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

Average Error: 2.0 → 2.7

Time: 9.2s

Precision: binary64

Cost: 7113

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \leq -3.35 \cdot 10^{-162} \lor \neg \left(t \leq 1.1 \cdot 10^{-284}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, z - t, t\right)\\ \mathbf{else}:\\ \;\;\;\;t + \frac{x \cdot z}{y}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -3.35e-162) (not (<= t 1.1e-284)))
   (fma (/ x y) (- z t) t)
   (+ t (/ (* x z) y))))

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 ((t <= -3.35e-162) || !(t <= 1.1e-284)) {
		tmp = fma((x / y), (z - t), t);
	} else {
		tmp = t + ((x * z) / y);
	}
	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 ((t <= -3.35e-162) || !(t <= 1.1e-284))
		tmp = fma(Float64(x / y), Float64(z - t), t);
	else
		tmp = Float64(t + Float64(Float64(x * z) / y));
	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[Or[LessEqual[t, -3.35e-162], N[Not[LessEqual[t, 1.1e-284]], $MachinePrecision]], N[(N[(x / y), $MachinePrecision] * N[(z - t), $MachinePrecision] + t), $MachinePrecision], N[(t + N[(N[(x * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Target

Original	2.0
Target	2.1
Herbie	2.7

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

Derivation

1. Split input into 2 regimes

2. if t < -3.3499999999999998e-162 or 1.1e-284 < t

   1. Initial program 1.4

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

   2. Simplified 1.4

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

      [Start] 1.4	\[ \frac{x}{y} \cdot \left(z - t\right) + t \]
      fma-def [=>] 1.4	\[ \color{blue}{\mathsf{fma}\left(\frac{x}{y}, z - t, t\right)} \]

   if -3.3499999999999998e-162 < t < 1.1e-284

   1. Initial program 4.9

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

   2. Taylor expanded in z around inf 9.4

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

4. Final simplification 2.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.35 \cdot 10^{-162} \lor \neg \left(t \leq 1.1 \cdot 10^{-284}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, z - t, t\right)\\ \mathbf{else}:\\ \;\;\;\;t + \frac{x \cdot z}{y}\\ \end{array} \]

Alternatives

Alternative 1
Error	18.9
Cost	1241

\[\begin{array}{l} t_1 := t \cdot \left(1 - \frac{x}{y}\right)\\ \mathbf{if}\;t \leq -1.26 \cdot 10^{-76}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{-277}:\\ \;\;\;\;\frac{x}{\frac{y}{z}}\\ \mathbf{elif}\;t \leq 1.95 \cdot 10^{-243}:\\ \;\;\;\;t\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{-192} \lor \neg \left(t \leq 1.06 \cdot 10^{-114}\right) \land t \leq 1.85 \cdot 10^{-73}:\\ \;\;\;\;\frac{x}{y} \cdot z\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	19.0
Cost	1240

\[\begin{array}{l} t_1 := x \cdot \frac{z - t}{y}\\ t_2 := t \cdot \left(1 - \frac{x}{y}\right)\\ \mathbf{if}\;t \leq -2.2 \cdot 10^{-76}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -5.2 \cdot 10^{-117}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -4.4 \cdot 10^{-143}:\\ \;\;\;\;t\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{-273}:\\ \;\;\;\;\frac{x \cdot z}{y}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-243}:\\ \;\;\;\;t\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{+59}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	18.9
Cost	1240

\[\begin{array}{l} t_1 := x \cdot \frac{z - t}{y}\\ t_2 := t \cdot \left(1 - \frac{x}{y}\right)\\ \mathbf{if}\;t \leq -2.05 \cdot 10^{-76}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -2.25 \cdot 10^{-122}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -3 \cdot 10^{-143}:\\ \;\;\;\;t\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-272}:\\ \;\;\;\;\left(x \cdot z\right) \cdot \frac{1}{y}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-243}:\\ \;\;\;\;t\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{+59}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	22.9
Cost	1164

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -2 \cdot 10^{-94}:\\ \;\;\;\;\frac{z}{\frac{y}{x}}\\ \mathbf{elif}\;\frac{x}{y} \leq 10:\\ \;\;\;\;t\\ \mathbf{elif}\;\frac{x}{y} \leq 5 \cdot 10^{+240}:\\ \;\;\;\;\frac{-t}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{z}{y}\\ \end{array} \]

Alternative 5
Error	22.9
Cost	1164

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -2 \cdot 10^{-94}:\\ \;\;\;\;\frac{z}{\frac{y}{x}}\\ \mathbf{elif}\;\frac{x}{y} \leq 10:\\ \;\;\;\;t\\ \mathbf{elif}\;\frac{x}{y} \leq 5 \cdot 10^{+240}:\\ \;\;\;\;\frac{x}{y} \cdot \left(-t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{z}{y}\\ \end{array} \]

Alternative 6
Error	5.0
Cost	969

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -4 \cdot 10^{+29} \lor \neg \left(\frac{x}{y} \leq 10\right):\\ \;\;\;\;x \cdot \frac{z - t}{y}\\ \mathbf{else}:\\ \;\;\;\;t + \frac{z}{\frac{y}{x}}\\ \end{array} \]

Alternative 7
Error	5.4
Cost	969

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -50000000000000 \lor \neg \left(\frac{x}{y} \leq 10^{-17}\right):\\ \;\;\;\;\frac{x \cdot \left(z - t\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;t + \frac{z}{\frac{y}{x}}\\ \end{array} \]

Alternative 8
Error	5.4
Cost	968

\[\begin{array}{l} t_1 := x \cdot \left(z - t\right)\\ \mathbf{if}\;\frac{x}{y} \leq -50000000000000:\\ \;\;\;\;\frac{1}{y} \cdot t_1\\ \mathbf{elif}\;\frac{x}{y} \leq 10^{-17}:\\ \;\;\;\;t + \frac{z}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{y}\\ \end{array} \]

Alternative 9
Error	23.1
Cost	841

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -1.05 \cdot 10^{-94} \lor \neg \left(\frac{x}{y} \leq 3.1 \cdot 10^{-17}\right):\\ \;\;\;\;\frac{x}{y} \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 10
Error	2.7
Cost	841

\[\begin{array}{l} \mathbf{if}\;t \leq -2.4 \cdot 10^{-163} \lor \neg \left(t \leq 1.6 \cdot 10^{-283}\right):\\ \;\;\;\;t + \frac{x}{y} \cdot \left(z - t\right)\\ \mathbf{else}:\\ \;\;\;\;t + \frac{x \cdot z}{y}\\ \end{array} \]

Alternative 11
Error	23.5
Cost	840

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -2 \cdot 10^{-94}:\\ \;\;\;\;\frac{x}{y} \cdot z\\ \mathbf{elif}\;\frac{x}{y} \leq 10^{-17}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{z}{y}\\ \end{array} \]

Alternative 12
Error	23.5
Cost	840

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -2 \cdot 10^{-94}:\\ \;\;\;\;\frac{x}{y} \cdot z\\ \mathbf{elif}\;\frac{x}{y} \leq 10^{-17}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{z}}\\ \end{array} \]

Alternative 13
Error	23.5
Cost	840

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -2 \cdot 10^{-94}:\\ \;\;\;\;\frac{z}{\frac{y}{x}}\\ \mathbf{elif}\;\frac{x}{y} \leq 10^{-17}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{z}}\\ \end{array} \]

Alternative 14
Error	31.3
Cost	64

\[t \]

Reproduce

herbie shell --seed 2023031 
(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))