Logistic function from Lakshay Garg

Average Error: 29.3 → 1.2

Time: 9.5s

Precision: binary64

Cost: 20744

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -5:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{2}{2 + \mathsf{expm1}\left(-2 \cdot x\right)} + -1}}\\ \mathbf{elif}\;-2 \cdot x \leq 10^{-24}:\\ \;\;\;\;-0.05396825396825397 \cdot {x}^{7} + \left(-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -5.0)
   (/ 1.0 (/ 1.0 (+ (/ 2.0 (+ 2.0 (expm1 (* -2.0 x)))) -1.0)))
   (if (<= (* -2.0 x) 1e-24)
     (+
      (* -0.05396825396825397 (pow x 7.0))
      (+
       (* -0.3333333333333333 (pow x 3.0))
       (+ x (* 0.13333333333333333 (pow x 5.0)))))
     -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 tmp;
	if ((-2.0 * x) <= -5.0) {
		tmp = 1.0 / (1.0 / ((2.0 / (2.0 + expm1((-2.0 * x)))) + -1.0));
	} else if ((-2.0 * x) <= 1e-24) {
		tmp = (-0.05396825396825397 * pow(x, 7.0)) + ((-0.3333333333333333 * pow(x, 3.0)) + (x + (0.13333333333333333 * pow(x, 5.0))));
	} 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 tmp;
	if ((-2.0 * x) <= -5.0) {
		tmp = 1.0 / (1.0 / ((2.0 / (2.0 + Math.expm1((-2.0 * x)))) + -1.0));
	} else if ((-2.0 * x) <= 1e-24) {
		tmp = (-0.05396825396825397 * Math.pow(x, 7.0)) + ((-0.3333333333333333 * Math.pow(x, 3.0)) + (x + (0.13333333333333333 * Math.pow(x, 5.0))));
	} 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):
	tmp = 0
	if (-2.0 * x) <= -5.0:
		tmp = 1.0 / (1.0 / ((2.0 / (2.0 + math.expm1((-2.0 * x)))) + -1.0))
	elif (-2.0 * x) <= 1e-24:
		tmp = (-0.05396825396825397 * math.pow(x, 7.0)) + ((-0.3333333333333333 * math.pow(x, 3.0)) + (x + (0.13333333333333333 * math.pow(x, 5.0))))
	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)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -5.0)
		tmp = Float64(1.0 / Float64(1.0 / Float64(Float64(2.0 / Float64(2.0 + expm1(Float64(-2.0 * x)))) + -1.0)));
	elseif (Float64(-2.0 * x) <= 1e-24)
		tmp = Float64(Float64(-0.05396825396825397 * (x ^ 7.0)) + Float64(Float64(-0.3333333333333333 * (x ^ 3.0)) + Float64(x + Float64(0.13333333333333333 * (x ^ 5.0)))));
	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_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -5.0], N[(1.0 / N[(1.0 / N[(N[(2.0 / N[(2.0 + N[(Exp[N[(-2.0 * x), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 1e-24], N[(N[(-0.05396825396825397 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x + N[(0.13333333333333333 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 -2 x) < -5

   1. Initial program 0.0

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

   2. Simplified 0.0

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

      [Start] 0.0	\[ \frac{2}{1 + e^{-2 \cdot x}} - 1 \]
      sub-neg [=>] 0.0	\[ \color{blue}{\frac{2}{1 + e^{-2 \cdot x}} + \left(-1\right)} \]
      *-commutative [=>] 0.0	\[ \frac{2}{1 + e^{\color{blue}{x \cdot -2}}} + \left(-1\right) \]
      exp-prod [=>] 0.0	\[ \frac{2}{1 + \color{blue}{{\left(e^{x}\right)}^{-2}}} + \left(-1\right) \]
      metadata-eval [=>] 0.0	\[ \frac{2}{1 + {\left(e^{x}\right)}^{-2}} + \color{blue}{-1} \]

   3. Applied egg-rr 0.0

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

   4. Taylor expanded in x around inf 0.0

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

   5. Simplified 0.0

      \[\leadsto \frac{1}{\frac{1}{\color{blue}{\frac{2}{2 + \mathsf{expm1}\left(x \cdot -2\right)} + -1}}} \]
      Proof

      [Start] 0.0	\[ \frac{1}{\frac{1}{2 \cdot \frac{1}{\frac{1}{{\left(e^{x}\right)}^{2}} + 1} - 1}} \]
      sub-neg [=>] 0.0	\[ \frac{1}{\frac{1}{\color{blue}{2 \cdot \frac{1}{\frac{1}{{\left(e^{x}\right)}^{2}} + 1} + \left(-1\right)}}} \]

   if -5 < (*.f64 -2 x) < 9.99999999999999924e-25

   1. Initial program 59.5

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

   2. Simplified 59.5

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

      [Start] 59.5	\[ \frac{2}{1 + e^{-2 \cdot x}} - 1 \]
      sub-neg [=>] 59.5	\[ \color{blue}{\frac{2}{1 + e^{-2 \cdot x}} + \left(-1\right)} \]
      *-commutative [=>] 59.5	\[ \frac{2}{1 + e^{\color{blue}{x \cdot -2}}} + \left(-1\right) \]
      exp-prod [=>] 59.5	\[ \frac{2}{1 + \color{blue}{{\left(e^{x}\right)}^{-2}}} + \left(-1\right) \]
      metadata-eval [=>] 59.5	\[ \frac{2}{1 + {\left(e^{x}\right)}^{-2}} + \color{blue}{-1} \]

   3. Taylor expanded in x around 0 0.1

      \[\leadsto \color{blue}{-0.05396825396825397 \cdot {x}^{7} + \left(-0.3333333333333333 \cdot {x}^{3} + \left(0.13333333333333333 \cdot {x}^{5} + x\right)\right)} \]

   if 9.99999999999999924e-25 < (*.f64 -2 x)

   1. Initial program 2.8

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

   2. Taylor expanded in x around 0 4.7

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

   3. Simplified 4.7

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

      [Start] 4.7	\[ \frac{2}{2 + -2 \cdot x} - 1 \]
      *-commutative [=>] 4.7	\[ \frac{2}{2 + \color{blue}{x \cdot -2}} - 1 \]

   4. Taylor expanded in x around inf 4.2

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

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -5:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{2}{2 + \mathsf{expm1}\left(-2 \cdot x\right)} + -1}}\\ \mathbf{elif}\;-2 \cdot x \leq 10^{-24}:\\ \;\;\;\;-0.05396825396825397 \cdot {x}^{7} + \left(-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternatives

Alternative 1
Error	1.1
Cost	14024

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.02:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{2}{2 + \mathsf{expm1}\left(-2 \cdot x\right)} + -1}}\\ \mathbf{elif}\;-2 \cdot x \leq 10^{-24}:\\ \;\;\;\;-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 2
Error	1.1
Cost	7492

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.002:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{2}{2 + \mathsf{expm1}\left(-2 \cdot x\right)} + -1}}\\ \mathbf{elif}\;-2 \cdot x \leq 10^{-24}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 3
Error	1.1
Cost	7304

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.002:\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{elif}\;-2 \cdot x \leq 10^{-24}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 4
Error	15.5
Cost	708

\[\begin{array}{l} \mathbf{if}\;x \leq -1.7:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot 0.3333333333333333 + \frac{1}{x}}\\ \end{array} \]

Alternative 5
Error	15.6
Cost	196

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

Alternative 6
Error	46.6
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023083 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))