Logistic function from Lakshay Garg

Percentage Accurate: 54.2% → 99.7%

Time: 11.1s

Precision: binary64

Cost: 32776

Math

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

↓

\[\begin{array}{l} t_0 := {\left(e^{-2}\right)}^{x}\\ \mathbf{if}\;-2 \cdot x \leq -0.1:\\ \;\;\;\;\log \left(e^{\frac{2}{t_0 + 1} + -1}\right)\\ \mathbf{elif}\;-2 \cdot x \leq 2 \cdot 10^{-12}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(t_0\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (pow (exp -2.0) x)))
   (if (<= (* -2.0 x) -0.1)
     (log (exp (+ (/ 2.0 (+ t_0 1.0)) -1.0)))
     (if (<= (* -2.0 x) 2e-12)
       (+ x (* -0.3333333333333333 (pow x 3.0)))
       (expm1 (- (log 2.0) (log1p 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 = pow(exp(-2.0), x);
	double tmp;
	if ((-2.0 * x) <= -0.1) {
		tmp = log(exp(((2.0 / (t_0 + 1.0)) + -1.0)));
	} else if ((-2.0 * x) <= 2e-12) {
		tmp = x + (-0.3333333333333333 * pow(x, 3.0));
	} else {
		tmp = expm1((log(2.0) - log1p(t_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.pow(Math.exp(-2.0), x);
	double tmp;
	if ((-2.0 * x) <= -0.1) {
		tmp = Math.log(Math.exp(((2.0 / (t_0 + 1.0)) + -1.0)));
	} else if ((-2.0 * x) <= 2e-12) {
		tmp = x + (-0.3333333333333333 * Math.pow(x, 3.0));
	} else {
		tmp = Math.expm1((Math.log(2.0) - Math.log1p(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 = math.pow(math.exp(-2.0), x)
	tmp = 0
	if (-2.0 * x) <= -0.1:
		tmp = math.log(math.exp(((2.0 / (t_0 + 1.0)) + -1.0)))
	elif (-2.0 * x) <= 2e-12:
		tmp = x + (-0.3333333333333333 * math.pow(x, 3.0))
	else:
		tmp = math.expm1((math.log(2.0) - math.log1p(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 = exp(-2.0) ^ x
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.1)
		tmp = log(exp(Float64(Float64(2.0 / Float64(t_0 + 1.0)) + -1.0)));
	elseif (Float64(-2.0 * x) <= 2e-12)
		tmp = Float64(x + Float64(-0.3333333333333333 * (x ^ 3.0)));
	else
		tmp = expm1(Float64(log(2.0) - log1p(t_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[Power[N[Exp[-2.0], $MachinePrecision], x], $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.1], N[Log[N[Exp[N[(N[(2.0 / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 2e-12], N[(x + N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Exp[N[(N[Log[2.0], $MachinePrecision] - N[Log[1 + t$95$0], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.9%

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

   2. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{0 + \mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)} \]
      Step-by-step derivation

      [Start] 99.9%	\[ \frac{2}{1 + e^{-2 \cdot x}} - 1 \]
      add-log-exp [=>] 99.9%	\[ \color{blue}{\log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right)} \]
      *-un-lft-identity [=>] 99.9%	\[ \log \color{blue}{\left(1 \cdot e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right)} \]
      log-prod [=>] 99.9%	\[ \color{blue}{\log 1 + \log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right)} \]
      metadata-eval [=>] 99.9%	\[ \color{blue}{0} + \log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right) \]
      add-log-exp [<=] 99.9%	\[ 0 + \color{blue}{\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)} \]
      add-exp-log [=>] 99.9%	\[ 0 + \left(\color{blue}{e^{\log \left(\frac{2}{1 + e^{-2 \cdot x}}\right)}} - 1\right) \]
      expm1-def [=>] 99.9%	\[ 0 + \color{blue}{\mathsf{expm1}\left(\log \left(\frac{2}{1 + e^{-2 \cdot x}}\right)\right)} \]
      log-div [=>] 99.9%	\[ 0 + \mathsf{expm1}\left(\color{blue}{\log 2 - \log \left(1 + e^{-2 \cdot x}\right)}\right) \]
      log1p-udef [<=] 99.9%	\[ 0 + \mathsf{expm1}\left(\log 2 - \color{blue}{\mathsf{log1p}\left(e^{-2 \cdot x}\right)}\right) \]
      exp-prod [=>] 99.9%	\[ 0 + \mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(\color{blue}{{\left(e^{-2}\right)}^{x}}\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)} \]
      Step-by-step derivation

      [Start] 99.9%	\[ 0 + \mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right) \]
      +-lft-identity [=>] 99.9%	\[ \color{blue}{\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)} \]

   4. Applied egg-rr 99.9%

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

      [Start] 99.9%	\[ \mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right) \]
      add-log-exp [=>] 99.9%	\[ \color{blue}{\log \left(e^{\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)}\right)} \]
      expm1-udef [=>] 99.9%	\[ \log \left(e^{\color{blue}{e^{\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)} - 1}}\right) \]
      sub-neg [=>] 99.9%	\[ \log \left(e^{\color{blue}{e^{\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)} + \left(-1\right)}}\right) \]
      exp-diff [=>] 99.9%	\[ \log \left(e^{\color{blue}{\frac{e^{\log 2}}{e^{\mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)}}} + \left(-1\right)}\right) \]
      add-exp-log [<=] 99.9%	\[ \log \left(e^{\frac{\color{blue}{2}}{e^{\mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)}} + \left(-1\right)}\right) \]
      log1p-udef [=>] 99.9%	\[ \log \left(e^{\frac{2}{e^{\color{blue}{\log \left(1 + {\left(e^{-2}\right)}^{x}\right)}}} + \left(-1\right)}\right) \]
      add-exp-log [<=] 99.9%	\[ \log \left(e^{\frac{2}{\color{blue}{1 + {\left(e^{-2}\right)}^{x}}} + \left(-1\right)}\right) \]
      +-commutative [=>] 99.9%	\[ \log \left(e^{\frac{2}{\color{blue}{{\left(e^{-2}\right)}^{x} + 1}} + \left(-1\right)}\right) \]
      metadata-eval [=>] 99.9%	\[ \log \left(e^{\frac{2}{{\left(e^{-2}\right)}^{x} + 1} + \color{blue}{-1}}\right) \]

   if -0.10000000000000001 < (*.f64 -2 x) < 1.99999999999999996e-12

   1. Initial program 8.0%

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

   2. Taylor expanded in x around 0 100.0%

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

   if 1.99999999999999996e-12 < (*.f64 -2 x)

   1. Initial program 99.9%

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

   2. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0 + \mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)} \]
      Step-by-step derivation

      [Start] 99.9%	\[ \frac{2}{1 + e^{-2 \cdot x}} - 1 \]
      add-log-exp [=>] 99.9%	\[ \color{blue}{\log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right)} \]
      *-un-lft-identity [=>] 99.9%	\[ \log \color{blue}{\left(1 \cdot e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right)} \]
      log-prod [=>] 99.9%	\[ \color{blue}{\log 1 + \log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right)} \]
      metadata-eval [=>] 99.9%	\[ \color{blue}{0} + \log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right) \]
      add-log-exp [<=] 99.9%	\[ 0 + \color{blue}{\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)} \]
      add-exp-log [=>] 99.9%	\[ 0 + \left(\color{blue}{e^{\log \left(\frac{2}{1 + e^{-2 \cdot x}}\right)}} - 1\right) \]
      expm1-def [=>] 99.9%	\[ 0 + \color{blue}{\mathsf{expm1}\left(\log \left(\frac{2}{1 + e^{-2 \cdot x}}\right)\right)} \]
      log-div [=>] 99.9%	\[ 0 + \mathsf{expm1}\left(\color{blue}{\log 2 - \log \left(1 + e^{-2 \cdot x}\right)}\right) \]
      log1p-udef [<=] 100.0%	\[ 0 + \mathsf{expm1}\left(\log 2 - \color{blue}{\mathsf{log1p}\left(e^{-2 \cdot x}\right)}\right) \]
      exp-prod [=>] 100.0%	\[ 0 + \mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(\color{blue}{{\left(e^{-2}\right)}^{x}}\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)} \]
      Step-by-step derivation

      [Start] 100.0%	\[ 0 + \mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right) \]
      +-lft-identity [=>] 100.0%	\[ \color{blue}{\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.1:\\ \;\;\;\;\log \left(e^{\frac{2}{{\left(e^{-2}\right)}^{x} + 1} + -1}\right)\\ \mathbf{elif}\;-2 \cdot x \leq 2 \cdot 10^{-12}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.6%
Cost	26372

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.1:\\ \;\;\;\;\log \left(e^{\frac{2}{{\left(e^{-2}\right)}^{x} + 1} + -1}\right)\\ \mathbf{elif}\;-2 \cdot x \leq 2 \cdot 10^{-12}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{2}{1 + {\left(e^{x}\right)}^{-2}}\\ \end{array} \]

Alternative 2
Accuracy	99.6%
Cost	26308

\[\begin{array}{l} t_0 := 1 + {\left(e^{x}\right)}^{-2}\\ \mathbf{if}\;-2 \cdot x \leq -0.1:\\ \;\;\;\;\mathsf{expm1}\left(-\log \left(t_0 \cdot 0.5\right)\right)\\ \mathbf{elif}\;-2 \cdot x \leq 2 \cdot 10^{-12}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{2}{t_0}\\ \end{array} \]

Alternative 3
Accuracy	99.6%
Cost	13832

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.1:\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{elif}\;-2 \cdot x \leq 2 \cdot 10^{-12}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{2}{1 + {\left(e^{x}\right)}^{-2}}\\ \end{array} \]

Alternative 4
Accuracy	99.6%
Cost	7497

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.1 \lor \neg \left(-2 \cdot x \leq 2 \cdot 10^{-12}\right):\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \end{array} \]

Alternative 5
Accuracy	78.6%
Cost	708

\[\begin{array}{l} \mathbf{if}\;x \leq -0.68:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2\right) \cdot \frac{1}{x + 2}\\ \end{array} \]

Alternative 6
Accuracy	79.1%
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 7
Accuracy	79.1%
Cost	328

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

Alternative 8
Accuracy	32.4%
Cost	196

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

Alternative 9
Accuracy	27.3%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023164 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))