Average Error: 2.2 → 2.2
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 r425937 = x;
        double r425938 = y;
        double r425939 = r425937 / r425938;
        double r425940 = z;
        double r425941 = t;
        double r425942 = r425940 - r425941;
        double r425943 = r425939 * r425942;
        double r425944 = r425943 + r425941;
        return r425944;
}
↓
double f(double x, double y, double z, double t) {
        double r425945 = x;
        double r425946 = y;
        double r425947 = r425945 / r425946;
        double r425948 = z;
        double r425949 = t;
        double r425950 = r425948 - r425949;
        double r425951 = fma(r425947, r425950, r425949);
        return r425951;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original2.2
Target2.4
Herbie2.2
\[\begin{array}{l} \mathbf{if}\;z \lt 2.7594565545626922 \cdot 10^{-282}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \mathbf{elif}\;z \lt 2.326994450874436 \cdot 10^{-110}:\\ \;\;\;\;x \cdot \frac{z - t}{y} + t\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \end{array}\]

Derivation

  1. Initial program 2.2

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

  2. Simplified2.2

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

  3. Final simplification2.2

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

Reproduce

herbie shell --seed 2020083 +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))