Average Error: 2.0 → 1.8
Time: 14.3s
Precision: 64
\[\frac{x}{y} \cdot \left(z - t\right) + t\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 3.758228562227714 \cdot 10^{40} \lor \neg \left(x \le 2.3059419046843504 \cdot 10^{219}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, z - t, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{z - t}} + t\\ \end{array}\]
\frac{x}{y} \cdot \left(z - t\right) + t
↓
\begin{array}{l}
\mathbf{if}\;x \le 3.758228562227714 \cdot 10^{40} \lor \neg \left(x \le 2.3059419046843504 \cdot 10^{219}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{y}, z - t, t\right)\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r635409 = x;
        double r635410 = y;
        double r635411 = r635409 / r635410;
        double r635412 = z;
        double r635413 = t;
        double r635414 = r635412 - r635413;
        double r635415 = r635411 * r635414;
        double r635416 = r635415 + r635413;
        return r635416;
}
↓
double f(double x, double y, double z, double t) {
        double r635417 = x;
        double r635418 = 3.758228562227714e+40;
        bool r635419 = r635417 <= r635418;
        double r635420 = 2.3059419046843504e+219;
        bool r635421 = r635417 <= r635420;
        double r635422 = !r635421;
        bool r635423 = r635419 || r635422;
        double r635424 = y;
        double r635425 = r635417 / r635424;
        double r635426 = z;
        double r635427 = t;
        double r635428 = r635426 - r635427;
        double r635429 = fma(r635425, r635428, r635427);
        double r635430 = r635424 / r635428;
        double r635431 = r635417 / r635430;
        double r635432 = r635431 + r635427;
        double r635433 = r635423 ? r635429 : r635432;
        return r635433;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original2.0
Target2.3
Herbie1.8
\[\begin{array}{l} \mathbf{if}\;z \lt 2.7594565545626922 \cdot 10^{-282}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \mathbf{elif}\;z \lt 2.326994450874436 \cdot 10^{-110}:\\ \;\;\;\;x \cdot \frac{z - t}{y} + t\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < 3.758228562227714e+40 or 2.3059419046843504e+219 < x

    1. Initial program 1.8

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

    2. Simplified1.8

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

    if 3.758228562227714e+40 < x < 2.3059419046843504e+219

    1. Initial program 3.4

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

    2. Simplified3.4

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

    4. Applied fma-udef3.4

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

    5. Simplified1.5

      \[\leadsto \color{blue}{\frac{x}{\frac{y}{z - t}}} + t\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 3.758228562227714 \cdot 10^{40} \lor \neg \left(x \le 2.3059419046843504 \cdot 10^{219}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, z - t, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{z - t}} + t\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 +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 2.759456554562692e-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))