Logistic function from Lakshay Garg

Percentage Accurate: 53.9% → 99.7%

Time: 8.7s

Alternatives: 6

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: 53.9% 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: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{if}\;-2 \cdot x \leq -10:\\ \;\;\;\;\frac{1}{\frac{1}{t\_0}}\\ \mathbf{elif}\;-2 \cdot x \leq 4 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \left(1 + -0.3333333333333333 \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) -1.0)))
   (if (<= (* -2.0 x) -10.0)
     (/ 1.0 (/ 1.0 t_0))
     (if (<= (* -2.0 x) 4e-8)
       (* x (+ 1.0 (* -0.3333333333333333 (* x x))))
       t_0))))

C

double code(double x, double y) {
	double t_0 = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	double tmp;
	if ((-2.0 * x) <= -10.0) {
		tmp = 1.0 / (1.0 / t_0);
	} else if ((-2.0 * x) <= 4e-8) {
		tmp = x * (1.0 + (-0.3333333333333333 * (x * x)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (2.0d0 / (1.0d0 + exp(((-2.0d0) * x)))) + (-1.0d0)
    if (((-2.0d0) * x) <= (-10.0d0)) then
        tmp = 1.0d0 / (1.0d0 / t_0)
    else if (((-2.0d0) * x) <= 4d-8) then
        tmp = x * (1.0d0 + ((-0.3333333333333333d0) * (x * x)))
    else
        tmp = t_0
    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)))) + -1.0;
	double tmp;
	if ((-2.0 * x) <= -10.0) {
		tmp = 1.0 / (1.0 / t_0);
	} else if ((-2.0 * x) <= 4e-8) {
		tmp = x * (1.0 + (-0.3333333333333333 * (x * x)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (2.0 / (1.0 + math.exp((-2.0 * x)))) + -1.0
	tmp = 0
	if (-2.0 * x) <= -10.0:
		tmp = 1.0 / (1.0 / t_0)
	elif (-2.0 * x) <= 4e-8:
		tmp = x * (1.0 + (-0.3333333333333333 * (x * x)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) + -1.0)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -10.0)
		tmp = Float64(1.0 / Float64(1.0 / t_0));
	elseif (Float64(-2.0 * x) <= 4e-8)
		tmp = Float64(x * Float64(1.0 + Float64(-0.3333333333333333 * Float64(x * x))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	tmp = 0.0;
	if ((-2.0 * x) <= -10.0)
		tmp = 1.0 / (1.0 / t_0);
	elseif ((-2.0 * x) <= 4e-8)
		tmp = x * (1.0 + (-0.3333333333333333 * (x * x)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -10.0], N[(1.0 / N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 4e-8], N[(x * N[(1.0 + N[(-0.3333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -10

   1. Initial program 100.0%

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

      1. flip-- N/A

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

      2. clear-num N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\frac{2}{1 + e^{-2 \cdot x}} + 1}{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1 \cdot 1}\right)}\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}}}\right)\right) \]

      5. flip-- N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{\frac{2}{1 + e^{-2 \cdot x}} - \color{blue}{1}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)}\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \left(\frac{2}{1 + e^{-2 \cdot x}} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(\frac{2}{1 + e^{-2 \cdot x}}\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(1 + e^{-2 \cdot x}\right)\right), \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(1, \left(e^{-2 \cdot x}\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right) \]

      11. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(-2 \cdot x\right)\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-2, x\right)\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right) \]

      13. metadata-eval 100.0%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-2, x\right)\right)\right)\right), -1\right)\right)\right) \]

   4. Applied egg-rr 100.0%

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

   if -10 < (*.f64 #s(literal -2 binary64) x) < 4.0000000000000001e-8

   1. Initial program 7.7%

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

   3. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + \frac{-1}{3} \cdot {x}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{3} \cdot {x}^{2}\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{3}, \color{blue}{\left({x}^{2}\right)}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{3}, \left(x \cdot \color{blue}{x}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right) \]

   5. Simplified 100.0%

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

   if 4.0000000000000001e-8 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -10:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{2}{1 + e^{-2 \cdot x}} + -1}}\\ \mathbf{elif}\;-2 \cdot x \leq 4 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \left(1 + -0.3333333333333333 \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{if}\;-2 \cdot x \leq -10:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;-2 \cdot x \leq 4 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \left(1 + -0.3333333333333333 \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) -1.0)))
   (if (<= (* -2.0 x) -10.0)
     t_0
     (if (<= (* -2.0 x) 4e-8)
       (* x (+ 1.0 (* -0.3333333333333333 (* x x))))
       t_0))))

C

double code(double x, double y) {
	double t_0 = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	double tmp;
	if ((-2.0 * x) <= -10.0) {
		tmp = t_0;
	} else if ((-2.0 * x) <= 4e-8) {
		tmp = x * (1.0 + (-0.3333333333333333 * (x * x)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (2.0d0 / (1.0d0 + exp(((-2.0d0) * x)))) + (-1.0d0)
    if (((-2.0d0) * x) <= (-10.0d0)) then
        tmp = t_0
    else if (((-2.0d0) * x) <= 4d-8) then
        tmp = x * (1.0d0 + ((-0.3333333333333333d0) * (x * x)))
    else
        tmp = t_0
    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)))) + -1.0;
	double tmp;
	if ((-2.0 * x) <= -10.0) {
		tmp = t_0;
	} else if ((-2.0 * x) <= 4e-8) {
		tmp = x * (1.0 + (-0.3333333333333333 * (x * x)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (2.0 / (1.0 + math.exp((-2.0 * x)))) + -1.0
	tmp = 0
	if (-2.0 * x) <= -10.0:
		tmp = t_0
	elif (-2.0 * x) <= 4e-8:
		tmp = x * (1.0 + (-0.3333333333333333 * (x * x)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) + -1.0)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -10.0)
		tmp = t_0;
	elseif (Float64(-2.0 * x) <= 4e-8)
		tmp = Float64(x * Float64(1.0 + Float64(-0.3333333333333333 * Float64(x * x))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	tmp = 0.0;
	if ((-2.0 * x) <= -10.0)
		tmp = t_0;
	elseif ((-2.0 * x) <= 4e-8)
		tmp = x * (1.0 + (-0.3333333333333333 * (x * x)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -10.0], t$95$0, If[LessEqual[N[(-2.0 * x), $MachinePrecision], 4e-8], N[(x * N[(1.0 + N[(-0.3333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -10 or 4.0000000000000001e-8 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 100.0%

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

   if -10 < (*.f64 #s(literal -2 binary64) x) < 4.0000000000000001e-8

   1. Initial program 7.7%

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

   3. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + \frac{-1}{3} \cdot {x}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{3} \cdot {x}^{2}\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{3}, \color{blue}{\left({x}^{2}\right)}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{3}, \left(x \cdot \color{blue}{x}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right) \]

   5. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -10:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 4 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \left(1 + -0.3333333333333333 \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.2% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.66:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + \left(x \cdot x\right) \cdot 0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.66) -1.0 (/ x (+ 1.0 (* (* x x) 0.3333333333333333)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.66) {
		tmp = -1.0;
	} else {
		tmp = x / (1.0 + ((x * x) * 0.3333333333333333));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.66d0)) then
        tmp = -1.0d0
    else
        tmp = x / (1.0d0 + ((x * x) * 0.3333333333333333d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.66) {
		tmp = -1.0;
	} else {
		tmp = x / (1.0 + ((x * x) * 0.3333333333333333));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.66:
		tmp = -1.0
	else:
		tmp = x / (1.0 + ((x * x) * 0.3333333333333333))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.66)
		tmp = -1.0;
	else
		tmp = Float64(x / Float64(1.0 + Float64(Float64(x * x) * 0.3333333333333333)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.66)
		tmp = -1.0;
	else
		tmp = x / (1.0 + ((x * x) * 0.3333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.66], -1.0, N[(x / N[(1.0 + N[(N[(x * x), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.65999999999999992

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \color{blue}{\left(2 + -2 \cdot x\right)}\right), 1\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \left(-2 \cdot x\right)\right)\right), 1\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \left(x \cdot -2\right)\right)\right), 1\right) \]

      3. *-lowering-*.f64 99.3%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), 1\right) \]

   5. Simplified 99.3%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-1} \]
   7. Step-by-step derivation

   8. Simplified 100.0%

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

   if -1.65999999999999992 < x

   1. Initial program 39.0%

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

      1. flip-- N/A

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

      2. clear-num N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\frac{2}{1 + e^{-2 \cdot x}} + 1}{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1 \cdot 1}\right)}\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}}}\right)\right) \]

      5. flip-- N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{\frac{2}{1 + e^{-2 \cdot x}} - \color{blue}{1}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)}\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \left(\frac{2}{1 + e^{-2 \cdot x}} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(\frac{2}{1 + e^{-2 \cdot x}}\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(1 + e^{-2 \cdot x}\right)\right), \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(1, \left(e^{-2 \cdot x}\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right) \]

      11. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(-2 \cdot x\right)\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-2, x\right)\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right) \]

      13. metadata-eval 39.0%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-2, x\right)\right)\right)\right), -1\right)\right)\right) \]

   4. Applied egg-rr 39.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1 + \frac{1}{3} \cdot {x}^{2}}{x}\right)}\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(1 + \frac{1}{3} \cdot {x}^{2}\right), \color{blue}{x}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{3} \cdot {x}^{2}\right)\right), x\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \frac{1}{3}\right)\right), x\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \frac{1}{3}\right)\right), x\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \frac{1}{3}\right)\right)\right), x\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{3}\right)\right)\right), x\right)\right) \]

      7. *-lowering-*.f64 67.9%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{3}\right)\right)\right), x\right)\right) \]

   7. Simplified 67.9%

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

      1. clear-num N/A

         \[\leadsto \frac{x}{\color{blue}{1 + x \cdot \left(x \cdot \frac{1}{3}\right)}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(1 + x \cdot \left(x \cdot \frac{1}{3}\right)\right)}\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(x \cdot \frac{1}{3}\right)\right)}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \color{blue}{\frac{1}{3}}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{1}{3} \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left(x \cdot x\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 68.1%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right) \]

   9. Applied egg-rr 68.1%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.66:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + \left(x \cdot x\right) \cdot 0.3333333333333333}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.1% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.66:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot 0.3333333333333333 + \frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.66) -1.0 (/ 1.0 (+ (* x 0.3333333333333333) (/ 1.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.66) {
		tmp = -1.0;
	} else {
		tmp = 1.0 / ((x * 0.3333333333333333) + (1.0 / 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 <= (-1.66d0)) then
        tmp = -1.0d0
    else
        tmp = 1.0d0 / ((x * 0.3333333333333333d0) + (1.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.66) {
		tmp = -1.0;
	} else {
		tmp = 1.0 / ((x * 0.3333333333333333) + (1.0 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.66:
		tmp = -1.0
	else:
		tmp = 1.0 / ((x * 0.3333333333333333) + (1.0 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.66)
		tmp = -1.0;
	else
		tmp = Float64(1.0 / Float64(Float64(x * 0.3333333333333333) + Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.66)
		tmp = -1.0;
	else
		tmp = 1.0 / ((x * 0.3333333333333333) + (1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.66], -1.0, N[(1.0 / N[(N[(x * 0.3333333333333333), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.65999999999999992

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \color{blue}{\left(2 + -2 \cdot x\right)}\right), 1\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \left(-2 \cdot x\right)\right)\right), 1\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \left(x \cdot -2\right)\right)\right), 1\right) \]

      3. *-lowering-*.f64 99.3%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), 1\right) \]

   5. Simplified 99.3%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-1} \]
   7. Step-by-step derivation

   8. Simplified 100.0%

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

   if -1.65999999999999992 < x

   1. Initial program 39.0%

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

      1. flip-- N/A

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

      2. clear-num N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\frac{2}{1 + e^{-2 \cdot x}} + 1}{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1 \cdot 1}\right)}\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}}}\right)\right) \]

      5. flip-- N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{\frac{2}{1 + e^{-2 \cdot x}} - \color{blue}{1}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)}\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \left(\frac{2}{1 + e^{-2 \cdot x}} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(\frac{2}{1 + e^{-2 \cdot x}}\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(1 + e^{-2 \cdot x}\right)\right), \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(1, \left(e^{-2 \cdot x}\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right) \]

      11. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(-2 \cdot x\right)\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-2, x\right)\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right) \]

      13. metadata-eval 39.0%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-2, x\right)\right)\right)\right), -1\right)\right)\right) \]

   4. Applied egg-rr 39.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1 + \frac{1}{3} \cdot {x}^{2}}{x}\right)}\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(1 + \frac{1}{3} \cdot {x}^{2}\right), \color{blue}{x}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{3} \cdot {x}^{2}\right)\right), x\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \frac{1}{3}\right)\right), x\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \frac{1}{3}\right)\right), x\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \frac{1}{3}\right)\right)\right), x\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{3}\right)\right)\right), x\right)\right) \]

      7. *-lowering-*.f64 67.9%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{3}\right)\right)\right), x\right)\right) \]

   7. Simplified 67.9%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1 + \frac{1}{3} \cdot {x}^{2}}{x}\right)}\right) \]
   9. Step-by-step derivation

      1. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{1}{{x}^{2}} \cdot {x}^{2} + \frac{1}{3} \cdot {x}^{2}}{x}\right)\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{{x}^{2} \cdot \left(\frac{1}{{x}^{2}} + \frac{1}{3}\right)}{x}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\left(x \cdot x\right) \cdot \left(\frac{1}{{x}^{2}} + \frac{1}{3}\right)}{x}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\left(x \cdot x\right) \cdot \left(\frac{1}{3} + \frac{1}{{x}^{2}}\right)}{x}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{x \cdot \left(x \cdot \left(\frac{1}{3} + \frac{1}{{x}^{2}}\right)\right)}{x}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\left(x \cdot \left(\frac{1}{3} + \frac{1}{{x}^{2}}\right)\right) \cdot x}{x}\right)\right) \]

      7. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(x \cdot \left(\frac{1}{3} + \frac{1}{{x}^{2}}\right)\right) \cdot \color{blue}{\frac{x}{x}}\right)\right) \]

      8. *-inverses N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(x \cdot \left(\frac{1}{3} + \frac{1}{{x}^{2}}\right)\right) \cdot 1\right)\right) \]

      9. *-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot \color{blue}{\left(\frac{1}{3} + \frac{1}{{x}^{2}}\right)}\right)\right) \]

      10. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{3} \cdot x + \color{blue}{\frac{1}{{x}^{2}} \cdot x}\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(\frac{1}{3} \cdot x\right), \color{blue}{\left(\frac{1}{{x}^{2}} \cdot x\right)}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(x \cdot \frac{1}{3}\right), \left(\color{blue}{\frac{1}{{x}^{2}}} \cdot x\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{3}\right), \left(\color{blue}{\frac{1}{{x}^{2}}} \cdot x\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{3}\right), \left(\frac{1}{x \cdot x} \cdot x\right)\right)\right) \]

      15. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{3}\right), \left(\frac{\frac{1}{x}}{x} \cdot x\right)\right)\right) \]

      16. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{3}\right), \left(\frac{\frac{1}{x} \cdot x}{\color{blue}{x}}\right)\right)\right) \]

      17. lft-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{3}\right), \left(\frac{1}{x}\right)\right)\right) \]

      18. /-lowering-/.f64 68.0%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{3}\right), \mathsf{/.f64}\left(1, \color{blue}{x}\right)\right)\right) \]

   10. Simplified 68.0%

       \[\leadsto \frac{1}{\color{blue}{x \cdot 0.3333333333333333 + \frac{1}{x}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 75.9% accurate, 18.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \color{blue}{\left(2 + -2 \cdot x\right)}\right), 1\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \left(-2 \cdot x\right)\right)\right), 1\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \left(x \cdot -2\right)\right)\right), 1\right) \]

      3. *-lowering-*.f64 99.3%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), 1\right) \]

   5. Simplified 99.3%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-1} \]
   7. Step-by-step derivation

   8. Simplified 100.0%

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

   if -1 < x

   1. Initial program 39.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 67.8%

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

Alternative 6: 27.2% 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 57.1%

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

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \color{blue}{\left(2 + -2 \cdot x\right)}\right), 1\right) \]
4. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \left(-2 \cdot x\right)\right)\right), 1\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \left(x \cdot -2\right)\right)\right), 1\right) \]

   3. *-lowering-*.f64 33.2%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), 1\right) \]

5. Simplified 33.2%

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

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{-1} \]
7. Step-by-step derivation

8. Simplified 31.8%

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

Reproduce

herbie shell --seed 2024161 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))