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

↓

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

\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 r22418555 = x;
        double r22418556 = y;
        double r22418557 = r22418555 / r22418556;
        double r22418558 = z;
        double r22418559 = t;
        double r22418560 = r22418558 - r22418559;
        double r22418561 = r22418557 * r22418560;
        double r22418562 = r22418561 + r22418559;
        return r22418562;
}

↓

double f(double x, double y, double z, double t) {
        double r22418563 = x;
        double r22418564 = y;
        double r22418565 = r22418563 / r22418564;
        double r22418566 = z;
        double r22418567 = t;
        double r22418568 = r22418566 - r22418567;
        double r22418569 = fma(r22418565, r22418568, r22418567);
        return r22418569;
}

Error

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

Target

Original	2.1
Target	2.2
Herbie	2.1

\[\begin{array}{l} \mathbf{if}\;z \lt 2.759456554562692 \cdot 10^{-282}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \mathbf{elif}\;z \lt 2.326994450874436 \cdot 10^{-110}:\\ \;\;\;\;x \cdot \frac{z - t}{y} + t\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \end{array}\]

Derivation

Initial program 2.1
\[\frac{x}{y} \cdot \left(z - t\right) + t\]
Simplified2.1
\[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{y}, z - t, t\right)}\]
Final simplification2.1
\[\leadsto \mathsf{fma}\left(\frac{x}{y}, z - t, t\right)\]

Reproduce

herbie shell --seed 2019163 +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))