Average Error: 16.9 → 13.3
Time: 22.0s
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 r429691 = x;
        double r429692 = y;
        double r429693 = z;
        double r429694 = r429692 * r429693;
        double r429695 = t;
        double r429696 = r429694 / r429695;
        double r429697 = r429691 + r429696;
        double r429698 = a;
        double r429699 = 1.0;
        double r429700 = r429698 + r429699;
        double r429701 = b;
        double r429702 = r429692 * r429701;
        double r429703 = r429702 / r429695;
        double r429704 = r429700 + r429703;
        double r429705 = r429697 / r429704;
        return r429705;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r429706 = t;
        double r429707 = -2.328380613761226e+75;
        bool r429708 = r429706 <= r429707;
        double r429709 = 8.239083667014754e-61;
        bool r429710 = r429706 <= r429709;
        double r429711 = !r429710;
        bool r429712 = r429708 || r429711;
        double r429713 = y;
        double r429714 = r429713 / r429706;
        double r429715 = z;
        double r429716 = x;
        double r429717 = fma(r429714, r429715, r429716);
        double r429718 = b;
        double r429719 = a;
        double r429720 = fma(r429714, r429718, r429719);
        double r429721 = 1.0;
        double r429722 = r429720 + r429721;
        double r429723 = r429717 / r429722;
        double r429724 = r429713 * r429715;
        double r429725 = r429724 / r429706;
        double r429726 = r429716 + r429725;
        double r429727 = r429719 + r429721;
        double r429728 = r429713 * r429718;
        double r429729 = r429728 / r429706;
        double r429730 = r429727 + r429729;
        double r429731 = r429726 / r429730;
        double r429732 = r429712 ? r429723 : r429731;
        return r429732;
}

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))))