Logistic function from Lakshay Garg

Percentage Accurate: 54.0% → 99.9%

Time: 8.2s

Alternatives: 9

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: 54.0% 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: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -5:\\ \;\;\;\;\left|-1 + \frac{2}{1 + {\left(e^{x}\right)}^{2}}\right|\\ \mathbf{elif}\;-2 \cdot x \leq 0.002:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left(0.13333333333333333 \cdot \left(x \cdot x\right) - 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -5.0)
   (fabs (+ -1.0 (/ 2.0 (+ 1.0 (pow (exp x) 2.0)))))
   (if (<= (* -2.0 x) 0.002)
     (*
      x
      (+
       1.0
       (* (pow x 2.0) (- (* 0.13333333333333333 (* x x)) 0.3333333333333333))))
     (+ -1.0 (/ 2.0 (+ 1.0 (exp (* -2.0 x))))))))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -5.0) {
		tmp = fabs((-1.0 + (2.0 / (1.0 + pow(exp(x), 2.0)))));
	} else if ((-2.0 * x) <= 0.002) {
		tmp = x * (1.0 + (pow(x, 2.0) * ((0.13333333333333333 * (x * x)) - 0.3333333333333333)));
	} else {
		tmp = -1.0 + (2.0 / (1.0 + exp((-2.0 * x))));
	}
	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)) then
        tmp = abs(((-1.0d0) + (2.0d0 / (1.0d0 + (exp(x) ** 2.0d0)))))
    else if (((-2.0d0) * x) <= 0.002d0) then
        tmp = x * (1.0d0 + ((x ** 2.0d0) * ((0.13333333333333333d0 * (x * x)) - 0.3333333333333333d0)))
    else
        tmp = (-1.0d0) + (2.0d0 / (1.0d0 + exp(((-2.0d0) * x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -5.0) {
		tmp = Math.abs((-1.0 + (2.0 / (1.0 + Math.pow(Math.exp(x), 2.0)))));
	} else if ((-2.0 * x) <= 0.002) {
		tmp = x * (1.0 + (Math.pow(x, 2.0) * ((0.13333333333333333 * (x * x)) - 0.3333333333333333)));
	} else {
		tmp = -1.0 + (2.0 / (1.0 + Math.exp((-2.0 * x))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (-2.0 * x) <= -5.0:
		tmp = math.fabs((-1.0 + (2.0 / (1.0 + math.pow(math.exp(x), 2.0)))))
	elif (-2.0 * x) <= 0.002:
		tmp = x * (1.0 + (math.pow(x, 2.0) * ((0.13333333333333333 * (x * x)) - 0.3333333333333333)))
	else:
		tmp = -1.0 + (2.0 / (1.0 + math.exp((-2.0 * x))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -5.0)
		tmp = abs(Float64(-1.0 + Float64(2.0 / Float64(1.0 + (exp(x) ^ 2.0)))));
	elseif (Float64(-2.0 * x) <= 0.002)
		tmp = Float64(x * Float64(1.0 + Float64((x ^ 2.0) * Float64(Float64(0.13333333333333333 * Float64(x * x)) - 0.3333333333333333))));
	else
		tmp = Float64(-1.0 + Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((-2.0 * x) <= -5.0)
		tmp = abs((-1.0 + (2.0 / (1.0 + (exp(x) ^ 2.0)))));
	elseif ((-2.0 * x) <= 0.002)
		tmp = x * (1.0 + ((x ^ 2.0) * ((0.13333333333333333 * (x * x)) - 0.3333333333333333)));
	else
		tmp = -1.0 + (2.0 / (1.0 + exp((-2.0 * x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -5.0], N[Abs[N[(-1.0 + N[(2.0 / N[(1.0 + N[Power[N[Exp[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.002], N[(x * N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(0.13333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -5

   1. Initial program 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. expm1-log1p-u 96.4%

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

      3. *-rgt-identity 96.4%

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

      4. exp-prod 96.4%

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

   4. Applied egg-rr 96.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)\right)} - 1 \]
   5. Step-by-step derivation

      1. pow-exp 96.4%

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

      2. *-commutative 96.4%

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

   6. Applied egg-rr 96.4%

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

      1. add-sqr-sqrt 96.4%

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

      2. sqrt-unprod 96.4%

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

      3. pow2 96.4%

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

   8. Applied egg-rr 100.0%

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

      1. unpow2 100.0%

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

      2. rem-sqrt-square 100.0%

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

      3. +-commutative 100.0%

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

   10. Simplified 100.0%

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

   if -5 < (*.f64 #s(literal -2 binary64) x) < 2e-3

   1. Initial program 7.3%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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. unpow2 100.0%

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

   5. Applied egg-rr 100.0%

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

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

   1. Initial program 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -5:\\ \;\;\;\;\left|-1 + \frac{2}{1 + {\left(e^{x}\right)}^{2}}\right|\\ \mathbf{elif}\;-2 \cdot x \leq 0.002:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left(0.13333333333333333 \cdot \left(x \cdot x\right) - 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -5 \lor \neg \left(-2 \cdot x \leq 0.002\right):\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left(0.13333333333333333 \cdot \left(x \cdot x\right) - 0.3333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* -2.0 x) -5.0) (not (<= (* -2.0 x) 0.002)))
   (+ -1.0 (/ 2.0 (+ 1.0 (exp (* -2.0 x)))))
   (*
    x
    (+
     1.0
     (* (pow x 2.0) (- (* 0.13333333333333333 (* x x)) 0.3333333333333333))))))

C

double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -5.0) || !((-2.0 * x) <= 0.002)) {
		tmp = -1.0 + (2.0 / (1.0 + exp((-2.0 * x))));
	} else {
		tmp = x * (1.0 + (pow(x, 2.0) * ((0.13333333333333333 * (x * x)) - 0.3333333333333333)));
	}
	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) <= 0.002d0))) then
        tmp = (-1.0d0) + (2.0d0 / (1.0d0 + exp(((-2.0d0) * x))))
    else
        tmp = x * (1.0d0 + ((x ** 2.0d0) * ((0.13333333333333333d0 * (x * x)) - 0.3333333333333333d0)))
    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) <= 0.002)) {
		tmp = -1.0 + (2.0 / (1.0 + Math.exp((-2.0 * x))));
	} else {
		tmp = x * (1.0 + (Math.pow(x, 2.0) * ((0.13333333333333333 * (x * x)) - 0.3333333333333333)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((-2.0 * x) <= -5.0) or not ((-2.0 * x) <= 0.002):
		tmp = -1.0 + (2.0 / (1.0 + math.exp((-2.0 * x))))
	else:
		tmp = x * (1.0 + (math.pow(x, 2.0) * ((0.13333333333333333 * (x * x)) - 0.3333333333333333)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((Float64(-2.0 * x) <= -5.0) || !(Float64(-2.0 * x) <= 0.002))
		tmp = Float64(-1.0 + Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))));
	else
		tmp = Float64(x * Float64(1.0 + Float64((x ^ 2.0) * Float64(Float64(0.13333333333333333 * Float64(x * x)) - 0.3333333333333333))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((-2.0 * x) <= -5.0) || ~(((-2.0 * x) <= 0.002)))
		tmp = -1.0 + (2.0 / (1.0 + exp((-2.0 * x))));
	else
		tmp = x * (1.0 + ((x ^ 2.0) * ((0.13333333333333333 * (x * x)) - 0.3333333333333333)));
	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], 0.002]], $MachinePrecision]], N[(-1.0 + N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(0.13333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -5 or 2e-3 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 100.0%

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

   if -5 < (*.f64 #s(literal -2 binary64) x) < 2e-3

   1. Initial program 7.3%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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. unpow2 100.0%

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

   5. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -5 \lor \neg \left(-2 \cdot x \leq 0.002\right):\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left(0.13333333333333333 \cdot \left(x \cdot x\right) - 0.3333333333333333\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -5 \lor \neg \left(-2 \cdot x \leq 0.002\right):\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* -2.0 x) -5.0) (not (<= (* -2.0 x) 0.002)))
   (+ -1.0 (/ 2.0 (+ 1.0 (exp (* -2.0 x)))))
   (+ 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) <= 0.002)) {
		tmp = -1.0 + (2.0 / (1.0 + exp((-2.0 * x))));
	} 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) <= 0.002d0))) then
        tmp = (-1.0d0) + (2.0d0 / (1.0d0 + exp(((-2.0d0) * x))))
    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) <= 0.002)) {
		tmp = -1.0 + (2.0 / (1.0 + Math.exp((-2.0 * x))));
	} 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) <= 0.002):
		tmp = -1.0 + (2.0 / (1.0 + math.exp((-2.0 * x))))
	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) <= 0.002))
		tmp = Float64(-1.0 + Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))));
	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) <= 0.002)))
		tmp = -1.0 + (2.0 / (1.0 + exp((-2.0 * x))));
	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], 0.002]], $MachinePrecision]], N[(-1.0 + N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -5 or 2e-3 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 100.0%

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

   if -5 < (*.f64 #s(literal -2 binary64) x) < 2e-3

   1. Initial program 7.3%

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

   3. Taylor expanded in x around 0 99.9%

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

      1. distribute-rgt-in 99.9%

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

      2. *-lft-identity 99.9%

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

      3. associate-*l* 99.9%

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

      4. unpow2 99.9%

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

      5. unpow3 99.9%

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

   5. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -5 \lor \neg \left(-2 \cdot x \leq 0.002\right):\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.1% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.4 \cdot 10^{-9}:\\ \;\;\;\;-1 + \frac{2}{2 + x \cdot \left(x \cdot \left(2 + x \cdot -1.3333333333333333\right) - 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{x + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -7.4e-9)
   (+
    -1.0
    (/ 2.0 (+ 2.0 (* x (- (* x (+ 2.0 (* x -1.3333333333333333))) 2.0)))))
   (/ (* x 2.0) (+ x 2.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -7.4e-9) {
		tmp = -1.0 + (2.0 / (2.0 + (x * ((x * (2.0 + (x * -1.3333333333333333))) - 2.0))));
	} else {
		tmp = (x * 2.0) / (x + 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-7.4d-9)) then
        tmp = (-1.0d0) + (2.0d0 / (2.0d0 + (x * ((x * (2.0d0 + (x * (-1.3333333333333333d0)))) - 2.0d0))))
    else
        tmp = (x * 2.0d0) / (x + 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -7.4e-9) {
		tmp = -1.0 + (2.0 / (2.0 + (x * ((x * (2.0 + (x * -1.3333333333333333))) - 2.0))));
	} else {
		tmp = (x * 2.0) / (x + 2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -7.4e-9:
		tmp = -1.0 + (2.0 / (2.0 + (x * ((x * (2.0 + (x * -1.3333333333333333))) - 2.0))))
	else:
		tmp = (x * 2.0) / (x + 2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -7.4e-9)
		tmp = Float64(-1.0 + Float64(2.0 / Float64(2.0 + Float64(x * Float64(Float64(x * Float64(2.0 + Float64(x * -1.3333333333333333))) - 2.0)))));
	else
		tmp = Float64(Float64(x * 2.0) / Float64(x + 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -7.4e-9)
		tmp = -1.0 + (2.0 / (2.0 + (x * ((x * (2.0 + (x * -1.3333333333333333))) - 2.0))));
	else
		tmp = (x * 2.0) / (x + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -7.4e-9], N[(-1.0 + N[(2.0 / N[(2.0 + N[(x * N[(N[(x * N[(2.0 + N[(x * -1.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 2.0), $MachinePrecision] / N[(x + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.4e-9

   1. Initial program 99.3%

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

   3. Taylor expanded in x around 0 97.8%

      \[\leadsto \frac{2}{\color{blue}{2 + x \cdot \left(x \cdot \left(2 + -1.3333333333333333 \cdot x\right) - 2\right)}} - 1 \]

   if -7.4e-9 < x

   1. Initial program 35.5%

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

   3. Taylor expanded in x around 0 6.0%

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

      1. +-commutative 6.0%

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

   5. Simplified 6.0%

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

      1. flip-- 5.9%

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

      2. metadata-eval 5.9%

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

      3. difference-of-sqr-1 5.9%

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

      4. associate-+l+ 5.9%

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

      5. metadata-eval 5.9%

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

      6. associate--l+ 70.4%

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

      7. metadata-eval 70.4%

         \[\leadsto \frac{\left(x + 2\right) \cdot \left(x + \color{blue}{0}\right)}{\left(x + 1\right) + 1} \]

      8. +-rgt-identity 70.4%

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

      9. associate-+l+ 70.4%

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

      10. metadata-eval 70.4%

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

   7. Applied egg-rr 70.4%

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

   8. Taylor expanded in x around 0 74.4%

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

      1. *-commutative 74.4%

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

   10. Simplified 74.4%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.4 \cdot 10^{-9}:\\ \;\;\;\;-1 + \frac{2}{2 + x \cdot \left(x \cdot \left(2 + x \cdot -1.3333333333333333\right) - 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{x + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.6% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 2.55:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;2 - \frac{4}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0) -1.0 (if (<= x 2.55) x (- 2.0 (/ 4.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = -1.0;
	} else if (x <= 2.55) {
		tmp = x;
	} else {
		tmp = 2.0 - (4.0 / 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 if (x <= 2.55d0) then
        tmp = x
    else
        tmp = 2.0d0 - (4.0d0 / 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 if (x <= 2.55) {
		tmp = x;
	} else {
		tmp = 2.0 - (4.0 / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = -1.0
	elif x <= 2.55:
		tmp = x
	else:
		tmp = 2.0 - (4.0 / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = -1.0;
	elseif (x <= 2.55)
		tmp = x;
	else
		tmp = Float64(2.0 - Float64(4.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = -1.0;
	elseif (x <= 2.55)
		tmp = x;
	else
		tmp = 2.0 - (4.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], -1.0, If[LessEqual[x, 2.55], x, N[(2.0 - N[(4.0 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 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 97.8%

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

   4. Taylor expanded in x around inf 98.9%

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

   if -1 < x < 2.5499999999999998

   1. Initial program 7.3%

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

   3. Taylor expanded in x around 0 99.5%

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

   if 2.5499999999999998 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 5.5%

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

      1. +-commutative 5.5%

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

   5. Simplified 5.5%

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

      1. flip-- 5.2%

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

      2. metadata-eval 5.2%

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

      3. difference-of-sqr-1 5.2%

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

      4. associate-+l+ 5.2%

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

      5. metadata-eval 5.2%

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

      6. associate--l+ 5.2%

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

      7. metadata-eval 5.2%

         \[\leadsto \frac{\left(x + 2\right) \cdot \left(x + \color{blue}{0}\right)}{\left(x + 1\right) + 1} \]

      8. +-rgt-identity 5.2%

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

      9. associate-+l+ 5.2%

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

      10. metadata-eval 5.2%

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

   7. Applied egg-rr 5.2%

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

   8. Taylor expanded in x around 0 18.8%

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

      1. *-commutative 18.8%

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

   10. Simplified 18.8%

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

   11. Taylor expanded in x around inf 18.8%

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

       1. associate-*r/ 18.8%

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

       2. metadata-eval 18.8%

          \[\leadsto 2 - \frac{\color{blue}{4}}{x} \]

   13. Simplified 18.8%

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

Alternative 6: 78.1% accurate, 9.1× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.68) -1.0 (/ (* x 2.0) (+ x 2.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.68) {
		tmp = -1.0;
	} else {
		tmp = (x * 2.0) / (x + 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-0.68d0)) then
        tmp = -1.0d0
    else
        tmp = (x * 2.0d0) / (x + 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.68) {
		tmp = -1.0;
	} else {
		tmp = (x * 2.0) / (x + 2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.68:
		tmp = -1.0
	else:
		tmp = (x * 2.0) / (x + 2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.68)
		tmp = -1.0;
	else
		tmp = Float64(Float64(x * 2.0) / Float64(x + 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.68)
		tmp = -1.0;
	else
		tmp = (x * 2.0) / (x + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.68], -1.0, N[(N[(x * 2.0), $MachinePrecision] / N[(x + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.680000000000000049

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 97.8%

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

   4. Taylor expanded in x around inf 98.9%

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

   if -0.680000000000000049 < x

   1. Initial program 35.9%

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

   3. Taylor expanded in x around 0 6.6%

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

      1. +-commutative 6.6%

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

   5. Simplified 6.6%

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

      1. flip-- 6.5%

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

      2. metadata-eval 6.5%

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

      3. difference-of-sqr-1 6.5%

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

      4. associate-+l+ 6.5%

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

      5. metadata-eval 6.5%

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

      6. associate--l+ 70.3%

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

      7. metadata-eval 70.3%

         \[\leadsto \frac{\left(x + 2\right) \cdot \left(x + \color{blue}{0}\right)}{\left(x + 1\right) + 1} \]

      8. +-rgt-identity 70.3%

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

      9. associate-+l+ 70.3%

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

      10. metadata-eval 70.3%

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

   7. Applied egg-rr 70.3%

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

   8. Taylor expanded in x around 0 74.1%

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

      1. *-commutative 74.1%

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

   10. Simplified 74.1%

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

Alternative 7: 78.1% accurate, 9.1× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.68) -1.0 (* x (/ 2.0 (+ x 2.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.68) {
		tmp = -1.0;
	} else {
		tmp = x * (2.0 / (x + 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-0.68d0)) then
        tmp = -1.0d0
    else
        tmp = x * (2.0d0 / (x + 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.68) {
		tmp = -1.0;
	} else {
		tmp = x * (2.0 / (x + 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.68:
		tmp = -1.0
	else:
		tmp = x * (2.0 / (x + 2.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.68)
		tmp = -1.0;
	else
		tmp = Float64(x * Float64(2.0 / Float64(x + 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.68)
		tmp = -1.0;
	else
		tmp = x * (2.0 / (x + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.68], -1.0, N[(x * N[(2.0 / N[(x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.680000000000000049

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 97.8%

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

   4. Taylor expanded in x around inf 98.9%

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

   if -0.680000000000000049 < x

   1. Initial program 35.9%

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

   3. Taylor expanded in x around 0 6.6%

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

      1. +-commutative 6.6%

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

   5. Simplified 6.6%

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

      1. flip-- 6.5%

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

      2. metadata-eval 6.5%

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

      3. difference-of-sqr-1 6.5%

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

      4. associate-+l+ 6.5%

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

      5. metadata-eval 6.5%

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

      6. associate--l+ 70.3%

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

      7. metadata-eval 70.3%

         \[\leadsto \frac{\left(x + 2\right) \cdot \left(x + \color{blue}{0}\right)}{\left(x + 1\right) + 1} \]

      8. +-rgt-identity 70.3%

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

      9. associate-+l+ 70.3%

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

      10. metadata-eval 70.3%

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

   7. Applied egg-rr 70.3%

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

   8. Taylor expanded in x around 0 74.1%

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

      1. *-commutative 74.1%

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

   10. Simplified 74.1%

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

       1. associate-/l* 74.1%

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

   12. Applied egg-rr 74.1%

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

Alternative 8: 75.1% 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 97.8%

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

   4. Taylor expanded in x around inf 98.9%

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

   if -1 < x

   1. Initial program 35.9%

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

   3. Taylor expanded in x around 0 70.4%

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

Alternative 9: 26.4% 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 51.4%

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

3. Taylor expanded in x around 0 27.7%

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

4. Taylor expanded in x around inf 26.2%

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

Reproduce

herbie shell --seed 2024129 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))