Logistic function from Lakshay Garg

Percentage Accurate: 53.5% → 99.6%

Time: 7.1s

Alternatives: 5

Speedup: 18.1×

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: 53.5% 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, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -10:\\ \;\;\;\;\left(\left(1 + \frac{2}{1 + e^{-2 \cdot x}}\right) + -1\right) + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.0005:\\ \;\;\;\;x + \mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -10.0)
   (+ (+ (+ 1.0 (/ 2.0 (+ 1.0 (exp (* -2.0 x))))) -1.0) -1.0)
   (if (<= (* -2.0 x) 0.0005)
     (+
      x
      (* (fma (* x x) 0.13333333333333333 -0.3333333333333333) (pow x 3.0)))
     -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -10.0) {
		tmp = ((1.0 + (2.0 / (1.0 + exp((-2.0 * x))))) + -1.0) + -1.0;
	} else if ((-2.0 * x) <= 0.0005) {
		tmp = x + (fma((x * x), 0.13333333333333333, -0.3333333333333333) * pow(x, 3.0));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -10.0)
		tmp = Float64(Float64(Float64(1.0 + Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x))))) + -1.0) + -1.0);
	elseif (Float64(-2.0 * x) <= 0.0005)
		tmp = Float64(x + Float64(fma(Float64(x * x), 0.13333333333333333, -0.3333333333333333) * (x ^ 3.0)));
	else
		tmp = -1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -10.0], N[(N[(N[(1.0 + N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.0005], N[(x + N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333 + -0.3333333333333333), $MachinePrecision] * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -10

   1. Initial program 100.0%

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

      1. expm1-log1p-u 96.4%

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

      2. expm1-undefine 96.4%

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

      3. log1p-undefine 97.6%

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

      4. +-commutative 97.6%

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

      5. add-exp-log 100.0%

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

      6. +-commutative 100.0%

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

      7. exp-prod 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

   if -10 < (*.f64 #s(literal -2 binary64) x) < 5.0000000000000001e-4

   1. Initial program 7.9%

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

   3. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.13333333333333333 \cdot {x}^{2} - 0.3333333333333333\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \color{blue}{1 \cdot x + \left({x}^{2} \cdot \left(0.13333333333333333 \cdot {x}^{2} - 0.3333333333333333\right)\right) \cdot x} \]

      2. *-lft-identity 100.0%

         \[\leadsto \color{blue}{x} + \left({x}^{2} \cdot \left(0.13333333333333333 \cdot {x}^{2} - 0.3333333333333333\right)\right) \cdot x \]

      3. *-commutative 100.0%

         \[\leadsto x + \color{blue}{\left(\left(0.13333333333333333 \cdot {x}^{2} - 0.3333333333333333\right) \cdot {x}^{2}\right)} \cdot x \]

      4. associate-*l* 100.0%

         \[\leadsto x + \color{blue}{\left(0.13333333333333333 \cdot {x}^{2} - 0.3333333333333333\right) \cdot \left({x}^{2} \cdot x\right)} \]

      5. *-commutative 100.0%

         \[\leadsto x + \left(\color{blue}{{x}^{2} \cdot 0.13333333333333333} - 0.3333333333333333\right) \cdot \left({x}^{2} \cdot x\right) \]

      6. fma-neg 100.0%

         \[\leadsto x + \color{blue}{\mathsf{fma}\left({x}^{2}, 0.13333333333333333, -0.3333333333333333\right)} \cdot \left({x}^{2} \cdot x\right) \]

      7. metadata-eval 100.0%

         \[\leadsto x + \mathsf{fma}\left({x}^{2}, 0.13333333333333333, \color{blue}{-0.3333333333333333}\right) \cdot \left({x}^{2} \cdot x\right) \]

      8. unpow2 100.0%

         \[\leadsto x + \mathsf{fma}\left({x}^{2}, 0.13333333333333333, -0.3333333333333333\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot x\right) \]

      9. unpow3 100.0%

         \[\leadsto x + \mathsf{fma}\left({x}^{2}, 0.13333333333333333, -0.3333333333333333\right) \cdot \color{blue}{{x}^{3}} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{x + \mathsf{fma}\left({x}^{2}, 0.13333333333333333, -0.3333333333333333\right) \cdot {x}^{3}} \]
   6. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto x + \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.13333333333333333, -0.3333333333333333\right) \cdot {x}^{3} \]

   7. Applied egg-rr 100.0%

      \[\leadsto x + \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.13333333333333333, -0.3333333333333333\right) \cdot {x}^{3} \]

   if 5.0000000000000001e-4 < (*.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 98.0%

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

      1. *-commutative 98.0%

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

   5. Simplified 98.0%

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

   6. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{-1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -10:\\ \;\;\;\;\left(\left(1 + \frac{2}{1 + e^{-2 \cdot x}}\right) + -1\right) + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.0005:\\ \;\;\;\;x + \mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -10:\\ \;\;\;\;\left(\left(1 + \frac{2}{1 + e^{-2 \cdot x}}\right) + -1\right) + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.0005:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -10.0)
   (+ (+ (+ 1.0 (/ 2.0 (+ 1.0 (exp (* -2.0 x))))) -1.0) -1.0)
   (if (<= (* -2.0 x) 0.0005) (+ x (* -0.3333333333333333 (pow x 3.0))) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -10.0) {
		tmp = ((1.0 + (2.0 / (1.0 + exp((-2.0 * x))))) + -1.0) + -1.0;
	} else if ((-2.0 * x) <= 0.0005) {
		tmp = x + (-0.3333333333333333 * pow(x, 3.0));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((-2.0d0) * x) <= (-10.0d0)) then
        tmp = ((1.0d0 + (2.0d0 / (1.0d0 + exp(((-2.0d0) * x))))) + (-1.0d0)) + (-1.0d0)
    else if (((-2.0d0) * x) <= 0.0005d0) then
        tmp = x + ((-0.3333333333333333d0) * (x ** 3.0d0))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -10.0) {
		tmp = ((1.0 + (2.0 / (1.0 + Math.exp((-2.0 * x))))) + -1.0) + -1.0;
	} else if ((-2.0 * x) <= 0.0005) {
		tmp = x + (-0.3333333333333333 * Math.pow(x, 3.0));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (-2.0 * x) <= -10.0:
		tmp = ((1.0 + (2.0 / (1.0 + math.exp((-2.0 * x))))) + -1.0) + -1.0
	elif (-2.0 * x) <= 0.0005:
		tmp = x + (-0.3333333333333333 * math.pow(x, 3.0))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -10.0)
		tmp = Float64(Float64(Float64(1.0 + Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x))))) + -1.0) + -1.0);
	elseif (Float64(-2.0 * x) <= 0.0005)
		tmp = Float64(x + Float64(-0.3333333333333333 * (x ^ 3.0)));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((-2.0 * x) <= -10.0)
		tmp = ((1.0 + (2.0 / (1.0 + exp((-2.0 * x))))) + -1.0) + -1.0;
	elseif ((-2.0 * x) <= 0.0005)
		tmp = x + (-0.3333333333333333 * (x ^ 3.0));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -10.0], N[(N[(N[(1.0 + N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.0005], N[(x + N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -10

   1. Initial program 100.0%

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

      1. expm1-log1p-u 96.4%

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

      2. expm1-undefine 96.4%

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

      3. log1p-undefine 97.6%

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

      4. +-commutative 97.6%

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

      5. add-exp-log 100.0%

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

      6. +-commutative 100.0%

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

      7. exp-prod 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

   if -10 < (*.f64 #s(literal -2 binary64) x) < 5.0000000000000001e-4

   1. Initial program 7.9%

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

   3. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{x \cdot \left(1 + -0.3333333333333333 \cdot {x}^{2}\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \color{blue}{1 \cdot x + \left(-0.3333333333333333 \cdot {x}^{2}\right) \cdot x} \]

      2. *-lft-identity 100.0%

         \[\leadsto \color{blue}{x} + \left(-0.3333333333333333 \cdot {x}^{2}\right) \cdot x \]

      3. associate-*l* 100.0%

         \[\leadsto x + \color{blue}{-0.3333333333333333 \cdot \left({x}^{2} \cdot x\right)} \]

      4. unpow2 100.0%

         \[\leadsto x + -0.3333333333333333 \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot x\right) \]

      5. unpow3 100.0%

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

   5. Simplified 100.0%

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

   if 5.0000000000000001e-4 < (*.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 98.0%

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

      1. *-commutative 98.0%

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

   5. Simplified 98.0%

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

   6. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{-1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -10:\\ \;\;\;\;\left(\left(1 + \frac{2}{1 + e^{-2 \cdot x}}\right) + -1\right) + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.0005:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -10:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.0005:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -10.0)
   (+ (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) -1.0)
   (if (<= (* -2.0 x) 0.0005) (+ x (* -0.3333333333333333 (pow x 3.0))) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -10.0) {
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	} else if ((-2.0 * x) <= 0.0005) {
		tmp = x + (-0.3333333333333333 * pow(x, 3.0));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((-2.0d0) * x) <= (-10.0d0)) then
        tmp = (2.0d0 / (1.0d0 + exp(((-2.0d0) * x)))) + (-1.0d0)
    else if (((-2.0d0) * x) <= 0.0005d0) then
        tmp = x + ((-0.3333333333333333d0) * (x ** 3.0d0))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -10.0) {
		tmp = (2.0 / (1.0 + Math.exp((-2.0 * x)))) + -1.0;
	} else if ((-2.0 * x) <= 0.0005) {
		tmp = x + (-0.3333333333333333 * Math.pow(x, 3.0));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (-2.0 * x) <= -10.0:
		tmp = (2.0 / (1.0 + math.exp((-2.0 * x)))) + -1.0
	elif (-2.0 * x) <= 0.0005:
		tmp = x + (-0.3333333333333333 * math.pow(x, 3.0))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -10.0)
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) + -1.0);
	elseif (Float64(-2.0 * x) <= 0.0005)
		tmp = Float64(x + Float64(-0.3333333333333333 * (x ^ 3.0)));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((-2.0 * x) <= -10.0)
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	elseif ((-2.0 * x) <= 0.0005)
		tmp = x + (-0.3333333333333333 * (x ^ 3.0));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -10.0], N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.0005], N[(x + N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -10

   1. Initial program 100.0%

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

   if -10 < (*.f64 #s(literal -2 binary64) x) < 5.0000000000000001e-4

   1. Initial program 7.9%

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

   3. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{x \cdot \left(1 + -0.3333333333333333 \cdot {x}^{2}\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \color{blue}{1 \cdot x + \left(-0.3333333333333333 \cdot {x}^{2}\right) \cdot x} \]

      2. *-lft-identity 100.0%

         \[\leadsto \color{blue}{x} + \left(-0.3333333333333333 \cdot {x}^{2}\right) \cdot x \]

      3. associate-*l* 100.0%

         \[\leadsto x + \color{blue}{-0.3333333333333333 \cdot \left({x}^{2} \cdot x\right)} \]

      4. unpow2 100.0%

         \[\leadsto x + -0.3333333333333333 \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot x\right) \]

      5. unpow3 100.0%

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

   5. Simplified 100.0%

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

   if 5.0000000000000001e-4 < (*.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 98.0%

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

      1. *-commutative 98.0%

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

   5. Simplified 98.0%

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

   6. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{-1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -10:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.0005:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.9% accurate, 18.1× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (if (<= x -1.0) -1.0 x))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = -1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = -1.0d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = -1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = -1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = -1.0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = -1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], -1.0, x]

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 98.0%

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

      1. *-commutative 98.0%

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

   5. Simplified 98.0%

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

   6. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{-1} \]

   if -1 < x

   1. Initial program 39.8%

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

   3. Taylor expanded in x around 0 67.0%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 26.7% accurate, 109.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 -1.0)

C

double code(double x, double y) {
	return -1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = -1.0d0
end function

Java

public static double code(double x, double y) {
	return -1.0;
}

Python

def code(x, y):
	return -1.0

Julia

function code(x, y)
	return -1.0
end

MATLAB

function tmp = code(x, y)
	tmp = -1.0;
end

Wolfram

code[x_, y_] := -1.0

Derivation

1. Initial program 57.2%

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

3. Taylor expanded in x around 0 32.2%

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

   1. *-commutative 32.2%

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

5. Simplified 32.2%

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

6. Taylor expanded in x around inf 31.0%

   \[\leadsto \color{blue}{-1} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024113 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))