Average Error: 16.4 → 12.8
Time: 18.7s
Precision: 64
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -304539584024975310848 \lor \neg \left(t \le 2.064065812686929528257053629271084316454 \cdot 10^{-53}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\mathsf{fma}\left(\frac{y}{t}, b, a\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \end{array}\]
\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
↓
\begin{array}{l}
\mathbf{if}\;t \le -304539584024975310848 \lor \neg \left(t \le 2.064065812686929528257053629271084316454 \cdot 10^{-53}\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 r600749 = x;
        double r600750 = y;
        double r600751 = z;
        double r600752 = r600750 * r600751;
        double r600753 = t;
        double r600754 = r600752 / r600753;
        double r600755 = r600749 + r600754;
        double r600756 = a;
        double r600757 = 1.0;
        double r600758 = r600756 + r600757;
        double r600759 = b;
        double r600760 = r600750 * r600759;
        double r600761 = r600760 / r600753;
        double r600762 = r600758 + r600761;
        double r600763 = r600755 / r600762;
        return r600763;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r600764 = t;
        double r600765 = -3.045395840249753e+20;
        bool r600766 = r600764 <= r600765;
        double r600767 = 2.0640658126869295e-53;
        bool r600768 = r600764 <= r600767;
        double r600769 = !r600768;
        bool r600770 = r600766 || r600769;
        double r600771 = y;
        double r600772 = r600771 / r600764;
        double r600773 = z;
        double r600774 = x;
        double r600775 = fma(r600772, r600773, r600774);
        double r600776 = b;
        double r600777 = a;
        double r600778 = fma(r600772, r600776, r600777);
        double r600779 = 1.0;
        double r600780 = r600778 + r600779;
        double r600781 = r600775 / r600780;
        double r600782 = r600771 * r600773;
        double r600783 = r600782 / r600764;
        double r600784 = r600774 + r600783;
        double r600785 = r600777 + r600779;
        double r600786 = r600771 * r600776;
        double r600787 = r600786 / r600764;
        double r600788 = r600785 + r600787;
        double r600789 = r600784 / r600788;
        double r600790 = r600770 ? r600781 : r600789;
        return r600790;
}

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

Original16.4
Target13.2
Herbie12.8
\[\begin{array}{l} \mathbf{if}\;t \lt -1.365908536631008841640163147697088508132 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t \lt 3.036967103737245906066829435890093573122 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -3.045395840249753e+20 or 2.0640658126869295e-53 < t

    1. Initial program 11.3

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

    2. Simplified4.7

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

    if -3.045395840249753e+20 < t < 2.0640658126869295e-53

    1. Initial program 22.3

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

  4. Final simplification12.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -304539584024975310848 \lor \neg \left(t \le 2.064065812686929528257053629271084316454 \cdot 10^{-53}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\mathsf{fma}\left(\frac{y}{t}, b, a\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +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))))