Average Error: 1.8 → 1.8
Time: 2.5s
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 r481471 = x;
        double r481472 = y;
        double r481473 = r481471 / r481472;
        double r481474 = z;
        double r481475 = t;
        double r481476 = r481474 - r481475;
        double r481477 = r481473 * r481476;
        double r481478 = r481477 + r481475;
        return r481478;
}
↓
double f(double x, double y, double z, double t) {
        double r481479 = x;
        double r481480 = y;
        double r481481 = r481479 / r481480;
        double r481482 = z;
        double r481483 = t;
        double r481484 = r481482 - r481483;
        double r481485 = fma(r481481, r481484, r481483);
        return r481485;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original1.8
Target2.3
Herbie1.8
\[\begin{array}{l} \mathbf{if}\;z \lt 2.7594565545626922 \cdot 10^{-282}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \mathbf{elif}\;z \lt 2.326994450874436 \cdot 10^{-110}:\\ \;\;\;\;x \cdot \frac{z - t}{y} + t\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \end{array}\]

Derivation

  1. Initial program 1.8

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

  2. Simplified1.8

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

  3. Final simplification1.8

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

Reproduce

herbie shell --seed 2020025 +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))