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

↓

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

\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 r310328 = x;
        double r310329 = y;
        double r310330 = r310328 / r310329;
        double r310331 = z;
        double r310332 = t;
        double r310333 = r310331 - r310332;
        double r310334 = r310330 * r310333;
        double r310335 = r310334 + r310332;
        return r310335;
}

↓

double f(double x, double y, double z, double t) {
        double r310336 = x;
        double r310337 = y;
        double r310338 = r310336 / r310337;
        double r310339 = z;
        double r310340 = t;
        double r310341 = r310339 - r310340;
        double r310342 = fma(r310338, r310341, r310340);
        return r310342;
}

Error

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

Target

Original	2.1
Target	2.2
Herbie	2.1

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

Derivation

Initial program 2.1
\[\frac{x}{y} \cdot \left(z - t\right) + t\]
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 2019322 +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))