Logistic function from Lakshay Garg

Percentage Accurate: 53.4% → 99.9%

Time: 8.7s

Alternatives: 10

Speedup: 9.9×

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: 53.4% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.2 \lor \neg \left(-2 \cdot x \leq 0.0001\right):\\ \;\;\;\;\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(e^{-2 \cdot x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left(\left(x \cdot x\right) \cdot \left(0.13333333333333333 + \left(x \cdot x\right) \cdot -0.05396825396825397\right) - 0.3333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* -2.0 x) -0.2) (not (<= (* -2.0 x) 0.0001)))
   (expm1 (- (log 2.0) (log1p (exp (* -2.0 x)))))
   (*
    x
    (+
     1.0
     (*
      (pow x 2.0)
      (-
       (* (* x x) (+ 0.13333333333333333 (* (* x x) -0.05396825396825397)))
       0.3333333333333333))))))

C

double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -0.2) || !((-2.0 * x) <= 0.0001)) {
		tmp = expm1((log(2.0) - log1p(exp((-2.0 * x)))));
	} else {
		tmp = x * (1.0 + (pow(x, 2.0) * (((x * x) * (0.13333333333333333 + ((x * x) * -0.05396825396825397))) - 0.3333333333333333)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -0.2) || !((-2.0 * x) <= 0.0001)) {
		tmp = Math.expm1((Math.log(2.0) - Math.log1p(Math.exp((-2.0 * x)))));
	} else {
		tmp = x * (1.0 + (Math.pow(x, 2.0) * (((x * x) * (0.13333333333333333 + ((x * x) * -0.05396825396825397))) - 0.3333333333333333)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((-2.0 * x) <= -0.2) or not ((-2.0 * x) <= 0.0001):
		tmp = math.expm1((math.log(2.0) - math.log1p(math.exp((-2.0 * x)))))
	else:
		tmp = x * (1.0 + (math.pow(x, 2.0) * (((x * x) * (0.13333333333333333 + ((x * x) * -0.05396825396825397))) - 0.3333333333333333)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((Float64(-2.0 * x) <= -0.2) || !(Float64(-2.0 * x) <= 0.0001))
		tmp = expm1(Float64(log(2.0) - log1p(exp(Float64(-2.0 * x)))));
	else
		tmp = Float64(x * Float64(1.0 + Float64((x ^ 2.0) * Float64(Float64(Float64(x * x) * Float64(0.13333333333333333 + Float64(Float64(x * x) * -0.05396825396825397))) - 0.3333333333333333))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.2], N[Not[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.0001]], $MachinePrecision]], N[(Exp[N[(N[Log[2.0], $MachinePrecision] - N[Log[1 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision], N[(x * N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * N[(0.13333333333333333 + N[(N[(x * x), $MachinePrecision] * -0.05396825396825397), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -0.20000000000000001 or 1.00000000000000005e-4 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 100.0%

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

      1. add-exp-log 100.0%

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

      2. expm1-define 100.0%

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

      3. log-div 100.0%

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

      4. log1p-define 100.0%

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

      5. exp-prod 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

   if -0.20000000000000001 < (*.f64 #s(literal -2 binary64) x) < 1.00000000000000005e-4

   1. Initial program 8.9%

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

   3. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.13333333333333333 + -0.05396825396825397 \cdot {x}^{2}\right) - 0.3333333333333333\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.13333333333333333 + -0.05396825396825397 \cdot \color{blue}{\left(x \cdot x\right)}\right) - 0.3333333333333333\right)\right) \]

   5. Applied egg-rr 100.0%

      \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.13333333333333333 + -0.05396825396825397 \cdot \color{blue}{\left(x \cdot x\right)}\right) - 0.3333333333333333\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.13333333333333333 + -0.05396825396825397 \cdot \color{blue}{\left(x \cdot x\right)}\right) - 0.3333333333333333\right)\right) \]

   7. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.2 \lor \neg \left(-2 \cdot x \leq 0.0001\right):\\ \;\;\;\;\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(e^{-2 \cdot x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left(\left(x \cdot x\right) \cdot \left(0.13333333333333333 + \left(x \cdot x\right) \cdot -0.05396825396825397\right) - 0.3333333333333333\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{e^{-2 \cdot x} + 1} + -1\\ \mathbf{if}\;-2 \cdot x \leq -0.2:\\ \;\;\;\;e^{\log t\_0}\\ \mathbf{elif}\;-2 \cdot x \leq 0.0001:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left(\left(x \cdot x\right) \cdot \left(0.13333333333333333 + \left(x \cdot x\right) \cdot -0.05396825396825397\right) - 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ 2.0 (+ (exp (* -2.0 x)) 1.0)) -1.0)))
   (if (<= (* -2.0 x) -0.2)
     (exp (log t_0))
     (if (<= (* -2.0 x) 0.0001)
       (*
        x
        (+
         1.0
         (*
          (pow x 2.0)
          (-
           (* (* x x) (+ 0.13333333333333333 (* (* x x) -0.05396825396825397)))
           0.3333333333333333))))
       t_0))))

C

double code(double x, double y) {
	double t_0 = (2.0 / (exp((-2.0 * x)) + 1.0)) + -1.0;
	double tmp;
	if ((-2.0 * x) <= -0.2) {
		tmp = exp(log(t_0));
	} else if ((-2.0 * x) <= 0.0001) {
		tmp = x * (1.0 + (pow(x, 2.0) * (((x * x) * (0.13333333333333333 + ((x * x) * -0.05396825396825397))) - 0.3333333333333333)));
	} else {
		tmp = 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 / (exp(((-2.0d0) * x)) + 1.0d0)) + (-1.0d0)
    if (((-2.0d0) * x) <= (-0.2d0)) then
        tmp = exp(log(t_0))
    else if (((-2.0d0) * x) <= 0.0001d0) then
        tmp = x * (1.0d0 + ((x ** 2.0d0) * (((x * x) * (0.13333333333333333d0 + ((x * x) * (-0.05396825396825397d0)))) - 0.3333333333333333d0)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (2.0 / (Math.exp((-2.0 * x)) + 1.0)) + -1.0;
	double tmp;
	if ((-2.0 * x) <= -0.2) {
		tmp = Math.exp(Math.log(t_0));
	} else if ((-2.0 * x) <= 0.0001) {
		tmp = x * (1.0 + (Math.pow(x, 2.0) * (((x * x) * (0.13333333333333333 + ((x * x) * -0.05396825396825397))) - 0.3333333333333333)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (2.0 / (math.exp((-2.0 * x)) + 1.0)) + -1.0
	tmp = 0
	if (-2.0 * x) <= -0.2:
		tmp = math.exp(math.log(t_0))
	elif (-2.0 * x) <= 0.0001:
		tmp = x * (1.0 + (math.pow(x, 2.0) * (((x * x) * (0.13333333333333333 + ((x * x) * -0.05396825396825397))) - 0.3333333333333333)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(2.0 / Float64(exp(Float64(-2.0 * x)) + 1.0)) + -1.0)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.2)
		tmp = exp(log(t_0));
	elseif (Float64(-2.0 * x) <= 0.0001)
		tmp = Float64(x * Float64(1.0 + Float64((x ^ 2.0) * Float64(Float64(Float64(x * x) * Float64(0.13333333333333333 + Float64(Float64(x * x) * -0.05396825396825397))) - 0.3333333333333333))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (2.0 / (exp((-2.0 * x)) + 1.0)) + -1.0;
	tmp = 0.0;
	if ((-2.0 * x) <= -0.2)
		tmp = exp(log(t_0));
	elseif ((-2.0 * x) <= 0.0001)
		tmp = x * (1.0 + ((x ^ 2.0) * (((x * x) * (0.13333333333333333 + ((x * x) * -0.05396825396825397))) - 0.3333333333333333)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(2.0 / N[(N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.2], N[Exp[N[Log[t$95$0], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.0001], N[(x * N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * N[(0.13333333333333333 + N[(N[(x * x), $MachinePrecision] * -0.05396825396825397), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -0.20000000000000001

   1. Initial program 100.0%

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

      1. add-exp-log 100.0%

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

      2. sub-neg 100.0%

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

      3. exp-prod 100.0%

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

      4. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

   if -0.20000000000000001 < (*.f64 #s(literal -2 binary64) x) < 1.00000000000000005e-4

   1. Initial program 8.9%

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

   3. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.13333333333333333 + -0.05396825396825397 \cdot {x}^{2}\right) - 0.3333333333333333\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.13333333333333333 + -0.05396825396825397 \cdot \color{blue}{\left(x \cdot x\right)}\right) - 0.3333333333333333\right)\right) \]

   5. Applied egg-rr 100.0%

      \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.13333333333333333 + -0.05396825396825397 \cdot \color{blue}{\left(x \cdot x\right)}\right) - 0.3333333333333333\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.13333333333333333 + -0.05396825396825397 \cdot \color{blue}{\left(x \cdot x\right)}\right) - 0.3333333333333333\right)\right) \]

   7. Applied egg-rr 100.0%

      \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(0.13333333333333333 + -0.05396825396825397 \cdot \left(x \cdot x\right)\right) - 0.3333333333333333\right)\right) \]

   if 1.00000000000000005e-4 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.2:\\ \;\;\;\;e^{\log \left(\frac{2}{e^{-2 \cdot x} + 1} + -1\right)}\\ \mathbf{elif}\;-2 \cdot x \leq 0.0001:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left(\left(x \cdot x\right) \cdot \left(0.13333333333333333 + \left(x \cdot x\right) \cdot -0.05396825396825397\right) - 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.2 \lor \neg \left(-2 \cdot x \leq 0.0001\right):\\ \;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} + -1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left(\left(x \cdot x\right) \cdot \left(0.13333333333333333 + \left(x \cdot x\right) \cdot -0.05396825396825397\right) - 0.3333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* -2.0 x) -0.2) (not (<= (* -2.0 x) 0.0001)))
   (+ (/ 2.0 (+ (exp (* -2.0 x)) 1.0)) -1.0)
   (*
    x
    (+
     1.0
     (*
      (pow x 2.0)
      (-
       (* (* x x) (+ 0.13333333333333333 (* (* x x) -0.05396825396825397)))
       0.3333333333333333))))))

C

double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -0.2) || !((-2.0 * x) <= 0.0001)) {
		tmp = (2.0 / (exp((-2.0 * x)) + 1.0)) + -1.0;
	} else {
		tmp = x * (1.0 + (pow(x, 2.0) * (((x * x) * (0.13333333333333333 + ((x * x) * -0.05396825396825397))) - 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) <= (-0.2d0)) .or. (.not. (((-2.0d0) * x) <= 0.0001d0))) then
        tmp = (2.0d0 / (exp(((-2.0d0) * x)) + 1.0d0)) + (-1.0d0)
    else
        tmp = x * (1.0d0 + ((x ** 2.0d0) * (((x * x) * (0.13333333333333333d0 + ((x * x) * (-0.05396825396825397d0)))) - 0.3333333333333333d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -0.2) || !((-2.0 * x) <= 0.0001)) {
		tmp = (2.0 / (Math.exp((-2.0 * x)) + 1.0)) + -1.0;
	} else {
		tmp = x * (1.0 + (Math.pow(x, 2.0) * (((x * x) * (0.13333333333333333 + ((x * x) * -0.05396825396825397))) - 0.3333333333333333)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((-2.0 * x) <= -0.2) or not ((-2.0 * x) <= 0.0001):
		tmp = (2.0 / (math.exp((-2.0 * x)) + 1.0)) + -1.0
	else:
		tmp = x * (1.0 + (math.pow(x, 2.0) * (((x * x) * (0.13333333333333333 + ((x * x) * -0.05396825396825397))) - 0.3333333333333333)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((Float64(-2.0 * x) <= -0.2) || !(Float64(-2.0 * x) <= 0.0001))
		tmp = Float64(Float64(2.0 / Float64(exp(Float64(-2.0 * x)) + 1.0)) + -1.0);
	else
		tmp = Float64(x * Float64(1.0 + Float64((x ^ 2.0) * Float64(Float64(Float64(x * x) * Float64(0.13333333333333333 + Float64(Float64(x * x) * -0.05396825396825397))) - 0.3333333333333333))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((-2.0 * x) <= -0.2) || ~(((-2.0 * x) <= 0.0001)))
		tmp = (2.0 / (exp((-2.0 * x)) + 1.0)) + -1.0;
	else
		tmp = x * (1.0 + ((x ^ 2.0) * (((x * x) * (0.13333333333333333 + ((x * x) * -0.05396825396825397))) - 0.3333333333333333)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.2], N[Not[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.0001]], $MachinePrecision]], N[(N[(2.0 / N[(N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(x * N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * N[(0.13333333333333333 + N[(N[(x * x), $MachinePrecision] * -0.05396825396825397), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -0.20000000000000001 or 1.00000000000000005e-4 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 100.0%

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

   if -0.20000000000000001 < (*.f64 #s(literal -2 binary64) x) < 1.00000000000000005e-4

   1. Initial program 8.9%

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

   3. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.13333333333333333 + -0.05396825396825397 \cdot {x}^{2}\right) - 0.3333333333333333\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.13333333333333333 + -0.05396825396825397 \cdot \color{blue}{\left(x \cdot x\right)}\right) - 0.3333333333333333\right)\right) \]

   5. Applied egg-rr 100.0%

      \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.13333333333333333 + -0.05396825396825397 \cdot \color{blue}{\left(x \cdot x\right)}\right) - 0.3333333333333333\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.13333333333333333 + -0.05396825396825397 \cdot \color{blue}{\left(x \cdot x\right)}\right) - 0.3333333333333333\right)\right) \]

   7. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.2 \lor \neg \left(-2 \cdot x \leq 0.0001\right):\\ \;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} + -1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left(\left(x \cdot x\right) \cdot \left(0.13333333333333333 + \left(x \cdot x\right) \cdot -0.05396825396825397\right) - 0.3333333333333333\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.02 \lor \neg \left(-2 \cdot x \leq 0.0001\right):\\ \;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} + -1\\ \mathbf{else}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* -2.0 x) -0.02) (not (<= (* -2.0 x) 0.0001)))
   (+ (/ 2.0 (+ (exp (* -2.0 x)) 1.0)) -1.0)
   (+ x (* -0.3333333333333333 (pow x 3.0)))))

C

double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -0.02) || !((-2.0 * x) <= 0.0001)) {
		tmp = (2.0 / (exp((-2.0 * x)) + 1.0)) + -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) <= (-0.02d0)) .or. (.not. (((-2.0d0) * x) <= 0.0001d0))) then
        tmp = (2.0d0 / (exp(((-2.0d0) * x)) + 1.0d0)) + (-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) <= -0.02) || !((-2.0 * x) <= 0.0001)) {
		tmp = (2.0 / (Math.exp((-2.0 * x)) + 1.0)) + -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) <= -0.02) or not ((-2.0 * x) <= 0.0001):
		tmp = (2.0 / (math.exp((-2.0 * x)) + 1.0)) + -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) <= -0.02) || !(Float64(-2.0 * x) <= 0.0001))
		tmp = Float64(Float64(2.0 / Float64(exp(Float64(-2.0 * x)) + 1.0)) + -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) <= -0.02) || ~(((-2.0 * x) <= 0.0001)))
		tmp = (2.0 / (exp((-2.0 * x)) + 1.0)) + -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], -0.02], N[Not[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.0001]], $MachinePrecision]], N[(N[(2.0 / N[(N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision] + 1.0), $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 #s(literal -2 binary64) x) < -0.0200000000000000004 or 1.00000000000000005e-4 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 99.9%

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

   if -0.0200000000000000004 < (*.f64 #s(literal -2 binary64) x) < 1.00000000000000005e-4

   1. Initial program 8.2%

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

   3. Taylor expanded in x around 0 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. *-lft-identity 100.0%

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

      3. associate-*l* 100.0%

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

      4. unpow2 100.0%

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

      5. unpow3 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.02 \lor \neg \left(-2 \cdot x \leq 0.0001\right):\\ \;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} + -1\\ \mathbf{else}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.1% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 2.5:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;2 - \frac{4}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0) -1.0 (if (<= x 2.5) 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 <= 2.5) {
		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 <= 2.5d0) 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 <= 2.5) {
		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 <= 2.5:
		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 <= 2.5)
		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 <= 2.5)
		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, 2.5], 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 96.4%

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

   4. Taylor expanded in x around inf 98.6%

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

   if -1 < x < 2.5

   1. Initial program 10.8%

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

   3. Taylor expanded in x around 0 97.7%

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

   if 2.5 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 5.3%

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

      1. +-commutative 5.3%

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

   5. Simplified 5.3%

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

      1. flip-- 5.0%

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

      2. metadata-eval 5.0%

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

      3. difference-of-sqr-1 5.0%

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

      4. associate-+l+ 5.0%

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

      5. metadata-eval 5.0%

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

      6. associate--l+ 5.0%

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

      7. metadata-eval 5.0%

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

      8. +-rgt-identity 5.0%

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

      9. associate-+l+ 5.0%

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

      10. metadata-eval 5.0%

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

   7. Applied egg-rr 5.0%

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

   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: 78.6% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.66:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{x + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.66) -1.0 (/ (* x 2.0) (+ x 2.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.66) {
		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 <= (-0.66d0)) 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 <= -0.66) {
		tmp = -1.0;
	} else {
		tmp = (x * 2.0) / (x + 2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.66:
		tmp = -1.0
	else:
		tmp = (x * 2.0) / (x + 2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.66)
		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 <= -0.66)
		tmp = -1.0;
	else
		tmp = (x * 2.0) / (x + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.66], -1.0, N[(N[(x * 2.0), $MachinePrecision] / N[(x + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.660000000000000031

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 96.4%

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

   4. Taylor expanded in x around inf 98.6%

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

   if -0.660000000000000031 < x

   1. Initial program 37.1%

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

   3. Taylor expanded in x around 0 7.7%

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

      1. +-commutative 7.7%

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

   5. Simplified 7.7%

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

      1. flip-- 7.7%

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

      2. metadata-eval 7.7%

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

      3. difference-of-sqr-1 7.7%

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

      4. associate-+l+ 7.7%

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

      5. metadata-eval 7.7%

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

      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+ 70.4%

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

      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 73.5%

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

      1. *-commutative 73.5%

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

   10. Simplified 73.5%

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

Alternative 7: 78.6% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.66:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{x + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.66) -1.0 (* x (/ 2.0 (+ x 2.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.66) {
		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 <= (-0.66d0)) 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 <= -0.66) {
		tmp = -1.0;
	} else {
		tmp = x * (2.0 / (x + 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.66:
		tmp = -1.0
	else:
		tmp = x * (2.0 / (x + 2.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.66)
		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 <= -0.66)
		tmp = -1.0;
	else
		tmp = x * (2.0 / (x + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.66], -1.0, N[(x * N[(2.0 / N[(x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.660000000000000031

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 96.4%

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

   4. Taylor expanded in x around inf 98.6%

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

   if -0.660000000000000031 < x

   1. Initial program 37.1%

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

   3. Taylor expanded in x around 0 7.7%

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

      1. +-commutative 7.7%

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

   5. Simplified 7.7%

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

      1. flip-- 7.7%

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

      2. metadata-eval 7.7%

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

      3. difference-of-sqr-1 7.7%

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

      4. associate-+l+ 7.7%

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

      5. metadata-eval 7.7%

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

      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+ 70.4%

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

      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 73.5%

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

      1. *-commutative 73.5%

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

   10. Simplified 73.5%

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

       1. associate-/l* 73.5%

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

   12. Applied egg-rr 73.5%

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

Alternative 8: 79.1% accurate, 9.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], -1.0, If[LessEqual[x, 2.0], x, 2.0]]

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 96.4%

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

   4. Taylor expanded in x around inf 98.6%

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

   if -1 < x < 2

   1. Initial program 10.8%

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

   3. Taylor expanded in x around 0 97.7%

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

   if 2 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 5.3%

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

      1. +-commutative 5.3%

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

   5. Simplified 5.3%

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

      1. flip-- 5.0%

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

      2. metadata-eval 5.0%

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

      3. difference-of-sqr-1 5.0%

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

      4. associate-+l+ 5.0%

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

      5. metadata-eval 5.0%

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

      6. associate--l+ 5.0%

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

      7. metadata-eval 5.0%

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

      8. +-rgt-identity 5.0%

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

      9. associate-+l+ 5.0%

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

      10. metadata-eval 5.0%

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

   7. Applied egg-rr 5.0%

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

   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%

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

Alternative 9: 31.5% accurate, 18.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{-308}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 1.1e-308) -1.0 2.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.1e-308) {
		tmp = -1.0;
	} else {
		tmp = 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 <= 1.1d-308) then
        tmp = -1.0d0
    else
        tmp = 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.1e-308) {
		tmp = -1.0;
	} else {
		tmp = 2.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.1e-308:
		tmp = -1.0
	else:
		tmp = 2.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.1e-308)
		tmp = -1.0;
	else
		tmp = 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.1e-308)
		tmp = -1.0;
	else
		tmp = 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.1e-308], -1.0, 2.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001e-308

   1. Initial program 50.5%

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

   3. Taylor expanded in x around 0 47.6%

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

   4. Taylor expanded in x around inf 47.1%

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

   if 1.1000000000000001e-308 < x

   1. Initial program 51.2%

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

   3. Taylor expanded in x around 0 7.3%

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

      1. +-commutative 7.3%

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

   5. Simplified 7.3%

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

      1. flip-- 7.1%

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

      2. metadata-eval 7.1%

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

      3. difference-of-sqr-1 7.1%

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

      4. associate-+l+ 7.1%

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

      5. metadata-eval 7.1%

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

      6. associate--l+ 55.7%

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

      7. metadata-eval 55.7%

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

      8. +-rgt-identity 55.7%

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

      9. associate-+l+ 55.8%

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

      10. metadata-eval 55.8%

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

   7. Applied egg-rr 55.8%

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

   8. Taylor expanded in x around 0 61.4%

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

      1. *-commutative 61.4%

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

   10. Simplified 61.4%

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

   11. Taylor expanded in x around inf 11.4%

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

Alternative 10: 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 50.8%

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

3. Taylor expanded in x around 0 26.2%

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

4. Taylor expanded in x around inf 24.2%

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

Reproduce

herbie shell --seed 2024165 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))