Average Error: 2.0 → 2.0
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 r447868 = x;
        double r447869 = y;
        double r447870 = r447868 / r447869;
        double r447871 = z;
        double r447872 = t;
        double r447873 = r447871 - r447872;
        double r447874 = r447870 * r447873;
        double r447875 = r447874 + r447872;
        return r447875;
}
↓
double f(double x, double y, double z, double t) {
        double r447876 = x;
        double r447877 = y;
        double r447878 = r447876 / r447877;
        double r447879 = z;
        double r447880 = t;
        double r447881 = r447879 - r447880;
        double r447882 = fma(r447878, r447881, r447880);
        return r447882;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original2.0
Target2.3
Herbie2.0
\[\begin{array}{l} \mathbf{if}\;z \lt 2.7594565545626922 \cdot 10^{-282}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \mathbf{elif}\;z \lt 2.326994450874436 \cdot 10^{-110}:\\ \;\;\;\;x \cdot \frac{z - t}{y} + t\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \end{array}\]

Derivation

  1. Initial program 2.0

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

  2. Simplified2.0

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

  3. Final simplification2.0

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

Reproduce

herbie shell --seed 2020047 +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))