Logistic function from Lakshay Garg

Percentage Accurate: 53.7% → 99.0%

Time: 8.6s

Alternatives: 7

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: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;-1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{expm1}\left(x \cdot -4\right)}{\mathsf{expm1}\left(-2 \cdot x\right)}} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ 2.0 (+ 1.0 (exp (* -2.0 x))))))
   (if (<= t_0 0.0)
     -1.0
     (if (<= t_0 1.0)
       (+ x (* -0.3333333333333333 (pow x 3.0)))
       (+ (/ 2.0 (/ (expm1 (* x -4.0)) (expm1 (* -2.0 x)))) -1.0)))))

C

double code(double x, double y) {
	double t_0 = 2.0 / (1.0 + exp((-2.0 * x)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = -1.0;
	} else if (t_0 <= 1.0) {
		tmp = x + (-0.3333333333333333 * pow(x, 3.0));
	} else {
		tmp = (2.0 / (expm1((x * -4.0)) / expm1((-2.0 * x)))) + -1.0;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = 2.0 / (1.0 + Math.exp((-2.0 * x)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = -1.0;
	} else if (t_0 <= 1.0) {
		tmp = x + (-0.3333333333333333 * Math.pow(x, 3.0));
	} else {
		tmp = (2.0 / (Math.expm1((x * -4.0)) / Math.expm1((-2.0 * x)))) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 2.0 / (1.0 + math.exp((-2.0 * x)))
	tmp = 0
	if t_0 <= 0.0:
		tmp = -1.0
	elif t_0 <= 1.0:
		tmp = x + (-0.3333333333333333 * math.pow(x, 3.0))
	else:
		tmp = (2.0 / (math.expm1((x * -4.0)) / math.expm1((-2.0 * x)))) + -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 (t_0 <= 0.0)
		tmp = -1.0;
	elseif (t_0 <= 1.0)
		tmp = Float64(x + Float64(-0.3333333333333333 * (x ^ 3.0)));
	else
		tmp = Float64(Float64(2.0 / Float64(expm1(Float64(x * -4.0)) / expm1(Float64(-2.0 * x)))) + -1.0);
	end
	return 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[t$95$0, 0.0], -1.0, If[LessEqual[t$95$0, 1.0], N[(x + N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 2 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (*.f64 #s(literal -2 binary64) x)))) < 0.0

   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 0.0 < (/.f64 #s(literal 2 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (*.f64 #s(literal -2 binary64) x)))) < 1

   1. Initial program 7.8%

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

      1. expm1-log1p-u 7.7%

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

      2. expm1-undefine 7.7%

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

      3. log1p-undefine 7.7%

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

      4. +-commutative 7.7%

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

      5. add-exp-log 7.7%

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

      6. +-commutative 7.7%

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

      7. exp-prod 7.7%

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

   4. Applied egg-rr 7.7%

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

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{x \cdot \left(1 + -0.3333333333333333 \cdot {x}^{2}\right)} \]
   6. 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%

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

   7. Simplified 100.0%

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

   if 1 < (/.f64 #s(literal 2 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (*.f64 #s(literal -2 binary64) x))))

   1. Initial program 98.0%

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

      1. flip-+ 96.7%

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

      2. metadata-eval 96.7%

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

      3. div-sub 96.7%

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

      4. exp-prod 96.7%

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

      5. pow2 96.7%

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

      6. *-commutative 96.7%

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

      7. exp-prod 96.7%

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

      8. pow-pow 96.7%

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

      9. metadata-eval 96.7%

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

      10. exp-prod 96.7%

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

   4. Applied egg-rr 96.7%

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

      1. div-sub 96.7%

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

      2. remove-double-neg 96.7%

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

      3. distribute-neg-frac2 96.7%

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

      4. distribute-frac-neg 96.7%

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

      5. sub-neg 96.7%

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

      6. +-commutative 96.7%

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

      7. distribute-neg-in 96.7%

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

      8. remove-double-neg 96.7%

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

      9. exp-prod 96.7%

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

      10. *-commutative 96.7%

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

      11. metadata-eval 96.7%

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

      12. associate-*r* 96.7%

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

      13. sub-neg 96.7%

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

      14. sub-neg 96.7%

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

      15. +-commutative 96.7%

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

      16. distribute-neg-in 96.7%

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

      17. remove-double-neg 96.7%

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

      18. sub-neg 96.7%

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

      19. exp-prod 96.7%

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

      20. expm1-undefine 96.8%

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

   6. Simplified 98.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{expm1}\left(x \cdot -4\right)}{\mathsf{expm1}\left(x \cdot -2\right)}}} - 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} \leq 0:\\ \;\;\;\;-1\\ \mathbf{elif}\;\frac{2}{1 + e^{-2 \cdot x}} \leq 1:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{expm1}\left(x \cdot -4\right)}{\mathsf{expm1}\left(-2 \cdot x\right)}} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;-1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ 2.0 (+ 1.0 (exp (* -2.0 x))))))
   (if (<= t_0 0.0)
     -1.0
     (if (<= t_0 1.0)
       (+ x (* -0.3333333333333333 (pow x 3.0)))
       (+ t_0 -1.0)))))

C

double code(double x, double y) {
	double t_0 = 2.0 / (1.0 + exp((-2.0 * x)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = -1.0;
	} else if (t_0 <= 1.0) {
		tmp = x + (-0.3333333333333333 * pow(x, 3.0));
	} else {
		tmp = t_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 (t_0 <= 0.0d0) then
        tmp = -1.0d0
    else if (t_0 <= 1.0d0) then
        tmp = x + ((-0.3333333333333333d0) * (x ** 3.0d0))
    else
        tmp = t_0 + (-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 (t_0 <= 0.0) {
		tmp = -1.0;
	} else if (t_0 <= 1.0) {
		tmp = x + (-0.3333333333333333 * Math.pow(x, 3.0));
	} else {
		tmp = t_0 + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 2.0 / (1.0 + math.exp((-2.0 * x)))
	tmp = 0
	if t_0 <= 0.0:
		tmp = -1.0
	elif t_0 <= 1.0:
		tmp = x + (-0.3333333333333333 * math.pow(x, 3.0))
	else:
		tmp = t_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 (t_0 <= 0.0)
		tmp = -1.0;
	elseif (t_0 <= 1.0)
		tmp = Float64(x + Float64(-0.3333333333333333 * (x ^ 3.0)));
	else
		tmp = Float64(t_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 (t_0 <= 0.0)
		tmp = -1.0;
	elseif (t_0 <= 1.0)
		tmp = x + (-0.3333333333333333 * (x ^ 3.0));
	else
		tmp = t_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[t$95$0, 0.0], -1.0, If[LessEqual[t$95$0, 1.0], N[(x + N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 2 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (*.f64 #s(literal -2 binary64) x)))) < 0.0

   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 0.0 < (/.f64 #s(literal 2 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (*.f64 #s(literal -2 binary64) x)))) < 1

   1. Initial program 7.8%

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

      1. expm1-log1p-u 7.7%

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

      2. expm1-undefine 7.7%

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

      3. log1p-undefine 7.7%

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

      4. +-commutative 7.7%

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

      5. add-exp-log 7.7%

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

      6. +-commutative 7.7%

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

      7. exp-prod 7.7%

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

   4. Applied egg-rr 7.7%

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

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{x \cdot \left(1 + -0.3333333333333333 \cdot {x}^{2}\right)} \]
   6. 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%

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

   7. Simplified 100.0%

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

   if 1 < (/.f64 #s(literal 2 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (*.f64 #s(literal -2 binary64) x))))

   1. Initial program 98.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} \leq 0:\\ \;\;\;\;-1\\ \mathbf{elif}\;\frac{2}{1 + e^{-2 \cdot x}} \leq 1:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.55 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{2 + x \cdot \left(x \cdot \left(2 + x \cdot -1.3333333333333333\right) - 2\right)} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot x}{2 + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.55e-5)
   (+
    (/ 2.0 (+ 2.0 (* x (- (* x (+ 2.0 (* x -1.3333333333333333))) 2.0))))
    -1.0)
   (/ (* 2.0 x) (+ 2.0 x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.55e-5) {
		tmp = (2.0 / (2.0 + (x * ((x * (2.0 + (x * -1.3333333333333333))) - 2.0)))) + -1.0;
	} else {
		tmp = (2.0 * x) / (2.0 + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.55d-5)) then
        tmp = (2.0d0 / (2.0d0 + (x * ((x * (2.0d0 + (x * (-1.3333333333333333d0)))) - 2.0d0)))) + (-1.0d0)
    else
        tmp = (2.0d0 * x) / (2.0d0 + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.55e-5) {
		tmp = (2.0 / (2.0 + (x * ((x * (2.0 + (x * -1.3333333333333333))) - 2.0)))) + -1.0;
	} else {
		tmp = (2.0 * x) / (2.0 + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.55e-5:
		tmp = (2.0 / (2.0 + (x * ((x * (2.0 + (x * -1.3333333333333333))) - 2.0)))) + -1.0
	else:
		tmp = (2.0 * x) / (2.0 + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.55e-5)
		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(2.0 * x) / Float64(2.0 + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.55e-5)
		tmp = (2.0 / (2.0 + (x * ((x * (2.0 + (x * -1.3333333333333333))) - 2.0)))) + -1.0;
	else
		tmp = (2.0 * x) / (2.0 + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.55e-5], 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[(2.0 * x), $MachinePrecision] / N[(2.0 + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.54999999999999998e-5

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 99.7%

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

   if -2.54999999999999998e-5 < x

   1. Initial program 37.5%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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-- 7.3%

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

      2. pow2 7.3%

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

      3. +-commutative 7.3%

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

      4. metadata-eval 7.3%

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

      5. +-commutative 7.3%

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

   7. Applied egg-rr 7.3%

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

      1. unpow2 7.3%

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

      2. difference-of-sqr-1 7.3%

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

      3. +-commutative 7.3%

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

      4. associate-+l+ 7.3%

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

      5. metadata-eval 7.3%

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

      6. sub-neg 7.3%

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

      7. metadata-eval 7.3%

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

      8. +-commutative 7.3%

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

      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 72.9%

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

       1. *-commutative 72.9%

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

   12. Simplified 72.9%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.55 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{2 + x \cdot \left(x \cdot \left(2 + x \cdot -1.3333333333333333\right) - 2\right)} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot x}{2 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.4% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.55 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{2 + x \cdot \left(2 \cdot x - 2\right)} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot x}{2 + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.55e-5)
   (+ (/ 2.0 (+ 2.0 (* x (- (* 2.0 x) 2.0)))) -1.0)
   (/ (* 2.0 x) (+ 2.0 x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.55e-5) {
		tmp = (2.0 / (2.0 + (x * ((2.0 * x) - 2.0)))) + -1.0;
	} else {
		tmp = (2.0 * x) / (2.0 + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.55d-5)) then
        tmp = (2.0d0 / (2.0d0 + (x * ((2.0d0 * x) - 2.0d0)))) + (-1.0d0)
    else
        tmp = (2.0d0 * x) / (2.0d0 + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.55e-5) {
		tmp = (2.0 / (2.0 + (x * ((2.0 * x) - 2.0)))) + -1.0;
	} else {
		tmp = (2.0 * x) / (2.0 + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.55e-5:
		tmp = (2.0 / (2.0 + (x * ((2.0 * x) - 2.0)))) + -1.0
	else:
		tmp = (2.0 * x) / (2.0 + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.55e-5)
		tmp = Float64(Float64(2.0 / Float64(2.0 + Float64(x * Float64(Float64(2.0 * x) - 2.0)))) + -1.0);
	else
		tmp = Float64(Float64(2.0 * x) / Float64(2.0 + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.55e-5)
		tmp = (2.0 / (2.0 + (x * ((2.0 * x) - 2.0)))) + -1.0;
	else
		tmp = (2.0 * x) / (2.0 + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.55e-5], N[(N[(2.0 / N[(2.0 + N[(x * N[(N[(2.0 * x), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(2.0 * x), $MachinePrecision] / N[(2.0 + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.54999999999999998e-5

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 99.5%

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

   if -2.54999999999999998e-5 < x

   1. Initial program 37.5%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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-- 7.3%

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

      2. pow2 7.3%

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

      3. +-commutative 7.3%

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

      4. metadata-eval 7.3%

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

      5. +-commutative 7.3%

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

   7. Applied egg-rr 7.3%

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

      1. unpow2 7.3%

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

      2. difference-of-sqr-1 7.3%

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

      3. +-commutative 7.3%

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

      4. associate-+l+ 7.3%

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

      5. metadata-eval 7.3%

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

      6. sub-neg 7.3%

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

      7. metadata-eval 7.3%

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

      8. +-commutative 7.3%

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

      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 72.9%

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

       1. *-commutative 72.9%

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

   12. Simplified 72.9%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.55 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{2 + x \cdot \left(2 \cdot x - 2\right)} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot x}{2 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.9% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-27}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot x}{2 + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0) -1.0 (if (<= x -3e-27) x (/ (* 2.0 x) (+ 2.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = -1.0;
	} else if (x <= -3e-27) {
		tmp = x;
	} else {
		tmp = (2.0 * x) / (2.0 + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = -1.0d0
    else if (x <= (-3d-27)) then
        tmp = x
    else
        tmp = (2.0d0 * x) / (2.0d0 + 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 <= -3e-27) {
		tmp = x;
	} else {
		tmp = (2.0 * x) / (2.0 + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = -1.0
	elif x <= -3e-27:
		tmp = x
	else:
		tmp = (2.0 * x) / (2.0 + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = -1.0;
	elseif (x <= -3e-27)
		tmp = x;
	else
		tmp = Float64(Float64(2.0 * x) / Float64(2.0 + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = -1.0;
	elseif (x <= -3e-27)
		tmp = x;
	else
		tmp = (2.0 * x) / (2.0 + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], -1.0, If[LessEqual[x, -3e-27], x, N[(N[(2.0 * x), $MachinePrecision] / N[(2.0 + 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 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 < -3.0000000000000001e-27

   1. Initial program 46.1%

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

   3. Taylor expanded in x around 0 87.8%

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

   if -3.0000000000000001e-27 < x

   1. Initial program 37.4%

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

   3. Taylor expanded in x around 0 6.3%

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

      1. +-commutative 6.3%

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

   5. Simplified 6.3%

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

      1. flip-- 6.2%

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

      2. pow2 6.2%

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

      3. +-commutative 6.2%

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

      4. metadata-eval 6.2%

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

      5. +-commutative 6.2%

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

   7. Applied egg-rr 6.2%

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

      1. unpow2 6.2%

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

      2. difference-of-sqr-1 6.2%

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

      3. +-commutative 6.2%

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

      4. associate-+l+ 6.2%

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

      5. metadata-eval 6.2%

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

      6. sub-neg 6.2%

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

      7. metadata-eval 6.2%

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

      8. +-commutative 6.2%

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

      9. associate-+r+ 68.8%

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

      10. metadata-eval 68.8%

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

      11. +-lft-identity 68.8%

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

      12. +-commutative 68.8%

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

      13. associate-+l+ 68.8%

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

      14. metadata-eval 68.8%

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

   9. Simplified 68.8%

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

   10. Taylor expanded in x around 0 72.8%

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

       1. *-commutative 72.8%

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

   12. Simplified 72.8%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-27}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot x}{2 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.4% accurate, 18.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+32}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -2.9e+32) -1.0 x))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.9e+32) {
		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 <= (-2.9d+32)) 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 <= -2.9e+32) {
		tmp = -1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.9e+32:
		tmp = -1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.9e+32)
		tmp = -1.0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.9e+32)
		tmp = -1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.9e+32], -1.0, x]

Derivation

1. Split input into 2 regimes

2. if x < -2.90000000000000003e32

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

      1. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 100.0%

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

   if -2.90000000000000003e32 < x

   1. Initial program 39.0%

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

   3. Taylor expanded in x around 0 68.3%

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

Alternative 7: 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))