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

↓

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

\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 r17444746 = x;
        double r17444747 = y;
        double r17444748 = r17444746 / r17444747;
        double r17444749 = z;
        double r17444750 = t;
        double r17444751 = r17444749 - r17444750;
        double r17444752 = r17444748 * r17444751;
        double r17444753 = r17444752 + r17444750;
        return r17444753;
}

↓

double f(double x, double y, double z, double t) {
        double r17444754 = x;
        double r17444755 = y;
        double r17444756 = r17444754 / r17444755;
        double r17444757 = z;
        double r17444758 = t;
        double r17444759 = r17444757 - r17444758;
        double r17444760 = fma(r17444756, r17444759, r17444758);
        return r17444760;
}

Error

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

Target

Original	2.1
Target	2.4
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 2019172 +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))