Average Error: 16.4 → 12.8
Time: 18.0s
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 r435254 = x;
        double r435255 = y;
        double r435256 = z;
        double r435257 = r435255 * r435256;
        double r435258 = t;
        double r435259 = r435257 / r435258;
        double r435260 = r435254 + r435259;
        double r435261 = a;
        double r435262 = 1.0;
        double r435263 = r435261 + r435262;
        double r435264 = b;
        double r435265 = r435255 * r435264;
        double r435266 = r435265 / r435258;
        double r435267 = r435263 + r435266;
        double r435268 = r435260 / r435267;
        return r435268;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r435269 = t;
        double r435270 = -3.045395840249753e+20;
        bool r435271 = r435269 <= r435270;
        double r435272 = 2.0640658126869295e-53;
        bool r435273 = r435269 <= r435272;
        double r435274 = !r435273;
        bool r435275 = r435271 || r435274;
        double r435276 = y;
        double r435277 = r435276 / r435269;
        double r435278 = z;
        double r435279 = x;
        double r435280 = fma(r435277, r435278, r435279);
        double r435281 = b;
        double r435282 = a;
        double r435283 = fma(r435277, r435281, r435282);
        double r435284 = 1.0;
        double r435285 = r435283 + r435284;
        double r435286 = r435280 / r435285;
        double r435287 = r435276 * r435278;
        double r435288 = r435287 / r435269;
        double r435289 = r435279 + r435288;
        double r435290 = r435282 + r435284;
        double r435291 = r435276 * r435281;
        double r435292 = r435291 / r435269;
        double r435293 = r435290 + r435292;
        double r435294 = r435289 / r435293;
        double r435295 = r435275 ? r435286 : r435294;
        return r435295;
}

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))))