Average Error: 2.1 → 1.5
Time: 14.3s
Precision: 64
\[\frac{x}{y} \cdot \left(z - t\right) + t\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} = -\infty:\\ \;\;\;\;\frac{z - t}{y} \cdot x + t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, z - t, t\right)\\ \end{array}\]
\frac{x}{y} \cdot \left(z - t\right) + t
↓
\begin{array}{l}
\mathbf{if}\;\frac{x}{y} = -\infty:\\
\;\;\;\;\frac{z - t}{y} \cdot x + t\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{y}, z - t, t\right)\\

\end{array}
double f(double x, double y, double z, double t) {
        double r385895 = x;
        double r385896 = y;
        double r385897 = r385895 / r385896;
        double r385898 = z;
        double r385899 = t;
        double r385900 = r385898 - r385899;
        double r385901 = r385897 * r385900;
        double r385902 = r385901 + r385899;
        return r385902;
}
↓
double f(double x, double y, double z, double t) {
        double r385903 = x;
        double r385904 = y;
        double r385905 = r385903 / r385904;
        double r385906 = -inf.0;
        bool r385907 = r385905 <= r385906;
        double r385908 = z;
        double r385909 = t;
        double r385910 = r385908 - r385909;
        double r385911 = r385910 / r385904;
        double r385912 = r385911 * r385903;
        double r385913 = r385912 + r385909;
        double r385914 = fma(r385905, r385910, r385909);
        double r385915 = r385907 ? r385913 : r385914;
        return r385915;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original2.1
Target2.3
Herbie1.5
\[\begin{array}{l} \mathbf{if}\;z \lt 2.759456554562692182563154937894909044548 \cdot 10^{-282}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \mathbf{elif}\;z \lt 2.32699445087443595687739933019129648094 \cdot 10^{-110}:\\ \;\;\;\;x \cdot \frac{z - t}{y} + t\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ x y) < -inf.0

    1. Initial program 64.0

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

    2. Simplified64.0

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

    4. Applied fma-udef64.0

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

    5. Simplified0.3

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

    if -inf.0 < (/ x y)

    1. Initial program 1.5

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

    2. Simplified1.5

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

  4. Final simplification1.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} = -\infty:\\ \;\;\;\;\frac{z - t}{y} \cdot x + t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, z - t, t\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019305 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.Signal.Multichannel:$cget from hsignal-0.2.7.1"
  :precision binary64

  :herbie-target
  (if (< z 2.7594565545626922e-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))