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

Average Accuracy: 96.6% → 96.7%

Time: 9.6s

Precision: binary64

Cost: 7369

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -2 \cdot 10^{-206} \lor \neg \left(\frac{x}{y} \leq 5 \cdot 10^{-124}\right):\\ \;\;\;\;t + \frac{x}{y} \cdot \left(z - t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{z - t}{y}, t\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (/ x y) -2e-206) (not (<= (/ x y) 5e-124)))
   (+ t (* (/ x y) (- z t)))
   (fma x (/ (- z t) y) 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 (((x / y) <= -2e-206) || !((x / y) <= 5e-124)) {
		tmp = t + ((x / y) * (z - t));
	} else {
		tmp = fma(x, ((z - t) / y), 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 ((Float64(x / y) <= -2e-206) || !(Float64(x / y) <= 5e-124))
		tmp = Float64(t + Float64(Float64(x / y) * Float64(z - t)));
	else
		tmp = fma(x, Float64(Float64(z - t) / y), 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[Or[LessEqual[N[(x / y), $MachinePrecision], -2e-206], N[Not[LessEqual[N[(x / y), $MachinePrecision], 5e-124]], $MachinePrecision]], N[(t + N[(N[(x / y), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(z - t), $MachinePrecision] / y), $MachinePrecision] + t), $MachinePrecision]]

Target

Original	96.6%
Target	96.2%
Herbie	96.7%

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < -2.00000000000000006e-206 or 5.0000000000000003e-124 < (/.f64 x y)

   1. Initial program 95.9%

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

   if -2.00000000000000006e-206 < (/.f64 x y) < 5.0000000000000003e-124

   1. Initial program 97.8%

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

   2. Simplified 98.1%

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

      [Start] 97.8	\[ \frac{x}{y} \cdot \left(z - t\right) + t \]
      associate-*l/ [=>] 98.2	\[ \color{blue}{\frac{x \cdot \left(z - t\right)}{y}} + t \]
      associate-*r/ [<=] 98.1	\[ \color{blue}{x \cdot \frac{z - t}{y}} + t \]
      fma-def [=>] 98.1	\[ \color{blue}{\mathsf{fma}\left(x, \frac{z - t}{y}, t\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -2 \cdot 10^{-206} \lor \neg \left(\frac{x}{y} \leq 5 \cdot 10^{-124}\right):\\ \;\;\;\;t + \frac{x}{y} \cdot \left(z - t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{z - t}{y}, t\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	65.4%
Cost	1164

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -5 \cdot 10^{-70}:\\ \;\;\;\;\frac{z}{\frac{y}{x}}\\ \mathbf{elif}\;\frac{x}{y} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;t\\ \mathbf{elif}\;\frac{x}{y} \leq 4 \cdot 10^{+76}:\\ \;\;\;\;\frac{x}{y} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{-t}{\frac{y}{x}}\\ \end{array} \]

Alternative 2
Accuracy	96.2%
Cost	1097

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -1 \cdot 10^{-67} \lor \neg \left(\frac{x}{y} \leq 5 \cdot 10^{-124}\right):\\ \;\;\;\;t + \frac{x}{y} \cdot \left(z - t\right)\\ \mathbf{else}:\\ \;\;\;\;t + \frac{x}{\frac{y}{z}}\\ \end{array} \]

Alternative 3
Accuracy	78.5%
Cost	969

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -2 \cdot 10^{-18} \lor \neg \left(\frac{x}{y} \leq 10^{-6}\right):\\ \;\;\;\;x \cdot \frac{z - t}{y}\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 4
Accuracy	80.4%
Cost	969

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -5 \cdot 10^{-70} \lor \neg \left(\frac{x}{y} \leq 2 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right)\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 5
Accuracy	93.2%
Cost	969

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -2000 \lor \neg \left(\frac{x}{y} \leq 2 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right)\\ \mathbf{else}:\\ \;\;\;\;t + \frac{x}{\frac{y}{z}}\\ \end{array} \]

Alternative 6
Accuracy	64.8%
Cost	841

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -5 \cdot 10^{-70} \lor \neg \left(\frac{x}{y} \leq 2 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{x}{y} \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 7
Accuracy	64.9%
Cost	840

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -5 \cdot 10^{-70}:\\ \;\;\;\;\frac{z}{\frac{y}{x}}\\ \mathbf{elif}\;\frac{x}{y} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot z\\ \end{array} \]

Alternative 8
Accuracy	50.0%
Cost	64

\[t \]

Reproduce

herbie shell --seed 2023135 
(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))