Logistic function from Lakshay Garg

Percentage Accurate: 54.2% → 99.1%

Time: 9.6s

Alternatives: 4

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: 54.2% 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: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-2 \cdot x}\\ t_1 := -1 - t\_0\\ \mathbf{if}\;-2 \cdot x \leq -10000000:\\ \;\;\;\;\frac{2}{1 + t\_0} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 10^{-10}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{4}{{t\_1}^{2}}}{-1 + \frac{2}{t\_1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (exp (* -2.0 x))) (t_1 (- -1.0 t_0)))
   (if (<= (* -2.0 x) -10000000.0)
     (+ (/ 2.0 (+ 1.0 t_0)) -1.0)
     (if (<= (* -2.0 x) 1e-10)
       x
       (/ (- 1.0 (/ 4.0 (pow t_1 2.0))) (+ -1.0 (/ 2.0 t_1)))))))

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) <= -10000000.0) {
		tmp = (2.0 / (1.0 + t_0)) + -1.0;
	} else if ((-2.0 * x) <= 1e-10) {
		tmp = x;
	} else {
		tmp = (1.0 - (4.0 / pow(t_1, 2.0))) / (-1.0 + (2.0 / t_1));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(((-2.0d0) * x))
    t_1 = (-1.0d0) - t_0
    if (((-2.0d0) * x) <= (-10000000.0d0)) then
        tmp = (2.0d0 / (1.0d0 + t_0)) + (-1.0d0)
    else if (((-2.0d0) * x) <= 1d-10) then
        tmp = x
    else
        tmp = (1.0d0 - (4.0d0 / (t_1 ** 2.0d0))) / ((-1.0d0) + (2.0d0 / t_1))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.exp((-2.0 * x));
	double t_1 = -1.0 - t_0;
	double tmp;
	if ((-2.0 * x) <= -10000000.0) {
		tmp = (2.0 / (1.0 + t_0)) + -1.0;
	} else if ((-2.0 * x) <= 1e-10) {
		tmp = x;
	} else {
		tmp = (1.0 - (4.0 / Math.pow(t_1, 2.0))) / (-1.0 + (2.0 / t_1));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.exp((-2.0 * x))
	t_1 = -1.0 - t_0
	tmp = 0
	if (-2.0 * x) <= -10000000.0:
		tmp = (2.0 / (1.0 + t_0)) + -1.0
	elif (-2.0 * x) <= 1e-10:
		tmp = x
	else:
		tmp = (1.0 - (4.0 / math.pow(t_1, 2.0))) / (-1.0 + (2.0 / t_1))
	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) <= -10000000.0)
		tmp = Float64(Float64(2.0 / Float64(1.0 + t_0)) + -1.0);
	elseif (Float64(-2.0 * x) <= 1e-10)
		tmp = x;
	else
		tmp = Float64(Float64(1.0 - Float64(4.0 / (t_1 ^ 2.0))) / Float64(-1.0 + Float64(2.0 / t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = exp((-2.0 * x));
	t_1 = -1.0 - t_0;
	tmp = 0.0;
	if ((-2.0 * x) <= -10000000.0)
		tmp = (2.0 / (1.0 + t_0)) + -1.0;
	elseif ((-2.0 * x) <= 1e-10)
		tmp = x;
	else
		tmp = (1.0 - (4.0 / (t_1 ^ 2.0))) / (-1.0 + (2.0 / t_1));
	end
	tmp_2 = 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], -10000000.0], N[(N[(2.0 / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 1e-10], x, N[(N[(1.0 - N[(4.0 / N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + N[(2.0 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -1e7

   1. Initial program 100.0%

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

   if -1e7 < (*.f64 #s(literal -2 binary64) x) < 1.00000000000000004e-10

   1. Initial program 5.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 100.0%

      \[\leadsto \color{blue}{x} \]

   if 1.00000000000000004e-10 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 99.9%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 \cdot 1 - \frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right), \color{blue}{\left(\left(\mathsf{neg}\left(1\right)\right) - \frac{2}{1 + e^{-2 \cdot x}}\right)}\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{1 - \frac{4}{{\left(-1 - e^{-2 \cdot x}\right)}^{2}}}{-1 - \frac{2}{1 + e^{-2 \cdot x}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -10000000:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 10^{-10}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{4}{{\left(-1 - e^{-2 \cdot x}\right)}^{2}}}{-1 + \frac{2}{-1 - e^{-2 \cdot x}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot \left(-0.3333333333333333 + \left(x \cdot x\right) \cdot \left(0.13333333333333333 + x \cdot \left(x \cdot -0.05396825396825397\right)\right)\right)\\ t_1 := \frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{if}\;-2 \cdot x \leq -10000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;-2 \cdot x \leq 0.05:\\ \;\;\;\;\frac{x \cdot \left(1 + t\_0 \cdot \left(t\_0 \cdot t\_0\right)\right)}{1 + t\_0 \cdot \left(-1 + t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (*
          (* x x)
          (+
           -0.3333333333333333
           (*
            (* x x)
            (+ 0.13333333333333333 (* x (* x -0.05396825396825397)))))))
        (t_1 (+ (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) -1.0)))
   (if (<= (* -2.0 x) -10000000.0)
     t_1
     (if (<= (* -2.0 x) 0.05)
       (/ (* x (+ 1.0 (* t_0 (* t_0 t_0)))) (+ 1.0 (* t_0 (+ -1.0 t_0))))
       t_1))))

C

double code(double x, double y) {
	double t_0 = (x * x) * (-0.3333333333333333 + ((x * x) * (0.13333333333333333 + (x * (x * -0.05396825396825397)))));
	double t_1 = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	double tmp;
	if ((-2.0 * x) <= -10000000.0) {
		tmp = t_1;
	} else if ((-2.0 * x) <= 0.05) {
		tmp = (x * (1.0 + (t_0 * (t_0 * t_0)))) / (1.0 + (t_0 * (-1.0 + t_0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x * x) * ((-0.3333333333333333d0) + ((x * x) * (0.13333333333333333d0 + (x * (x * (-0.05396825396825397d0))))))
    t_1 = (2.0d0 / (1.0d0 + exp(((-2.0d0) * x)))) + (-1.0d0)
    if (((-2.0d0) * x) <= (-10000000.0d0)) then
        tmp = t_1
    else if (((-2.0d0) * x) <= 0.05d0) then
        tmp = (x * (1.0d0 + (t_0 * (t_0 * t_0)))) / (1.0d0 + (t_0 * ((-1.0d0) + t_0)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * x) * (-0.3333333333333333 + ((x * x) * (0.13333333333333333 + (x * (x * -0.05396825396825397)))));
	double t_1 = (2.0 / (1.0 + Math.exp((-2.0 * x)))) + -1.0;
	double tmp;
	if ((-2.0 * x) <= -10000000.0) {
		tmp = t_1;
	} else if ((-2.0 * x) <= 0.05) {
		tmp = (x * (1.0 + (t_0 * (t_0 * t_0)))) / (1.0 + (t_0 * (-1.0 + t_0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * x) * (-0.3333333333333333 + ((x * x) * (0.13333333333333333 + (x * (x * -0.05396825396825397)))))
	t_1 = (2.0 / (1.0 + math.exp((-2.0 * x)))) + -1.0
	tmp = 0
	if (-2.0 * x) <= -10000000.0:
		tmp = t_1
	elif (-2.0 * x) <= 0.05:
		tmp = (x * (1.0 + (t_0 * (t_0 * t_0)))) / (1.0 + (t_0 * (-1.0 + t_0)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * x) * Float64(-0.3333333333333333 + Float64(Float64(x * x) * Float64(0.13333333333333333 + Float64(x * Float64(x * -0.05396825396825397))))))
	t_1 = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) + -1.0)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -10000000.0)
		tmp = t_1;
	elseif (Float64(-2.0 * x) <= 0.05)
		tmp = Float64(Float64(x * Float64(1.0 + Float64(t_0 * Float64(t_0 * t_0)))) / Float64(1.0 + Float64(t_0 * Float64(-1.0 + t_0))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * x) * (-0.3333333333333333 + ((x * x) * (0.13333333333333333 + (x * (x * -0.05396825396825397)))));
	t_1 = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	tmp = 0.0;
	if ((-2.0 * x) <= -10000000.0)
		tmp = t_1;
	elseif ((-2.0 * x) <= 0.05)
		tmp = (x * (1.0 + (t_0 * (t_0 * t_0)))) / (1.0 + (t_0 * (-1.0 + t_0)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * N[(-0.3333333333333333 + N[(N[(x * x), $MachinePrecision] * N[(0.13333333333333333 + N[(x * N[(x * -0.05396825396825397), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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], -10000000.0], t$95$1, If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.05], N[(N[(x * N[(1.0 + N[(t$95$0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$0 * N[(-1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -1e7 or 0.050000000000000003 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 100.0%

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

   if -1e7 < (*.f64 #s(literal -2 binary64) x) < 0.050000000000000003

   1. Initial program 6.6%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right)} - \frac{1}{3}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right)} - \frac{1}{3}\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) + \frac{-1}{3}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-1}{3} + \color{blue}{{x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{3}, \left({x}^{2} \cdot \left(\frac{-17}{315} \cdot {x}^{2} + \color{blue}{\frac{2}{15}}\right)\right)\right)\right)\right)\right) \]

      11. distribute-lft-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{3}, \left({x}^{2} \cdot \left(\frac{-17}{315} \cdot {x}^{2}\right) + \color{blue}{{x}^{2} \cdot \frac{2}{15}}\right)\right)\right)\right)\right) \]

      12. cancel-sign-sub N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{3}, \left({x}^{2} \cdot \left(\frac{-17}{315} \cdot {x}^{2}\right) - \color{blue}{\left(\mathsf{neg}\left({x}^{2}\right)\right) \cdot \frac{2}{15}}\right)\right)\right)\right)\right) \]

      13. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{3}, \left({x}^{2} \cdot \left(\frac{-17}{315} \cdot {x}^{2}\right) - \left(\mathsf{neg}\left({x}^{2} \cdot \frac{2}{15}\right)\right)\right)\right)\right)\right)\right) \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{3}, \left({x}^{2} \cdot \left(\frac{-17}{315} \cdot {x}^{2}\right) - {x}^{2} \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{2}{15}\right)\right)}\right)\right)\right)\right)\right) \]

      15. distribute-lft-out-- N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{3}, \left({x}^{2} \cdot \color{blue}{\left(\frac{-17}{315} \cdot {x}^{2} - \left(\mathsf{neg}\left(\frac{2}{15}\right)\right)\right)}\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{-17}{315} \cdot {x}^{2} - \left(\mathsf{neg}\left(\frac{2}{15}\right)\right)\right)}\right)\right)\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{-17}{315} \cdot {x}^{2}} - \left(\mathsf{neg}\left(\frac{2}{15}\right)\right)\right)\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{-17}{315} \cdot {x}^{2}} - \left(\mathsf{neg}\left(\frac{2}{15}\right)\right)\right)\right)\right)\right)\right)\right) \]

      19. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\left(\frac{-17}{315} \cdot {x}^{2}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{2}{15}\right)\right)}\right)\right)\right)\right)\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.3333333333333333 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot -0.05396825396825397\right) - -0.13333333333333333\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. flip3-+ N/A

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

      3. associate-*l/ N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left({1}^{3} + {\left(\left(x \cdot x\right) \cdot \left(\frac{-1}{3} + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{-17}{315}\right) - \frac{-2}{15}\right)\right)\right)}^{3}\right) \cdot x\right), \color{blue}{\left(1 \cdot 1 + \left(\left(\left(x \cdot x\right) \cdot \left(\frac{-1}{3} + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{-17}{315}\right) - \frac{-2}{15}\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-1}{3} + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{-17}{315}\right) - \frac{-2}{15}\right)\right)\right) - 1 \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-1}{3} + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{-17}{315}\right) - \frac{-2}{15}\right)\right)\right)\right)\right)}\right) \]

   7. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\left(1 + \left(\left(x \cdot x\right) \cdot \left(-0.3333333333333333 + \left(x \cdot x\right) \cdot \left(0.13333333333333333 + x \cdot \left(x \cdot -0.05396825396825397\right)\right)\right)\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(-0.3333333333333333 + \left(x \cdot x\right) \cdot \left(0.13333333333333333 + x \cdot \left(x \cdot -0.05396825396825397\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(-0.3333333333333333 + \left(x \cdot x\right) \cdot \left(0.13333333333333333 + x \cdot \left(x \cdot -0.05396825396825397\right)\right)\right)\right)\right)\right) \cdot x}{1 + \left(\left(x \cdot x\right) \cdot \left(-0.3333333333333333 + \left(x \cdot x\right) \cdot \left(0.13333333333333333 + x \cdot \left(x \cdot -0.05396825396825397\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(-0.3333333333333333 + \left(x \cdot x\right) \cdot \left(0.13333333333333333 + x \cdot \left(x \cdot -0.05396825396825397\right)\right)\right) - 1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -10000000:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.05:\\ \;\;\;\;\frac{x \cdot \left(1 + \left(\left(x \cdot x\right) \cdot \left(-0.3333333333333333 + \left(x \cdot x\right) \cdot \left(0.13333333333333333 + x \cdot \left(x \cdot -0.05396825396825397\right)\right)\right)\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(-0.3333333333333333 + \left(x \cdot x\right) \cdot \left(0.13333333333333333 + x \cdot \left(x \cdot -0.05396825396825397\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(-0.3333333333333333 + \left(x \cdot x\right) \cdot \left(0.13333333333333333 + x \cdot \left(x \cdot -0.05396825396825397\right)\right)\right)\right)\right)\right)}{1 + \left(\left(x \cdot x\right) \cdot \left(-0.3333333333333333 + \left(x \cdot x\right) \cdot \left(0.13333333333333333 + x \cdot \left(x \cdot -0.05396825396825397\right)\right)\right)\right) \cdot \left(-1 + \left(x \cdot x\right) \cdot \left(-0.3333333333333333 + \left(x \cdot x\right) \cdot \left(0.13333333333333333 + x \cdot \left(x \cdot -0.05396825396825397\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.3% 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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \color{blue}{\left(2 + x \cdot \left(2 \cdot x - 2\right)\right)}\right), 1\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \left(x \cdot \left(2 \cdot x - 2\right)\right)\right)\right), 1\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left(2 \cdot x - 2\right)\right)\right)\right), 1\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left(2 \cdot x + \left(\mathsf{neg}\left(2\right)\right)\right)\right)\right)\right), 1\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left(2 \cdot x + -2\right)\right)\right)\right), 1\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left(-2 + 2 \cdot x\right)\right)\right)\right), 1\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-2, \left(2 \cdot x\right)\right)\right)\right)\right), 1\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-2, \left(x \cdot 2\right)\right)\right)\right)\right), 1\right) \]

      8. *-lowering-*.f64 99.1%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(x, 2\right)\right)\right)\right)\right), 1\right) \]

   5. Simplified 99.1%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-1} \]
   7. Step-by-step derivation

   8. Simplified 100.0%

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

   if -1 < x

   1. Initial program 40.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 66.0%

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

Alternative 4: 27.6% 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 57.3%

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

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \color{blue}{\left(2 + x \cdot \left(2 \cdot x - 2\right)\right)}\right), 1\right) \]
4. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \left(x \cdot \left(2 \cdot x - 2\right)\right)\right)\right), 1\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left(2 \cdot x - 2\right)\right)\right)\right), 1\right) \]

   3. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left(2 \cdot x + \left(\mathsf{neg}\left(2\right)\right)\right)\right)\right)\right), 1\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left(2 \cdot x + -2\right)\right)\right)\right), 1\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left(-2 + 2 \cdot x\right)\right)\right)\right), 1\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-2, \left(2 \cdot x\right)\right)\right)\right)\right), 1\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-2, \left(x \cdot 2\right)\right)\right)\right)\right), 1\right) \]

   8. *-lowering-*.f64 31.9%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(x, 2\right)\right)\right)\right)\right), 1\right) \]

5. Simplified 31.9%

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

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{-1} \]
7. Step-by-step derivation

8. Simplified 31.1%

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

Reproduce

herbie shell --seed 2024152 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))