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

Average Error: 2.2 → 2.5

Time: 11.1s

Precision: binary64

Cost: 7628

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -5 \cdot 10^{+265}:\\ \;\;\;\;\frac{x \cdot \left(z - t\right)}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq -100:\\ \;\;\;\;t + \frac{x}{y} \cdot \left(z - t\right)\\ \mathbf{elif}\;\frac{x}{y} \leq 0:\\ \;\;\;\;t + \frac{x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, z - t, t\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ x y) -5e+265)
   (/ (* x (- z t)) y)
   (if (<= (/ x y) -100.0)
     (+ t (* (/ x y) (- z t)))
     (if (<= (/ x y) 0.0) (+ 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+265) {
		tmp = (x * (z - t)) / y;
	} else if ((x / y) <= -100.0) {
		tmp = t + ((x / y) * (z - t));
	} else if ((x / y) <= 0.0) {
		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+265)
		tmp = Float64(Float64(x * Float64(z - t)) / y);
	elseif (Float64(x / y) <= -100.0)
		tmp = Float64(t + Float64(Float64(x / y) * Float64(z - t)));
	elseif (Float64(x / y) <= 0.0)
		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+265], N[(N[(x * N[(z - t), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[N[(x / y), $MachinePrecision], -100.0], N[(t + N[(N[(x / y), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x / y), $MachinePrecision], 0.0], N[(t + N[(N[(x * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(z - t), $MachinePrecision] + t), $MachinePrecision]]]]

Target

Original	2.2
Target	2.3
Herbie	2.5

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 x y) < -5.0000000000000002e265

   1. Initial program 39.7

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

   2. Simplified 0.7

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

      [Start] 39.7	\[ \frac{x}{y} \cdot \left(z - t\right) + t \]
      associate-*l/ [=>] 0.3	\[ \color{blue}{\frac{x \cdot \left(z - t\right)}{y}} + t \]
      associate-*r/ [<=] 0.7	\[ \color{blue}{x \cdot \frac{z - t}{y}} + t \]
      fma-def [=>] 0.7	\[ \color{blue}{\mathsf{fma}\left(x, \frac{z - t}{y}, t\right)} \]

   3. Taylor expanded in x around -inf 0.3

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

   if -5.0000000000000002e265 < (/.f64 x y) < -100

   1. Initial program 0.2

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

   if -100 < (/.f64 x y) < 0.0

   1. Initial program 1.2

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

   2. Taylor expanded in z around inf 3.8

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

   if 0.0 < (/.f64 x y)

   1. Initial program 2.0

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

   2. Simplified 2.0

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

      [Start] 2.0	\[ \frac{x}{y} \cdot \left(z - t\right) + t \]
      fma-def [=>] 2.0	\[ \color{blue}{\mathsf{fma}\left(\frac{x}{y}, z - t, t\right)} \]

3. Recombined 4 regimes into one program.

4. Final simplification 2.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -5 \cdot 10^{+265}:\\ \;\;\;\;\frac{x \cdot \left(z - t\right)}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq -100:\\ \;\;\;\;t + \frac{x}{y} \cdot \left(z - t\right)\\ \mathbf{elif}\;\frac{x}{y} \leq 0:\\ \;\;\;\;t + \frac{x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, z - t, t\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	22.3
Cost	1944

\[\begin{array}{l} t_1 := \frac{-t}{\frac{y}{x}}\\ t_2 := \frac{z}{\frac{y}{x}}\\ t_3 := \frac{x}{\frac{y}{z}}\\ \mathbf{if}\;\frac{x}{y} \leq -1.5 \cdot 10^{+128}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\frac{x}{y} \leq -5 \cdot 10^{+23}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{x}{y} \leq -2 \cdot 10^{-36}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{x}{y} \leq 5 \cdot 10^{-62}:\\ \;\;\;\;t\\ \mathbf{elif}\;\frac{x}{y} \leq 5 \cdot 10^{+33}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{x}{y} \leq 5 \cdot 10^{+94}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 2
Error	22.3
Cost	1944

\[\begin{array}{l} t_1 := \frac{z}{\frac{y}{x}}\\ t_2 := \frac{x}{\frac{y}{z}}\\ \mathbf{if}\;\frac{x}{y} \leq -1.5 \cdot 10^{+128}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{x}{y} \leq -5 \cdot 10^{+23}:\\ \;\;\;\;\frac{x}{y} \cdot \left(-t\right)\\ \mathbf{elif}\;\frac{x}{y} \leq -2 \cdot 10^{-36}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{x}{y} \leq 5 \cdot 10^{-62}:\\ \;\;\;\;t\\ \mathbf{elif}\;\frac{x}{y} \leq 5 \cdot 10^{+33}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{x}{y} \leq 5 \cdot 10^{+94}:\\ \;\;\;\;\frac{-t}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	2.6
Cost	1488

\[\begin{array}{l} t_1 := t + \frac{z}{\frac{y}{x}}\\ t_2 := \frac{x}{y} \cdot \left(z - t\right)\\ \mathbf{if}\;\frac{x}{y} \leq -100:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{x}{y} \leq -1 \cdot 10^{-152}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{x}{y} \leq 5 \cdot 10^{-288}:\\ \;\;\;\;t + \frac{x}{\frac{y}{z}}\\ \mathbf{elif}\;\frac{x}{y} \leq 0.001:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	2.5
Cost	1357

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -5 \cdot 10^{+265}:\\ \;\;\;\;\frac{x \cdot \left(z - t\right)}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq -100 \lor \neg \left(\frac{x}{y} \leq 0\right):\\ \;\;\;\;t + \frac{x}{y} \cdot \left(z - t\right)\\ \mathbf{else}:\\ \;\;\;\;t + \frac{x \cdot z}{y}\\ \end{array} \]

Alternative 5
Error	4.4
Cost	1229

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -5 \cdot 10^{+265}:\\ \;\;\;\;\frac{x \cdot \left(z - t\right)}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq -100 \lor \neg \left(\frac{x}{y} \leq 2 \cdot 10^{-22}\right):\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right)\\ \mathbf{else}:\\ \;\;\;\;t + \frac{x \cdot z}{y}\\ \end{array} \]

Alternative 6
Error	16.7
Cost	978

\[\begin{array}{l} \mathbf{if}\;t \leq -2.1 \cdot 10^{+26} \lor \neg \left(t \leq -2.8 \cdot 10^{-15}\right) \land \left(t \leq -3.2 \cdot 10^{-144} \lor \neg \left(t \leq 6.6 \cdot 10^{-149}\right)\right):\\ \;\;\;\;t \cdot \left(1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{z - t}{y}\\ \end{array} \]

Alternative 7
Error	14.6
Cost	969

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -2 \cdot 10^{-36} \lor \neg \left(\frac{x}{y} \leq 2 \cdot 10^{-22}\right):\\ \;\;\;\;x \cdot \frac{z - t}{y}\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 8
Error	13.1
Cost	969

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -4 \cdot 10^{-69} \lor \neg \left(\frac{x}{y} \leq 5 \cdot 10^{-62}\right):\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(1 - \frac{x}{y}\right)\\ \end{array} \]

Alternative 9
Error	4.3
Cost	969

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -100 \lor \neg \left(\frac{x}{y} \leq 0.001\right):\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right)\\ \mathbf{else}:\\ \;\;\;\;t + \frac{x}{\frac{y}{z}}\\ \end{array} \]

Alternative 10
Error	5.1
Cost	969

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -100 \lor \neg \left(\frac{x}{y} \leq 2 \cdot 10^{-22}\right):\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right)\\ \mathbf{else}:\\ \;\;\;\;t + \frac{x \cdot z}{y}\\ \end{array} \]

Alternative 11
Error	22.6
Cost	841

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -1.48 \cdot 10^{-36} \lor \neg \left(\frac{x}{y} \leq 2.45 \cdot 10^{-60}\right):\\ \;\;\;\;\frac{x}{y} \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 12
Error	23.1
Cost	840

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -2 \cdot 10^{-36}:\\ \;\;\;\;\frac{x}{\frac{y}{z}}\\ \mathbf{elif}\;\frac{x}{y} \leq 5 \cdot 10^{-62}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot z\\ \end{array} \]

Alternative 13
Error	23.1
Cost	840

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -2 \cdot 10^{-36}:\\ \;\;\;\;\frac{x}{\frac{y}{z}}\\ \mathbf{elif}\;\frac{x}{y} \leq 5 \cdot 10^{-62}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\frac{y}{x}}\\ \end{array} \]

Alternative 14
Error	32.0
Cost	64

\[t \]

Reproduce

herbie shell --seed 2023059 
(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))