Average Error: 16.9 → 13.3
Time: 20.2s
Precision: 64
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -2.328380613761226163125891047288073973053 \cdot 10^{75} \lor \neg \left(t \le 8.239083667014753748570637214436972745483 \cdot 10^{-61}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\mathsf{fma}\left(\frac{y}{t}, b, a\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \end{array}\]
\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
↓
\begin{array}{l}
\mathbf{if}\;t \le -2.328380613761226163125891047288073973053 \cdot 10^{75} \lor \neg \left(t \le 8.239083667014753748570637214436972745483 \cdot 10^{-61}\right):\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\mathsf{fma}\left(\frac{y}{t}, b, a\right) + 1}\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r488996 = x;
        double r488997 = y;
        double r488998 = z;
        double r488999 = r488997 * r488998;
        double r489000 = t;
        double r489001 = r488999 / r489000;
        double r489002 = r488996 + r489001;
        double r489003 = a;
        double r489004 = 1.0;
        double r489005 = r489003 + r489004;
        double r489006 = b;
        double r489007 = r488997 * r489006;
        double r489008 = r489007 / r489000;
        double r489009 = r489005 + r489008;
        double r489010 = r489002 / r489009;
        return r489010;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r489011 = t;
        double r489012 = -2.328380613761226e+75;
        bool r489013 = r489011 <= r489012;
        double r489014 = 8.239083667014754e-61;
        bool r489015 = r489011 <= r489014;
        double r489016 = !r489015;
        bool r489017 = r489013 || r489016;
        double r489018 = y;
        double r489019 = r489018 / r489011;
        double r489020 = z;
        double r489021 = x;
        double r489022 = fma(r489019, r489020, r489021);
        double r489023 = b;
        double r489024 = a;
        double r489025 = fma(r489019, r489023, r489024);
        double r489026 = 1.0;
        double r489027 = r489025 + r489026;
        double r489028 = r489022 / r489027;
        double r489029 = r489018 * r489020;
        double r489030 = r489029 / r489011;
        double r489031 = r489021 + r489030;
        double r489032 = r489024 + r489026;
        double r489033 = r489018 * r489023;
        double r489034 = r489033 / r489011;
        double r489035 = r489032 + r489034;
        double r489036 = r489031 / r489035;
        double r489037 = r489017 ? r489028 : r489036;
        return r489037;
}

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

Target

Original16.9
Target13.1
Herbie13.3
\[\begin{array}{l} \mathbf{if}\;t \lt -1.365908536631008841640163147697088508132 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t \lt 3.036967103737245906066829435890093573122 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -2.328380613761226e+75 or 8.239083667014754e-61 < t

    1. Initial program 11.8

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

    2. Simplified4.6

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

    if -2.328380613761226e+75 < t < 8.239083667014754e-61

    1. Initial program 22.0

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

  4. Final simplification13.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.328380613761226163125891047288073973053 \cdot 10^{75} \lor \neg \left(t \le 8.239083667014753748570637214436972745483 \cdot 10^{-61}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\mathsf{fma}\left(\frac{y}{t}, b, a\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +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))))