Logistic function from Lakshay Garg

Average Error: 29.2 → 0.6

Time: 12.6s

Precision: binary64

Cost: 58756

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{log1p}\left(e^{-2 \cdot x}\right)\\ \mathbf{if}\;-2 \cdot x \leq -0.001:\\ \;\;\;\;\mathsf{expm1}\left(\frac{{t_0}^{2} - {\left(\mathsf{log1p}\left(1\right)\right)}^{2}}{\left(-\mathsf{log1p}\left(1\right)\right) - t_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (log1p (exp (* -2.0 x)))))
   (if (<= (* -2.0 x) -0.001)
     (expm1
      (/ (- (pow t_0 2.0) (pow (log1p 1.0) 2.0)) (- (- (log1p 1.0)) t_0)))
     (expm1 x))))

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 = log1p(exp((-2.0 * x)));
	double tmp;
	if ((-2.0 * x) <= -0.001) {
		tmp = expm1(((pow(t_0, 2.0) - pow(log1p(1.0), 2.0)) / (-log1p(1.0) - t_0)));
	} else {
		tmp = expm1(x);
	}
	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.log1p(Math.exp((-2.0 * x)));
	double tmp;
	if ((-2.0 * x) <= -0.001) {
		tmp = Math.expm1(((Math.pow(t_0, 2.0) - Math.pow(Math.log1p(1.0), 2.0)) / (-Math.log1p(1.0) - t_0)));
	} else {
		tmp = Math.expm1(x);
	}
	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.log1p(math.exp((-2.0 * x)))
	tmp = 0
	if (-2.0 * x) <= -0.001:
		tmp = math.expm1(((math.pow(t_0, 2.0) - math.pow(math.log1p(1.0), 2.0)) / (-math.log1p(1.0) - t_0)))
	else:
		tmp = math.expm1(x)
	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 = log1p(exp(Float64(-2.0 * x)))
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.001)
		tmp = expm1(Float64(Float64((t_0 ^ 2.0) - (log1p(1.0) ^ 2.0)) / Float64(Float64(-log1p(1.0)) - t_0)));
	else
		tmp = expm1(x);
	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[Log[1 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.001], N[(Exp[N[(N[(N[Power[t$95$0, 2.0], $MachinePrecision] - N[Power[N[Log[1 + 1.0], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[((-N[Log[1 + 1.0], $MachinePrecision]) - t$95$0), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision], N[(Exp[x] - 1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 -2 x) < -1e-3

   1. Initial program 0.0

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

   2. Applied egg-rr 0.0

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

   3. Applied egg-rr 1.6

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

   4. Applied egg-rr 0.0

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\frac{\left(-{\left(\mathsf{log1p}\left(1\right)\right)}^{2}\right) + {\left(\mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)}^{2}}{-\left(\mathsf{log1p}\left(1\right) + \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)}}\right) \]

   5. Simplified 0.0

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\frac{{\left(\mathsf{log1p}\left(e^{-2 \cdot x}\right)\right)}^{2} + \left(-{\left(\mathsf{log1p}\left(1\right)\right)}^{2}\right)}{-\left(\mathsf{log1p}\left(1\right) + \mathsf{log1p}\left(e^{-2 \cdot x}\right)\right)}}\right) \]
      Proof

      [Start] 0.0	\[ \mathsf{expm1}\left(\frac{\left(-{\left(\mathsf{log1p}\left(1\right)\right)}^{2}\right) + {\left(\mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)}^{2}}{-\left(\mathsf{log1p}\left(1\right) + \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)}\right) \]
      /-rgt-identity [<=] 0.0	\[ \mathsf{expm1}\left(\frac{\left(-{\left(\mathsf{log1p}\left(1\right)\right)}^{2}\right) + {\left(\mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)}^{2}}{\color{blue}{\frac{-\left(\mathsf{log1p}\left(1\right) + \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)}{1}}}\right) \]
      +-commutative [=>] 0.0	\[ \mathsf{expm1}\left(\frac{\color{blue}{{\left(\mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)}^{2} + \left(-{\left(\mathsf{log1p}\left(1\right)\right)}^{2}\right)}}{\frac{-\left(\mathsf{log1p}\left(1\right) + \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)}{1}}\right) \]
      exp-prod [<=] 0.0	\[ \mathsf{expm1}\left(\frac{{\left(\mathsf{log1p}\left(\color{blue}{e^{-2 \cdot x}}\right)\right)}^{2} + \left(-{\left(\mathsf{log1p}\left(1\right)\right)}^{2}\right)}{\frac{-\left(\mathsf{log1p}\left(1\right) + \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)}{1}}\right) \]
      /-rgt-identity [=>] 0.0	\[ \mathsf{expm1}\left(\frac{{\left(\mathsf{log1p}\left(e^{-2 \cdot x}\right)\right)}^{2} + \left(-{\left(\mathsf{log1p}\left(1\right)\right)}^{2}\right)}{\color{blue}{-\left(\mathsf{log1p}\left(1\right) + \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)}}\right) \]
      exp-prod [<=] 0.0	\[ \mathsf{expm1}\left(\frac{{\left(\mathsf{log1p}\left(e^{-2 \cdot x}\right)\right)}^{2} + \left(-{\left(\mathsf{log1p}\left(1\right)\right)}^{2}\right)}{-\left(\mathsf{log1p}\left(1\right) + \mathsf{log1p}\left(\color{blue}{e^{-2 \cdot x}}\right)\right)}\right) \]

   if -1e-3 < (*.f64 -2 x)

   1. Initial program 38.7

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

   2. Applied egg-rr 38.6

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

   3. Taylor expanded in x around 0 0.8

      \[\leadsto \mathsf{expm1}\left(\color{blue}{x}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.001:\\ \;\;\;\;\mathsf{expm1}\left(\frac{{\left(\mathsf{log1p}\left(e^{-2 \cdot x}\right)\right)}^{2} - {\left(\mathsf{log1p}\left(1\right)\right)}^{2}}{\left(-\mathsf{log1p}\left(1\right)\right) - \mathsf{log1p}\left(e^{-2 \cdot x}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.6
Cost	27396

\[\begin{array}{l} t_0 := \frac{2}{2 + \mathsf{expm1}\left(-2 \cdot x\right)}\\ \mathbf{if}\;\frac{2}{e^{-2 \cdot x} + 1} \leq 1.0005:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + {t_0}^{2}}{1 + t_0}\\ \end{array} \]

Alternative 2
Error	0.6
Cost	26180

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.001:\\ \;\;\;\;\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(e^{-2 \cdot x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \]

Alternative 3
Error	0.6
Cost	20164

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.001:\\ \;\;\;\;{\left({\left(-1 + \frac{2}{e^{-2 \cdot x} + 1}\right)}^{3}\right)}^{0.3333333333333333}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \]

Alternative 4
Error	0.6
Cost	7236

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.001:\\ \;\;\;\;-1 + \frac{2}{2 + \mathsf{expm1}\left(-2 \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \]

Alternative 5
Error	13.4
Cost	6596

\[\begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2\right) \cdot \frac{1}{x + 2}\\ \end{array} \]

Alternative 6
Error	13.4
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 7
Error	15.1
Cost	196

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

Alternative 8
Error	46.0
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023187 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))