Average Error: 1.9 → 1.9
Time: 2.7s
Precision: 64
\[\frac{x}{y} \cdot \left(z - t\right) + t\]
↓
\[\mathsf{fma}\left(\frac{x}{y}, z - t, t\right)\]
\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 r452713 = x;
        double r452714 = y;
        double r452715 = r452713 / r452714;
        double r452716 = z;
        double r452717 = t;
        double r452718 = r452716 - r452717;
        double r452719 = r452715 * r452718;
        double r452720 = r452719 + r452717;
        return r452720;
}
↓
double f(double x, double y, double z, double t) {
        double r452721 = x;
        double r452722 = y;
        double r452723 = r452721 / r452722;
        double r452724 = z;
        double r452725 = t;
        double r452726 = r452724 - r452725;
        double r452727 = fma(r452723, r452726, r452725);
        return r452727;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original1.9
Target2.2
Herbie1.9
\[\begin{array}{l} \mathbf{if}\;z \lt 2.7594565545626922 \cdot 10^{-282}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \mathbf{elif}\;z \lt 2.326994450874436 \cdot 10^{-110}:\\ \;\;\;\;x \cdot \frac{z - t}{y} + t\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \end{array}\]

Derivation

  1. Initial program 1.9

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

  2. Simplified1.9

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

  3. Final simplification1.9

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

Reproduce

herbie shell --seed 2020081 +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))