Average Error: 2.1 → 2.1
Time: 2.6s
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 r461818 = x;
        double r461819 = y;
        double r461820 = r461818 / r461819;
        double r461821 = z;
        double r461822 = t;
        double r461823 = r461821 - r461822;
        double r461824 = r461820 * r461823;
        double r461825 = r461824 + r461822;
        return r461825;
}
↓
double f(double x, double y, double z, double t) {
        double r461826 = x;
        double r461827 = y;
        double r461828 = r461826 / r461827;
        double r461829 = z;
        double r461830 = t;
        double r461831 = r461829 - r461830;
        double r461832 = fma(r461828, r461831, r461830);
        return r461832;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original2.1
Target2.4
Herbie2.1
\[\begin{array}{l} \mathbf{if}\;z \lt 2.7594565545626922 \cdot 10^{-282}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \mathbf{elif}\;z \lt 2.326994450874436 \cdot 10^{-110}:\\ \;\;\;\;x \cdot \frac{z - t}{y} + t\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \end{array}\]

Derivation

  1. Initial program 2.1

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

  2. Simplified2.1

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

  3. Final simplification2.1

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

Reproduce

herbie shell --seed 2020020 +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))