Average Error: 16.5 → 13.1
Time: 36.2s
Precision: 64
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -6.084229636969371216885683145647115259784 \cdot 10^{78}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z, x\right) \cdot \frac{1}{\mathsf{fma}\left(b, \frac{y}{t}, a\right) + 1}\\ \mathbf{elif}\;t \le 1092165316512745.5:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\frac{b \cdot y}{t} + \left(1 + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z, x\right) \cdot \frac{1}{\mathsf{fma}\left(b, \frac{y}{t}, a\right) + 1}\\ \end{array}\]
\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
↓
\begin{array}{l}
\mathbf{if}\;t \le -6.084229636969371216885683145647115259784 \cdot 10^{78}:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z, x\right) \cdot \frac{1}{\mathsf{fma}\left(b, \frac{y}{t}, a\right) + 1}\\

\mathbf{elif}\;t \le 1092165316512745.5:\\
\;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\frac{b \cdot y}{t} + \left(1 + a\right)}\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r33475202 = x;
        double r33475203 = y;
        double r33475204 = z;
        double r33475205 = r33475203 * r33475204;
        double r33475206 = t;
        double r33475207 = r33475205 / r33475206;
        double r33475208 = r33475202 + r33475207;
        double r33475209 = a;
        double r33475210 = 1.0;
        double r33475211 = r33475209 + r33475210;
        double r33475212 = b;
        double r33475213 = r33475203 * r33475212;
        double r33475214 = r33475213 / r33475206;
        double r33475215 = r33475211 + r33475214;
        double r33475216 = r33475208 / r33475215;
        return r33475216;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r33475217 = t;
        double r33475218 = -6.084229636969371e+78;
        bool r33475219 = r33475217 <= r33475218;
        double r33475220 = y;
        double r33475221 = r33475220 / r33475217;
        double r33475222 = z;
        double r33475223 = x;
        double r33475224 = fma(r33475221, r33475222, r33475223);
        double r33475225 = 1.0;
        double r33475226 = b;
        double r33475227 = a;
        double r33475228 = fma(r33475226, r33475221, r33475227);
        double r33475229 = 1.0;
        double r33475230 = r33475228 + r33475229;
        double r33475231 = r33475225 / r33475230;
        double r33475232 = r33475224 * r33475231;
        double r33475233 = 1092165316512745.5;
        bool r33475234 = r33475217 <= r33475233;
        double r33475235 = r33475220 * r33475222;
        double r33475236 = r33475235 / r33475217;
        double r33475237 = r33475223 + r33475236;
        double r33475238 = r33475226 * r33475220;
        double r33475239 = r33475238 / r33475217;
        double r33475240 = r33475229 + r33475227;
        double r33475241 = r33475239 + r33475240;
        double r33475242 = r33475237 / r33475241;
        double r33475243 = r33475234 ? r33475242 : r33475232;
        double r33475244 = r33475219 ? r33475232 : r33475243;
        return r33475244;
}

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

Original16.5
Target13.4
Herbie13.1
\[\begin{array}{l} \mathbf{if}\;t \lt -1.365908536631008841640163147697088508132 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t \lt 3.036967103737245906066829435890093573122 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -6.084229636969371e+78 or 1092165316512745.5 < t

    1. Initial program 11.3

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

    2. Simplified3.4

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

    4. Applied div-inv3.5

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

    if -6.084229636969371e+78 < t < 1092165316512745.5

    1. Initial program 20.5

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

  4. Final simplification13.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -6.084229636969371216885683145647115259784 \cdot 10^{78}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z, x\right) \cdot \frac{1}{\mathsf{fma}\left(b, \frac{y}{t}, a\right) + 1}\\ \mathbf{elif}\;t \le 1092165316512745.5:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\frac{b \cdot y}{t} + \left(1 + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z, x\right) \cdot \frac{1}{\mathsf{fma}\left(b, \frac{y}{t}, a\right) + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 +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))))