Average Error: 2.1 → 1.1
Time: 15.9s
Precision: 64
\[\frac{x}{y} \cdot \left(z - t\right) + t\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \le -2.2202714348581967929360518371166962138 \cdot 10^{152}:\\ \;\;\;\;t + x \cdot \frac{z - t}{y}\\ \mathbf{elif}\;\frac{x}{y} \le 1.029063798245920862442624041354209055291 \cdot 10^{133}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, z - t, t\right)\\ \mathbf{else}:\\ \;\;\;\;t + x \cdot \frac{z - t}{y}\\ \end{array}\]
\frac{x}{y} \cdot \left(z - t\right) + t
↓
\begin{array}{l}
\mathbf{if}\;\frac{x}{y} \le -2.2202714348581967929360518371166962138 \cdot 10^{152}:\\
\;\;\;\;t + x \cdot \frac{z - t}{y}\\

\mathbf{elif}\;\frac{x}{y} \le 1.029063798245920862442624041354209055291 \cdot 10^{133}:\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{y}, z - t, t\right)\\

\mathbf{else}:\\
\;\;\;\;t + x \cdot \frac{z - t}{y}\\

\end{array}
double f(double x, double y, double z, double t) {
        double r18542107 = x;
        double r18542108 = y;
        double r18542109 = r18542107 / r18542108;
        double r18542110 = z;
        double r18542111 = t;
        double r18542112 = r18542110 - r18542111;
        double r18542113 = r18542109 * r18542112;
        double r18542114 = r18542113 + r18542111;
        return r18542114;
}
↓
double f(double x, double y, double z, double t) {
        double r18542115 = x;
        double r18542116 = y;
        double r18542117 = r18542115 / r18542116;
        double r18542118 = -2.220271434858197e+152;
        bool r18542119 = r18542117 <= r18542118;
        double r18542120 = t;
        double r18542121 = z;
        double r18542122 = r18542121 - r18542120;
        double r18542123 = r18542122 / r18542116;
        double r18542124 = r18542115 * r18542123;
        double r18542125 = r18542120 + r18542124;
        double r18542126 = 1.0290637982459209e+133;
        bool r18542127 = r18542117 <= r18542126;
        double r18542128 = fma(r18542117, r18542122, r18542120);
        double r18542129 = r18542127 ? r18542128 : r18542125;
        double r18542130 = r18542119 ? r18542125 : r18542129;
        return r18542130;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original2.1
Target2.3
Herbie1.1
\[\begin{array}{l} \mathbf{if}\;z \lt 2.759456554562692182563154937894909044548 \cdot 10^{-282}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \mathbf{elif}\;z \lt 2.32699445087443595687739933019129648094 \cdot 10^{-110}:\\ \;\;\;\;x \cdot \frac{z - t}{y} + t\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ x y) < -2.220271434858197e+152 or 1.0290637982459209e+133 < (/ x y)

    1. Initial program 11.4

      \[\frac{x}{y} \cdot \left(z - t\right) + t\]
    2. Using strategy rm

    3. Applied div-inv11.5

      \[\leadsto \color{blue}{\left(x \cdot \frac{1}{y}\right)} \cdot \left(z - t\right) + t\]

    4. Applied associate-*l*3.1

      \[\leadsto \color{blue}{x \cdot \left(\frac{1}{y} \cdot \left(z - t\right)\right)} + t\]

    5. Simplified3.0

      \[\leadsto x \cdot \color{blue}{\frac{z - t}{y}} + t\]

    if -2.220271434858197e+152 < (/ x y) < 1.0290637982459209e+133

    1. Initial program 0.9

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

    2. Simplified0.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{y}, z - t, t\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \le -2.2202714348581967929360518371166962138 \cdot 10^{152}:\\ \;\;\;\;t + x \cdot \frac{z - t}{y}\\ \mathbf{elif}\;\frac{x}{y} \le 1.029063798245920862442624041354209055291 \cdot 10^{133}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, z - t, t\right)\\ \mathbf{else}:\\ \;\;\;\;t + x \cdot \frac{z - t}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.Signal.Multichannel:$cget from hsignal-0.2.7.1"

  :herbie-target
  (if (< z 2.759456554562692e-282) (+ (* (/ x y) (- z t)) t) (if (< z 2.326994450874436e-110) (+ (* x (/ (- z t) y)) t) (+ (* (/ x y) (- z t)) t)))

  (+ (* (/ x y) (- z t)) t))