Logistic function from Lakshay Garg

Percentage Accurate: 53.6% → 99.5%

Time: 6.8s

Alternatives: 5

Speedup: 5.3×

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: 53.6% 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: 99.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -2000:\\ \;\;\;\;1\\ \mathbf{elif}\;-2 \cdot x \leq 0.005:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -2000.0)
   1.0
   (if (<= (* -2.0 x) 0.005)
     (fma
      (fma (* x x) 0.13333333333333333 -0.3333333333333333)
      (* x (* x x))
      x)
     -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -2000.0) {
		tmp = 1.0;
	} else if ((-2.0 * x) <= 0.005) {
		tmp = fma(fma((x * x), 0.13333333333333333, -0.3333333333333333), (x * (x * x)), x);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -2000.0)
		tmp = 1.0;
	elseif (Float64(-2.0 * x) <= 0.005)
		tmp = fma(fma(Float64(x * x), 0.13333333333333333, -0.3333333333333333), Float64(x * Float64(x * x)), x);
	else
		tmp = -1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -2000.0], 1.0, If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.005], N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333 + -0.3333333333333333), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -2e3

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. lower--.f64 N/A

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

      4. count-2 N/A

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

      5. lower-+.f64 1.6

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

   5. Applied rewrites 1.6%

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

   7. Applied rewrites 96.2%

      \[\leadsto \color{blue}{\frac{2}{8 + \left(x + x\right)} \cdot \left(\left(x + x\right) + 4\right)} - 1 \]

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 100.0%

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

   if -2e3 < (*.f64 #s(literal -2 binary64) x) < 0.0050000000000000001

   1. Initial program 10.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \color{blue}{1 \cdot x + \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right) \cdot x} \]

      2. *-lft-identity N/A

         \[\leadsto \color{blue}{x} + \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right) \cdot x \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right) \cdot x + x} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)} + x \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) \cdot \left(x \cdot {x}^{2}\right)} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x \cdot {x}^{2}, x\right)} \]

      8. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{2}{15} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, x \cdot {x}^{2}, x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \frac{2}{15}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), x \cdot {x}^{2}, x\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{2} \cdot \frac{2}{15} + \color{blue}{\frac{-1}{3}}, x \cdot {x}^{2}, x\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{2}{15}, \frac{-1}{3}\right)}, x \cdot {x}^{2}, x\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{2}{15}, \frac{-1}{3}\right), x \cdot {x}^{2}, x\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{2}{15}, \frac{-1}{3}\right), x \cdot {x}^{2}, x\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{2}{15}, \frac{-1}{3}\right), \color{blue}{x \cdot {x}^{2}}, x\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{2}{15}, \frac{-1}{3}\right), x \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \]

      16. lower-*.f64 100.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right), x \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)} \]

   if 0.0050000000000000001 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. lower--.f64 N/A

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

      4. count-2 N/A

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

      5. lower-+.f64 98.8

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 100.0%

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

Alternative 2: 99.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -2000:\\ \;\;\;\;1\\ \mathbf{elif}\;-2 \cdot x \leq 0.005:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -2000.0)
   1.0
   (if (<= (* -2.0 x) 0.005) (fma -0.3333333333333333 (* x (* x x)) x) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -2000.0) {
		tmp = 1.0;
	} else if ((-2.0 * x) <= 0.005) {
		tmp = fma(-0.3333333333333333, (x * (x * x)), x);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -2000.0)
		tmp = 1.0;
	elseif (Float64(-2.0 * x) <= 0.005)
		tmp = fma(-0.3333333333333333, Float64(x * Float64(x * x)), x);
	else
		tmp = -1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -2000.0], 1.0, If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.005], N[(-0.3333333333333333 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -2e3

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. lower--.f64 N/A

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

      4. count-2 N/A

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

      5. lower-+.f64 1.6

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

   5. Applied rewrites 1.6%

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

   7. Applied rewrites 96.2%

      \[\leadsto \color{blue}{\frac{2}{8 + \left(x + x\right)} \cdot \left(\left(x + x\right) + 4\right)} - 1 \]

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 100.0%

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

   if -2e3 < (*.f64 #s(literal -2 binary64) x) < 0.0050000000000000001

   1. Initial program 10.0%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 99.8

          \[\leadsto \mathsf{fma}\left(-0.3333333333333333, x \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.3333333333333333, x \cdot \left(x \cdot x\right), x\right)} \]

   if 0.0050000000000000001 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. lower--.f64 N/A

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

      4. count-2 N/A

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

      5. lower-+.f64 98.8

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 100.0%

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

Alternative 3: 99.3% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -2000:\\ \;\;\;\;1\\ \mathbf{elif}\;-2 \cdot x \leq 0.005:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -2000.0) 1.0 (if (<= (* -2.0 x) 0.005) x -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -2000.0) {
		tmp = 1.0;
	} else if ((-2.0 * x) <= 0.005) {
		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 (((-2.0d0) * x) <= (-2000.0d0)) then
        tmp = 1.0d0
    else if (((-2.0d0) * x) <= 0.005d0) 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 ((-2.0 * x) <= -2000.0) {
		tmp = 1.0;
	} else if ((-2.0 * x) <= 0.005) {
		tmp = x;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (-2.0 * x) <= -2000.0:
		tmp = 1.0
	elif (-2.0 * x) <= 0.005:
		tmp = x
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -2000.0)
		tmp = 1.0;
	elseif (Float64(-2.0 * x) <= 0.005)
		tmp = x;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((-2.0 * x) <= -2000.0)
		tmp = 1.0;
	elseif ((-2.0 * x) <= 0.005)
		tmp = x;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -2000.0], 1.0, If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.005], x, -1.0]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -2e3

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. lower--.f64 N/A

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

      4. count-2 N/A

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

      5. lower-+.f64 1.6

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

   5. Applied rewrites 1.6%

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

   7. Applied rewrites 96.2%

      \[\leadsto \color{blue}{\frac{2}{8 + \left(x + x\right)} \cdot \left(\left(x + x\right) + 4\right)} - 1 \]

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 100.0%

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

   if -2e3 < (*.f64 #s(literal -2 binary64) x) < 0.0050000000000000001

   1. Initial program 10.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 9.5

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

   5. Applied rewrites 9.5%

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

      1. associate--l+ N/A

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

      2. metadata-eval N/A

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

      3. +-rgt-identity 98.8

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

   7. Applied rewrites 98.8%

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

   if 0.0050000000000000001 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. lower--.f64 N/A

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

      4. count-2 N/A

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

      5. lower-+.f64 98.8

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 100.0%

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

Alternative 4: 52.1% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* -2.0 x) -5e-310) 1.0 -1.0))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * 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 (((-2.0d0) * 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 ((-2.0 * x) <= -5e-310) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (-2.0 * x) <= -5e-310:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * 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 ((-2.0 * x) <= -5e-310)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -5e-310], 1.0, -1.0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -4.999999999999985e-310

   1. Initial program 56.9%

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

   3. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. lower--.f64 N/A

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

      4. count-2 N/A

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

      5. lower-+.f64 5.2

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

   5. Applied rewrites 5.2%

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

   7. Applied rewrites 53.2%

      \[\leadsto \color{blue}{\frac{2}{8 + \left(x + x\right)} \cdot \left(\left(x + x\right) + 4\right)} - 1 \]

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 54.6%

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

   if -4.999999999999985e-310 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 51.7%

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

   3. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. lower--.f64 N/A

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

      4. count-2 N/A

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

      5. lower-+.f64 50.4

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

   5. Applied rewrites 50.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 49.5%

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

Alternative 5: 27.6% accurate, 123.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return -1.0

Julia

function code(x, y)
	return -1.0
end

MATLAB

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

Wolfram

code[x_, y_] := -1.0

Derivation

1. Initial program 54.3%

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

3. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. cancel-sign-sub-inv N/A

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

   3. lower--.f64 N/A

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

   4. count-2 N/A

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

   5. lower-+.f64 27.6

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

5. Applied rewrites 27.6%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 25.5%

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

Reproduce

herbie shell --seed 2024219 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))