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

↓

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

\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 r22175844 = x;
        double r22175845 = y;
        double r22175846 = r22175844 / r22175845;
        double r22175847 = z;
        double r22175848 = t;
        double r22175849 = r22175847 - r22175848;
        double r22175850 = r22175846 * r22175849;
        double r22175851 = r22175850 + r22175848;
        return r22175851;
}

↓

double f(double x, double y, double z, double t) {
        double r22175852 = x;
        double r22175853 = y;
        double r22175854 = r22175852 / r22175853;
        double r22175855 = z;
        double r22175856 = t;
        double r22175857 = r22175855 - r22175856;
        double r22175858 = fma(r22175854, r22175857, r22175856);
        return r22175858;
}

Error

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

Target

Original	2.0
Target	2.0
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 2019192 +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))