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

Average Error: 1.9 → 1.7

Time: 12.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -4.87632604569760037637408890268880098973 \cdot 10^{75} \lor \neg \left(y \le -5.330647358840705908227051774940756230968 \cdot 10^{-307}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, z - t, t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x \cdot z}{y} - \frac{t \cdot x}{y}\right) + t\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r331403 = x;
        double r331404 = y;
        double r331405 = r331403 / r331404;
        double r331406 = z;
        double r331407 = t;
        double r331408 = r331406 - r331407;
        double r331409 = r331405 * r331408;
        double r331410 = r331409 + r331407;
        return r331410;
}

↓

double f(double x, double y, double z, double t) {
        double r331411 = y;
        double r331412 = -4.8763260456976e+75;
        bool r331413 = r331411 <= r331412;
        double r331414 = -5.330647358840706e-307;
        bool r331415 = r331411 <= r331414;
        double r331416 = !r331415;
        bool r331417 = r331413 || r331416;
        double r331418 = x;
        double r331419 = r331418 / r331411;
        double r331420 = z;
        double r331421 = t;
        double r331422 = r331420 - r331421;
        double r331423 = fma(r331419, r331422, r331421);
        double r331424 = r331418 * r331420;
        double r331425 = r331424 / r331411;
        double r331426 = r331421 * r331418;
        double r331427 = r331426 / r331411;
        double r331428 = r331425 - r331427;
        double r331429 = r331428 + r331421;
        double r331430 = r331417 ? r331423 : r331429;
        return r331430;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	1.9
Target	2.3
Herbie	1.7

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

Derivation

1. Split input into 2 regimes

2. if y < -4.8763260456976e+75 or -5.330647358840706e-307 < y

   1. Initial program 1.6

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

   2. Simplified 1.6

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

   if -4.8763260456976e+75 < y < -5.330647358840706e-307

   1. Initial program 2.8

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

   2. Simplified 2.8

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

   4. Applied fma-udef 2.8

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

   5. Simplified 11.7

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

   6. Taylor expanded around 0 2.1

      \[\leadsto \color{blue}{\left(\frac{x \cdot z}{y} - \frac{t \cdot x}{y}\right)} + t\]
3. Recombined 2 regimes into one program.

4. Final simplification 1.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -4.87632604569760037637408890268880098973 \cdot 10^{75} \lor \neg \left(y \le -5.330647358840705908227051774940756230968 \cdot 10^{-307}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, z - t, t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x \cdot z}{y} - \frac{t \cdot x}{y}\right) + t\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 +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 2.7594565545626922e-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))