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

Average Error: 2.0 → 1.4

Time: 3.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -2.2024189491739263 \cdot 10^{-21}:\\ \;\;\;\;t + \frac{x}{\frac{y}{z - t}}\\ \mathbf{elif}\;y \leq 2.8858824852870217 \cdot 10^{-56}:\\ \;\;\;\;t + \left(x \cdot \left(z - t\right)\right) \cdot \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, z - t, t\right)\\ \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (+ (* (/ x y) (- z t)) t))

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= y -2.2024189491739263e-21)
   (+ t (/ x (/ y (- z t))))
   (if (<= y 2.8858824852870217e-56)
     (+ t (* (* x (- z t)) (/ 1.0 y)))
     (fma (/ x y) (- z t) t))))

C

double code(double x, double y, double z, double t) {
	return ((x / y) * (z - t)) + t;
}

↓

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -2.2024189491739263e-21) {
		tmp = t + (x / (y / (z - t)));
	} else if (y <= 2.8858824852870217e-56) {
		tmp = t + ((x * (z - t)) * (1.0 / y));
	} else {
		tmp = fma((x / y), (z - t), t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x / y) * Float64(z - t)) + t)
end

↓

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -2.2024189491739263e-21)
		tmp = Float64(t + Float64(x / Float64(y / Float64(z - t))));
	elseif (y <= 2.8858824852870217e-56)
		tmp = Float64(t + Float64(Float64(x * Float64(z - t)) * Float64(1.0 / y)));
	else
		tmp = fma(Float64(x / y), Float64(z - t), t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x / y), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]

↓

code[x_, y_, z_, t_] := If[LessEqual[y, -2.2024189491739263e-21], N[(t + N[(x / N[(y / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.8858824852870217e-56], N[(t + N[(N[(x * N[(z - t), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(z - t), $MachinePrecision] + t), $MachinePrecision]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	2.0
Target	2.3
Herbie	1.4

\[\begin{array}{l} \mathbf{if}\;z < 2.759456554562692 \cdot 10^{-282}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \mathbf{elif}\;z < 2.326994450874436 \cdot 10^{-110}:\\ \;\;\;\;x \cdot \frac{z - t}{y} + t\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if y < -2.20241894917392626e-21

   1. Initial program 1.1

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

   2. Applied egg-rr 0.8

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

   if -2.20241894917392626e-21 < y < 2.8858824852870217e-56

   1. Initial program 3.8

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

   2. Applied egg-rr 4.5

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

   3. Applied egg-rr 2.1

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

   if 2.8858824852870217e-56 < y

   1. Initial program 1.2

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

   2. Applied egg-rr 1.2

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

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2024189491739263 \cdot 10^{-21}:\\ \;\;\;\;t + \frac{x}{\frac{y}{z - t}}\\ \mathbf{elif}\;y \leq 2.8858824852870217 \cdot 10^{-56}:\\ \;\;\;\;t + \left(x \cdot \left(z - t\right)\right) \cdot \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, z - t, t\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022150 
(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))