Numeric.Signal.Multichannel:$cget from hsignal-0.2.7.1

Average Error: 2.2 → 2.2

Time: 17.1s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r386025 = x;
        double r386026 = y;
        double r386027 = r386025 / r386026;
        double r386028 = z;
        double r386029 = t;
        double r386030 = r386028 - r386029;
        double r386031 = r386027 * r386030;
        double r386032 = r386031 + r386029;
        return r386032;
}

↓

double f(double x, double y, double z, double t) {
        double r386033 = x;
        double r386034 = y;
        double r386035 = r386033 / r386034;
        double r386036 = z;
        double r386037 = t;
        double r386038 = r386036 - r386037;
        double r386039 = fma(r386035, r386038, r386037);
        return r386039;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	2.2
Target	2.4
Herbie	2.2

\[\begin{array}{l} \mathbf{if}\;z \lt 2.759456554562692182563154937894909044548 \cdot 10^{-282}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \mathbf{elif}\;z \lt 2.32699445087443595687739933019129648094 \cdot 10^{-110}:\\ \;\;\;\;x \cdot \frac{z - t}{y} + t\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \end{array}\]

Derivation

1. Initial program 2.2

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

2. Simplified 2.2

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

3. Final simplification 2.2

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

Reproduce

herbie shell --seed 2019326 +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))