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

↓

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

\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 r318677 = x;
        double r318678 = y;
        double r318679 = r318677 / r318678;
        double r318680 = z;
        double r318681 = t;
        double r318682 = r318680 - r318681;
        double r318683 = r318679 * r318682;
        double r318684 = r318683 + r318681;
        return r318684;
}

↓

double f(double x, double y, double z, double t) {
        double r318685 = x;
        double r318686 = y;
        double r318687 = r318685 / r318686;
        double r318688 = z;
        double r318689 = t;
        double r318690 = r318688 - r318689;
        double r318691 = fma(r318687, r318690, r318689);
        return r318691;
}

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 2019304 +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 2.7594565545626922e-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))