Average Error: 16.5 → 12.9
Time: 20.3s
Precision: 64
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.746515370639549540143905675148851571849 \cdot 10^{-50} \lor \neg \left(t \le 5.787492129914299632539133765158045701988 \cdot 10^{-52}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\mathsf{fma}\left(\frac{y}{t}, b, a\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \end{array}\]
\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
↓
\begin{array}{l}
\mathbf{if}\;t \le -1.746515370639549540143905675148851571849 \cdot 10^{-50} \lor \neg \left(t \le 5.787492129914299632539133765158045701988 \cdot 10^{-52}\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 r495165 = x;
        double r495166 = y;
        double r495167 = z;
        double r495168 = r495166 * r495167;
        double r495169 = t;
        double r495170 = r495168 / r495169;
        double r495171 = r495165 + r495170;
        double r495172 = a;
        double r495173 = 1.0;
        double r495174 = r495172 + r495173;
        double r495175 = b;
        double r495176 = r495166 * r495175;
        double r495177 = r495176 / r495169;
        double r495178 = r495174 + r495177;
        double r495179 = r495171 / r495178;
        return r495179;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r495180 = t;
        double r495181 = -1.7465153706395495e-50;
        bool r495182 = r495180 <= r495181;
        double r495183 = 5.7874921299143e-52;
        bool r495184 = r495180 <= r495183;
        double r495185 = !r495184;
        bool r495186 = r495182 || r495185;
        double r495187 = y;
        double r495188 = r495187 / r495180;
        double r495189 = z;
        double r495190 = x;
        double r495191 = fma(r495188, r495189, r495190);
        double r495192 = b;
        double r495193 = a;
        double r495194 = fma(r495188, r495192, r495193);
        double r495195 = 1.0;
        double r495196 = r495194 + r495195;
        double r495197 = r495191 / r495196;
        double r495198 = r495187 * r495189;
        double r495199 = r495198 / r495180;
        double r495200 = r495190 + r495199;
        double r495201 = r495193 + r495195;
        double r495202 = r495187 * r495192;
        double r495203 = r495202 / r495180;
        double r495204 = r495201 + r495203;
        double r495205 = r495200 / r495204;
        double r495206 = r495186 ? r495197 : r495205;
        return r495206;
}

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
Target13.5
Herbie12.9
\[\begin{array}{l} \mathbf{if}\;t \lt -1.365908536631008841640163147697088508132 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t \lt 3.036967103737245906066829435890093573122 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -1.7465153706395495e-50 or 5.7874921299143e-52 < t

    1. Initial program 11.4

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

    2. Simplified5.4

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

    if -1.7465153706395495e-50 < t < 5.7874921299143e-52

    1. Initial program 23.8

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

  4. Final simplification12.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.746515370639549540143905675148851571849 \cdot 10^{-50} \lor \neg \left(t \le 5.787492129914299632539133765158045701988 \cdot 10^{-52}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\mathsf{fma}\left(\frac{y}{t}, b, a\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +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))))