Average Error: 16.0 → 12.8
Time: 11.3s
Precision: 64
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.6373364464566329 \cdot 10^{139} \lor \neg \left(t \le 2.28491505957256337 \cdot 10^{-15}\right):\\ \;\;\;\;\frac{1}{\left(1 + \mathsf{fma}\left(\frac{y}{t}, b, a\right)\right) \cdot \frac{1}{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \end{array}\]
\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 r693145 = x;
        double r693146 = y;
        double r693147 = z;
        double r693148 = r693146 * r693147;
        double r693149 = t;
        double r693150 = r693148 / r693149;
        double r693151 = r693145 + r693150;
        double r693152 = a;
        double r693153 = 1.0;
        double r693154 = r693152 + r693153;
        double r693155 = b;
        double r693156 = r693146 * r693155;
        double r693157 = r693156 / r693149;
        double r693158 = r693154 + r693157;
        double r693159 = r693151 / r693158;
        return r693159;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r693160 = t;
        double r693161 = -1.637336446456633e+139;
        bool r693162 = r693160 <= r693161;
        double r693163 = 2.2849150595725634e-15;
        bool r693164 = r693160 <= r693163;
        double r693165 = !r693164;
        bool r693166 = r693162 || r693165;
        double r693167 = 1.0;
        double r693168 = 1.0;
        double r693169 = y;
        double r693170 = r693169 / r693160;
        double r693171 = b;
        double r693172 = a;
        double r693173 = fma(r693170, r693171, r693172);
        double r693174 = r693168 + r693173;
        double r693175 = z;
        double r693176 = x;
        double r693177 = fma(r693170, r693175, r693176);
        double r693178 = r693167 / r693177;
        double r693179 = r693174 * r693178;
        double r693180 = r693167 / r693179;
        double r693181 = r693169 * r693175;
        double r693182 = r693181 / r693160;
        double r693183 = r693176 + r693182;
        double r693184 = r693172 + r693168;
        double r693185 = r693169 * r693171;
        double r693186 = r693185 / r693160;
        double r693187 = r693184 + r693186;
        double r693188 = r693183 / r693187;
        double r693189 = r693166 ? r693180 : r693188;
        return r693189;
}

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))))