Logistic function from Lakshay Garg

Average Error: 29.1 → 0.1

Time: 7.2s

Precision: binary64

Cost: 19972

Math

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

↓

\[\begin{array}{l} t_0 := e^{-2 \cdot x}\\ \mathbf{if}\;-2 \cdot x \leq -10:\\ \;\;\;\;e^{-\mathsf{log1p}\left(t_0\right)} \cdot 2 + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.0001:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + \frac{2}{t_0 + 1}\right) + -1\right) + -1\\ \end{array} \]

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) -10.0)
     (+ (* (exp (- (log1p t_0))) 2.0) -1.0)
     (if (<= (* -2.0 x) 0.0001)
       (+ x (* -0.3333333333333333 (pow x 3.0)))
       (+ (+ (+ 1.0 (/ 2.0 (+ t_0 1.0))) -1.0) -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) <= -10.0) {
		tmp = (exp(-log1p(t_0)) * 2.0) + -1.0;
	} else if ((-2.0 * x) <= 0.0001) {
		tmp = x + (-0.3333333333333333 * pow(x, 3.0));
	} else {
		tmp = ((1.0 + (2.0 / (t_0 + 1.0))) + -1.0) + -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) <= -10.0) {
		tmp = (Math.exp(-Math.log1p(t_0)) * 2.0) + -1.0;
	} else if ((-2.0 * x) <= 0.0001) {
		tmp = x + (-0.3333333333333333 * Math.pow(x, 3.0));
	} else {
		tmp = ((1.0 + (2.0 / (t_0 + 1.0))) + -1.0) + -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) <= -10.0:
		tmp = (math.exp(-math.log1p(t_0)) * 2.0) + -1.0
	elif (-2.0 * x) <= 0.0001:
		tmp = x + (-0.3333333333333333 * math.pow(x, 3.0))
	else:
		tmp = ((1.0 + (2.0 / (t_0 + 1.0))) + -1.0) + -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) <= -10.0)
		tmp = Float64(Float64(exp(Float64(-log1p(t_0))) * 2.0) + -1.0);
	elseif (Float64(-2.0 * x) <= 0.0001)
		tmp = Float64(x + Float64(-0.3333333333333333 * (x ^ 3.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(2.0 / Float64(t_0 + 1.0))) + -1.0) + -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], -10.0], N[(N[(N[Exp[(-N[Log[1 + t$95$0], $MachinePrecision])], $MachinePrecision] * 2.0), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.0001], N[(x + N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + N[(2.0 / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] + -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 -2 x) < -10

   1. Initial program 0.0

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

   2. Applied egg-rr 0.0

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

   3. Applied egg-rr 0.0

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

   4. Applied egg-rr 0.0

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

   5. Taylor expanded in x around inf 0.0

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

   6. Simplified 0.0

      \[\leadsto e^{-\color{blue}{\mathsf{log1p}\left(e^{x \cdot -2}\right)}} \cdot 2 - 1 \]
      Proof
      (log1p.f64 (exp.f64 (*.f64 x -2))): 0 points increase in error, 0 points decrease in error
      (log1p.f64 (exp.f64 (Rewrite<= *-commutative_binary64 (*.f64 -2 x)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= log1p-def_binary64 (log.f64 (+.f64 1 (exp.f64 (*.f64 -2 x))))): 1 points increase in error, 2 points decrease in error

   if -10 < (*.f64 -2 x) < 1.00000000000000005e-4

   1. Initial program 58.8

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

   2. Taylor expanded in x around 0 0.2

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

   if 1.00000000000000005e-4 < (*.f64 -2 x)

   1. Initial program 0.1

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

   2. Applied egg-rr 0.1

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

   3. Applied egg-rr 0.1

      \[\leadsto \left(\left(1 + \frac{2}{1 + \color{blue}{e^{x \cdot -2}}}\right) - 1\right) - 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -10:\\ \;\;\;\;e^{-\mathsf{log1p}\left(e^{-2 \cdot x}\right)} \cdot 2 + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.0001:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + \frac{2}{e^{-2 \cdot x} + 1}\right) + -1\right) + -1\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	7752

\[\begin{array}{l} t_0 := \frac{2}{e^{-2 \cdot x} + 1}\\ \mathbf{if}\;-2 \cdot x \leq -10:\\ \;\;\;\;t_0 + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.0001:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + t_0\right) + -1\right) + -1\\ \end{array} \]

Alternative 2
Error	0.1
Cost	7496

\[\begin{array}{l} t_0 := \frac{2}{e^{-2 \cdot x} + 1} + -1\\ \mathbf{if}\;-2 \cdot x \leq -10:\\ \;\;\;\;t_0\\ \mathbf{elif}\;-2 \cdot x \leq 0.0001:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	13.9
Cost	836

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

Alternative 4
Error	14.3
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -536894285.7933369:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 7.101387351013941 \cdot 10^{-13}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;2 + \frac{-4}{x}\\ \end{array} \]

Alternative 5
Error	14.3
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq -536894285.7933369:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{x + 2}{x}}\\ \end{array} \]

Alternative 6
Error	14.3
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -536894285.7933369:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 7.101387351013941 \cdot 10^{-13}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]

Alternative 7
Error	29.0
Cost	196

\[\begin{array}{l} \mathbf{if}\;x \leq 7.101387351013941 \cdot 10^{-13}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]

Alternative 8
Error	59.5
Cost	64

\[2 \]

Reproduce

herbie shell --seed 2022310 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))