Average Error: 16.4 → 12.8
Time: 22.3s
Precision: 64
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1.0\right) + \frac{y \cdot b}{t}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.1271786680970628 \cdot 10^{-58}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\mathsf{fma}\left(\frac{y}{t}, b, 1.0 + a\right)}\\ \mathbf{elif}\;t \le 1.924178015281595 \cdot 10^{-05}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\frac{b \cdot y}{t} + \left(1.0 + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\mathsf{fma}\left(\frac{y}{t}, b, 1.0 + a\right)}\\ \end{array}\]
\frac{x + \frac{y \cdot z}{t}}{\left(a + 1.0\right) + \frac{y \cdot b}{t}}
↓
\begin{array}{l}
\mathbf{if}\;t \le -1.1271786680970628 \cdot 10^{-58}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\mathsf{fma}\left(\frac{y}{t}, b, 1.0 + a\right)}\\

\mathbf{elif}\;t \le 1.924178015281595 \cdot 10^{-05}:\\
\;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\frac{b \cdot y}{t} + \left(1.0 + a\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\mathsf{fma}\left(\frac{y}{t}, b, 1.0 + a\right)}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r22707363 = x;
        double r22707364 = y;
        double r22707365 = z;
        double r22707366 = r22707364 * r22707365;
        double r22707367 = t;
        double r22707368 = r22707366 / r22707367;
        double r22707369 = r22707363 + r22707368;
        double r22707370 = a;
        double r22707371 = 1.0;
        double r22707372 = r22707370 + r22707371;
        double r22707373 = b;
        double r22707374 = r22707364 * r22707373;
        double r22707375 = r22707374 / r22707367;
        double r22707376 = r22707372 + r22707375;
        double r22707377 = r22707369 / r22707376;
        return r22707377;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r22707378 = t;
        double r22707379 = -1.1271786680970628e-58;
        bool r22707380 = r22707378 <= r22707379;
        double r22707381 = y;
        double r22707382 = r22707381 / r22707378;
        double r22707383 = z;
        double r22707384 = x;
        double r22707385 = fma(r22707382, r22707383, r22707384);
        double r22707386 = b;
        double r22707387 = 1.0;
        double r22707388 = a;
        double r22707389 = r22707387 + r22707388;
        double r22707390 = fma(r22707382, r22707386, r22707389);
        double r22707391 = r22707385 / r22707390;
        double r22707392 = 1.924178015281595e-05;
        bool r22707393 = r22707378 <= r22707392;
        double r22707394 = r22707381 * r22707383;
        double r22707395 = r22707394 / r22707378;
        double r22707396 = r22707384 + r22707395;
        double r22707397 = r22707386 * r22707381;
        double r22707398 = r22707397 / r22707378;
        double r22707399 = r22707398 + r22707389;
        double r22707400 = r22707396 / r22707399;
        double r22707401 = r22707393 ? r22707400 : r22707391;
        double r22707402 = r22707380 ? r22707391 : r22707401;
        return r22707402;
}

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
Target13.3
Herbie12.8
\[\begin{array}{l} \mathbf{if}\;t \lt -1.3659085366310088 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1.0\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t \lt 3.036967103737246 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1.0\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -1.1271786680970628e-58 or 1.924178015281595e-05 < t

    1. Initial program 11.3

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1.0\right) + \frac{y \cdot b}{t}}\]

    2. Simplified4.9

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\mathsf{fma}\left(\frac{y}{t}, b, a + 1.0\right)}}\]

    if -1.1271786680970628e-58 < t < 1.924178015281595e-05

    1. Initial program 22.9

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1.0\right) + \frac{y \cdot b}{t}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification12.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.1271786680970628 \cdot 10^{-58}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\mathsf{fma}\left(\frac{y}{t}, b, 1.0 + a\right)}\\ \mathbf{elif}\;t \le 1.924178015281595 \cdot 10^{-05}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\frac{b \cdot y}{t} + \left(1.0 + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\mathsf{fma}\left(\frac{y}{t}, b, 1.0 + a\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 +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 (* (+ x (* (/ y t) z)) (/ 1 (+ (+ a 1.0) (* (/ y t) b))))) (if (< t 3.036967103737246e-130) (/ z b) (* 1 (* (+ x (* (/ y t) z)) (/ 1 (+ (+ a 1.0) (* (/ y t) b)))))))

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