Logistic function from Lakshay Garg

Percentage Accurate: 53.8% → 99.6%

Time: 10.3s

Alternatives: 10

Speedup: 21.3×

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: 53.8% 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: 99.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + e^{-2 \cdot x}\\ \mathbf{if}\;-2 \cdot x \leq -1000:\\ \;\;\;\;\frac{2}{t_0} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-7}:\\ \;\;\;\;-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot \frac{1}{{t_0}^{2}} + -1}{1 + \frac{2}{{\left(\sqrt[3]{t_0}\right)}^{3}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (exp (* -2.0 x)))))
   (if (<= (* -2.0 x) -1000.0)
     (+ (/ 2.0 t_0) -1.0)
     (if (<= (* -2.0 x) 5e-7)
       (+
        (* -0.3333333333333333 (pow x 3.0))
        (+ x (* 0.13333333333333333 (pow x 5.0))))
       (/
        (+ (* 4.0 (/ 1.0 (pow t_0 2.0))) -1.0)
        (+ 1.0 (/ 2.0 (pow (cbrt t_0) 3.0))))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + exp((-2.0 * x));
	double tmp;
	if ((-2.0 * x) <= -1000.0) {
		tmp = (2.0 / t_0) + -1.0;
	} else if ((-2.0 * x) <= 5e-7) {
		tmp = (-0.3333333333333333 * pow(x, 3.0)) + (x + (0.13333333333333333 * pow(x, 5.0)));
	} else {
		tmp = ((4.0 * (1.0 / pow(t_0, 2.0))) + -1.0) / (1.0 + (2.0 / pow(cbrt(t_0), 3.0)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + Math.exp((-2.0 * x));
	double tmp;
	if ((-2.0 * x) <= -1000.0) {
		tmp = (2.0 / t_0) + -1.0;
	} else if ((-2.0 * x) <= 5e-7) {
		tmp = (-0.3333333333333333 * Math.pow(x, 3.0)) + (x + (0.13333333333333333 * Math.pow(x, 5.0)));
	} else {
		tmp = ((4.0 * (1.0 / Math.pow(t_0, 2.0))) + -1.0) / (1.0 + (2.0 / Math.pow(Math.cbrt(t_0), 3.0)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 + exp(Float64(-2.0 * x)))
	tmp = 0.0
	if (Float64(-2.0 * x) <= -1000.0)
		tmp = Float64(Float64(2.0 / t_0) + -1.0);
	elseif (Float64(-2.0 * x) <= 5e-7)
		tmp = Float64(Float64(-0.3333333333333333 * (x ^ 3.0)) + Float64(x + Float64(0.13333333333333333 * (x ^ 5.0))));
	else
		tmp = Float64(Float64(Float64(4.0 * Float64(1.0 / (t_0 ^ 2.0))) + -1.0) / Float64(1.0 + Float64(2.0 / (cbrt(t_0) ^ 3.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -1000.0], N[(N[(2.0 / t$95$0), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 5e-7], N[(N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x + N[(0.13333333333333333 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(4.0 * N[(1.0 / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / N[(1.0 + N[(2.0 / N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 -2 x) < -1e3

   1. Initial program 100.0%

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

   if -1e3 < (*.f64 -2 x) < 4.99999999999999977e-7

   1. Initial program 9.4%

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

   2. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot {x}^{3} + \left(0.13333333333333333 \cdot {x}^{5} + x\right)} \]

   if 4.99999999999999977e-7 < (*.f64 -2 x)

   1. Initial program 100.0%

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

      1. flip-- 100.0%

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

      2. div-inv 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. associate-*r/ 100.0%

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

      2. *-rgt-identity 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 100.0%

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

      1. add-cube-cbrt 100.0%

         \[\leadsto \frac{4 \cdot \frac{1}{{\left(1 + e^{-2 \cdot x}\right)}^{2}} - 1}{1 + \frac{2}{\color{blue}{\left(\sqrt[3]{1 + {\left(e^{-2}\right)}^{x}} \cdot \sqrt[3]{1 + {\left(e^{-2}\right)}^{x}}\right) \cdot \sqrt[3]{1 + {\left(e^{-2}\right)}^{x}}}}} \]

      2. pow3 100.0%

         \[\leadsto \frac{4 \cdot \frac{1}{{\left(1 + e^{-2 \cdot x}\right)}^{2}} - 1}{1 + \frac{2}{\color{blue}{{\left(\sqrt[3]{1 + {\left(e^{-2}\right)}^{x}}\right)}^{3}}}} \]

   8. Applied egg-rr 100.0%

      \[\leadsto \frac{4 \cdot \frac{1}{{\left(1 + e^{-2 \cdot x}\right)}^{2}} - 1}{1 + \frac{2}{\color{blue}{{\left(\sqrt[3]{1 + {\left(e^{-2}\right)}^{x}}\right)}^{3}}}} \]
   9. Step-by-step derivation

      1. *-un-lft-identity 100.0%

         \[\leadsto \frac{4 \cdot \frac{1}{{\left(1 + e^{-2 \cdot x}\right)}^{2}} - 1}{1 + \frac{2}{{\color{blue}{\left(1 \cdot \sqrt[3]{1 + {\left(e^{-2}\right)}^{x}}\right)}}^{3}}} \]

      2. *-commutative 100.0%

         \[\leadsto \frac{4 \cdot \frac{1}{{\left(1 + e^{-2 \cdot x}\right)}^{2}} - 1}{1 + \frac{2}{{\color{blue}{\left(\sqrt[3]{1 + {\left(e^{-2}\right)}^{x}} \cdot 1\right)}}^{3}}} \]

      3. pow-exp 100.0%

         \[\leadsto \frac{4 \cdot \frac{1}{{\left(1 + e^{-2 \cdot x}\right)}^{2}} - 1}{1 + \frac{2}{{\left(\sqrt[3]{1 + \color{blue}{e^{-2 \cdot x}}} \cdot 1\right)}^{3}}} \]

      4. *-commutative 100.0%

         \[\leadsto \frac{4 \cdot \frac{1}{{\left(1 + e^{-2 \cdot x}\right)}^{2}} - 1}{1 + \frac{2}{{\left(\sqrt[3]{1 + e^{\color{blue}{x \cdot -2}}} \cdot 1\right)}^{3}}} \]

   10. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -1000:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-7}:\\ \;\;\;\;-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot \frac{1}{{\left(1 + e^{-2 \cdot x}\right)}^{2}} + -1}{1 + \frac{2}{{\left(\sqrt[3]{1 + e^{-2 \cdot x}}\right)}^{3}}}\\ \end{array} \]

Alternative 2: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + e^{-2 \cdot x}\\ t_1 := \frac{2}{t_0}\\ \mathbf{if}\;-2 \cdot x \leq -1000:\\ \;\;\;\;t_1 + -1\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-7}:\\ \;\;\;\;-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{4}{{t_0}^{2}} + -1}{1 + t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (exp (* -2.0 x)))) (t_1 (/ 2.0 t_0)))
   (if (<= (* -2.0 x) -1000.0)
     (+ t_1 -1.0)
     (if (<= (* -2.0 x) 5e-7)
       (+
        (* -0.3333333333333333 (pow x 3.0))
        (+ x (* 0.13333333333333333 (pow x 5.0))))
       (/ (+ (/ 4.0 (pow t_0 2.0)) -1.0) (+ 1.0 t_1))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + exp((-2.0 * x));
	double t_1 = 2.0 / t_0;
	double tmp;
	if ((-2.0 * x) <= -1000.0) {
		tmp = t_1 + -1.0;
	} else if ((-2.0 * x) <= 5e-7) {
		tmp = (-0.3333333333333333 * pow(x, 3.0)) + (x + (0.13333333333333333 * pow(x, 5.0)));
	} else {
		tmp = ((4.0 / pow(t_0, 2.0)) + -1.0) / (1.0 + t_1);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + exp(((-2.0d0) * x))
    t_1 = 2.0d0 / t_0
    if (((-2.0d0) * x) <= (-1000.0d0)) then
        tmp = t_1 + (-1.0d0)
    else if (((-2.0d0) * x) <= 5d-7) then
        tmp = ((-0.3333333333333333d0) * (x ** 3.0d0)) + (x + (0.13333333333333333d0 * (x ** 5.0d0)))
    else
        tmp = ((4.0d0 / (t_0 ** 2.0d0)) + (-1.0d0)) / (1.0d0 + t_1)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + Math.exp((-2.0 * x));
	double t_1 = 2.0 / t_0;
	double tmp;
	if ((-2.0 * x) <= -1000.0) {
		tmp = t_1 + -1.0;
	} else if ((-2.0 * x) <= 5e-7) {
		tmp = (-0.3333333333333333 * Math.pow(x, 3.0)) + (x + (0.13333333333333333 * Math.pow(x, 5.0)));
	} else {
		tmp = ((4.0 / Math.pow(t_0, 2.0)) + -1.0) / (1.0 + t_1);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + math.exp((-2.0 * x))
	t_1 = 2.0 / t_0
	tmp = 0
	if (-2.0 * x) <= -1000.0:
		tmp = t_1 + -1.0
	elif (-2.0 * x) <= 5e-7:
		tmp = (-0.3333333333333333 * math.pow(x, 3.0)) + (x + (0.13333333333333333 * math.pow(x, 5.0)))
	else:
		tmp = ((4.0 / math.pow(t_0, 2.0)) + -1.0) / (1.0 + t_1)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + exp(Float64(-2.0 * x)))
	t_1 = Float64(2.0 / t_0)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -1000.0)
		tmp = Float64(t_1 + -1.0);
	elseif (Float64(-2.0 * x) <= 5e-7)
		tmp = Float64(Float64(-0.3333333333333333 * (x ^ 3.0)) + Float64(x + Float64(0.13333333333333333 * (x ^ 5.0))));
	else
		tmp = Float64(Float64(Float64(4.0 / (t_0 ^ 2.0)) + -1.0) / Float64(1.0 + t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + exp((-2.0 * x));
	t_1 = 2.0 / t_0;
	tmp = 0.0;
	if ((-2.0 * x) <= -1000.0)
		tmp = t_1 + -1.0;
	elseif ((-2.0 * x) <= 5e-7)
		tmp = (-0.3333333333333333 * (x ^ 3.0)) + (x + (0.13333333333333333 * (x ^ 5.0)));
	else
		tmp = ((4.0 / (t_0 ^ 2.0)) + -1.0) / (1.0 + t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / t$95$0), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -1000.0], N[(t$95$1 + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 5e-7], N[(N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x + N[(0.13333333333333333 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(4.0 / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 -2 x) < -1e3

   1. Initial program 100.0%

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

   if -1e3 < (*.f64 -2 x) < 4.99999999999999977e-7

   1. Initial program 9.4%

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

   2. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot {x}^{3} + \left(0.13333333333333333 \cdot {x}^{5} + x\right)} \]

   if 4.99999999999999977e-7 < (*.f64 -2 x)

   1. Initial program 100.0%

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

      1. flip-- 100.0%

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

      2. div-inv 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. associate-*r/ 100.0%

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

      2. *-rgt-identity 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 100.0%

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

   7. Taylor expanded in x around inf 100.0%

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

      1. sub-neg 100.0%

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

      2. un-div-inv 100.0%

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

      3. *-commutative 100.0%

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

      4. metadata-eval 100.0%

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

   9. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -1000:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-7}:\\ \;\;\;\;-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{4}{{\left(1 + e^{-2 \cdot x}\right)}^{2}} + -1}{1 + \frac{2}{1 + e^{-2 \cdot x}}}\\ \end{array} \]

Alternative 3: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -1000 \lor \neg \left(-2 \cdot x \leq 5 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* -2.0 x) -1000.0) (not (<= (* -2.0 x) 5e-7)))
   (+ (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) -1.0)
   (+
    (* -0.3333333333333333 (pow x 3.0))
    (+ x (* 0.13333333333333333 (pow x 5.0))))))

C

double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -1000.0) || !((-2.0 * x) <= 5e-7)) {
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	} else {
		tmp = (-0.3333333333333333 * pow(x, 3.0)) + (x + (0.13333333333333333 * pow(x, 5.0)));
	}
	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) <= (-1000.0d0)) .or. (.not. (((-2.0d0) * x) <= 5d-7))) then
        tmp = (2.0d0 / (1.0d0 + exp(((-2.0d0) * x)))) + (-1.0d0)
    else
        tmp = ((-0.3333333333333333d0) * (x ** 3.0d0)) + (x + (0.13333333333333333d0 * (x ** 5.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -1000.0) || !((-2.0 * x) <= 5e-7)) {
		tmp = (2.0 / (1.0 + Math.exp((-2.0 * x)))) + -1.0;
	} else {
		tmp = (-0.3333333333333333 * Math.pow(x, 3.0)) + (x + (0.13333333333333333 * Math.pow(x, 5.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((-2.0 * x) <= -1000.0) or not ((-2.0 * x) <= 5e-7):
		tmp = (2.0 / (1.0 + math.exp((-2.0 * x)))) + -1.0
	else:
		tmp = (-0.3333333333333333 * math.pow(x, 3.0)) + (x + (0.13333333333333333 * math.pow(x, 5.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((Float64(-2.0 * x) <= -1000.0) || !(Float64(-2.0 * x) <= 5e-7))
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) + -1.0);
	else
		tmp = Float64(Float64(-0.3333333333333333 * (x ^ 3.0)) + Float64(x + Float64(0.13333333333333333 * (x ^ 5.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((-2.0 * x) <= -1000.0) || ~(((-2.0 * x) <= 5e-7)))
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	else
		tmp = (-0.3333333333333333 * (x ^ 3.0)) + (x + (0.13333333333333333 * (x ^ 5.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[(-2.0 * x), $MachinePrecision], -1000.0], N[Not[LessEqual[N[(-2.0 * x), $MachinePrecision], 5e-7]], $MachinePrecision]], N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x + N[(0.13333333333333333 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 -2 x) < -1e3 or 4.99999999999999977e-7 < (*.f64 -2 x)

   1. Initial program 100.0%

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

   if -1e3 < (*.f64 -2 x) < 4.99999999999999977e-7

   1. Initial program 9.4%

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

   2. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot {x}^{3} + \left(0.13333333333333333 \cdot {x}^{5} + x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -1000 \lor \neg \left(-2 \cdot x \leq 5 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\\ \end{array} \]

Alternative 4: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.002 \lor \neg \left(-2 \cdot x \leq 5 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{else}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* -2.0 x) -0.002) (not (<= (* -2.0 x) 5e-7)))
   (+ (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) -1.0)
   (+ x (* -0.3333333333333333 (pow x 3.0)))))

C

double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -0.002) || !((-2.0 * x) <= 5e-7)) {
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	} else {
		tmp = x + (-0.3333333333333333 * pow(x, 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((((-2.0d0) * x) <= (-0.002d0)) .or. (.not. (((-2.0d0) * x) <= 5d-7))) then
        tmp = (2.0d0 / (1.0d0 + exp(((-2.0d0) * x)))) + (-1.0d0)
    else
        tmp = x + ((-0.3333333333333333d0) * (x ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -0.002) || !((-2.0 * x) <= 5e-7)) {
		tmp = (2.0 / (1.0 + Math.exp((-2.0 * x)))) + -1.0;
	} else {
		tmp = x + (-0.3333333333333333 * Math.pow(x, 3.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((-2.0 * x) <= -0.002) or not ((-2.0 * x) <= 5e-7):
		tmp = (2.0 / (1.0 + math.exp((-2.0 * x)))) + -1.0
	else:
		tmp = x + (-0.3333333333333333 * math.pow(x, 3.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((Float64(-2.0 * x) <= -0.002) || !(Float64(-2.0 * x) <= 5e-7))
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) + -1.0);
	else
		tmp = Float64(x + Float64(-0.3333333333333333 * (x ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((-2.0 * x) <= -0.002) || ~(((-2.0 * x) <= 5e-7)))
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	else
		tmp = x + (-0.3333333333333333 * (x ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.002], N[Not[LessEqual[N[(-2.0 * x), $MachinePrecision], 5e-7]], $MachinePrecision]], N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(x + N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 -2 x) < -2e-3 or 4.99999999999999977e-7 < (*.f64 -2 x)

   1. Initial program 99.9%

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

   if -2e-3 < (*.f64 -2 x) < 4.99999999999999977e-7

   1. Initial program 8.7%

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

   2. Taylor expanded in x around 0 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.002 \lor \neg \left(-2 \cdot x \leq 5 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{else}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \end{array} \]

Alternative 5: 79.4% accurate, 11.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = -1.0d0
    else if (x <= 2.55d0) then
        tmp = x
    else
        tmp = 2.0d0 - (4.0d0 / x)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 96.8%

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

      1. *-commutative 96.8%

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

   4. Simplified 96.8%

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

   5. Taylor expanded in x around inf 99.1%

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

   if -1 < x < 2.5499999999999998

   1. Initial program 10.8%

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

   2. Taylor expanded in x around 0 98.4%

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

   if 2.5499999999999998 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 5.8%

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

      1. +-commutative 5.8%

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

   4. Simplified 5.8%

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

      1. flip-- 5.5%

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

      2. pow2 5.5%

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

      3. +-commutative 5.5%

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

      4. metadata-eval 5.5%

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

      5. +-commutative 5.5%

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

   6. Applied egg-rr 5.5%

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

      1. unpow2 5.5%

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

      2. difference-of-sqr-1 5.5%

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

      3. +-commutative 5.5%

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

      4. associate-+r+ 5.5%

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

      5. metadata-eval 5.5%

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

      6. *-rgt-identity 5.5%

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

      7. fma-neg 5.5%

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

      8. metadata-eval 5.5%

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

      9. fma-def 5.5%

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

      10. *-rgt-identity 5.5%

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

      11. +-commutative 5.5%

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

      12. associate-+l+ 5.5%

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

      13. metadata-eval 5.5%

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

      14. +-rgt-identity 5.5%

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

      15. +-commutative 5.5%

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

      16. associate-+r+ 5.5%

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

      17. metadata-eval 5.5%

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

   8. Simplified 5.5%

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

   9. Taylor expanded in x around 0 18.8%

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

       1. *-commutative 18.8%

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

   11. Simplified 18.8%

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

   12. Taylor expanded in x around inf 18.8%

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

       1. associate-*r/ 18.8%

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

       2. metadata-eval 18.8%

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

   14. Simplified 18.8%

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

4. Final simplification 80.2%

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

Alternative 6: 78.8% accurate, 12.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -0.680000000000000049

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 95.8%

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

      1. *-commutative 95.8%

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

   4. Simplified 95.8%

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

   5. Taylor expanded in x around inf 98.1%

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

   if -0.680000000000000049 < x

   1. Initial program 39.2%

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

   2. Taylor expanded in x around 0 8.2%

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

      1. +-commutative 8.2%

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

   4. Simplified 8.2%

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

      1. flip-- 8.1%

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

      2. pow2 8.1%

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

      3. +-commutative 8.1%

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

      4. metadata-eval 8.1%

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

      5. +-commutative 8.1%

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

   6. Applied egg-rr 8.1%

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

      1. unpow2 8.1%

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

      2. difference-of-sqr-1 8.1%

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

      3. +-commutative 8.1%

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

      4. associate-+r+ 8.1%

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

      5. metadata-eval 8.1%

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

      6. *-rgt-identity 8.1%

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

      7. fma-neg 8.1%

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

      8. metadata-eval 8.1%

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

      9. fma-def 8.1%

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

      10. *-rgt-identity 8.1%

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

      11. +-commutative 8.1%

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

      12. associate-+l+ 68.7%

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

      13. metadata-eval 68.7%

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

      14. +-rgt-identity 68.7%

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

      15. +-commutative 68.7%

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

      16. associate-+r+ 68.7%

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

      17. metadata-eval 68.7%

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

   8. Simplified 68.7%

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

   9. Taylor expanded in x around 0 71.7%

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

       1. *-commutative 71.7%

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

   11. Simplified 71.7%

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

       1. expm1-log1p-u 71.7%

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

       2. expm1-udef 12.1%

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

       3. *-commutative 12.1%

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

       4. *-un-lft-identity 12.1%

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

       5. times-frac 12.1%

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

       6. metadata-eval 12.1%

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

       7. +-commutative 12.1%

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

   13. Applied egg-rr 12.1%

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

       1. expm1-def 71.7%

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

       2. expm1-log1p 71.7%

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

       3. *-commutative 71.7%

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

       4. associate-*l/ 71.7%

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

       5. associate-*r/ 71.7%

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

   15. Simplified 71.7%

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

4. Final simplification 79.3%

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

Alternative 7: 78.8% accurate, 12.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -0.680000000000000049

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 95.8%

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

      1. *-commutative 95.8%

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

   4. Simplified 95.8%

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

   5. Taylor expanded in x around inf 98.1%

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

   if -0.680000000000000049 < x

   1. Initial program 39.2%

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

   2. Taylor expanded in x around 0 8.2%

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

      1. +-commutative 8.2%

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

   4. Simplified 8.2%

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

      1. flip-- 8.1%

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

      2. pow2 8.1%

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

      3. +-commutative 8.1%

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

      4. metadata-eval 8.1%

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

      5. +-commutative 8.1%

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

   6. Applied egg-rr 8.1%

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

      1. unpow2 8.1%

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

      2. difference-of-sqr-1 8.1%

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

      3. +-commutative 8.1%

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

      4. associate-+r+ 8.1%

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

      5. metadata-eval 8.1%

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

      6. *-rgt-identity 8.1%

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

      7. fma-neg 8.1%

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

      8. metadata-eval 8.1%

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

      9. fma-def 8.1%

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

      10. *-rgt-identity 8.1%

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

      11. +-commutative 8.1%

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

      12. associate-+l+ 68.7%

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

      13. metadata-eval 68.7%

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

      14. +-rgt-identity 68.7%

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

      15. +-commutative 68.7%

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

      16. associate-+r+ 68.7%

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

      17. metadata-eval 68.7%

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

   8. Simplified 68.7%

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

   9. Taylor expanded in x around 0 71.7%

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

       1. *-commutative 71.7%

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

   11. Simplified 71.7%

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

4. Final simplification 79.3%

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

Alternative 8: 79.4% accurate, 21.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 96.8%

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

      1. *-commutative 96.8%

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

   4. Simplified 96.8%

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

   5. Taylor expanded in x around inf 99.1%

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

   if -1 < x < 2

   1. Initial program 10.8%

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

   2. Taylor expanded in x around 0 98.4%

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

   if 2 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 5.8%

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

      1. +-commutative 5.8%

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

   4. Simplified 5.8%

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

      1. flip-- 5.5%

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

      2. pow2 5.5%

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

      3. +-commutative 5.5%

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

      4. metadata-eval 5.5%

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

      5. +-commutative 5.5%

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

   6. Applied egg-rr 5.5%

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

      1. unpow2 5.5%

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

      2. difference-of-sqr-1 5.5%

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

      3. +-commutative 5.5%

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

      4. associate-+r+ 5.5%

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

      5. metadata-eval 5.5%

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

      6. *-rgt-identity 5.5%

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

      7. fma-neg 5.5%

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

      8. metadata-eval 5.5%

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

      9. fma-def 5.5%

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

      10. *-rgt-identity 5.5%

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

      11. +-commutative 5.5%

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

      12. associate-+l+ 5.5%

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

      13. metadata-eval 5.5%

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

      14. +-rgt-identity 5.5%

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

      15. +-commutative 5.5%

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

      16. associate-+r+ 5.5%

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

      17. metadata-eval 5.5%

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

   8. Simplified 5.5%

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

   9. Taylor expanded in x around 0 18.8%

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

       1. *-commutative 18.8%

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

   11. Simplified 18.8%

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

   12. Taylor expanded in x around inf 18.8%

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

4. Final simplification 80.2%

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

Alternative 9: 32.2% accurate, 35.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.2 \cdot 10^{-308}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 1.2e-308) -1.0 2.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.2e-308) {
		tmp = -1.0;
	} else {
		tmp = 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 1.2d-308) then
        tmp = -1.0d0
    else
        tmp = 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.2e-308) {
		tmp = -1.0;
	} else {
		tmp = 2.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.2e-308:
		tmp = -1.0
	else:
		tmp = 2.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.2e-308)
		tmp = -1.0;
	else
		tmp = 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.2e-308)
		tmp = -1.0;
	else
		tmp = 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.2e-308], -1.0, 2.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.1999999999999998e-308

   1. Initial program 55.3%

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

   2. Taylor expanded in x around 0 52.6%

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

      1. *-commutative 52.6%

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

   4. Simplified 52.6%

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

   5. Taylor expanded in x around inf 52.0%

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

   if 1.1999999999999998e-308 < x

   1. Initial program 58.8%

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

   2. Taylor expanded in x around 0 7.4%

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

      1. +-commutative 7.4%

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

   4. Simplified 7.4%

      \[\leadsto \color{blue}{\left(x + 1\right)} - 1 \]
   5. 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%

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

   6. Applied egg-rr 7.3%

      \[\leadsto \color{blue}{\frac{{\left(1 + x\right)}^{2} - 1}{\left(1 + x\right) + 1}} \]
   7. 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%

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

      4. associate-+r+ 7.3%

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

      5. metadata-eval 7.3%

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

      6. *-rgt-identity 7.3%

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

      7. fma-neg 7.3%

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

      8. metadata-eval 7.3%

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

      9. fma-def 7.3%

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

      10. *-rgt-identity 7.3%

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

      11. +-commutative 7.3%

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

      12. associate-+l+ 48.4%

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

      13. metadata-eval 48.4%

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

      14. +-rgt-identity 48.4%

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

      15. +-commutative 48.4%

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

      16. associate-+r+ 48.4%

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

      17. metadata-eval 48.4%

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

   8. Simplified 48.4%

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

   9. Taylor expanded in x around 0 54.7%

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

       1. *-commutative 54.7%

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

   11. Simplified 54.7%

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

   12. Taylor expanded in x around inf 12.6%

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

4. Final simplification 35.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.2 \cdot 10^{-308}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]

Alternative 10: 27.1% accurate, 109.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 -1.0)

C

double code(double x, double y) {
	return -1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = -1.0d0
end function

Java

public static double code(double x, double y) {
	return -1.0;
}

Python

def code(x, y):
	return -1.0

Julia

function code(x, y)
	return -1.0
end

MATLAB

function tmp = code(x, y)
	tmp = -1.0;
end

Wolfram

code[x_, y_] := -1.0

Derivation

1. Initial program 56.8%

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

2. Taylor expanded in x around 0 32.3%

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

   1. *-commutative 32.3%

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

4. Simplified 32.3%

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

5. Taylor expanded in x around inf 30.7%

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

6. Final simplification 30.7%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023174 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))