Logistic function from Lakshay Garg

Percentage Accurate: 53.3% → 99.3%

Time: 15.2s

Alternatives: 6

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: 53.3% 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: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + e^{-2 \cdot x}\\ t_1 := \frac{2}{t\_0}\\ \mathbf{if}\;-2 \cdot x \leq -5:\\ \;\;\;\;{\left(\sqrt{t\_1 + -1}\right)}^{2}\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-16}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\left(4 \cdot \frac{1}{{t\_0}^{2}} + -1\right) \cdot \frac{1}{1 + t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (exp (* -2.0 x)))) (t_1 (/ 2.0 t_0)))
   (if (<= (* -2.0 x) -5.0)
     (pow (sqrt (+ t_1 -1.0)) 2.0)
     (if (<= (* -2.0 x) 5e-16)
       (+ x (* -0.3333333333333333 (pow x 3.0)))
       (* (+ (* 4.0 (/ 1.0 (pow t_0 2.0))) -1.0) (/ 1.0 (+ 1.0 t_1)))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + exp((-2.0 * x));
	double t_1 = 2.0 / t_0;
	double tmp;
	if ((-2.0 * x) <= -5.0) {
		tmp = pow(sqrt((t_1 + -1.0)), 2.0);
	} else if ((-2.0 * x) <= 5e-16) {
		tmp = x + (-0.3333333333333333 * pow(x, 3.0));
	} else {
		tmp = ((4.0 * (1.0 / pow(t_0, 2.0))) + -1.0) * (1.0 / (1.0 + 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 = 1.0d0 + exp(((-2.0d0) * x))
    t_1 = 2.0d0 / t_0
    if (((-2.0d0) * x) <= (-5.0d0)) then
        tmp = sqrt((t_1 + (-1.0d0))) ** 2.0d0
    else if (((-2.0d0) * x) <= 5d-16) then
        tmp = x + ((-0.3333333333333333d0) * (x ** 3.0d0))
    else
        tmp = ((4.0d0 * (1.0d0 / (t_0 ** 2.0d0))) + (-1.0d0)) * (1.0d0 / (1.0d0 + t_1))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + Math.exp((-2.0 * x));
	double t_1 = 2.0 / t_0;
	double tmp;
	if ((-2.0 * x) <= -5.0) {
		tmp = Math.pow(Math.sqrt((t_1 + -1.0)), 2.0);
	} else if ((-2.0 * x) <= 5e-16) {
		tmp = x + (-0.3333333333333333 * Math.pow(x, 3.0));
	} else {
		tmp = ((4.0 * (1.0 / Math.pow(t_0, 2.0))) + -1.0) * (1.0 / (1.0 + t_1));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + math.exp((-2.0 * x))
	t_1 = 2.0 / t_0
	tmp = 0
	if (-2.0 * x) <= -5.0:
		tmp = math.pow(math.sqrt((t_1 + -1.0)), 2.0)
	elif (-2.0 * x) <= 5e-16:
		tmp = x + (-0.3333333333333333 * math.pow(x, 3.0))
	else:
		tmp = ((4.0 * (1.0 / math.pow(t_0, 2.0))) + -1.0) * (1.0 / (1.0 + t_1))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + exp(Float64(-2.0 * x)))
	t_1 = Float64(2.0 / t_0)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -5.0)
		tmp = sqrt(Float64(t_1 + -1.0)) ^ 2.0;
	elseif (Float64(-2.0 * x) <= 5e-16)
		tmp = Float64(x + Float64(-0.3333333333333333 * (x ^ 3.0)));
	else
		tmp = Float64(Float64(Float64(4.0 * Float64(1.0 / (t_0 ^ 2.0))) + -1.0) * Float64(1.0 / Float64(1.0 + t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + exp((-2.0 * x));
	t_1 = 2.0 / t_0;
	tmp = 0.0;
	if ((-2.0 * x) <= -5.0)
		tmp = sqrt((t_1 + -1.0)) ^ 2.0;
	elseif ((-2.0 * x) <= 5e-16)
		tmp = x + (-0.3333333333333333 * (x ^ 3.0));
	else
		tmp = ((4.0 * (1.0 / (t_0 ^ 2.0))) + -1.0) * (1.0 / (1.0 + t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / t$95$0), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -5.0], N[Power[N[Sqrt[N[(t$95$1 + -1.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 5e-16], N[(x + N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(4.0 * N[(1.0 / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] * N[(1.0 / N[(1.0 + t$95$1), $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. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 100.0%

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

      2. pow2 100.0%

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

      3. sub-neg 100.0%

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

      4. exp-prod 100.0%

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

      5. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

   if -5 < (*.f64 -2 x) < 5.0000000000000004e-16

   1. Initial program 7.0%

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

   3. Taylor expanded in x around 0 100.0%

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

   if 5.0000000000000004e-16 < (*.f64 -2 x)

   1. Initial program 100.0%

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

      1. flip-- 100.0%

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

      2. div-inv 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

   6. Taylor expanded in x around inf 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -5:\\ \;\;\;\;{\left(\sqrt{\frac{2}{1 + e^{-2 \cdot x}} + -1}\right)}^{2}\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-16}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\left(4 \cdot \frac{1}{{\left(1 + e^{-2 \cdot x}\right)}^{2}} + -1\right) \cdot \frac{1}{1 + \frac{2}{1 + e^{-2 \cdot x}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{if}\;-2 \cdot x \leq -5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-16}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{t\_0}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) -1.0)))
   (if (<= (* -2.0 x) -5.0)
     t_0
     (if (<= (* -2.0 x) 5e-16)
       (+ x (* -0.3333333333333333 (pow x 3.0)))
       (log (exp t_0))))))

C

double code(double x, double y) {
	double t_0 = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	double tmp;
	if ((-2.0 * x) <= -5.0) {
		tmp = t_0;
	} else if ((-2.0 * x) <= 5e-16) {
		tmp = x + (-0.3333333333333333 * pow(x, 3.0));
	} else {
		tmp = log(exp(t_0));
	}
	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) :: tmp
    t_0 = (2.0d0 / (1.0d0 + exp(((-2.0d0) * x)))) + (-1.0d0)
    if (((-2.0d0) * x) <= (-5.0d0)) then
        tmp = t_0
    else if (((-2.0d0) * x) <= 5d-16) then
        tmp = x + ((-0.3333333333333333d0) * (x ** 3.0d0))
    else
        tmp = log(exp(t_0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (2.0 / (1.0 + Math.exp((-2.0 * x)))) + -1.0;
	double tmp;
	if ((-2.0 * x) <= -5.0) {
		tmp = t_0;
	} else if ((-2.0 * x) <= 5e-16) {
		tmp = x + (-0.3333333333333333 * Math.pow(x, 3.0));
	} else {
		tmp = Math.log(Math.exp(t_0));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (2.0 / (1.0 + math.exp((-2.0 * x)))) + -1.0
	tmp = 0
	if (-2.0 * x) <= -5.0:
		tmp = t_0
	elif (-2.0 * x) <= 5e-16:
		tmp = x + (-0.3333333333333333 * math.pow(x, 3.0))
	else:
		tmp = math.log(math.exp(t_0))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) + -1.0)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -5.0)
		tmp = t_0;
	elseif (Float64(-2.0 * x) <= 5e-16)
		tmp = Float64(x + Float64(-0.3333333333333333 * (x ^ 3.0)));
	else
		tmp = log(exp(t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	tmp = 0.0;
	if ((-2.0 * x) <= -5.0)
		tmp = t_0;
	elseif ((-2.0 * x) <= 5e-16)
		tmp = x + (-0.3333333333333333 * (x ^ 3.0));
	else
		tmp = log(exp(t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$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], -5.0], t$95$0, If[LessEqual[N[(-2.0 * x), $MachinePrecision], 5e-16], N[(x + N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[Exp[t$95$0], $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. Add Preprocessing

   if -5 < (*.f64 -2 x) < 5.0000000000000004e-16

   1. Initial program 7.0%

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

   3. Taylor expanded in x around 0 100.0%

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

   if 5.0000000000000004e-16 < (*.f64 -2 x)

   1. Initial program 100.0%

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

      1. add-log-exp 100.0%

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

      2. sub-neg 100.0%

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

      3. exp-prod 100.0%

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

      4. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -5:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-16}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} + -1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{if}\;-2 \cdot x \leq -5:\\ \;\;\;\;{\left(\sqrt{t\_0}\right)}^{2}\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-16}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{t\_0}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) -1.0)))
   (if (<= (* -2.0 x) -5.0)
     (pow (sqrt t_0) 2.0)
     (if (<= (* -2.0 x) 5e-16)
       (+ x (* -0.3333333333333333 (pow x 3.0)))
       (log (exp t_0))))))

C

double code(double x, double y) {
	double t_0 = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	double tmp;
	if ((-2.0 * x) <= -5.0) {
		tmp = pow(sqrt(t_0), 2.0);
	} else if ((-2.0 * x) <= 5e-16) {
		tmp = x + (-0.3333333333333333 * pow(x, 3.0));
	} else {
		tmp = log(exp(t_0));
	}
	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) :: tmp
    t_0 = (2.0d0 / (1.0d0 + exp(((-2.0d0) * x)))) + (-1.0d0)
    if (((-2.0d0) * x) <= (-5.0d0)) then
        tmp = sqrt(t_0) ** 2.0d0
    else if (((-2.0d0) * x) <= 5d-16) then
        tmp = x + ((-0.3333333333333333d0) * (x ** 3.0d0))
    else
        tmp = log(exp(t_0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (2.0 / (1.0 + Math.exp((-2.0 * x)))) + -1.0;
	double tmp;
	if ((-2.0 * x) <= -5.0) {
		tmp = Math.pow(Math.sqrt(t_0), 2.0);
	} else if ((-2.0 * x) <= 5e-16) {
		tmp = x + (-0.3333333333333333 * Math.pow(x, 3.0));
	} else {
		tmp = Math.log(Math.exp(t_0));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (2.0 / (1.0 + math.exp((-2.0 * x)))) + -1.0
	tmp = 0
	if (-2.0 * x) <= -5.0:
		tmp = math.pow(math.sqrt(t_0), 2.0)
	elif (-2.0 * x) <= 5e-16:
		tmp = x + (-0.3333333333333333 * math.pow(x, 3.0))
	else:
		tmp = math.log(math.exp(t_0))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) + -1.0)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -5.0)
		tmp = sqrt(t_0) ^ 2.0;
	elseif (Float64(-2.0 * x) <= 5e-16)
		tmp = Float64(x + Float64(-0.3333333333333333 * (x ^ 3.0)));
	else
		tmp = log(exp(t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	tmp = 0.0;
	if ((-2.0 * x) <= -5.0)
		tmp = sqrt(t_0) ^ 2.0;
	elseif ((-2.0 * x) <= 5e-16)
		tmp = x + (-0.3333333333333333 * (x ^ 3.0));
	else
		tmp = log(exp(t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$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], -5.0], N[Power[N[Sqrt[t$95$0], $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 5e-16], N[(x + N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[Exp[t$95$0], $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. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 100.0%

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

      2. pow2 100.0%

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

      3. sub-neg 100.0%

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

      4. exp-prod 100.0%

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

      5. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

   if -5 < (*.f64 -2 x) < 5.0000000000000004e-16

   1. Initial program 7.0%

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

   3. Taylor expanded in x around 0 100.0%

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

   if 5.0000000000000004e-16 < (*.f64 -2 x)

   1. Initial program 100.0%

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

      1. add-log-exp 100.0%

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

      2. sub-neg 100.0%

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

      3. exp-prod 100.0%

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

      4. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -5:\\ \;\;\;\;{\left(\sqrt{\frac{2}{1 + e^{-2 \cdot x}} + -1}\right)}^{2}\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-16}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} + -1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -5 \lor \neg \left(-2 \cdot x \leq 5 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{else}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* -2.0 x) -5.0) (not (<= (* -2.0 x) 5e-16)))
   (+ (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) -1.0)
   (+ x (* -0.3333333333333333 (pow x 3.0)))))

C

double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -5.0) || !((-2.0 * x) <= 5e-16)) {
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	} else {
		tmp = x + (-0.3333333333333333 * pow(x, 3.0));
	}
	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) <= (-5.0d0)) .or. (.not. (((-2.0d0) * x) <= 5d-16))) then
        tmp = (2.0d0 / (1.0d0 + exp(((-2.0d0) * x)))) + (-1.0d0)
    else
        tmp = x + ((-0.3333333333333333d0) * (x ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -5.0) || !((-2.0 * x) <= 5e-16)) {
		tmp = (2.0 / (1.0 + Math.exp((-2.0 * x)))) + -1.0;
	} else {
		tmp = x + (-0.3333333333333333 * Math.pow(x, 3.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((-2.0 * x) <= -5.0) or not ((-2.0 * x) <= 5e-16):
		tmp = (2.0 / (1.0 + math.exp((-2.0 * x)))) + -1.0
	else:
		tmp = x + (-0.3333333333333333 * math.pow(x, 3.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((Float64(-2.0 * x) <= -5.0) || !(Float64(-2.0 * x) <= 5e-16))
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) + -1.0);
	else
		tmp = Float64(x + Float64(-0.3333333333333333 * (x ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((-2.0 * x) <= -5.0) || ~(((-2.0 * x) <= 5e-16)))
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	else
		tmp = x + (-0.3333333333333333 * (x ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[(-2.0 * x), $MachinePrecision], -5.0], N[Not[LessEqual[N[(-2.0 * x), $MachinePrecision], 5e-16]], $MachinePrecision]], N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(x + N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 -2 x) < -5 or 5.0000000000000004e-16 < (*.f64 -2 x)

   1. Initial program 100.0%

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

   if -5 < (*.f64 -2 x) < 5.0000000000000004e-16

   1. Initial program 7.0%

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

   3. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{x + -0.3333333333333333 \cdot {x}^{3}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -5 \lor \neg \left(-2 \cdot x \leq 5 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{else}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 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 97.5%

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

      1. *-commutative 97.5%

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

   5. Simplified 97.5%

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

   6. Taylor expanded in x around inf 98.9%

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

   if -1 < x

   1. Initial program 39.0%

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

   3. Taylor expanded in x around 0 67.3%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 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 56.4%

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

3. Taylor expanded in x around 0 31.4%

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

   1. *-commutative 31.4%

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

5. Simplified 31.4%

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

6. Taylor expanded in x around inf 30.3%

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

7. Final simplification 30.3%

   \[\leadsto -1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024031 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))