Average Error: 16.0 → 12.8
Time: 12.4s
Precision: 64
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.6373364464566329 \cdot 10^{139} \lor \neg \left(t \le 2.28491505957256337 \cdot 10^{-15}\right):\\ \;\;\;\;\frac{1}{\left(1 + \mathsf{fma}\left(\frac{y}{t}, b, a\right)\right) \cdot \frac{1}{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \end{array}\]
\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
↓
\begin{array}{l}
\mathbf{if}\;t \le -1.6373364464566329 \cdot 10^{139} \lor \neg \left(t \le 2.28491505957256337 \cdot 10^{-15}\right):\\
\;\;\;\;\frac{1}{\left(1 + \mathsf{fma}\left(\frac{y}{t}, b, a\right)\right) \cdot \frac{1}{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}}\\

\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 r636305 = x;
        double r636306 = y;
        double r636307 = z;
        double r636308 = r636306 * r636307;
        double r636309 = t;
        double r636310 = r636308 / r636309;
        double r636311 = r636305 + r636310;
        double r636312 = a;
        double r636313 = 1.0;
        double r636314 = r636312 + r636313;
        double r636315 = b;
        double r636316 = r636306 * r636315;
        double r636317 = r636316 / r636309;
        double r636318 = r636314 + r636317;
        double r636319 = r636311 / r636318;
        return r636319;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r636320 = t;
        double r636321 = -1.637336446456633e+139;
        bool r636322 = r636320 <= r636321;
        double r636323 = 2.2849150595725634e-15;
        bool r636324 = r636320 <= r636323;
        double r636325 = !r636324;
        bool r636326 = r636322 || r636325;
        double r636327 = 1.0;
        double r636328 = 1.0;
        double r636329 = y;
        double r636330 = r636329 / r636320;
        double r636331 = b;
        double r636332 = a;
        double r636333 = fma(r636330, r636331, r636332);
        double r636334 = r636328 + r636333;
        double r636335 = z;
        double r636336 = x;
        double r636337 = fma(r636330, r636335, r636336);
        double r636338 = r636327 / r636337;
        double r636339 = r636334 * r636338;
        double r636340 = r636327 / r636339;
        double r636341 = r636329 * r636335;
        double r636342 = r636341 / r636320;
        double r636343 = r636336 + r636342;
        double r636344 = r636332 + r636328;
        double r636345 = r636329 * r636331;
        double r636346 = r636345 / r636320;
        double r636347 = r636344 + r636346;
        double r636348 = r636343 / r636347;
        double r636349 = r636326 ? r636340 : r636348;
        return r636349;
}

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

Original16.0
Target12.9
Herbie12.8
\[\begin{array}{l} \mathbf{if}\;t \lt -1.3659085366310088 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t \lt 3.0369671037372459 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -1.637336446456633e+139 or 2.2849150595725634e-15 < t

    1. Initial program 11.8

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

    2. Simplified3.4

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

    4. Applied clear-num3.8

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

    6. Applied div-inv3.9

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

    if -1.637336446456633e+139 < t < 2.2849150595725634e-15

    1. Initial program 19.0

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

  4. Final simplification12.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.6373364464566329 \cdot 10^{139} \lor \neg \left(t \le 2.28491505957256337 \cdot 10^{-15}\right):\\ \;\;\;\;\frac{1}{\left(1 + \mathsf{fma}\left(\frac{y}{t}, b, a\right)\right) \cdot \frac{1}{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +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))))