Logistic function from Lakshay Garg

Percentage Accurate: 53.7% → 99.8%

Time: 10.1s

Alternatives: 11

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: 53.7% 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: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -5:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.0002:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.13333333333333333 + {x}^{2} \cdot -0.05396825396825397\right) - 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right) + -1\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -5.0)
   (+ (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) -1.0)
   (if (<= (* -2.0 x) 0.0002)
     (*
      x
      (+
       1.0
       (*
        (pow x 2.0)
        (-
         (*
          (pow x 2.0)
          (+ 0.13333333333333333 (* (pow x 2.0) -0.05396825396825397)))
         0.3333333333333333))))
     (+ (+ (+ 1.0 (/ 2.0 (+ 1.0 (pow (exp -2.0) x)))) -1.0) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -5.0) {
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	} else if ((-2.0 * x) <= 0.0002) {
		tmp = x * (1.0 + (pow(x, 2.0) * ((pow(x, 2.0) * (0.13333333333333333 + (pow(x, 2.0) * -0.05396825396825397))) - 0.3333333333333333)));
	} else {
		tmp = ((1.0 + (2.0 / (1.0 + pow(exp(-2.0), x)))) + -1.0) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((-2.0d0) * x) <= (-5.0d0)) then
        tmp = (2.0d0 / (1.0d0 + exp(((-2.0d0) * x)))) + (-1.0d0)
    else if (((-2.0d0) * x) <= 0.0002d0) then
        tmp = x * (1.0d0 + ((x ** 2.0d0) * (((x ** 2.0d0) * (0.13333333333333333d0 + ((x ** 2.0d0) * (-0.05396825396825397d0)))) - 0.3333333333333333d0)))
    else
        tmp = ((1.0d0 + (2.0d0 / (1.0d0 + (exp((-2.0d0)) ** x)))) + (-1.0d0)) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -5.0) {
		tmp = (2.0 / (1.0 + Math.exp((-2.0 * x)))) + -1.0;
	} else if ((-2.0 * x) <= 0.0002) {
		tmp = x * (1.0 + (Math.pow(x, 2.0) * ((Math.pow(x, 2.0) * (0.13333333333333333 + (Math.pow(x, 2.0) * -0.05396825396825397))) - 0.3333333333333333)));
	} else {
		tmp = ((1.0 + (2.0 / (1.0 + Math.pow(Math.exp(-2.0), x)))) + -1.0) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (-2.0 * x) <= -5.0:
		tmp = (2.0 / (1.0 + math.exp((-2.0 * x)))) + -1.0
	elif (-2.0 * x) <= 0.0002:
		tmp = x * (1.0 + (math.pow(x, 2.0) * ((math.pow(x, 2.0) * (0.13333333333333333 + (math.pow(x, 2.0) * -0.05396825396825397))) - 0.3333333333333333)))
	else:
		tmp = ((1.0 + (2.0 / (1.0 + math.pow(math.exp(-2.0), x)))) + -1.0) + -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -5.0)
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) + -1.0);
	elseif (Float64(-2.0 * x) <= 0.0002)
		tmp = Float64(x * Float64(1.0 + Float64((x ^ 2.0) * Float64(Float64((x ^ 2.0) * Float64(0.13333333333333333 + Float64((x ^ 2.0) * -0.05396825396825397))) - 0.3333333333333333))));
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(2.0 / Float64(1.0 + (exp(-2.0) ^ x)))) + -1.0) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((-2.0 * x) <= -5.0)
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	elseif ((-2.0 * x) <= 0.0002)
		tmp = x * (1.0 + ((x ^ 2.0) * (((x ^ 2.0) * (0.13333333333333333 + ((x ^ 2.0) * -0.05396825396825397))) - 0.3333333333333333)));
	else
		tmp = ((1.0 + (2.0 / (1.0 + (exp(-2.0) ^ x)))) + -1.0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -5.0], N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.0002], N[(x * N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(N[Power[x, 2.0], $MachinePrecision] * N[(0.13333333333333333 + N[(N[Power[x, 2.0], $MachinePrecision] * -0.05396825396825397), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + N[(2.0 / N[(1.0 + N[Power[N[Exp[-2.0], $MachinePrecision], x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] + -1.0), $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

   if -5 < (*.f64 #s(literal -2 binary64) x) < 2.0000000000000001e-4

   1. Initial program 9.0%

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

   3. Taylor expanded in x around 0 100.0%

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

   if 2.0000000000000001e-4 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-undefine 100.0%

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

      3. log1p-undefine 100.0%

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

      4. +-commutative 100.0%

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

      5. add-exp-log 100.0%

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

      6. +-commutative 100.0%

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

      7. exp-prod 100.0%

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

   4. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -5:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.0002:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.13333333333333333 + {x}^{2} \cdot -0.05396825396825397\right) - 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right) + -1\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{\mathsf{expm1}\left(x \cdot -4\right)}, \mathsf{expm1}\left(-2 \cdot x\right), -1\right)\\ \mathbf{elif}\;-2 \cdot x \leq 0.0002:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right) + -1\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -0.02)
   (fma (/ 2.0 (expm1 (* x -4.0))) (expm1 (* -2.0 x)) -1.0)
   (if (<= (* -2.0 x) 0.0002)
     (* x (+ 1.0 (* (pow x 2.0) -0.3333333333333333)))
     (+ (+ (+ 1.0 (/ 2.0 (+ 1.0 (pow (exp -2.0) x)))) -1.0) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.02) {
		tmp = fma((2.0 / expm1((x * -4.0))), expm1((-2.0 * x)), -1.0);
	} else if ((-2.0 * x) <= 0.0002) {
		tmp = x * (1.0 + (pow(x, 2.0) * -0.3333333333333333));
	} else {
		tmp = ((1.0 + (2.0 / (1.0 + pow(exp(-2.0), x)))) + -1.0) + -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.02)
		tmp = fma(Float64(2.0 / expm1(Float64(x * -4.0))), expm1(Float64(-2.0 * x)), -1.0);
	elseif (Float64(-2.0 * x) <= 0.0002)
		tmp = Float64(x * Float64(1.0 + Float64((x ^ 2.0) * -0.3333333333333333)));
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(2.0 / Float64(1.0 + (exp(-2.0) ^ x)))) + -1.0) + -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.02], N[(N[(2.0 / N[(Exp[N[(x * -4.0), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] * N[(Exp[N[(-2.0 * x), $MachinePrecision]] - 1), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.0002], N[(x * N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + N[(2.0 / N[(1.0 + N[Power[N[Exp[-2.0], $MachinePrecision], x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.8%

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

      1. flip-+ 99.8%

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

      2. metadata-eval 99.8%

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

      3. div-sub 99.8%

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

      4. exp-prod 99.8%

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

      5. pow2 99.8%

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

      6. *-commutative 99.8%

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

      7. exp-prod 99.7%

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

      8. pow-pow 99.7%

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

      9. metadata-eval 99.7%

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

      10. exp-prod 99.7%

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

   4. Applied egg-rr 99.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{1}{1 - {\left(e^{-2}\right)}^{x}} - \frac{{\left(e^{x}\right)}^{-4}}{1 - {\left(e^{-2}\right)}^{x}}}} - 1 \]
   5. Step-by-step derivation

      1. div-sub 99.7%

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

      2. remove-double-neg 99.7%

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

      3. distribute-neg-frac2 99.7%

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

      4. distribute-frac-neg 99.7%

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

      5. sub-neg 99.7%

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

      6. +-commutative 99.7%

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

      7. distribute-neg-in 99.7%

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

      8. remove-double-neg 99.7%

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

      9. exp-prod 99.7%

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

      10. *-commutative 99.7%

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

      11. metadata-eval 99.7%

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

      12. associate-*r* 99.7%

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

      13. sub-neg 99.7%

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

      14. expm1-undefine 99.7%

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

      15. sub-neg 99.7%

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

      16. +-commutative 99.7%

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

      17. distribute-neg-in 99.7%

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

      18. remove-double-neg 99.7%

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

      19. sub-neg 99.7%

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

   6. Simplified 99.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{expm1}\left(x \cdot -4\right)}{\mathsf{expm1}\left(x \cdot -2\right)}}} - 1 \]

   7. Taylor expanded in x around inf 99.8%

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

      1. expm1-define 99.9%

         \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(-2 \cdot x\right)}}{e^{-4 \cdot x} - 1} - 1 \]

      2. associate-*r/ 99.9%

         \[\leadsto \color{blue}{\frac{2 \cdot \mathsf{expm1}\left(-2 \cdot x\right)}{e^{-4 \cdot x} - 1}} - 1 \]

      3. expm1-define 99.9%

         \[\leadsto \frac{2 \cdot \mathsf{expm1}\left(-2 \cdot x\right)}{\color{blue}{\mathsf{expm1}\left(-4 \cdot x\right)}} - 1 \]

      4. *-commutative 99.9%

         \[\leadsto \frac{2 \cdot \mathsf{expm1}\left(-2 \cdot x\right)}{\mathsf{expm1}\left(\color{blue}{x \cdot -4}\right)} - 1 \]

      5. associate-*l/ 99.8%

         \[\leadsto \color{blue}{\frac{2}{\mathsf{expm1}\left(x \cdot -4\right)} \cdot \mathsf{expm1}\left(-2 \cdot x\right)} - 1 \]

      6. fma-neg 99.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{\mathsf{expm1}\left(x \cdot -4\right)}, \mathsf{expm1}\left(-2 \cdot x\right), -1\right)} \]

      7. metadata-eval 99.9%

         \[\leadsto \mathsf{fma}\left(\frac{2}{\mathsf{expm1}\left(x \cdot -4\right)}, \mathsf{expm1}\left(-2 \cdot x\right), \color{blue}{-1}\right) \]

      8. *-commutative 99.9%

         \[\leadsto \mathsf{fma}\left(\frac{2}{\mathsf{expm1}\left(x \cdot -4\right)}, \mathsf{expm1}\left(\color{blue}{x \cdot -2}\right), -1\right) \]

   9. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{\mathsf{expm1}\left(x \cdot -4\right)}, \mathsf{expm1}\left(x \cdot -2\right), -1\right)} \]

   if -0.0200000000000000004 < (*.f64 #s(literal -2 binary64) x) < 2.0000000000000001e-4

   1. Initial program 8.3%

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

   3. Taylor expanded in x around 0 100.0%

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

   if 2.0000000000000001e-4 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-undefine 100.0%

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

      3. log1p-undefine 100.0%

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

      4. +-commutative 100.0%

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

      5. add-exp-log 100.0%

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

      6. +-commutative 100.0%

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

      7. exp-prod 100.0%

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

   4. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{\mathsf{expm1}\left(x \cdot -4\right)}, \mathsf{expm1}\left(-2 \cdot x\right), -1\right)\\ \mathbf{elif}\;-2 \cdot x \leq 0.0002:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right) + -1\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.02:\\ \;\;\;\;\frac{2}{\frac{\mathsf{expm1}\left(x \cdot -4\right)}{\mathsf{expm1}\left(-2 \cdot x\right)}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.0002:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right) + -1\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -0.02)
   (+ (/ 2.0 (/ (expm1 (* x -4.0)) (expm1 (* -2.0 x)))) -1.0)
   (if (<= (* -2.0 x) 0.0002)
     (* x (+ 1.0 (* (pow x 2.0) -0.3333333333333333)))
     (+ (+ (+ 1.0 (/ 2.0 (+ 1.0 (pow (exp -2.0) x)))) -1.0) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.02) {
		tmp = (2.0 / (expm1((x * -4.0)) / expm1((-2.0 * x)))) + -1.0;
	} else if ((-2.0 * x) <= 0.0002) {
		tmp = x * (1.0 + (pow(x, 2.0) * -0.3333333333333333));
	} else {
		tmp = ((1.0 + (2.0 / (1.0 + pow(exp(-2.0), x)))) + -1.0) + -1.0;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.02) {
		tmp = (2.0 / (Math.expm1((x * -4.0)) / Math.expm1((-2.0 * x)))) + -1.0;
	} else if ((-2.0 * x) <= 0.0002) {
		tmp = x * (1.0 + (Math.pow(x, 2.0) * -0.3333333333333333));
	} else {
		tmp = ((1.0 + (2.0 / (1.0 + Math.pow(Math.exp(-2.0), x)))) + -1.0) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (-2.0 * x) <= -0.02:
		tmp = (2.0 / (math.expm1((x * -4.0)) / math.expm1((-2.0 * x)))) + -1.0
	elif (-2.0 * x) <= 0.0002:
		tmp = x * (1.0 + (math.pow(x, 2.0) * -0.3333333333333333))
	else:
		tmp = ((1.0 + (2.0 / (1.0 + math.pow(math.exp(-2.0), x)))) + -1.0) + -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.02)
		tmp = Float64(Float64(2.0 / Float64(expm1(Float64(x * -4.0)) / expm1(Float64(-2.0 * x)))) + -1.0);
	elseif (Float64(-2.0 * x) <= 0.0002)
		tmp = Float64(x * Float64(1.0 + Float64((x ^ 2.0) * -0.3333333333333333)));
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(2.0 / Float64(1.0 + (exp(-2.0) ^ x)))) + -1.0) + -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.02], N[(N[(2.0 / N[(N[(Exp[N[(x * -4.0), $MachinePrecision]] - 1), $MachinePrecision] / N[(Exp[N[(-2.0 * x), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.0002], N[(x * N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + N[(2.0 / N[(1.0 + N[Power[N[Exp[-2.0], $MachinePrecision], x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.8%

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

      1. flip-+ 99.8%

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

      2. metadata-eval 99.8%

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

      3. div-sub 99.8%

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

      4. exp-prod 99.8%

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

      5. pow2 99.8%

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

      6. *-commutative 99.8%

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

      7. exp-prod 99.7%

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

      8. pow-pow 99.7%

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

      9. metadata-eval 99.7%

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

      10. exp-prod 99.7%

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

   4. Applied egg-rr 99.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{1}{1 - {\left(e^{-2}\right)}^{x}} - \frac{{\left(e^{x}\right)}^{-4}}{1 - {\left(e^{-2}\right)}^{x}}}} - 1 \]
   5. Step-by-step derivation

      1. div-sub 99.7%

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

      2. remove-double-neg 99.7%

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

      3. distribute-neg-frac2 99.7%

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

      4. distribute-frac-neg 99.7%

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

      5. sub-neg 99.7%

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

      6. +-commutative 99.7%

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

      7. distribute-neg-in 99.7%

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

      8. remove-double-neg 99.7%

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

      9. exp-prod 99.7%

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

      10. *-commutative 99.7%

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

      11. metadata-eval 99.7%

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

      12. associate-*r* 99.7%

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

      13. sub-neg 99.7%

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

      14. expm1-undefine 99.7%

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

      15. sub-neg 99.7%

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

      16. +-commutative 99.7%

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

      17. distribute-neg-in 99.7%

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

      18. remove-double-neg 99.7%

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

      19. sub-neg 99.7%

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

   6. Simplified 99.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{expm1}\left(x \cdot -4\right)}{\mathsf{expm1}\left(x \cdot -2\right)}}} - 1 \]

   if -0.0200000000000000004 < (*.f64 #s(literal -2 binary64) x) < 2.0000000000000001e-4

   1. Initial program 8.3%

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

   3. Taylor expanded in x around 0 100.0%

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

   if 2.0000000000000001e-4 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-undefine 100.0%

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

      3. log1p-undefine 100.0%

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

      4. +-commutative 100.0%

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

      5. add-exp-log 100.0%

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

      6. +-commutative 100.0%

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

      7. exp-prod 100.0%

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

   4. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.02:\\ \;\;\;\;\frac{2}{\frac{\mathsf{expm1}\left(x \cdot -4\right)}{\mathsf{expm1}\left(-2 \cdot x\right)}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.0002:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right) + -1\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.02:\\ \;\;\;\;\frac{2}{\frac{\mathsf{expm1}\left(x \cdot -4\right)}{\mathsf{expm1}\left(-2 \cdot x\right)}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.0002:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + \frac{2}{1 + e^{-2 \cdot x}}\right) + -1\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -0.02)
   (+ (/ 2.0 (/ (expm1 (* x -4.0)) (expm1 (* -2.0 x)))) -1.0)
   (if (<= (* -2.0 x) 0.0002)
     (* x (+ 1.0 (* (pow x 2.0) -0.3333333333333333)))
     (+ (+ (+ 1.0 (/ 2.0 (+ 1.0 (exp (* -2.0 x))))) -1.0) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.02) {
		tmp = (2.0 / (expm1((x * -4.0)) / expm1((-2.0 * x)))) + -1.0;
	} else if ((-2.0 * x) <= 0.0002) {
		tmp = x * (1.0 + (pow(x, 2.0) * -0.3333333333333333));
	} else {
		tmp = ((1.0 + (2.0 / (1.0 + exp((-2.0 * x))))) + -1.0) + -1.0;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.02) {
		tmp = (2.0 / (Math.expm1((x * -4.0)) / Math.expm1((-2.0 * x)))) + -1.0;
	} else if ((-2.0 * x) <= 0.0002) {
		tmp = x * (1.0 + (Math.pow(x, 2.0) * -0.3333333333333333));
	} else {
		tmp = ((1.0 + (2.0 / (1.0 + Math.exp((-2.0 * x))))) + -1.0) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (-2.0 * x) <= -0.02:
		tmp = (2.0 / (math.expm1((x * -4.0)) / math.expm1((-2.0 * x)))) + -1.0
	elif (-2.0 * x) <= 0.0002:
		tmp = x * (1.0 + (math.pow(x, 2.0) * -0.3333333333333333))
	else:
		tmp = ((1.0 + (2.0 / (1.0 + math.exp((-2.0 * x))))) + -1.0) + -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.02)
		tmp = Float64(Float64(2.0 / Float64(expm1(Float64(x * -4.0)) / expm1(Float64(-2.0 * x)))) + -1.0);
	elseif (Float64(-2.0 * x) <= 0.0002)
		tmp = Float64(x * Float64(1.0 + Float64((x ^ 2.0) * -0.3333333333333333)));
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x))))) + -1.0) + -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.02], N[(N[(2.0 / N[(N[(Exp[N[(x * -4.0), $MachinePrecision]] - 1), $MachinePrecision] / N[(Exp[N[(-2.0 * x), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.0002], N[(x * N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.8%

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

      1. flip-+ 99.8%

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

      2. metadata-eval 99.8%

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

      3. div-sub 99.8%

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

      4. exp-prod 99.8%

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

      5. pow2 99.8%

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

      6. *-commutative 99.8%

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

      7. exp-prod 99.7%

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

      8. pow-pow 99.7%

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

      9. metadata-eval 99.7%

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

      10. exp-prod 99.7%

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

   4. Applied egg-rr 99.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{1}{1 - {\left(e^{-2}\right)}^{x}} - \frac{{\left(e^{x}\right)}^{-4}}{1 - {\left(e^{-2}\right)}^{x}}}} - 1 \]
   5. Step-by-step derivation

      1. div-sub 99.7%

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

      2. remove-double-neg 99.7%

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

      3. distribute-neg-frac2 99.7%

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

      4. distribute-frac-neg 99.7%

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

      5. sub-neg 99.7%

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

      6. +-commutative 99.7%

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

      7. distribute-neg-in 99.7%

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

      8. remove-double-neg 99.7%

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

      9. exp-prod 99.7%

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

      10. *-commutative 99.7%

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

      11. metadata-eval 99.7%

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

      12. associate-*r* 99.7%

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

      13. sub-neg 99.7%

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

      14. expm1-undefine 99.7%

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

      15. sub-neg 99.7%

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

      16. +-commutative 99.7%

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

      17. distribute-neg-in 99.7%

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

      18. remove-double-neg 99.7%

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

      19. sub-neg 99.7%

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

   6. Simplified 99.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{expm1}\left(x \cdot -4\right)}{\mathsf{expm1}\left(x \cdot -2\right)}}} - 1 \]

   if -0.0200000000000000004 < (*.f64 #s(literal -2 binary64) x) < 2.0000000000000001e-4

   1. Initial program 8.3%

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

   3. Taylor expanded in x around 0 100.0%

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

   if 2.0000000000000001e-4 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-undefine 100.0%

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

      3. log1p-undefine 100.0%

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

      4. +-commutative 100.0%

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

      5. add-exp-log 100.0%

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

      6. +-commutative 100.0%

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

      7. exp-prod 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.02:\\ \;\;\;\;\frac{2}{\frac{\mathsf{expm1}\left(x \cdot -4\right)}{\mathsf{expm1}\left(-2 \cdot x\right)}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.0002:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + \frac{2}{1 + e^{-2 \cdot x}}\right) + -1\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{if}\;-2 \cdot x \leq -0.02:\\ \;\;\;\;t\_0 + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.0002:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + t\_0\right) + -1\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ 2.0 (+ 1.0 (exp (* -2.0 x))))))
   (if (<= (* -2.0 x) -0.02)
     (+ t_0 -1.0)
     (if (<= (* -2.0 x) 0.0002)
       (* x (+ 1.0 (* (pow x 2.0) -0.3333333333333333)))
       (+ (+ (+ 1.0 t_0) -1.0) -1.0)))))

C

double code(double x, double y) {
	double t_0 = 2.0 / (1.0 + exp((-2.0 * x)));
	double tmp;
	if ((-2.0 * x) <= -0.02) {
		tmp = t_0 + -1.0;
	} else if ((-2.0 * x) <= 0.0002) {
		tmp = x * (1.0 + (pow(x, 2.0) * -0.3333333333333333));
	} else {
		tmp = ((1.0 + t_0) + -1.0) + -1.0;
	}
	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)))
    if (((-2.0d0) * x) <= (-0.02d0)) then
        tmp = t_0 + (-1.0d0)
    else if (((-2.0d0) * x) <= 0.0002d0) then
        tmp = x * (1.0d0 + ((x ** 2.0d0) * (-0.3333333333333333d0)))
    else
        tmp = ((1.0d0 + t_0) + (-1.0d0)) + (-1.0d0)
    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)));
	double tmp;
	if ((-2.0 * x) <= -0.02) {
		tmp = t_0 + -1.0;
	} else if ((-2.0 * x) <= 0.0002) {
		tmp = x * (1.0 + (Math.pow(x, 2.0) * -0.3333333333333333));
	} else {
		tmp = ((1.0 + t_0) + -1.0) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 2.0 / (1.0 + math.exp((-2.0 * x)))
	tmp = 0
	if (-2.0 * x) <= -0.02:
		tmp = t_0 + -1.0
	elif (-2.0 * x) <= 0.0002:
		tmp = x * (1.0 + (math.pow(x, 2.0) * -0.3333333333333333))
	else:
		tmp = ((1.0 + t_0) + -1.0) + -1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x))))
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.02)
		tmp = Float64(t_0 + -1.0);
	elseif (Float64(-2.0 * x) <= 0.0002)
		tmp = Float64(x * Float64(1.0 + Float64((x ^ 2.0) * -0.3333333333333333)));
	else
		tmp = Float64(Float64(Float64(1.0 + t_0) + -1.0) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 2.0 / (1.0 + exp((-2.0 * x)));
	tmp = 0.0;
	if ((-2.0 * x) <= -0.02)
		tmp = t_0 + -1.0;
	elseif ((-2.0 * x) <= 0.0002)
		tmp = x * (1.0 + ((x ^ 2.0) * -0.3333333333333333));
	else
		tmp = ((1.0 + t_0) + -1.0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.02], N[(t$95$0 + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.0002], N[(x * N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + t$95$0), $MachinePrecision] + -1.0), $MachinePrecision] + -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.8%

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

   if -0.0200000000000000004 < (*.f64 #s(literal -2 binary64) x) < 2.0000000000000001e-4

   1. Initial program 8.3%

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

   3. Taylor expanded in x around 0 100.0%

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

   if 2.0000000000000001e-4 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-undefine 100.0%

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

      3. log1p-undefine 100.0%

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

      4. +-commutative 100.0%

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

      5. add-exp-log 100.0%

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

      6. +-commutative 100.0%

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

      7. exp-prod 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.02:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.0002:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + \frac{2}{1 + e^{-2 \cdot x}}\right) + -1\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.02 \lor \neg \left(-2 \cdot x \leq 0.0002\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot -0.3333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* -2.0 x) -0.02) (not (<= (* -2.0 x) 0.0002)))
   (+ (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) -1.0)
   (* x (+ 1.0 (* (pow x 2.0) -0.3333333333333333)))))

C

double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -0.02) || !((-2.0 * x) <= 0.0002)) {
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	} else {
		tmp = x * (1.0 + (pow(x, 2.0) * -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) <= (-0.02d0)) .or. (.not. (((-2.0d0) * x) <= 0.0002d0))) then
        tmp = (2.0d0 / (1.0d0 + exp(((-2.0d0) * x)))) + (-1.0d0)
    else
        tmp = x * (1.0d0 + ((x ** 2.0d0) * (-0.3333333333333333d0)))
    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) <= 0.0002)) {
		tmp = (2.0 / (1.0 + Math.exp((-2.0 * x)))) + -1.0;
	} else {
		tmp = x * (1.0 + (Math.pow(x, 2.0) * -0.3333333333333333));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((-2.0 * x) <= -0.02) or not ((-2.0 * x) <= 0.0002):
		tmp = (2.0 / (1.0 + math.exp((-2.0 * x)))) + -1.0
	else:
		tmp = x * (1.0 + (math.pow(x, 2.0) * -0.3333333333333333))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((Float64(-2.0 * x) <= -0.02) || !(Float64(-2.0 * x) <= 0.0002))
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) + -1.0);
	else
		tmp = Float64(x * Float64(1.0 + Float64((x ^ 2.0) * -0.3333333333333333)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((-2.0 * x) <= -0.02) || ~(((-2.0 * x) <= 0.0002)))
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	else
		tmp = x * (1.0 + ((x ^ 2.0) * -0.3333333333333333));
	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], 0.0002]], $MachinePrecision]], N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(x * N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -0.0200000000000000004 or 2.0000000000000001e-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) < 2.0000000000000001e-4

   1. Initial program 8.3%

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

   3. Taylor expanded in x around 0 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.02 \lor \neg \left(-2 \cdot x \leq 0.0002\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot -0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.04 \cdot 10^{-8}:\\ \;\;\;\;\frac{2}{2 + x \cdot \left(x \cdot \left(2 + x \cdot -1.3333333333333333\right) - 2\right)} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{x + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.04e-8)
   (+
    (/ 2.0 (+ 2.0 (* x (- (* x (+ 2.0 (* x -1.3333333333333333))) 2.0))))
    -1.0)
   (/ (* x 2.0) (+ x 2.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.04e-8) {
		tmp = (2.0 / (2.0 + (x * ((x * (2.0 + (x * -1.3333333333333333))) - 2.0)))) + -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 <= (-1.04d-8)) then
        tmp = (2.0d0 / (2.0d0 + (x * ((x * (2.0d0 + (x * (-1.3333333333333333d0)))) - 2.0d0)))) + (-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 <= -1.04e-8) {
		tmp = (2.0 / (2.0 + (x * ((x * (2.0 + (x * -1.3333333333333333))) - 2.0)))) + -1.0;
	} else {
		tmp = (x * 2.0) / (x + 2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.04e-8:
		tmp = (2.0 / (2.0 + (x * ((x * (2.0 + (x * -1.3333333333333333))) - 2.0)))) + -1.0
	else:
		tmp = (x * 2.0) / (x + 2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.04e-8)
		tmp = Float64(Float64(2.0 / Float64(2.0 + Float64(x * Float64(Float64(x * Float64(2.0 + Float64(x * -1.3333333333333333))) - 2.0)))) + -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 <= -1.04e-8)
		tmp = (2.0 / (2.0 + (x * ((x * (2.0 + (x * -1.3333333333333333))) - 2.0)))) + -1.0;
	else
		tmp = (x * 2.0) / (x + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.04e-8], N[(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] + -1.0), $MachinePrecision], N[(N[(x * 2.0), $MachinePrecision] / N[(x + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.04e-8

   1. Initial program 99.1%

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

   3. Taylor expanded in x around 0 99.1%

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

   if -1.04e-8 < x

   1. Initial program 37.0%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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-- 6.6%

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

      2. pow2 6.6%

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

      3. +-commutative 6.6%

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

      4. metadata-eval 6.6%

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

      5. +-commutative 6.6%

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

   7. Applied egg-rr 6.6%

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

      1. unpow2 6.6%

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

      2. difference-of-sqr-1 6.6%

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

      3. +-commutative 6.6%

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

      4. associate-+l+ 6.6%

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

      5. metadata-eval 6.6%

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

      6. sub-neg 6.6%

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

      7. metadata-eval 6.6%

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

      8. +-commutative 6.6%

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

      9. associate-+r+ 69.4%

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

      10. metadata-eval 69.4%

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

      11. +-lft-identity 69.4%

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

      12. +-commutative 69.4%

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

      13. associate-+l+ 69.5%

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

      14. metadata-eval 69.5%

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

   9. Simplified 69.5%

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

   10. Taylor expanded in x around 0 73.2%

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

       1. *-commutative 73.2%

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

   12. Simplified 73.2%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.04 \cdot 10^{-8}:\\ \;\;\;\;\frac{2}{2 + x \cdot \left(x \cdot \left(2 + x \cdot -1.3333333333333333\right) - 2\right)} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{x + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.5% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.04 \cdot 10^{-8}:\\ \;\;\;\;\frac{2}{2 + x \cdot \left(x \cdot 2 - 2\right)} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{x + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.04e-8)
   (+ (/ 2.0 (+ 2.0 (* x (- (* x 2.0) 2.0)))) -1.0)
   (/ (* x 2.0) (+ x 2.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.04e-8) {
		tmp = (2.0 / (2.0 + (x * ((x * 2.0) - 2.0)))) + -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 <= (-1.04d-8)) then
        tmp = (2.0d0 / (2.0d0 + (x * ((x * 2.0d0) - 2.0d0)))) + (-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 <= -1.04e-8) {
		tmp = (2.0 / (2.0 + (x * ((x * 2.0) - 2.0)))) + -1.0;
	} else {
		tmp = (x * 2.0) / (x + 2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.04e-8:
		tmp = (2.0 / (2.0 + (x * ((x * 2.0) - 2.0)))) + -1.0
	else:
		tmp = (x * 2.0) / (x + 2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.04e-8)
		tmp = Float64(Float64(2.0 / Float64(2.0 + Float64(x * Float64(Float64(x * 2.0) - 2.0)))) + -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 <= -1.04e-8)
		tmp = (2.0 / (2.0 + (x * ((x * 2.0) - 2.0)))) + -1.0;
	else
		tmp = (x * 2.0) / (x + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.04e-8], N[(N[(2.0 / N[(2.0 + N[(x * N[(N[(x * 2.0), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(x * 2.0), $MachinePrecision] / N[(x + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.04e-8

   1. Initial program 99.1%

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

   3. Taylor expanded in x around 0 98.6%

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

   if -1.04e-8 < x

   1. Initial program 37.0%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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-- 6.6%

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

      2. pow2 6.6%

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

      3. +-commutative 6.6%

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

      4. metadata-eval 6.6%

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

      5. +-commutative 6.6%

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

   7. Applied egg-rr 6.6%

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

      1. unpow2 6.6%

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

      2. difference-of-sqr-1 6.6%

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

      3. +-commutative 6.6%

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

      4. associate-+l+ 6.6%

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

      5. metadata-eval 6.6%

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

      6. sub-neg 6.6%

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

      7. metadata-eval 6.6%

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

      8. +-commutative 6.6%

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

      9. associate-+r+ 69.4%

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

      10. metadata-eval 69.4%

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

      11. +-lft-identity 69.4%

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

      12. +-commutative 69.4%

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

      13. associate-+l+ 69.5%

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

      14. metadata-eval 69.5%

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

   9. Simplified 69.5%

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

   10. Taylor expanded in x around 0 73.2%

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

       1. *-commutative 73.2%

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

   12. Simplified 73.2%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.04 \cdot 10^{-8}:\\ \;\;\;\;\frac{2}{2 + x \cdot \left(x \cdot 2 - 2\right)} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{x + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.6% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.65:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{x + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.65) -1.0 (/ (* x 2.0) (+ x 2.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.65) {
		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 <= (-0.65d0)) 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 <= -0.65) {
		tmp = -1.0;
	} else {
		tmp = (x * 2.0) / (x + 2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.65:
		tmp = -1.0
	else:
		tmp = (x * 2.0) / (x + 2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.65)
		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 <= -0.65)
		tmp = -1.0;
	else
		tmp = (x * 2.0) / (x + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.65], -1.0, N[(N[(x * 2.0), $MachinePrecision] / N[(x + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.650000000000000022

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.5%

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

      1. *-commutative 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in x around inf 100.0%

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

   if -0.650000000000000022 < x

   1. Initial program 37.7%

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

   3. Taylor expanded in x around 0 7.6%

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

      1. +-commutative 7.6%

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

   5. Simplified 7.6%

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

      1. flip-- 7.6%

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

      2. pow2 7.6%

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

      3. +-commutative 7.6%

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

      4. metadata-eval 7.6%

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

      5. +-commutative 7.6%

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

   7. Applied egg-rr 7.6%

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

      1. unpow2 7.6%

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

      2. difference-of-sqr-1 7.6%

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

      3. +-commutative 7.6%

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

      4. associate-+l+ 7.6%

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

      5. metadata-eval 7.6%

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

      6. sub-neg 7.6%

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

      7. metadata-eval 7.6%

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

      8. +-commutative 7.6%

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

      9. associate-+r+ 69.5%

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

      10. metadata-eval 69.5%

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

      11. +-lft-identity 69.5%

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

      12. +-commutative 69.5%

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

      13. associate-+l+ 69.5%

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

      14. metadata-eval 69.5%

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

   9. Simplified 69.5%

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

   10. Taylor expanded in x around 0 72.7%

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

       1. *-commutative 72.7%

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

   12. Simplified 72.7%

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

Alternative 10: 75.9% accurate, 18.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.5%

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

      1. *-commutative 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in x around inf 100.0%

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

   if -1 < x

   1. Initial program 37.7%

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

   3. Taylor expanded in x around 0 69.6%

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

Alternative 11: 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 54.5%

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

3. Taylor expanded in x around 0 31.0%

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

   1. *-commutative 31.0%

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

5. Simplified 31.0%

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

6. Taylor expanded in x around inf 29.2%

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

Reproduce

herbie shell --seed 2024103 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))