Average Error: 2.1 → 1.8
Time: 16.1s
Precision: 64
\[\frac{x}{y} \cdot \left(z - t\right) + t\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -8.817139765258242920811999043847260264933 \cdot 10^{-176} \lor \neg \left(t \le 2.916381983573460547937387731151674530239 \cdot 10^{-154}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, z - t, t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{z - t}{y} + t\\ \end{array}\]
\frac{x}{y} \cdot \left(z - t\right) + t
↓
\begin{array}{l}
\mathbf{if}\;t \le -8.817139765258242920811999043847260264933 \cdot 10^{-176} \lor \neg \left(t \le 2.916381983573460547937387731151674530239 \cdot 10^{-154}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{y}, z - t, t\right)\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r374111 = x;
        double r374112 = y;
        double r374113 = r374111 / r374112;
        double r374114 = z;
        double r374115 = t;
        double r374116 = r374114 - r374115;
        double r374117 = r374113 * r374116;
        double r374118 = r374117 + r374115;
        return r374118;
}
↓
double f(double x, double y, double z, double t) {
        double r374119 = t;
        double r374120 = -8.817139765258243e-176;
        bool r374121 = r374119 <= r374120;
        double r374122 = 2.9163819835734605e-154;
        bool r374123 = r374119 <= r374122;
        double r374124 = !r374123;
        bool r374125 = r374121 || r374124;
        double r374126 = x;
        double r374127 = y;
        double r374128 = r374126 / r374127;
        double r374129 = z;
        double r374130 = r374129 - r374119;
        double r374131 = fma(r374128, r374130, r374119);
        double r374132 = r374130 / r374127;
        double r374133 = r374126 * r374132;
        double r374134 = r374133 + r374119;
        double r374135 = r374125 ? r374131 : r374134;
        return r374135;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original2.1
Target2.4
Herbie1.8
\[\begin{array}{l} \mathbf{if}\;z \lt 2.759456554562692182563154937894909044548 \cdot 10^{-282}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \mathbf{elif}\;z \lt 2.32699445087443595687739933019129648094 \cdot 10^{-110}:\\ \;\;\;\;x \cdot \frac{z - t}{y} + t\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -8.817139765258243e-176 or 2.9163819835734605e-154 < t

    1. Initial program 0.9

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

    2. Simplified0.9

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

    if -8.817139765258243e-176 < t < 2.9163819835734605e-154

    1. Initial program 5.6

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

    3. Applied div-inv5.6

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

    4. Applied associate-*l*4.7

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

    5. Simplified4.6

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

  4. Final simplification1.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -8.817139765258242920811999043847260264933 \cdot 10^{-176} \lor \neg \left(t \le 2.916381983573460547937387731151674530239 \cdot 10^{-154}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, z - t, t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{z - t}{y} + t\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 +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))