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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -3.89229228781631368 \cdot 10^{68}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z, x\right) \cdot \frac{1}{1 + \mathsf{fma}\left(\frac{y}{t}, b, a\right)}\\ \mathbf{elif}\;t \le 3.6584687480036831 \cdot 10^{-44}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 + \mathsf{fma}\left(\frac{y}{t}, b, a\right)}\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;t \le -3.89229228781631368 \cdot 10^{68}:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z, x\right) \cdot \frac{1}{1 + \mathsf{fma}\left(\frac{y}{t}, b, a\right)}\\

\mathbf{elif}\;t \le 3.6584687480036831 \cdot 10^{-44}:\\
\;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\

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

\end{array}

double f(double x, double y, double z, double t, double a, double b) {
        double r691072 = x;
        double r691073 = y;
        double r691074 = z;
        double r691075 = r691073 * r691074;
        double r691076 = t;
        double r691077 = r691075 / r691076;
        double r691078 = r691072 + r691077;
        double r691079 = a;
        double r691080 = 1.0;
        double r691081 = r691079 + r691080;
        double r691082 = b;
        double r691083 = r691073 * r691082;
        double r691084 = r691083 / r691076;
        double r691085 = r691081 + r691084;
        double r691086 = r691078 / r691085;
        return r691086;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r691087 = t;
        double r691088 = -3.8922922878163137e+68;
        bool r691089 = r691087 <= r691088;
        double r691090 = y;
        double r691091 = r691090 / r691087;
        double r691092 = z;
        double r691093 = x;
        double r691094 = fma(r691091, r691092, r691093);
        double r691095 = 1.0;
        double r691096 = 1.0;
        double r691097 = b;
        double r691098 = a;
        double r691099 = fma(r691091, r691097, r691098);
        double r691100 = r691096 + r691099;
        double r691101 = r691095 / r691100;
        double r691102 = r691094 * r691101;
        double r691103 = 3.658468748003683e-44;
        bool r691104 = r691087 <= r691103;
        double r691105 = r691090 * r691092;
        double r691106 = r691105 / r691087;
        double r691107 = r691093 + r691106;
        double r691108 = r691098 + r691096;
        double r691109 = r691090 * r691097;
        double r691110 = r691109 / r691087;
        double r691111 = r691108 + r691110;
        double r691112 = r691107 / r691111;
        double r691113 = r691094 / r691100;
        double r691114 = r691104 ? r691112 : r691113;
        double r691115 = r691089 ? r691102 : r691114;
        return r691115;
}

Original	16.8
Target	13.3
Herbie	13.2

Derivation

Split input into 3 regimes
if t < -3.8922922878163137e+68
1. Initial program 11.8
  \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
2. Simplified3.3
  \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 + \mathsf{fma}\left(\frac{y}{t}, b, a\right)}}\]
3. Using strategy rm
4. Applied div-inv3.4
  \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{t}, z, x\right) \cdot \frac{1}{1 + \mathsf{fma}\left(\frac{y}{t}, b, a\right)}}\]
if -3.8922922878163137e+68 < t < 3.658468748003683e-44
1. Initial program 21.4
  \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
2. Using strategy rm
3. Applied pow121.4
  \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\color{blue}{{\left(\left(a + 1\right) + \frac{y \cdot b}{t}\right)}^{1}}}\]
if 3.658468748003683e-44 < t
1. Initial program 12.4
  \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
2. Simplified6.0
  \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 + \mathsf{fma}\left(\frac{y}{t}, b, a\right)}}\]
Recombined 3 regimes into one program.
Final simplification13.2
\[\leadsto \begin{array}{l} \mathbf{if}\;t \le -3.89229228781631368 \cdot 10^{68}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z, x\right) \cdot \frac{1}{1 + \mathsf{fma}\left(\frac{y}{t}, b, a\right)}\\ \mathbf{elif}\;t \le 3.6584687480036831 \cdot 10^{-44}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 + \mathsf{fma}\left(\frac{y}{t}, b, a\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +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))))

Falling back on racket egraph because egg-herbie package not installed (more)

Error

Target

Derivation

`if t < -3.8922922878163137e+68`

`if -3.8922922878163137e+68 < t < 3.658468748003683e-44`

`if 3.658468748003683e-44 < t`

Reproduce