Logistic function from Lakshay Garg

Average Error: 29.3 → 1.0

Time: 10.1s

Precision: binary64

Cost: 46788

Math

\[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]

↓

\[\begin{array}{l} t_0 := e^{-2 \cdot x}\\ \mathbf{if}\;-2 \cdot x \leq -0.1:\\ \;\;\;\;\frac{-1 + \frac{-8}{{\left(-1 - t_0\right)}^{3}}}{1 + \left(\frac{2}{t_0 + 1} + \frac{4}{{\left(e^{2}\right)}^{\left(\mathsf{log1p}\left(t_0\right)\right)}}\right)}\\ \mathbf{elif}\;-2 \cdot x \leq 10^{-19}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (exp (* -2.0 x))))
   (if (<= (* -2.0 x) -0.1)
     (/
      (+ -1.0 (/ -8.0 (pow (- -1.0 t_0) 3.0)))
      (+ 1.0 (+ (/ 2.0 (+ t_0 1.0)) (/ 4.0 (pow (exp 2.0) (log1p t_0))))))
     (if (<= (* -2.0 x) 1e-19)
       (+ x (* -0.3333333333333333 (pow x 3.0)))
       -1.0))))

C

double code(double x, double y) {
	return (2.0 / (1.0 + exp((-2.0 * x)))) - 1.0;
}

↓

double code(double x, double y) {
	double t_0 = exp((-2.0 * x));
	double tmp;
	if ((-2.0 * x) <= -0.1) {
		tmp = (-1.0 + (-8.0 / pow((-1.0 - t_0), 3.0))) / (1.0 + ((2.0 / (t_0 + 1.0)) + (4.0 / pow(exp(2.0), log1p(t_0)))));
	} else if ((-2.0 * x) <= 1e-19) {
		tmp = x + (-0.3333333333333333 * pow(x, 3.0));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	return (2.0 / (1.0 + Math.exp((-2.0 * x)))) - 1.0;
}

↓

public static double code(double x, double y) {
	double t_0 = Math.exp((-2.0 * x));
	double tmp;
	if ((-2.0 * x) <= -0.1) {
		tmp = (-1.0 + (-8.0 / Math.pow((-1.0 - t_0), 3.0))) / (1.0 + ((2.0 / (t_0 + 1.0)) + (4.0 / Math.pow(Math.exp(2.0), Math.log1p(t_0)))));
	} else if ((-2.0 * x) <= 1e-19) {
		tmp = x + (-0.3333333333333333 * Math.pow(x, 3.0));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	return (2.0 / (1.0 + math.exp((-2.0 * x)))) - 1.0

↓

def code(x, y):
	t_0 = math.exp((-2.0 * x))
	tmp = 0
	if (-2.0 * x) <= -0.1:
		tmp = (-1.0 + (-8.0 / math.pow((-1.0 - t_0), 3.0))) / (1.0 + ((2.0 / (t_0 + 1.0)) + (4.0 / math.pow(math.exp(2.0), math.log1p(t_0)))))
	elif (-2.0 * x) <= 1e-19:
		tmp = x + (-0.3333333333333333 * math.pow(x, 3.0))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	return Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) - 1.0)
end

↓

function code(x, y)
	t_0 = exp(Float64(-2.0 * x))
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.1)
		tmp = Float64(Float64(-1.0 + Float64(-8.0 / (Float64(-1.0 - t_0) ^ 3.0))) / Float64(1.0 + Float64(Float64(2.0 / Float64(t_0 + 1.0)) + Float64(4.0 / (exp(2.0) ^ log1p(t_0))))));
	elseif (Float64(-2.0 * x) <= 1e-19)
		tmp = Float64(x + Float64(-0.3333333333333333 * (x ^ 3.0)));
	else
		tmp = -1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

↓

code[x_, y_] := Block[{t$95$0 = N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.1], N[(N[(-1.0 + N[(-8.0 / N[Power[N[(-1.0 - t$95$0), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(2.0 / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision] + N[(4.0 / N[Power[N[Exp[2.0], $MachinePrecision], N[Log[1 + t$95$0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 1e-19], N[(x + N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 -2 x) < -0.10000000000000001

   1. Initial program 0.0

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]

   2. Applied egg-rr 0.0

      \[\leadsto \color{blue}{\left(-1 + \frac{-8}{{\left(-1 - {\left(e^{-2}\right)}^{x}\right)}^{3}}\right) \cdot \frac{1}{1 + \left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}} + \frac{4}{{\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{2}}\right)}} \]

   3. Simplified 0.0

      \[\leadsto \color{blue}{\frac{\left(-1 + \frac{-8}{{\left(-1 - e^{x \cdot -2}\right)}^{3}}\right) \cdot 1}{1 + \left(\frac{2}{1 + e^{x \cdot -2}} + \frac{4}{{\left(1 + e^{x \cdot -2}\right)}^{2}}\right)}} \]
      Proof

      [Start] 0.0	\[ \left(-1 + \frac{-8}{{\left(-1 - {\left(e^{-2}\right)}^{x}\right)}^{3}}\right) \cdot \frac{1}{1 + \left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}} + \frac{4}{{\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{2}}\right)} \]
      associate-*r/ [=>] 0.0	\[ \color{blue}{\frac{\left(-1 + \frac{-8}{{\left(-1 - {\left(e^{-2}\right)}^{x}\right)}^{3}}\right) \cdot 1}{1 + \left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}} + \frac{4}{{\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{2}}\right)}} \]
      exp-prod [<=] 0.0	\[ \frac{\left(-1 + \frac{-8}{{\left(-1 - \color{blue}{e^{-2 \cdot x}}\right)}^{3}}\right) \cdot 1}{1 + \left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}} + \frac{4}{{\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{2}}\right)} \]
      *-commutative [=>] 0.0	\[ \frac{\left(-1 + \frac{-8}{{\left(-1 - e^{\color{blue}{x \cdot -2}}\right)}^{3}}\right) \cdot 1}{1 + \left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}} + \frac{4}{{\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{2}}\right)} \]
      unpow2 [=>] 0.0	\[ \frac{\left(-1 + \frac{-8}{{\left(-1 - e^{x \cdot -2}\right)}^{3}}\right) \cdot 1}{1 + \left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}} + \frac{4}{\color{blue}{\left(1 + {\left(e^{-2}\right)}^{x}\right) \cdot \left(1 + {\left(e^{-2}\right)}^{x}\right)}}\right)} \]
      associate-/r* [=>] 0.0	\[ \frac{\left(-1 + \frac{-8}{{\left(-1 - e^{x \cdot -2}\right)}^{3}}\right) \cdot 1}{1 + \left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}} + \color{blue}{\frac{\frac{4}{1 + {\left(e^{-2}\right)}^{x}}}{1 + {\left(e^{-2}\right)}^{x}}}\right)} \]
      /-rgt-identity [<=] 0.0	\[ \frac{\left(-1 + \frac{-8}{{\left(-1 - e^{x \cdot -2}\right)}^{3}}\right) \cdot 1}{1 + \left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}} + \frac{\frac{4}{1 + {\left(e^{-2}\right)}^{x}}}{\color{blue}{\frac{1 + {\left(e^{-2}\right)}^{x}}{1}}}\right)} \]
      associate-/r/ [=>] 0.0	\[ \frac{\left(-1 + \frac{-8}{{\left(-1 - e^{x \cdot -2}\right)}^{3}}\right) \cdot 1}{1 + \left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}} + \color{blue}{\frac{\frac{4}{1 + {\left(e^{-2}\right)}^{x}}}{1 + {\left(e^{-2}\right)}^{x}} \cdot 1}\right)} \]

   4. Applied egg-rr 0.0

      \[\leadsto \frac{\left(-1 + \frac{-8}{{\left(-1 - e^{x \cdot -2}\right)}^{3}}\right) \cdot 1}{1 + \left(\frac{2}{1 + e^{x \cdot -2}} + \frac{4}{\color{blue}{e^{2 \cdot \mathsf{log1p}\left({\left(e^{x}\right)}^{-2}\right)}}}\right)} \]

   5. Simplified 0.0

      \[\leadsto \frac{\left(-1 + \frac{-8}{{\left(-1 - e^{x \cdot -2}\right)}^{3}}\right) \cdot 1}{1 + \left(\frac{2}{1 + e^{x \cdot -2}} + \frac{4}{\color{blue}{{\left(e^{2}\right)}^{\left(\mathsf{log1p}\left(e^{x \cdot -2}\right)\right)}}}\right)} \]
      Proof

      [Start] 0.0	\[ \frac{\left(-1 + \frac{-8}{{\left(-1 - e^{x \cdot -2}\right)}^{3}}\right) \cdot 1}{1 + \left(\frac{2}{1 + e^{x \cdot -2}} + \frac{4}{e^{2 \cdot \mathsf{log1p}\left({\left(e^{x}\right)}^{-2}\right)}}\right)} \]
      exp-prod [=>] 0.0	\[ \frac{\left(-1 + \frac{-8}{{\left(-1 - e^{x \cdot -2}\right)}^{3}}\right) \cdot 1}{1 + \left(\frac{2}{1 + e^{x \cdot -2}} + \frac{4}{\color{blue}{{\left(e^{2}\right)}^{\left(\mathsf{log1p}\left({\left(e^{x}\right)}^{-2}\right)\right)}}}\right)} \]
      exp-prod [<=] 0.0	\[ \frac{\left(-1 + \frac{-8}{{\left(-1 - e^{x \cdot -2}\right)}^{3}}\right) \cdot 1}{1 + \left(\frac{2}{1 + e^{x \cdot -2}} + \frac{4}{{\left(e^{2}\right)}^{\left(\mathsf{log1p}\left(\color{blue}{e^{x \cdot -2}}\right)\right)}}\right)} \]

   if -0.10000000000000001 < (*.f64 -2 x) < 9.9999999999999998e-20

   1. Initial program 59.5

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]

   2. Taylor expanded in x around 0 0.1

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot {x}^{3} + x} \]

   if 9.9999999999999998e-20 < (*.f64 -2 x)

   1. Initial program 2.2

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]

   2. Taylor expanded in x around 0 4.2

      \[\leadsto \frac{2}{\color{blue}{2 + -2 \cdot x}} - 1 \]

   3. Simplified 4.2

      \[\leadsto \frac{2}{\color{blue}{2 + x \cdot -2}} - 1 \]
      Proof

      [Start] 4.2	\[ \frac{2}{2 + -2 \cdot x} - 1 \]
      *-commutative [=>] 4.2	\[ \frac{2}{2 + \color{blue}{x \cdot -2}} - 1 \]

   4. Taylor expanded in x around inf 3.8

      \[\leadsto \color{blue}{-1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.1:\\ \;\;\;\;\frac{-1 + \frac{-8}{{\left(-1 - e^{-2 \cdot x}\right)}^{3}}}{1 + \left(\frac{2}{e^{-2 \cdot x} + 1} + \frac{4}{{\left(e^{2}\right)}^{\left(\mathsf{log1p}\left(e^{-2 \cdot x}\right)\right)}}\right)}\\ \mathbf{elif}\;-2 \cdot x \leq 10^{-19}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternatives

Alternative 1
Error	1.0
Cost	34116

\[\begin{array}{l} t_0 := e^{-2 \cdot x} + 1\\ \mathbf{if}\;-2 \cdot x \leq -0.1:\\ \;\;\;\;\frac{-1 + \frac{8}{{t_0}^{3}}}{\frac{2}{t_0} + \left(1 + \frac{4}{{t_0}^{2}}\right)}\\ \mathbf{elif}\;-2 \cdot x \leq 10^{-19}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 2
Error	1.0
Cost	20036

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.1:\\ \;\;\;\;\log \left(e^{-1 + \frac{2}{e^{-2 \cdot x} + 1}}\right)\\ \mathbf{elif}\;-2 \cdot x \leq 10^{-19}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 3
Error	1.0
Cost	7304

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.1:\\ \;\;\;\;-1 + \frac{2}{e^{-2 \cdot x} + 1}\\ \mathbf{elif}\;-2 \cdot x \leq 10^{-19}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 4
Error	15.7
Cost	196

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	46.9
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023041 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))