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

Average Error: 2.2 → 1.7

Time: 9.8s

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -2.2597023320458045 \cdot 10^{76} \lor \neg \left(x \le 685516672564733568\right):\\ \;\;\;\;t + \left(\frac{\sqrt[3]{x}}{y} \cdot \left(z - t\right)\right) \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x \cdot z}{y} + \left(-\frac{t \cdot x}{y}\right)\right) + t\\ \end{array}\]

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	1.7

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

Derivation

1. Split input into 2 regimes

2. if x < -2.2597023320458045e76 or 685516672564733568 < x

   1. Initial program 4.8

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

   3. Applied *-un-lft-identity 4.8

      \[\leadsto \frac{x}{\color{blue}{1 \cdot y}} \cdot \left(z - t\right) + t\]

   4. Applied add-cube-cbrt 5.5

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{1 \cdot y} \cdot \left(z - t\right) + t\]

   5. Applied times-frac 5.5

      \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{1} \cdot \frac{\sqrt[3]{x}}{y}\right)} \cdot \left(z - t\right) + t\]

   6. Applied associate-*l* 2.3

      \[\leadsto \color{blue}{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{1} \cdot \left(\frac{\sqrt[3]{x}}{y} \cdot \left(z - t\right)\right)} + t\]

   if -2.2597023320458045e76 < x < 685516672564733568

   1. Initial program 1.0

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

   3. Applied sub-neg 1.0

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

   4. Applied distribute-lft-in 1.0

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

   5. Simplified 1.0

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

   6. Simplified 1.4

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

4. Final simplification 1.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.2597023320458045 \cdot 10^{76} \lor \neg \left(x \le 685516672564733568\right):\\ \;\;\;\;t + \left(\frac{\sqrt[3]{x}}{y} \cdot \left(z - t\right)\right) \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x \cdot z}{y} + \left(-\frac{t \cdot x}{y}\right)\right) + t\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(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))