Logistic function from Lakshay Garg

Average Accuracy: 54.2% → 99.9%

Time: 10.3s

Precision: binary64

Cost: 20744

Math

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

↓

\[\begin{array}{l} t_0 := \frac{2}{e^{-2 \cdot x} + 1} + -1\\ \mathbf{if}\;-2 \cdot x \leq -20:\\ \;\;\;\;\log \left(e^{t_0}\right)\\ \mathbf{elif}\;-2 \cdot x \leq 0.05:\\ \;\;\;\;-0.05396825396825397 \cdot {x}^{7} + \left(-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ 2.0 (+ (exp (* -2.0 x)) 1.0)) -1.0)))
   (if (<= (* -2.0 x) -20.0)
     (log (exp t_0))
     (if (<= (* -2.0 x) 0.05)
       (+
        (* -0.05396825396825397 (pow x 7.0))
        (+
         (* -0.3333333333333333 (pow x 3.0))
         (+ x (* 0.13333333333333333 (pow x 5.0)))))
       t_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 = (2.0 / (exp((-2.0 * x)) + 1.0)) + -1.0;
	double tmp;
	if ((-2.0 * x) <= -20.0) {
		tmp = log(exp(t_0));
	} else if ((-2.0 * x) <= 0.05) {
		tmp = (-0.05396825396825397 * pow(x, 7.0)) + ((-0.3333333333333333 * pow(x, 3.0)) + (x + (0.13333333333333333 * pow(x, 5.0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (2.0d0 / (1.0d0 + exp(((-2.0d0) * x)))) - 1.0d0
end function

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (2.0d0 / (exp(((-2.0d0) * x)) + 1.0d0)) + (-1.0d0)
    if (((-2.0d0) * x) <= (-20.0d0)) then
        tmp = log(exp(t_0))
    else if (((-2.0d0) * x) <= 0.05d0) then
        tmp = ((-0.05396825396825397d0) * (x ** 7.0d0)) + (((-0.3333333333333333d0) * (x ** 3.0d0)) + (x + (0.13333333333333333d0 * (x ** 5.0d0))))
    else
        tmp = t_0
    end if
    code = tmp
end function

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 = (2.0 / (Math.exp((-2.0 * x)) + 1.0)) + -1.0;
	double tmp;
	if ((-2.0 * x) <= -20.0) {
		tmp = Math.log(Math.exp(t_0));
	} else if ((-2.0 * x) <= 0.05) {
		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 = t_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 = (2.0 / (math.exp((-2.0 * x)) + 1.0)) + -1.0
	tmp = 0
	if (-2.0 * x) <= -20.0:
		tmp = math.log(math.exp(t_0))
	elif (-2.0 * x) <= 0.05:
		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 = t_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 = Float64(Float64(2.0 / Float64(exp(Float64(-2.0 * x)) + 1.0)) + -1.0)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -20.0)
		tmp = log(exp(t_0));
	elseif (Float64(-2.0 * x) <= 0.05)
		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 = t_0;
	end
	return tmp
end

MATLAB

function tmp = code(x, y)
	tmp = (2.0 / (1.0 + exp((-2.0 * x)))) - 1.0;
end

↓

function tmp_2 = code(x, y)
	t_0 = (2.0 / (exp((-2.0 * x)) + 1.0)) + -1.0;
	tmp = 0.0;
	if ((-2.0 * x) <= -20.0)
		tmp = log(exp(t_0));
	elseif ((-2.0 * x) <= 0.05)
		tmp = (-0.05396825396825397 * (x ^ 7.0)) + ((-0.3333333333333333 * (x ^ 3.0)) + (x + (0.13333333333333333 * (x ^ 5.0))));
	else
		tmp = t_0;
	end
	tmp_2 = 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[(N[(2.0 / N[(N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -20.0], N[Log[N[Exp[t$95$0], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.05], 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], t$95$0]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   2. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)}\right)}^{3}} \]
      Proof

      [Start] 100.0	\[ \frac{2}{1 + e^{-2 \cdot x}} - 1 \]
      add-cube-cbrt [=>] 100.0	\[ \color{blue}{\left(\sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1} \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right) \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1}} \]
      pow3 [=>] 100.0	\[ \color{blue}{{\left(\sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right)}^{3}} \]
      add-exp-log [=>] 100.0	\[ {\left(\sqrt[3]{\color{blue}{e^{\log \left(\frac{2}{1 + e^{-2 \cdot x}}\right)}} - 1}\right)}^{3} \]
      expm1-def [=>] 100.0	\[ {\left(\sqrt[3]{\color{blue}{\mathsf{expm1}\left(\log \left(\frac{2}{1 + e^{-2 \cdot x}}\right)\right)}}\right)}^{3} \]
      log-div [=>] 100.0	\[ {\left(\sqrt[3]{\mathsf{expm1}\left(\color{blue}{\log 2 - \log \left(1 + e^{-2 \cdot x}\right)}\right)}\right)}^{3} \]
      log1p-udef [<=] 100.0	\[ {\left(\sqrt[3]{\mathsf{expm1}\left(\log 2 - \color{blue}{\mathsf{log1p}\left(e^{-2 \cdot x}\right)}\right)}\right)}^{3} \]
      exp-prod [=>] 100.0	\[ {\left(\sqrt[3]{\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(\color{blue}{{\left(e^{-2}\right)}^{x}}\right)\right)}\right)}^{3} \]

   3. Applied egg-rr 100.0%

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

      [Start] 100.0	\[ {\left(\sqrt[3]{\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)}\right)}^{3} \]
      add-log-exp [=>] 100.0	\[ \color{blue}{\log \left(e^{{\left(\sqrt[3]{\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)}\right)}^{3}}\right)} \]
      rem-cube-cbrt [=>] 100.0	\[ \log \left(e^{\color{blue}{\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)}}\right) \]
      expm1-udef [=>] 100.0	\[ \log \left(e^{\color{blue}{e^{\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)} - 1}}\right) \]
      sub-neg [=>] 100.0	\[ \log \left(e^{\color{blue}{e^{\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)} + \left(-1\right)}}\right) \]
      exp-diff [=>] 100.0	\[ \log \left(e^{\color{blue}{\frac{e^{\log 2}}{e^{\mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)}}} + \left(-1\right)}\right) \]
      add-exp-log [<=] 100.0	\[ \log \left(e^{\frac{\color{blue}{2}}{e^{\mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)}} + \left(-1\right)}\right) \]
      log1p-udef [=>] 100.0	\[ \log \left(e^{\frac{2}{e^{\color{blue}{\log \left(1 + {\left(e^{-2}\right)}^{x}\right)}}} + \left(-1\right)}\right) \]
      add-exp-log [<=] 100.0	\[ \log \left(e^{\frac{2}{\color{blue}{1 + {\left(e^{-2}\right)}^{x}}} + \left(-1\right)}\right) \]
      +-commutative [=>] 100.0	\[ \log \left(e^{\frac{2}{\color{blue}{{\left(e^{-2}\right)}^{x} + 1}} + \left(-1\right)}\right) \]
      metadata-eval [=>] 100.0	\[ \log \left(e^{\frac{2}{{\left(e^{-2}\right)}^{x} + 1} + \color{blue}{-1}}\right) \]

   4. Taylor expanded in x around inf 100.0%

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

   if -20 < (*.f64 -2 x) < 0.050000000000000003

   1. Initial program 8.4%

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

   2. Taylor expanded in x around 0 99.8%

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

   if 0.050000000000000003 < (*.f64 -2 x)

   1. Initial program 100.0%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -20:\\ \;\;\;\;\log \left(e^{\frac{2}{e^{-2 \cdot x} + 1} + -1}\right)\\ \mathbf{elif}\;-2 \cdot x \leq 0.05:\\ \;\;\;\;-0.05396825396825397 \cdot {x}^{7} + \left(-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} + -1\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.9%
Cost	20297

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -20 \lor \neg \left(-2 \cdot x \leq 0.02\right):\\ \;\;\;\;\log \left(e^{\frac{2}{e^{-2 \cdot x} + 1} + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\\ \end{array} \]

Alternative 2
Accuracy	99.9%
Cost	14025

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -20 \lor \neg \left(-2 \cdot x \leq 0.02\right):\\ \;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} + -1\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\\ \end{array} \]

Alternative 3
Accuracy	99.9%
Cost	7497

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.005 \lor \neg \left(-2 \cdot x \leq 5 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} + -1\\ \mathbf{else}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \end{array} \]

Alternative 4
Accuracy	79.5%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 2.6:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;2 - \frac{4}{x}\\ \end{array} \]

Alternative 5
Accuracy	78.9%
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq -0.65:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{x + 2}\\ \end{array} \]

Alternative 6
Accuracy	79.4%
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]

Alternative 7
Accuracy	32.6%
Cost	196

\[\begin{array}{l} \mathbf{if}\;x \leq 1.2 \cdot 10^{-308}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]

Alternative 8
Accuracy	27.6%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023151 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))