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

↓

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

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

↓

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

double f(double x, double y, double z, double t) {
        double r16981621 = x;
        double r16981622 = y;
        double r16981623 = r16981621 / r16981622;
        double r16981624 = z;
        double r16981625 = t;
        double r16981626 = r16981624 - r16981625;
        double r16981627 = r16981623 * r16981626;
        double r16981628 = r16981627 + r16981625;
        return r16981628;
}

↓

double f(double x, double y, double z, double t) {
        double r16981629 = x;
        double r16981630 = y;
        double r16981631 = r16981629 / r16981630;
        double r16981632 = z;
        double r16981633 = t;
        double r16981634 = r16981632 - r16981633;
        double r16981635 = fma(r16981631, r16981634, r16981633);
        return r16981635;
}

Error

The X axis uses an exponential scale — Bits error versus `x`

Target

Original	2.0
Target	2.3
Herbie	2.0

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

Derivation

Initial program 2.0
\[\frac{x}{y} \cdot \left(z - t\right) + t\]
Simplified2.0
\[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{y}, z - t, t\right)}\]
Final simplification2.0
\[\leadsto \mathsf{fma}\left(\frac{x}{y}, z - t, t\right)\]

Reproduce

herbie shell --seed 2019171 +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))