 16.7 → 12.7
Time: 12.4s
Precision: 64
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.71685530057196864384990922906635199685 \cdot 10^{-78} \lor \neg \left(y \le 0.05603438225657088322950727388160885311663\right):\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + y \cdot \frac{b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left(a + 1\right) + \frac{y \cdot b}{t}}{x + \frac{y \cdot z}{t}}}\\ \end{array}\]
\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
↓
\begin{array}{l}
\mathbf{if}\;y \le -1.71685530057196864384990922906635199685 \cdot 10^{-78} \lor \neg \left(y \le 0.05603438225657088322950727388160885311663\right):\\
\;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + y \cdot \frac{b}{t}}\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r458821 = x;
        double r458822 = y;
        double r458823 = z;
        double r458824 = r458822 * r458823;
        double r458825 = t;
        double r458826 = r458824 / r458825;
        double r458827 = r458821 + r458826;
        double r458828 = a;
        double r458829 = 1.0;
        double r458830 = r458828 + r458829;
        double r458831 = b;
        double r458832 = r458822 * r458831;
        double r458833 = r458832 / r458825;
        double r458834 = r458830 + r458833;
        double r458835 = r458827 / r458834;
        return r458835;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r458836 = y;
        double r458837 = -1.7168553005719686e-78;
        bool r458838 = r458836 <= r458837;
        double r458839 = 0.05603438225657088;
        bool r458840 = r458836 <= r458839;
        double r458841 = !r458840;
        bool r458842 = r458838 || r458841;
        double r458843 = x;
        double r458844 = z;
        double r458845 = t;
        double r458846 = r458844 / r458845;
        double r458847 = r458836 * r458846;
        double r458848 = r458843 + r458847;
        double r458849 = a;
        double r458850 = 1.0;
        double r458851 = r458849 + r458850;
        double r458852 = b;
        double r458853 = r458852 / r458845;
        double r458854 = r458836 * r458853;
        double r458855 = r458851 + r458854;
        double r458856 = r458848 / r458855;
        double r458857 = 1.0;
        double r458858 = r458836 * r458852;
        double r458859 = r458858 / r458845;
        double r458860 = r458851 + r458859;
        double r458861 = r458836 * r458844;
        double r458862 = r458861 / r458845;
        double r458863 = r458843 + r458862;
        double r458864 = r458860 / r458863;
        double r458865 = r458857 / r458864;
        double r458866 = r458842 ? r458856 : r458865;
        return r458866;
}

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

Original16.7
Target13.1
Herbie12.7
\[\begin{array}{l} \mathbf{if}\;t \lt -1.365908536631008841640163147697088508132 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t \lt 3.036967103737245906066829435890093573122 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -1.7168553005719686e-78 or 0.05603438225657088 < y

    1. Initial program 27.3

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

    3. Applied associate-/l*23.9

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

    5. Applied *-un-lft-identity23.9

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

    6. Applied times-frac19.8

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

    7. Simplified19.8

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

    9. Applied div-inv19.9

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

    10. Simplified19.8

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

    if -1.7168553005719686e-78 < y < 0.05603438225657088

    1. Initial program 3.5

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

    3. Applied clear-num3.9

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(a + 1\right) + \frac{y \cdot b}{t}}{x + \frac{y \cdot z}{t}}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification12.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.71685530057196864384990922906635199685 \cdot 10^{-78} \lor \neg \left(y \le 0.05603438225657088322950727388160885311663\right):\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\left(a + 1\right) + y \cdot \frac{b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left(a + 1\right) + \frac{y \cdot b}{t}}{x + \frac{y \cdot z}{t}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 
(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))))