Logistic function from Lakshay Garg

Percentage Accurate: 54.3% → 99.2%

Time: 3.6s

Alternatives: 4

Speedup: 9.9×

Specification

Math

\[\begin{array}{l} \\ \frac{2}{1 + e^{-2 \cdot x}} - 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))

C

double code(double x, double y) {
	return (2.0 / (1.0 + exp((-2.0 * x)))) - 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (2.0d0 / (1.0d0 + exp(((-2.0d0) * x)))) - 1.0d0
end function

Java

public static double code(double x, double y) {
	return (2.0 / (1.0 + Math.exp((-2.0 * x)))) - 1.0;
}

Python

def code(x, y):
	return (2.0 / (1.0 + math.exp((-2.0 * x)))) - 1.0

Julia

function code(x, y)
	return Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) - 1.0)
end

MATLAB

function tmp = code(x, y)
	tmp = (2.0 / (1.0 + exp((-2.0 * x)))) - 1.0;
end

Wolfram

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

Initial Program: 54.3% 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: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -500:\\ \;\;\;\;1\\ \mathbf{elif}\;-2 \cdot x \leq 2 \cdot 10^{-5}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -500.0)
   1.0
   (if (<= (* -2.0 x) 2e-5) (+ x (* -0.3333333333333333 (pow x 3.0))) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -500.0) {
		tmp = 1.0;
	} else if ((-2.0 * x) <= 2e-5) {
		tmp = x + (-0.3333333333333333 * pow(x, 3.0));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((-2.0d0) * x) <= (-500.0d0)) then
        tmp = 1.0d0
    else if (((-2.0d0) * x) <= 2d-5) then
        tmp = x + ((-0.3333333333333333d0) * (x ** 3.0d0))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -500.0) {
		tmp = 1.0;
	} else if ((-2.0 * x) <= 2e-5) {
		tmp = x + (-0.3333333333333333 * Math.pow(x, 3.0));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (-2.0 * x) <= -500.0:
		tmp = 1.0
	elif (-2.0 * x) <= 2e-5:
		tmp = x + (-0.3333333333333333 * math.pow(x, 3.0))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -500.0)
		tmp = 1.0;
	elseif (Float64(-2.0 * x) <= 2e-5)
		tmp = Float64(x + Float64(-0.3333333333333333 * (x ^ 3.0)));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((-2.0 * x) <= -500.0)
		tmp = 1.0;
	elseif ((-2.0 * x) <= 2e-5)
		tmp = x + (-0.3333333333333333 * (x ^ 3.0));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -500.0], 1.0, If[LessEqual[N[(-2.0 * x), $MachinePrecision], 2e-5], N[(x + N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 -2 x) < -500

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 5.7%

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

      1. +-commutative 5.7%

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

   5. Simplified 5.7%

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

      1. associate--l+ 5.7%

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

      2. metadata-eval 5.7%

         \[\leadsto x + \color{blue}{0} \]

      3. flip-+ 5.3%

         \[\leadsto \color{blue}{\frac{x \cdot x - 0 \cdot 0}{x - 0}} \]

      4. metadata-eval 5.3%

         \[\leadsto \frac{x \cdot x - \color{blue}{0}}{x - 0} \]

      5. fma-neg 5.3%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x, -0\right)}}{x - 0} \]

      6. metadata-eval 5.3%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, \color{blue}{0}\right)}{x - 0} \]

      7. *-rgt-identity 5.3%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, 0\right)}{\color{blue}{x \cdot 1} - 0} \]

      8. fma-neg 5.3%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, 0\right)}{\color{blue}{\mathsf{fma}\left(x, 1, -0\right)}} \]

      9. metadata-eval 5.3%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, 0\right)}{\mathsf{fma}\left(x, 1, \color{blue}{0}\right)} \]

      10. fma-def 5.3%

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

      11. *-rgt-identity 5.3%

          \[\leadsto \frac{\mathsf{fma}\left(x, x, 0\right)}{\color{blue}{x} + 0} \]

      12. +-rgt-identity 5.3%

          \[\leadsto \frac{\mathsf{fma}\left(x, x, 0\right)}{\color{blue}{x}} \]

   7. Applied egg-rr 5.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x, 0\right)}{x}} \]
   8. Step-by-step derivation

      1. fma-udef 5.3%

         \[\leadsto \frac{\color{blue}{x \cdot x + 0}}{x} \]

      2. unpow2 5.3%

         \[\leadsto \frac{\color{blue}{{x}^{2}} + 0}{x} \]

      3. +-rgt-identity 5.3%

         \[\leadsto \frac{\color{blue}{{x}^{2}}}{x} \]

   9. Simplified 5.3%

      \[\leadsto \color{blue}{\frac{{x}^{2}}{x}} \]

   11. Applied egg-rr 100.0%

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

   if -500 < (*.f64 -2 x) < 2.00000000000000016e-5

   1. Initial program 7.9%

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

   3. Taylor expanded in x around 0 100.0%

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

   if 2.00000000000000016e-5 < (*.f64 -2 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 4.9%

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

      1. +-commutative 4.9%

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

   5. Simplified 4.9%

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

      1. associate--l+ 4.9%

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

      2. metadata-eval 4.9%

         \[\leadsto x + \color{blue}{0} \]

      3. flip-+ 4.5%

         \[\leadsto \color{blue}{\frac{x \cdot x - 0 \cdot 0}{x - 0}} \]

      4. metadata-eval 4.5%

         \[\leadsto \frac{x \cdot x - \color{blue}{0}}{x - 0} \]

      5. fma-neg 4.5%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x, -0\right)}}{x - 0} \]

      6. metadata-eval 4.5%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, \color{blue}{0}\right)}{x - 0} \]

      7. *-rgt-identity 4.5%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, 0\right)}{\color{blue}{x \cdot 1} - 0} \]

      8. fma-neg 4.5%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, 0\right)}{\color{blue}{\mathsf{fma}\left(x, 1, -0\right)}} \]

      9. metadata-eval 4.5%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, 0\right)}{\mathsf{fma}\left(x, 1, \color{blue}{0}\right)} \]

      10. fma-def 4.5%

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

      11. *-rgt-identity 4.5%

          \[\leadsto \frac{\mathsf{fma}\left(x, x, 0\right)}{\color{blue}{x} + 0} \]

      12. +-rgt-identity 4.5%

          \[\leadsto \frac{\mathsf{fma}\left(x, x, 0\right)}{\color{blue}{x}} \]

   7. Applied egg-rr 4.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x, 0\right)}{x}} \]
   8. Step-by-step derivation

      1. fma-udef 4.5%

         \[\leadsto \frac{\color{blue}{x \cdot x + 0}}{x} \]

      2. unpow2 4.5%

         \[\leadsto \frac{\color{blue}{{x}^{2}} + 0}{x} \]

      3. +-rgt-identity 4.5%

         \[\leadsto \frac{\color{blue}{{x}^{2}}}{x} \]

   9. Simplified 4.5%

      \[\leadsto \color{blue}{\frac{{x}^{2}}{x}} \]

   11. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{-1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -500:\\ \;\;\;\;1\\ \mathbf{elif}\;-2 \cdot x \leq 2 \cdot 10^{-5}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 9.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 4.9%

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

      1. +-commutative 4.9%

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

   5. Simplified 4.9%

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

      1. associate--l+ 4.9%

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

      2. metadata-eval 4.9%

         \[\leadsto x + \color{blue}{0} \]

      3. flip-+ 4.5%

         \[\leadsto \color{blue}{\frac{x \cdot x - 0 \cdot 0}{x - 0}} \]

      4. metadata-eval 4.5%

         \[\leadsto \frac{x \cdot x - \color{blue}{0}}{x - 0} \]

      5. fma-neg 4.5%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x, -0\right)}}{x - 0} \]

      6. metadata-eval 4.5%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, \color{blue}{0}\right)}{x - 0} \]

      7. *-rgt-identity 4.5%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, 0\right)}{\color{blue}{x \cdot 1} - 0} \]

      8. fma-neg 4.5%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, 0\right)}{\color{blue}{\mathsf{fma}\left(x, 1, -0\right)}} \]

      9. metadata-eval 4.5%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, 0\right)}{\mathsf{fma}\left(x, 1, \color{blue}{0}\right)} \]

      10. fma-def 4.5%

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

      11. *-rgt-identity 4.5%

          \[\leadsto \frac{\mathsf{fma}\left(x, x, 0\right)}{\color{blue}{x} + 0} \]

      12. +-rgt-identity 4.5%

          \[\leadsto \frac{\mathsf{fma}\left(x, x, 0\right)}{\color{blue}{x}} \]

   7. Applied egg-rr 4.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x, 0\right)}{x}} \]
   8. Step-by-step derivation

      1. fma-udef 4.5%

         \[\leadsto \frac{\color{blue}{x \cdot x + 0}}{x} \]

      2. unpow2 4.5%

         \[\leadsto \frac{\color{blue}{{x}^{2}} + 0}{x} \]

      3. +-rgt-identity 4.5%

         \[\leadsto \frac{\color{blue}{{x}^{2}}}{x} \]

   9. Simplified 4.5%

      \[\leadsto \color{blue}{\frac{{x}^{2}}{x}} \]

   11. Applied egg-rr 100.0%

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

   if -1 < x < 1

   1. Initial program 7.9%

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

   3. Taylor expanded in x around 0 99.6%

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

   if 1 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 5.7%

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

      1. +-commutative 5.7%

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

   5. Simplified 5.7%

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

      1. associate--l+ 5.7%

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

      2. metadata-eval 5.7%

         \[\leadsto x + \color{blue}{0} \]

      3. flip-+ 5.3%

         \[\leadsto \color{blue}{\frac{x \cdot x - 0 \cdot 0}{x - 0}} \]

      4. metadata-eval 5.3%

         \[\leadsto \frac{x \cdot x - \color{blue}{0}}{x - 0} \]

      5. fma-neg 5.3%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x, -0\right)}}{x - 0} \]

      6. metadata-eval 5.3%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, \color{blue}{0}\right)}{x - 0} \]

      7. *-rgt-identity 5.3%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, 0\right)}{\color{blue}{x \cdot 1} - 0} \]

      8. fma-neg 5.3%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, 0\right)}{\color{blue}{\mathsf{fma}\left(x, 1, -0\right)}} \]

      9. metadata-eval 5.3%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, 0\right)}{\mathsf{fma}\left(x, 1, \color{blue}{0}\right)} \]

      10. fma-def 5.3%

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

      11. *-rgt-identity 5.3%

          \[\leadsto \frac{\mathsf{fma}\left(x, x, 0\right)}{\color{blue}{x} + 0} \]

      12. +-rgt-identity 5.3%

          \[\leadsto \frac{\mathsf{fma}\left(x, x, 0\right)}{\color{blue}{x}} \]

   7. Applied egg-rr 5.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x, 0\right)}{x}} \]
   8. Step-by-step derivation

      1. fma-udef 5.3%

         \[\leadsto \frac{\color{blue}{x \cdot x + 0}}{x} \]

      2. unpow2 5.3%

         \[\leadsto \frac{\color{blue}{{x}^{2}} + 0}{x} \]

      3. +-rgt-identity 5.3%

         \[\leadsto \frac{\color{blue}{{x}^{2}}}{x} \]

   9. Simplified 5.3%

      \[\leadsto \color{blue}{\frac{{x}^{2}}{x}} \]

   11. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.8%

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

Alternative 3: 52.7% accurate, 18.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -5e-310) -1.0 1.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5e-310) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-5d-310)) then
        tmp = -1.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -5e-310) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5e-310:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5e-310)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5e-310], -1.0, 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 50.2%

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

   3. Taylor expanded in x around 0 6.4%

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

      1. +-commutative 6.4%

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

   5. Simplified 6.4%

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

      1. associate--l+ 56.0%

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

      2. metadata-eval 56.0%

         \[\leadsto x + \color{blue}{0} \]

      3. flip-+ 34.1%

         \[\leadsto \color{blue}{\frac{x \cdot x - 0 \cdot 0}{x - 0}} \]

      4. metadata-eval 34.1%

         \[\leadsto \frac{x \cdot x - \color{blue}{0}}{x - 0} \]

      5. fma-neg 34.1%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x, -0\right)}}{x - 0} \]

      6. metadata-eval 34.1%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, \color{blue}{0}\right)}{x - 0} \]

      7. *-rgt-identity 34.1%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, 0\right)}{\color{blue}{x \cdot 1} - 0} \]

      8. fma-neg 34.1%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, 0\right)}{\color{blue}{\mathsf{fma}\left(x, 1, -0\right)}} \]

      9. metadata-eval 34.1%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, 0\right)}{\mathsf{fma}\left(x, 1, \color{blue}{0}\right)} \]

      10. fma-def 34.1%

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

      11. *-rgt-identity 34.1%

          \[\leadsto \frac{\mathsf{fma}\left(x, x, 0\right)}{\color{blue}{x} + 0} \]

      12. +-rgt-identity 34.1%

          \[\leadsto \frac{\mathsf{fma}\left(x, x, 0\right)}{\color{blue}{x}} \]

   7. Applied egg-rr 34.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x, 0\right)}{x}} \]
   8. Step-by-step derivation

      1. fma-udef 34.1%

         \[\leadsto \frac{\color{blue}{x \cdot x + 0}}{x} \]

      2. unpow2 34.1%

         \[\leadsto \frac{\color{blue}{{x}^{2}} + 0}{x} \]

      3. +-rgt-identity 34.1%

         \[\leadsto \frac{\color{blue}{{x}^{2}}}{x} \]

   9. Simplified 34.1%

      \[\leadsto \color{blue}{\frac{{x}^{2}}{x}} \]

   11. Applied egg-rr 48.9%

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

   if -4.999999999999985e-310 < x

   1. Initial program 49.2%

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

   3. Taylor expanded in x around 0 6.8%

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

      1. +-commutative 6.8%

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

   5. Simplified 6.8%

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

      1. associate--l+ 57.5%

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

      2. metadata-eval 57.5%

         \[\leadsto x + \color{blue}{0} \]

      3. flip-+ 36.1%

         \[\leadsto \color{blue}{\frac{x \cdot x - 0 \cdot 0}{x - 0}} \]

      4. metadata-eval 36.1%

         \[\leadsto \frac{x \cdot x - \color{blue}{0}}{x - 0} \]

      5. fma-neg 36.1%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x, -0\right)}}{x - 0} \]

      6. metadata-eval 36.1%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, \color{blue}{0}\right)}{x - 0} \]

      7. *-rgt-identity 36.1%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, 0\right)}{\color{blue}{x \cdot 1} - 0} \]

      8. fma-neg 36.1%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, 0\right)}{\color{blue}{\mathsf{fma}\left(x, 1, -0\right)}} \]

      9. metadata-eval 36.1%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, 0\right)}{\mathsf{fma}\left(x, 1, \color{blue}{0}\right)} \]

      10. fma-def 36.1%

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

      11. *-rgt-identity 36.1%

          \[\leadsto \frac{\mathsf{fma}\left(x, x, 0\right)}{\color{blue}{x} + 0} \]

      12. +-rgt-identity 36.1%

          \[\leadsto \frac{\mathsf{fma}\left(x, x, 0\right)}{\color{blue}{x}} \]

   7. Applied egg-rr 36.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x, 0\right)}{x}} \]
   8. Step-by-step derivation

      1. fma-udef 36.1%

         \[\leadsto \frac{\color{blue}{x \cdot x + 0}}{x} \]

      2. unpow2 36.1%

         \[\leadsto \frac{\color{blue}{{x}^{2}} + 0}{x} \]

      3. +-rgt-identity 36.1%

         \[\leadsto \frac{\color{blue}{{x}^{2}}}{x} \]

   9. Simplified 36.1%

      \[\leadsto \color{blue}{\frac{{x}^{2}}{x}} \]

   11. Applied egg-rr 47.8%

       \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 26.8% 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 49.6%

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

3. Taylor expanded in x around 0 6.7%

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

   1. +-commutative 6.7%

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

5. Simplified 6.7%

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

   1. associate--l+ 56.9%

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

   2. metadata-eval 56.9%

      \[\leadsto x + \color{blue}{0} \]

   3. flip-+ 35.2%

      \[\leadsto \color{blue}{\frac{x \cdot x - 0 \cdot 0}{x - 0}} \]

   4. metadata-eval 35.2%

      \[\leadsto \frac{x \cdot x - \color{blue}{0}}{x - 0} \]

   5. fma-neg 35.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x, -0\right)}}{x - 0} \]

   6. metadata-eval 35.2%

      \[\leadsto \frac{\mathsf{fma}\left(x, x, \color{blue}{0}\right)}{x - 0} \]

   7. *-rgt-identity 35.2%

      \[\leadsto \frac{\mathsf{fma}\left(x, x, 0\right)}{\color{blue}{x \cdot 1} - 0} \]

   8. fma-neg 35.2%

      \[\leadsto \frac{\mathsf{fma}\left(x, x, 0\right)}{\color{blue}{\mathsf{fma}\left(x, 1, -0\right)}} \]

   9. metadata-eval 35.2%

      \[\leadsto \frac{\mathsf{fma}\left(x, x, 0\right)}{\mathsf{fma}\left(x, 1, \color{blue}{0}\right)} \]

   10. fma-def 35.2%

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

   11. *-rgt-identity 35.2%

       \[\leadsto \frac{\mathsf{fma}\left(x, x, 0\right)}{\color{blue}{x} + 0} \]

   12. +-rgt-identity 35.2%

       \[\leadsto \frac{\mathsf{fma}\left(x, x, 0\right)}{\color{blue}{x}} \]

7. Applied egg-rr 35.2%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x, 0\right)}{x}} \]
8. Step-by-step derivation

   1. fma-udef 35.2%

      \[\leadsto \frac{\color{blue}{x \cdot x + 0}}{x} \]

   2. unpow2 35.2%

      \[\leadsto \frac{\color{blue}{{x}^{2}} + 0}{x} \]

   3. +-rgt-identity 35.2%

      \[\leadsto \frac{\color{blue}{{x}^{2}}}{x} \]

9. Simplified 35.2%

   \[\leadsto \color{blue}{\frac{{x}^{2}}{x}} \]

11. Applied egg-rr 22.3%

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

12. Final simplification 22.3%

    \[\leadsto -1 \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024027 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))