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

Average Accuracy: 96.9% → 97.4%

Time: 12.8s

Precision: binary64

Cost: 7368

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -5 \cdot 10^{-133}:\\ \;\;\;\;t + \frac{x}{y} \cdot \left(z - t\right)\\ \mathbf{elif}\;\frac{x}{y} \leq 4 \cdot 10^{-196}:\\ \;\;\;\;t + \frac{x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, z - t, t\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ x y) -5e-133)
   (+ t (* (/ x y) (- z t)))
   (if (<= (/ x y) 4e-196) (+ t (/ (* x z) 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 ((x / y) <= -5e-133) {
		tmp = t + ((x / y) * (z - t));
	} else if ((x / y) <= 4e-196) {
		tmp = t + ((x * z) / 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 (Float64(x / y) <= -5e-133)
		tmp = Float64(t + Float64(Float64(x / y) * Float64(z - t)));
	elseif (Float64(x / y) <= 4e-196)
		tmp = Float64(t + Float64(Float64(x * z) / 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[N[(x / y), $MachinePrecision], -5e-133], N[(t + N[(N[(x / y), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x / y), $MachinePrecision], 4e-196], N[(t + N[(N[(x * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(z - t), $MachinePrecision] + t), $MachinePrecision]]]

Target

Original	96.9%
Target	96.4%
Herbie	97.4%

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 x y) < -4.9999999999999999e-133

   1. Initial program 95.8%

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

   if -4.9999999999999999e-133 < (/.f64 x y) < 4.0000000000000002e-196

   1. Initial program 97.5%

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

   2. Taylor expanded in z around inf 98.9%

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

   if 4.0000000000000002e-196 < (/.f64 x y)

   1. Initial program 97.2%

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

   2. Simplified 97.2%

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

      [Start] 97.2	\[ \frac{x}{y} \cdot \left(z - t\right) + t \]
      fma-def [=>] 97.2	\[ \color{blue}{\mathsf{fma}\left(\frac{x}{y}, z - t, t\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -5 \cdot 10^{-133}:\\ \;\;\;\;t + \frac{x}{y} \cdot \left(z - t\right)\\ \mathbf{elif}\;\frac{x}{y} \leq 4 \cdot 10^{-196}:\\ \;\;\;\;t + \frac{x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, z - t, t\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	64.7%
Cost	1684

\[\begin{array}{l} t_1 := \frac{x}{y} \cdot z\\ \mathbf{if}\;\frac{x}{y} \leq -1 \cdot 10^{-17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{x}{y} \leq 5 \cdot 10^{-115}:\\ \;\;\;\;t\\ \mathbf{elif}\;\frac{x}{y} \leq 2 \cdot 10^{+116}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{x}{y} \leq 5 \cdot 10^{+153}:\\ \;\;\;\;t \cdot \frac{-x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 10^{+267}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{y} \cdot \left(-x\right)\\ \end{array} \]

Alternative 2
Accuracy	64.7%
Cost	1684

\[\begin{array}{l} t_1 := \frac{x}{y} \cdot z\\ \mathbf{if}\;\frac{x}{y} \leq -1 \cdot 10^{-17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{x}{y} \leq 5 \cdot 10^{-115}:\\ \;\;\;\;t\\ \mathbf{elif}\;\frac{x}{y} \leq 2 \cdot 10^{+116}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{x}{y} \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\frac{t}{-\frac{y}{x}}\\ \mathbf{elif}\;\frac{x}{y} \leq 10^{+267}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{y} \cdot \left(-x\right)\\ \end{array} \]

Alternative 3
Accuracy	64.7%
Cost	1684

\[\begin{array}{l} t_1 := \frac{x}{y} \cdot z\\ \mathbf{if}\;\frac{x}{y} \leq -1 \cdot 10^{-17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{x}{y} \leq 5 \cdot 10^{-115}:\\ \;\;\;\;t\\ \mathbf{elif}\;\frac{x}{y} \leq 2 \cdot 10^{+116}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{x}{y} \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\frac{t}{-\frac{y}{x}}\\ \mathbf{elif}\;\frac{x}{y} \leq 10^{+267}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-\frac{y}{t}}\\ \end{array} \]

Alternative 4
Accuracy	90.5%
Cost	1488

\[\begin{array}{l} t_1 := \frac{x \cdot \left(z - t\right)}{y}\\ \mathbf{if}\;\frac{x}{y} \leq -1 \cdot 10^{+108}:\\ \;\;\;\;\frac{x}{\frac{y}{z - t}}\\ \mathbf{elif}\;\frac{x}{y} \leq -4 \cdot 10^{+38}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{x}{y} \leq -5 \cdot 10^{-8}:\\ \;\;\;\;t - \frac{x}{y} \cdot t\\ \mathbf{elif}\;\frac{x}{y} \leq 1:\\ \;\;\;\;t + \frac{x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Accuracy	64.6%
Cost	1424

\[\begin{array}{l} t_1 := \frac{x}{y} \cdot z\\ \mathbf{if}\;\frac{x}{y} \leq -1 \cdot 10^{-17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{x}{y} \leq 5 \cdot 10^{-115}:\\ \;\;\;\;t\\ \mathbf{elif}\;\frac{x}{y} \leq 2 \cdot 10^{+116}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{x}{y} \leq 5 \cdot 10^{+153}:\\ \;\;\;\;t \cdot \frac{-x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{z}}\\ \end{array} \]

Alternative 6
Accuracy	97.3%
Cost	1097

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -5 \cdot 10^{-133} \lor \neg \left(\frac{x}{y} \leq 4 \cdot 10^{-196}\right):\\ \;\;\;\;t + \frac{x}{y} \cdot \left(z - t\right)\\ \mathbf{else}:\\ \;\;\;\;t + \frac{x \cdot z}{y}\\ \end{array} \]

Alternative 7
Accuracy	92.5%
Cost	969

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -5000000000 \lor \neg \left(\frac{x}{y} \leq 10^{+38}\right):\\ \;\;\;\;\frac{x}{\frac{y}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;t + \frac{z}{\frac{y}{x}}\\ \end{array} \]

Alternative 8
Accuracy	64.1%
Cost	841

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -1 \cdot 10^{-17} \lor \neg \left(\frac{x}{y} \leq 5 \cdot 10^{-115}\right):\\ \;\;\;\;\frac{x}{y} \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 9
Accuracy	71.5%
Cost	713

\[\begin{array}{l} \mathbf{if}\;t \leq -6.1 \cdot 10^{-111} \lor \neg \left(t \leq 4.3 \cdot 10^{-46}\right):\\ \;\;\;\;t \cdot \left(1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot z\\ \end{array} \]

Alternative 10
Accuracy	87.7%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{-58} \lor \neg \left(z \leq 3.4 \cdot 10^{-110}\right):\\ \;\;\;\;t + \frac{z}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(1 - \frac{x}{y}\right)\\ \end{array} \]

Alternative 11
Accuracy	85.8%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-59}:\\ \;\;\;\;t + \frac{x \cdot z}{y}\\ \mathbf{elif}\;z \leq 3.05 \cdot 10^{-109}:\\ \;\;\;\;t \cdot \left(1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;t + \frac{z}{\frac{y}{x}}\\ \end{array} \]

Alternative 12
Accuracy	85.8%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{-58}:\\ \;\;\;\;t + \frac{x \cdot z}{y}\\ \mathbf{elif}\;z \leq 9.8 \cdot 10^{-110}:\\ \;\;\;\;t - \frac{x}{y} \cdot t\\ \mathbf{else}:\\ \;\;\;\;t + \frac{z}{\frac{y}{x}}\\ \end{array} \]

Alternative 13
Accuracy	85.9%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{-58}:\\ \;\;\;\;t + \frac{x \cdot z}{y}\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{-109}:\\ \;\;\;\;t - \frac{t}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;t + \frac{z}{\frac{y}{x}}\\ \end{array} \]

Alternative 14
Accuracy	50.6%
Cost	64

\[t \]

Reproduce

herbie shell --seed 2023147 
(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))