Average Error: 2.3 → 2.3
Time: 3.0s
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 r540175 = x;
        double r540176 = y;
        double r540177 = r540175 / r540176;
        double r540178 = z;
        double r540179 = t;
        double r540180 = r540178 - r540179;
        double r540181 = r540177 * r540180;
        double r540182 = r540181 + r540179;
        return r540182;
}
↓
double f(double x, double y, double z, double t) {
        double r540183 = x;
        double r540184 = y;
        double r540185 = r540183 / r540184;
        double r540186 = z;
        double r540187 = t;
        double r540188 = r540186 - r540187;
        double r540189 = fma(r540185, r540188, r540187);
        return r540189;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original2.3
Target2.4
Herbie2.3
\[\begin{array}{l} \mathbf{if}\;z \lt 2.7594565545626922 \cdot 10^{-282}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \mathbf{elif}\;z \lt 2.326994450874436 \cdot 10^{-110}:\\ \;\;\;\;x \cdot \frac{z - t}{y} + t\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \end{array}\]

Derivation

  1. Initial program 2.3

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

  2. Simplified2.3

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

  3. Final simplification2.3

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

Reproduce

herbie shell --seed 2020057 +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))