Average Error: 16.7 → 13.3
Time: 19.1s
Precision: 64
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -12951009003417.23046875 \lor \neg \left(t \le 5.561447179273978400673729013888102999816 \cdot 10^{112}\right):\\ \;\;\;\;\frac{y \cdot \frac{z}{t} + x}{\mathsf{fma}\left(\frac{y}{t}, b, a\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \end{array}\]
\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
↓
\begin{array}{l}
\mathbf{if}\;t \le -12951009003417.23046875 \lor \neg \left(t \le 5.561447179273978400673729013888102999816 \cdot 10^{112}\right):\\
\;\;\;\;\frac{y \cdot \frac{z}{t} + x}{\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 r500335 = x;
        double r500336 = y;
        double r500337 = z;
        double r500338 = r500336 * r500337;
        double r500339 = t;
        double r500340 = r500338 / r500339;
        double r500341 = r500335 + r500340;
        double r500342 = a;
        double r500343 = 1.0;
        double r500344 = r500342 + r500343;
        double r500345 = b;
        double r500346 = r500336 * r500345;
        double r500347 = r500346 / r500339;
        double r500348 = r500344 + r500347;
        double r500349 = r500341 / r500348;
        return r500349;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r500350 = t;
        double r500351 = -12951009003417.23;
        bool r500352 = r500350 <= r500351;
        double r500353 = 5.561447179273978e+112;
        bool r500354 = r500350 <= r500353;
        double r500355 = !r500354;
        bool r500356 = r500352 || r500355;
        double r500357 = y;
        double r500358 = z;
        double r500359 = r500358 / r500350;
        double r500360 = r500357 * r500359;
        double r500361 = x;
        double r500362 = r500360 + r500361;
        double r500363 = r500357 / r500350;
        double r500364 = b;
        double r500365 = a;
        double r500366 = fma(r500363, r500364, r500365);
        double r500367 = 1.0;
        double r500368 = r500366 + r500367;
        double r500369 = r500362 / r500368;
        double r500370 = r500357 * r500358;
        double r500371 = r500370 / r500350;
        double r500372 = r500361 + r500371;
        double r500373 = r500365 + r500367;
        double r500374 = r500357 * r500364;
        double r500375 = r500374 / r500350;
        double r500376 = r500373 + r500375;
        double r500377 = r500372 / r500376;
        double r500378 = r500356 ? r500369 : r500377;
        return r500378;
}

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

Original16.7
Target13.0
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 < -12951009003417.23 or 5.561447179273978e+112 < t

    1. Initial program 11.4

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

    2. Simplified3.2

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

    4. Applied fma-udef3.2

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

    6. Applied div-inv3.2

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

    7. Applied associate-*l*3.0

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

    8. Simplified3.0

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

    if -12951009003417.23 < t < 5.561447179273978e+112

    1. Initial program 20.3

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

  4. Final simplification13.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -12951009003417.23046875 \lor \neg \left(t \le 5.561447179273978400673729013888102999816 \cdot 10^{112}\right):\\ \;\;\;\;\frac{y \cdot \frac{z}{t} + x}{\mathsf{fma}\left(\frac{y}{t}, b, a\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 +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 3.0369671037372459e-130) (/ z b) (* 1 (* (+ x (* (/ y t) z)) (/ 1 (+ (+ a 1) (* (/ y t) b)))))))

  (/ (+ x (/ (* y z) t)) (+ (+ a 1) (/ (* y b) t))))