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

Average Error: 2.3 → 2.8

Time: 39.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -1.214276496349715308350212003616713772243 \cdot 10^{208}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, z - t, t\right)\\ \mathbf{elif}\;y \le 3.810917599276301682261457087969902172823 \cdot 10^{77}:\\ \;\;\;\;\frac{x \cdot \left(z - t\right)}{y} + t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{y}, x, t\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r23927472 = x;
        double r23927473 = y;
        double r23927474 = r23927472 / r23927473;
        double r23927475 = z;
        double r23927476 = t;
        double r23927477 = r23927475 - r23927476;
        double r23927478 = r23927474 * r23927477;
        double r23927479 = r23927478 + r23927476;
        return r23927479;
}

↓

double f(double x, double y, double z, double t) {
        double r23927480 = y;
        double r23927481 = -1.2142764963497153e+208;
        bool r23927482 = r23927480 <= r23927481;
        double r23927483 = x;
        double r23927484 = r23927483 / r23927480;
        double r23927485 = z;
        double r23927486 = t;
        double r23927487 = r23927485 - r23927486;
        double r23927488 = fma(r23927484, r23927487, r23927486);
        double r23927489 = 3.8109175992763017e+77;
        bool r23927490 = r23927480 <= r23927489;
        double r23927491 = r23927483 * r23927487;
        double r23927492 = r23927491 / r23927480;
        double r23927493 = r23927492 + r23927486;
        double r23927494 = r23927487 / r23927480;
        double r23927495 = fma(r23927494, r23927483, r23927486);
        double r23927496 = r23927490 ? r23927493 : r23927495;
        double r23927497 = r23927482 ? r23927488 : r23927496;
        return r23927497;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	2.3
Target	2.5
Herbie	2.8

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

Derivation

1. Split input into 3 regimes

2. if y < -1.2142764963497153e+208

   1. Initial program 1.8

      \[\frac{x}{y} \cdot \left(z - t\right) + t\]
   2. Using strategy rm

   3. Applied fma-def 1.8

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

   if -1.2142764963497153e+208 < y < 3.8109175992763017e+77

   1. Initial program 2.8

      \[\frac{x}{y} \cdot \left(z - t\right) + t\]
   2. Using strategy rm

   3. Applied associate-*l/ 3.5

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

   if 3.8109175992763017e+77 < y

   1. Initial program 1.1

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

   2. Simplified 1.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - t}{y}, x, t\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 2.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.214276496349715308350212003616713772243 \cdot 10^{208}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, z - t, t\right)\\ \mathbf{elif}\;y \le 3.810917599276301682261457087969902172823 \cdot 10^{77}:\\ \;\;\;\;\frac{x \cdot \left(z - t\right)}{y} + t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{y}, x, t\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.Signal.Multichannel:$cget from hsignal-0.2.7.1"

  :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))