Average Error: 17.2 → 13.4
Time: 22.9s
Precision: 64
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -2.40419363033911193 \cdot 10^{81} \lor \neg \left(t \le 7.1851806179354163 \cdot 10^{-56}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\mathsf{fma}\left(\frac{y}{t}, b, a\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \end{array}\]
\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
↓
\begin{array}{l}
\mathbf{if}\;t \le -2.40419363033911193 \cdot 10^{81} \lor \neg \left(t \le 7.1851806179354163 \cdot 10^{-56}\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 r433646 = x;
        double r433647 = y;
        double r433648 = z;
        double r433649 = r433647 * r433648;
        double r433650 = t;
        double r433651 = r433649 / r433650;
        double r433652 = r433646 + r433651;
        double r433653 = a;
        double r433654 = 1.0;
        double r433655 = r433653 + r433654;
        double r433656 = b;
        double r433657 = r433647 * r433656;
        double r433658 = r433657 / r433650;
        double r433659 = r433655 + r433658;
        double r433660 = r433652 / r433659;
        return r433660;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r433661 = t;
        double r433662 = -2.404193630339112e+81;
        bool r433663 = r433661 <= r433662;
        double r433664 = 7.185180617935416e-56;
        bool r433665 = r433661 <= r433664;
        double r433666 = !r433665;
        bool r433667 = r433663 || r433666;
        double r433668 = y;
        double r433669 = r433668 / r433661;
        double r433670 = z;
        double r433671 = x;
        double r433672 = fma(r433669, r433670, r433671);
        double r433673 = b;
        double r433674 = a;
        double r433675 = fma(r433669, r433673, r433674);
        double r433676 = 1.0;
        double r433677 = r433675 + r433676;
        double r433678 = r433672 / r433677;
        double r433679 = r433668 * r433670;
        double r433680 = r433679 / r433661;
        double r433681 = r433671 + r433680;
        double r433682 = r433674 + r433676;
        double r433683 = r433668 * r433673;
        double r433684 = r433683 / r433661;
        double r433685 = r433682 + r433684;
        double r433686 = r433681 / r433685;
        double r433687 = r433667 ? r433678 : r433686;
        return r433687;
}

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

Original17.2
Target13.7
Herbie13.4
\[\begin{array}{l} \mathbf{if}\;t \lt -1.3659085366310088 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t \lt 3.0369671037372459 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -2.404193630339112e+81 or 7.185180617935416e-56 < t

    1. Initial program 12.3

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

    2. Simplified4.4

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

    if -2.404193630339112e+81 < t < 7.185180617935416e-56

    1. Initial program 21.8

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

  4. Final simplification13.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.40419363033911193 \cdot 10^{81} \lor \neg \left(t \le 7.1851806179354163 \cdot 10^{-56}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\mathsf{fma}\left(\frac{y}{t}, b, a\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 +o rules:numerics
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, B"

  :herbie-target
  (if (< t -1.3659085366310088e-271) (* 1.0 (* (+ x (* (/ y t) z)) (/ 1.0 (+ (+ a 1.0) (* (/ y t) b))))) (if (< t 3.036967103737246e-130) (/ z b) (* 1.0 (* (+ x (* (/ y t) z)) (/ 1.0 (+ (+ a 1.0) (* (/ y t) b)))))))

  (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))