Logistic function from Lakshay Garg

Average Accuracy: 53.2% → 99.9%

Time: 12.8s

Precision: binary64

Cost: 20744

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -5:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{expm1}\left(-2 \cdot x\right), \frac{2}{\mathsf{expm1}\left(x \cdot -4\right)}, -1\right)\\ \mathbf{elif}\;-2 \cdot x \leq 10^{-6}:\\ \;\;\;\;-0.05396825396825397 \cdot {x}^{7} + \left(-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \end{array} \]

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)
   (fma (expm1 (* -2.0 x)) (/ 2.0 (expm1 (* x -4.0))) -1.0)
   (if (<= (* -2.0 x) 1e-6)
     (+
      (* -0.05396825396825397 (pow x 7.0))
      (+
       (* -0.3333333333333333 (pow x 3.0))
       (+ x (* 0.13333333333333333 (pow x 5.0)))))
     (+ -1.0 (/ 2.0 (+ 1.0 (exp (* -2.0 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 tmp;
	if ((-2.0 * x) <= -5.0) {
		tmp = fma(expm1((-2.0 * x)), (2.0 / expm1((x * -4.0))), -1.0);
	} else if ((-2.0 * x) <= 1e-6) {
		tmp = (-0.05396825396825397 * pow(x, 7.0)) + ((-0.3333333333333333 * pow(x, 3.0)) + (x + (0.13333333333333333 * pow(x, 5.0))));
	} else {
		tmp = -1.0 + (2.0 / (1.0 + exp((-2.0 * 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)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -5.0)
		tmp = fma(expm1(Float64(-2.0 * x)), Float64(2.0 / expm1(Float64(x * -4.0))), -1.0);
	elseif (Float64(-2.0 * x) <= 1e-6)
		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 = Float64(-1.0 + Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * 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_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -5.0], N[(N[(Exp[N[(-2.0 * x), $MachinePrecision]] - 1), $MachinePrecision] * N[(2.0 / N[(Exp[N[(x * -4.0), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 1e-6], 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], N[(-1.0 + N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   2. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(\frac{2}{1 - e^{x \cdot -4}} \cdot 1 + \frac{2}{1 - e^{x \cdot -4}} \cdot \left(-{\left(e^{-2}\right)}^{x}\right)\right)} - 1 \]
      Proof

      [Start] 100.0	\[ \frac{2}{1 + e^{-2 \cdot x}} - 1 \]
      flip-+ [=>] 100.0	\[ \frac{2}{\color{blue}{\frac{1 \cdot 1 - e^{-2 \cdot x} \cdot e^{-2 \cdot x}}{1 - e^{-2 \cdot x}}}} - 1 \]
      associate-/r/ [=>] 100.0	\[ \color{blue}{\frac{2}{1 \cdot 1 - e^{-2 \cdot x} \cdot e^{-2 \cdot x}} \cdot \left(1 - e^{-2 \cdot x}\right)} - 1 \]
      sub-neg [=>] 100.0	\[ \frac{2}{1 \cdot 1 - e^{-2 \cdot x} \cdot e^{-2 \cdot x}} \cdot \color{blue}{\left(1 + \left(-e^{-2 \cdot x}\right)\right)} - 1 \]
      distribute-lft-in [=>] 100.0	\[ \color{blue}{\left(\frac{2}{1 \cdot 1 - e^{-2 \cdot x} \cdot e^{-2 \cdot x}} \cdot 1 + \frac{2}{1 \cdot 1 - e^{-2 \cdot x} \cdot e^{-2 \cdot x}} \cdot \left(-e^{-2 \cdot x}\right)\right)} - 1 \]
      metadata-eval [=>] 100.0	\[ \left(\frac{2}{\color{blue}{1} - e^{-2 \cdot x} \cdot e^{-2 \cdot x}} \cdot 1 + \frac{2}{1 \cdot 1 - e^{-2 \cdot x} \cdot e^{-2 \cdot x}} \cdot \left(-e^{-2 \cdot x}\right)\right) - 1 \]
      prod-exp [=>] 100.0	\[ \left(\frac{2}{1 - \color{blue}{e^{-2 \cdot x + -2 \cdot x}}} \cdot 1 + \frac{2}{1 \cdot 1 - e^{-2 \cdot x} \cdot e^{-2 \cdot x}} \cdot \left(-e^{-2 \cdot x}\right)\right) - 1 \]
      *-commutative [=>] 100.0	\[ \left(\frac{2}{1 - e^{\color{blue}{x \cdot -2} + -2 \cdot x}} \cdot 1 + \frac{2}{1 \cdot 1 - e^{-2 \cdot x} \cdot e^{-2 \cdot x}} \cdot \left(-e^{-2 \cdot x}\right)\right) - 1 \]
      *-commutative [=>] 100.0	\[ \left(\frac{2}{1 - e^{x \cdot -2 + \color{blue}{x \cdot -2}}} \cdot 1 + \frac{2}{1 \cdot 1 - e^{-2 \cdot x} \cdot e^{-2 \cdot x}} \cdot \left(-e^{-2 \cdot x}\right)\right) - 1 \]
      distribute-lft-out [=>] 100.0	\[ \left(\frac{2}{1 - e^{\color{blue}{x \cdot \left(-2 + -2\right)}}} \cdot 1 + \frac{2}{1 \cdot 1 - e^{-2 \cdot x} \cdot e^{-2 \cdot x}} \cdot \left(-e^{-2 \cdot x}\right)\right) - 1 \]
      metadata-eval [=>] 100.0	\[ \left(\frac{2}{1 - e^{x \cdot \color{blue}{-4}}} \cdot 1 + \frac{2}{1 \cdot 1 - e^{-2 \cdot x} \cdot e^{-2 \cdot x}} \cdot \left(-e^{-2 \cdot x}\right)\right) - 1 \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{2 \cdot \frac{\mathsf{expm1}\left(x \cdot -2\right)}{\mathsf{expm1}\left(x \cdot -4\right)}} - 1 \]
      Proof

      [Start] 100.0	\[ \left(\frac{2}{1 - e^{x \cdot -4}} \cdot 1 + \frac{2}{1 - e^{x \cdot -4}} \cdot \left(-{\left(e^{-2}\right)}^{x}\right)\right) - 1 \]
      distribute-lft-out [=>] 100.0	\[ \color{blue}{\frac{2}{1 - e^{x \cdot -4}} \cdot \left(1 + \left(-{\left(e^{-2}\right)}^{x}\right)\right)} - 1 \]
      metadata-eval [<=] 100.0	\[ \frac{\color{blue}{\frac{-2}{-1}}}{1 - e^{x \cdot -4}} \cdot \left(1 + \left(-{\left(e^{-2}\right)}^{x}\right)\right) - 1 \]
      associate-/r* [<=] 100.0	\[ \color{blue}{\frac{-2}{-1 \cdot \left(1 - e^{x \cdot -4}\right)}} \cdot \left(1 + \left(-{\left(e^{-2}\right)}^{x}\right)\right) - 1 \]
      neg-mul-1 [<=] 100.0	\[ \frac{-2}{\color{blue}{-\left(1 - e^{x \cdot -4}\right)}} \cdot \left(1 + \left(-{\left(e^{-2}\right)}^{x}\right)\right) - 1 \]
      neg-sub0 [=>] 100.0	\[ \frac{-2}{\color{blue}{0 - \left(1 - e^{x \cdot -4}\right)}} \cdot \left(1 + \left(-{\left(e^{-2}\right)}^{x}\right)\right) - 1 \]
      associate--r- [=>] 100.0	\[ \frac{-2}{\color{blue}{\left(0 - 1\right) + e^{x \cdot -4}}} \cdot \left(1 + \left(-{\left(e^{-2}\right)}^{x}\right)\right) - 1 \]
      metadata-eval [=>] 100.0	\[ \frac{-2}{\color{blue}{-1} + e^{x \cdot -4}} \cdot \left(1 + \left(-{\left(e^{-2}\right)}^{x}\right)\right) - 1 \]
      +-commutative [<=] 100.0	\[ \frac{-2}{\color{blue}{e^{x \cdot -4} + -1}} \cdot \left(1 + \left(-{\left(e^{-2}\right)}^{x}\right)\right) - 1 \]
      metadata-eval [<=] 100.0	\[ \frac{-2}{e^{x \cdot -4} + \color{blue}{\left(-1\right)}} \cdot \left(1 + \left(-{\left(e^{-2}\right)}^{x}\right)\right) - 1 \]
      sub-neg [<=] 100.0	\[ \frac{-2}{\color{blue}{e^{x \cdot -4} - 1}} \cdot \left(1 + \left(-{\left(e^{-2}\right)}^{x}\right)\right) - 1 \]
      sub-neg [<=] 100.0	\[ \frac{-2}{e^{x \cdot -4} - 1} \cdot \color{blue}{\left(1 - {\left(e^{-2}\right)}^{x}\right)} - 1 \]
      associate-*l/ [=>] 100.0	\[ \color{blue}{\frac{-2 \cdot \left(1 - {\left(e^{-2}\right)}^{x}\right)}{e^{x \cdot -4} - 1}} - 1 \]
      metadata-eval [<=] 100.0	\[ \frac{\color{blue}{\left(2 \cdot -1\right)} \cdot \left(1 - {\left(e^{-2}\right)}^{x}\right)}{e^{x \cdot -4} - 1} - 1 \]
      associate-*r* [<=] 100.0	\[ \frac{\color{blue}{2 \cdot \left(-1 \cdot \left(1 - {\left(e^{-2}\right)}^{x}\right)\right)}}{e^{x \cdot -4} - 1} - 1 \]
      neg-mul-1 [<=] 100.0	\[ \frac{2 \cdot \color{blue}{\left(-\left(1 - {\left(e^{-2}\right)}^{x}\right)\right)}}{e^{x \cdot -4} - 1} - 1 \]
      *-lft-identity [<=] 100.0	\[ \frac{2 \cdot \left(-\left(1 - {\left(e^{-2}\right)}^{x}\right)\right)}{\color{blue}{1 \cdot \left(e^{x \cdot -4} - 1\right)}} - 1 \]
      times-frac [=>] 100.0	\[ \color{blue}{\frac{2}{1} \cdot \frac{-\left(1 - {\left(e^{-2}\right)}^{x}\right)}{e^{x \cdot -4} - 1}} - 1 \]

   4. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{2 \cdot \frac{e^{-2 \cdot x} - 1}{e^{-4 \cdot x} - 1} - 1} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{expm1}\left(x \cdot -2\right), \frac{2}{\mathsf{expm1}\left(x \cdot -4\right)}, -1\right)} \]
      Proof

      [Start] 100.0	\[ 2 \cdot \frac{e^{-2 \cdot x} - 1}{e^{-4 \cdot x} - 1} - 1 \]
      *-commutative [<=] 100.0	\[ 2 \cdot \frac{e^{\color{blue}{x \cdot -2}} - 1}{e^{-4 \cdot x} - 1} - 1 \]
      expm1-def [=>] 100.0	\[ 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(x \cdot -2\right)}}{e^{-4 \cdot x} - 1} - 1 \]
      *-rgt-identity [<=] 100.0	\[ 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(x \cdot -2\right) \cdot 1}}{e^{-4 \cdot x} - 1} - 1 \]
      *-commutative [=>] 100.0	\[ 2 \cdot \frac{\mathsf{expm1}\left(x \cdot -2\right) \cdot 1}{e^{\color{blue}{x \cdot -4}} - 1} - 1 \]
      expm1-def [=>] 100.0	\[ 2 \cdot \frac{\mathsf{expm1}\left(x \cdot -2\right) \cdot 1}{\color{blue}{\mathsf{expm1}\left(x \cdot -4\right)}} - 1 \]
      associate-*r/ [<=] 100.0	\[ 2 \cdot \color{blue}{\left(\mathsf{expm1}\left(x \cdot -2\right) \cdot \frac{1}{\mathsf{expm1}\left(x \cdot -4\right)}\right)} - 1 \]
      associate-*r/ [=>] 100.0	\[ 2 \cdot \color{blue}{\frac{\mathsf{expm1}\left(x \cdot -2\right) \cdot 1}{\mathsf{expm1}\left(x \cdot -4\right)}} - 1 \]
      *-rgt-identity [=>] 100.0	\[ 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(x \cdot -2\right)}}{\mathsf{expm1}\left(x \cdot -4\right)} - 1 \]
      associate-*r/ [=>] 100.0	\[ \color{blue}{\frac{2 \cdot \mathsf{expm1}\left(x \cdot -2\right)}{\mathsf{expm1}\left(x \cdot -4\right)}} - 1 \]
      associate-*l/ [<=] 100.0	\[ \color{blue}{\frac{2}{\mathsf{expm1}\left(x \cdot -4\right)} \cdot \mathsf{expm1}\left(x \cdot -2\right)} - 1 \]
      *-commutative [<=] 100.0	\[ \color{blue}{\mathsf{expm1}\left(x \cdot -2\right) \cdot \frac{2}{\mathsf{expm1}\left(x \cdot -4\right)}} - 1 \]
      sub-neg [=>] 100.0	\[ \color{blue}{\mathsf{expm1}\left(x \cdot -2\right) \cdot \frac{2}{\mathsf{expm1}\left(x \cdot -4\right)} + \left(-1\right)} \]
      metadata-eval [=>] 100.0	\[ \mathsf{expm1}\left(x \cdot -2\right) \cdot \frac{2}{\mathsf{expm1}\left(x \cdot -4\right)} + \color{blue}{-1} \]

   if -5 < (*.f64 -2 x) < 9.99999999999999955e-7

   1. Initial program 7.6%

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

   2. Taylor expanded in x around 0 99.9%

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

   if 9.99999999999999955e-7 < (*.f64 -2 x)

   1. Initial program 99.7%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -5:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{expm1}\left(-2 \cdot x\right), \frac{2}{\mathsf{expm1}\left(x \cdot -4\right)}, -1\right)\\ \mathbf{elif}\;-2 \cdot x \leq 10^{-6}:\\ \;\;\;\;-0.05396825396825397 \cdot {x}^{7} + \left(-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.9%
Cost	20036

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{expm1}\left(-2 \cdot x\right), \frac{2}{\mathsf{expm1}\left(x \cdot -4\right)}, -1\right)\\ \mathbf{elif}\;-2 \cdot x \leq 10^{-6}:\\ \;\;\;\;-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \end{array} \]

Alternative 2
Accuracy	99.9%
Cost	14025

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.02 \lor \neg \left(-2 \cdot x \leq 10^{-6}\right):\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\\ \end{array} \]

Alternative 3
Accuracy	99.9%
Cost	7497

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.02 \lor \neg \left(-2 \cdot x \leq 10^{-6}\right):\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \end{array} \]

Alternative 4
Accuracy	79.3%
Cost	708

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

Alternative 5
Accuracy	76.8%
Cost	196

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

Alternative 6
Accuracy	27.2%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023137 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))