Logistic function from Lakshay Garg

Percentage Accurate: 54.5% → 99.6%

Time: 11.2s

Alternatives: 20

Speedup: 4.7×

Specification

Math

\[\begin{array}{l} \\ \frac{2}{1 + e^{-2 \cdot x}} - 1 \end{array} \]

FPCore

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

C

double code(double x, double y) {
	return (2.0 / (1.0 + exp((-2.0 * x)))) - 1.0;
}

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

Java

public static double code(double x, double y) {
	return (2.0 / (1.0 + Math.exp((-2.0 * x)))) - 1.0;
}

Python

def code(x, y):
	return (2.0 / (1.0 + math.exp((-2.0 * x)))) - 1.0

Julia

function code(x, y)
	return Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) - 1.0)
end

MATLAB

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

Initial Program: 54.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{2}{1 + e^{-2 \cdot x}} - 1 \end{array} \]

FPCore

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

C

double code(double x, double y) {
	return (2.0 / (1.0 + exp((-2.0 * x)))) - 1.0;
}

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

Java

public static double code(double x, double y) {
	return (2.0 / (1.0 + Math.exp((-2.0 * x)))) - 1.0;
}

Python

def code(x, y):
	return (2.0 / (1.0 + math.exp((-2.0 * x)))) - 1.0

Julia

function code(x, y)
	return Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) - 1.0)
end

MATLAB

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

Alternative 1: 99.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;-2 \cdot x \leq -40:\\ \;\;\;\;\frac{2}{2 + \frac{x + x}{\left(\left(x + x\right) - \left(x + x\right)\right) - \left(x + x\right)}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.05396825396825397, 0.13333333333333333\right), -0.3333333333333333\right), t\_0, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(8, t\_0, 2\right)} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (if (<= (* -2.0 x) -40.0)
     (+ (/ 2.0 (+ 2.0 (/ (+ x x) (- (- (+ x x) (+ x x)) (+ x x))))) -1.0)
     (if (<= (* -2.0 x) 2.0)
       (fma
        (fma
         (* x x)
         (fma (* x x) -0.05396825396825397 0.13333333333333333)
         -0.3333333333333333)
        t_0
        x)
       (+ (/ 2.0 (fma 8.0 t_0 2.0)) -1.0)))))

C

double code(double x, double y) {
	double t_0 = x * (x * x);
	double tmp;
	if ((-2.0 * x) <= -40.0) {
		tmp = (2.0 / (2.0 + ((x + x) / (((x + x) - (x + x)) - (x + x))))) + -1.0;
	} else if ((-2.0 * x) <= 2.0) {
		tmp = fma(fma((x * x), fma((x * x), -0.05396825396825397, 0.13333333333333333), -0.3333333333333333), t_0, x);
	} else {
		tmp = (2.0 / fma(8.0, t_0, 2.0)) + -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (Float64(-2.0 * x) <= -40.0)
		tmp = Float64(Float64(2.0 / Float64(2.0 + Float64(Float64(x + x) / Float64(Float64(Float64(x + x) - Float64(x + x)) - Float64(x + x))))) + -1.0);
	elseif (Float64(-2.0 * x) <= 2.0)
		tmp = fma(fma(Float64(x * x), fma(Float64(x * x), -0.05396825396825397, 0.13333333333333333), -0.3333333333333333), t_0, x);
	else
		tmp = Float64(Float64(2.0 / fma(8.0, t_0, 2.0)) + -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -40.0], N[(N[(2.0 / N[(2.0 + N[(N[(x + x), $MachinePrecision] / N[(N[(N[(x + x), $MachinePrecision] - N[(x + x), $MachinePrecision]), $MachinePrecision] - N[(x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 2.0], N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.05396825396825397 + 0.13333333333333333), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * t$95$0 + x), $MachinePrecision], N[(N[(2.0 / N[(8.0 * t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -40

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{2}{\color{blue}{2 + -2 \cdot x}} - 1 \]
   4. Step-by-step derivation

      1. metadata-eval N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. lower--.f64 N/A

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

      4. count-2 N/A

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

      5. lower-+.f64 1.6

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

   5. Applied rewrites 1.6%

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

   7. Applied rewrites 100.0%

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

   if -40 < (*.f64 #s(literal -2 binary64) x) < 2

   1. Initial program 10.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right) + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)\right) \cdot x + 1 \cdot x} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)\right)} + 1 \cdot x \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)} + 1 \cdot x \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right) \cdot \left(x \cdot {x}^{2}\right)} + 1 \cdot x \]

      6. *-lft-identity N/A

         \[\leadsto \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right) \cdot \left(x \cdot {x}^{2}\right) + \color{blue}{x} \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}, x \cdot {x}^{2}, x\right)} \]

   5. Applied rewrites 98.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.05396825396825397, 0.13333333333333333\right), -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)} \]

   if 2 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{2}{\color{blue}{2 + -2 \cdot x}} - 1 \]
   4. Step-by-step derivation

      1. metadata-eval N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. lower--.f64 N/A

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

      4. count-2 N/A

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

      5. lower-+.f64 96.1

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

   5. Applied rewrites 96.1%

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

   7. Applied rewrites 98.8%

      \[\leadsto \frac{2}{\mathsf{fma}\left(8, \color{blue}{x \cdot \left(x \cdot x\right)}, 2\right)} - 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -40:\\ \;\;\;\;\frac{2}{2 + \frac{x + x}{\left(\left(x + x\right) - \left(x + x\right)\right) - \left(x + x\right)}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.05396825396825397, 0.13333333333333333\right), -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(8, x \cdot \left(x \cdot x\right), 2\right)} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-2 \cdot x}\\ t_1 := 1 + t\_0\\ \mathbf{if}\;-2 \cdot x \leq -0.001:\\ \;\;\;\;\frac{{\left(t\_1 \cdot 0.5\right)}^{-4} + -1}{\left(1 + \frac{-2}{-1 - t\_0}\right) \cdot \mathsf{fma}\left(4, {t\_1}^{-2}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(-\left(-x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (exp (* -2.0 x))) (t_1 (+ 1.0 t_0)))
   (if (<= (* -2.0 x) -0.001)
     (/
      (+ (pow (* t_1 0.5) -4.0) -1.0)
      (* (+ 1.0 (/ -2.0 (- -1.0 t_0))) (fma 4.0 (pow t_1 -2.0) 1.0)))
     (expm1 (- (- x))))))

C

double code(double x, double y) {
	double t_0 = exp((-2.0 * x));
	double t_1 = 1.0 + t_0;
	double tmp;
	if ((-2.0 * x) <= -0.001) {
		tmp = (pow((t_1 * 0.5), -4.0) + -1.0) / ((1.0 + (-2.0 / (-1.0 - t_0))) * fma(4.0, pow(t_1, -2.0), 1.0));
	} else {
		tmp = expm1(-(-x));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = exp(Float64(-2.0 * x))
	t_1 = Float64(1.0 + t_0)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.001)
		tmp = Float64(Float64((Float64(t_1 * 0.5) ^ -4.0) + -1.0) / Float64(Float64(1.0 + Float64(-2.0 / Float64(-1.0 - t_0))) * fma(4.0, (t_1 ^ -2.0), 1.0)));
	else
		tmp = expm1(Float64(-Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + t$95$0), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.001], N[(N[(N[Power[N[(t$95$1 * 0.5), $MachinePrecision], -4.0], $MachinePrecision] + -1.0), $MachinePrecision] / N[(N[(1.0 + N[(-2.0 / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(4.0 * N[Power[t$95$1, -2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Exp[(-(-x))] - 1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -1e-3

   1. Initial program 99.9%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. metadata-eval N/A

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

      4. flip-- N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{{\left(\left(1 + e^{-2 \cdot x}\right) \cdot 0.5\right)}^{-4} + -1}{\left(1 - \frac{-2}{1 + e^{-2 \cdot x}}\right) \cdot \mathsf{fma}\left(4, {\left(1 + e^{-2 \cdot x}\right)}^{-2}, 1\right)}} \]

   if -1e-3 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 38.4%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. inv-pow N/A

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

      5. metadata-eval N/A

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

      6. pow-to-exp N/A

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

      7. lower-expm1.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. div-inv N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 38.5

          \[\leadsto \mathsf{expm1}\left(\log \left(\left(1 + e^{-2 \cdot x}\right) \cdot 0.5\right) \cdot \color{blue}{-1}\right) \]

   4. Applied rewrites 38.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\log \left(\left(1 + e^{-2 \cdot x}\right) \cdot 0.5\right) \cdot -1\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\left(-1 \cdot x\right)} \cdot -1\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot -1\right) \]

      2. lower-neg.f64 98.8

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

   7. Applied rewrites 98.8%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\left(-x\right)} \cdot -1\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot -1}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{expm1}\left(\color{blue}{-1 \cdot \left(\mathsf{neg}\left(x\right)\right)}\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)}\right) \]

      4. lower-neg.f64 98.8

         \[\leadsto \mathsf{expm1}\left(\color{blue}{-\left(-x\right)}\right) \]

   9. Applied rewrites 98.8%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.001:\\ \;\;\;\;\frac{{\left(\left(1 + e^{-2 \cdot x}\right) \cdot 0.5\right)}^{-4} + -1}{\left(1 + \frac{-2}{-1 - e^{-2 \cdot x}}\right) \cdot \mathsf{fma}\left(4, {\left(1 + e^{-2 \cdot x}\right)}^{-2}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(-\left(-x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024228 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))