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

↓

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

\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 r98062959 = x;
        double r98062960 = y;
        double r98062961 = r98062959 / r98062960;
        double r98062962 = z;
        double r98062963 = t;
        double r98062964 = r98062962 - r98062963;
        double r98062965 = r98062961 * r98062964;
        double r98062966 = r98062965 + r98062963;
        return r98062966;
}

↓

double f(double x, double y, double z, double t) {
        double r98062967 = x;
        double r98062968 = y;
        double r98062969 = r98062967 / r98062968;
        double r98062970 = z;
        double r98062971 = t;
        double r98062972 = r98062970 - r98062971;
        double r98062973 = fma(r98062969, r98062972, r98062971);
        return r98062973;
}

Error

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

Target

Original	2.3
Target	2.5
Herbie	2.3

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

Derivation

Initial program 2.3
\[\frac{x}{y} \cdot \left(z - t\right) + t\]
Simplified2.3
\[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{y}, z - t, t\right)}\]
Final simplification2.3
\[\leadsto \mathsf{fma}\left(\frac{x}{y}, z - t, t\right)\]

Reproduce

herbie shell --seed 2019173 +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))