Average Error: 1.8 → 1.8
Time: 2.8s
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 r423350 = x;
        double r423351 = y;
        double r423352 = r423350 / r423351;
        double r423353 = z;
        double r423354 = t;
        double r423355 = r423353 - r423354;
        double r423356 = r423352 * r423355;
        double r423357 = r423356 + r423354;
        return r423357;
}
↓
double f(double x, double y, double z, double t) {
        double r423358 = x;
        double r423359 = y;
        double r423360 = r423358 / r423359;
        double r423361 = z;
        double r423362 = t;
        double r423363 = r423361 - r423362;
        double r423364 = fma(r423360, r423363, r423362);
        return r423364;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original1.8
Target2.0
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 2020064 +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))