Average Error: 17.0 → 15.0
Time: 21.1s
Precision: 64
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le 3.614569668922552776109548998076496091155 \cdot 10^{-31}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(1 + a\right) + \frac{1}{\frac{t}{y \cdot b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{y}{t}, b, 1 + a\right)} \cdot \mathsf{fma}\left(\frac{y}{t}, z, x\right)\\ \end{array}\]
\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
↓
\begin{array}{l}
\mathbf{if}\;t \le 3.614569668922552776109548998076496091155 \cdot 10^{-31}:\\
\;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(1 + a\right) + \frac{1}{\frac{t}{y \cdot b}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{y}{t}, b, 1 + a\right)} \cdot \mathsf{fma}\left(\frac{y}{t}, z, x\right)\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r36228258 = x;
        double r36228259 = y;
        double r36228260 = z;
        double r36228261 = r36228259 * r36228260;
        double r36228262 = t;
        double r36228263 = r36228261 / r36228262;
        double r36228264 = r36228258 + r36228263;
        double r36228265 = a;
        double r36228266 = 1.0;
        double r36228267 = r36228265 + r36228266;
        double r36228268 = b;
        double r36228269 = r36228259 * r36228268;
        double r36228270 = r36228269 / r36228262;
        double r36228271 = r36228267 + r36228270;
        double r36228272 = r36228264 / r36228271;
        return r36228272;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r36228273 = t;
        double r36228274 = 3.6145696689225528e-31;
        bool r36228275 = r36228273 <= r36228274;
        double r36228276 = x;
        double r36228277 = y;
        double r36228278 = z;
        double r36228279 = r36228277 * r36228278;
        double r36228280 = r36228279 / r36228273;
        double r36228281 = r36228276 + r36228280;
        double r36228282 = 1.0;
        double r36228283 = a;
        double r36228284 = r36228282 + r36228283;
        double r36228285 = 1.0;
        double r36228286 = b;
        double r36228287 = r36228277 * r36228286;
        double r36228288 = r36228273 / r36228287;
        double r36228289 = r36228285 / r36228288;
        double r36228290 = r36228284 + r36228289;
        double r36228291 = r36228281 / r36228290;
        double r36228292 = r36228277 / r36228273;
        double r36228293 = fma(r36228292, r36228286, r36228284);
        double r36228294 = r36228285 / r36228293;
        double r36228295 = fma(r36228292, r36228278, r36228276);
        double r36228296 = r36228294 * r36228295;
        double r36228297 = r36228275 ? r36228291 : r36228296;
        return r36228297;
}

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

Original17.0
Target13.5
Herbie15.0
\[\begin{array}{l} \mathbf{if}\;t \lt -1.365908536631008841640163147697088508132 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t \lt 3.036967103737245906066829435890093573122 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < 3.6145696689225528e-31

    1. Initial program 19.1

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
    2. Using strategy rm

    3. Applied clear-num19.1

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

    if 3.6145696689225528e-31 < t

    1. Initial program 11.7

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

    2. Simplified4.4

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

    4. Applied div-inv4.5

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

  4. Final simplification15.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 3.614569668922552776109548998076496091155 \cdot 10^{-31}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(1 + a\right) + \frac{1}{\frac{t}{y \cdot b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{y}{t}, b, 1 + a\right)} \cdot \mathsf{fma}\left(\frac{y}{t}, z, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +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))))