Logistic function from Lakshay Garg

Percentage Accurate: 54.2% → 99.3%

Time: 4.5s

Alternatives: 7

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: 54.2% 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: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -20 \lor \neg \left(-2 \cdot x \leq 2 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* -2.0 x) -20.0) (not (<= (* -2.0 x) 2e-16)))
   (+ (/ 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) <= -20.0) || !((-2.0 * x) <= 2e-16)) {
		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) <= (-20.0d0)) .or. (.not. (((-2.0d0) * x) <= 2d-16))) 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) <= -20.0) || !((-2.0 * x) <= 2e-16)) {
		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) <= -20.0) or not ((-2.0 * x) <= 2e-16):
		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) <= -20.0) || !(Float64(-2.0 * x) <= 2e-16))
		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) <= -20.0) || ~(((-2.0 * x) <= 2e-16)))
		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], -20.0], N[Not[LessEqual[N[(-2.0 * x), $MachinePrecision], 2e-16]], $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) < -20 or 2e-16 < (*.f64 -2 x)

   1. Initial program 100.0%

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

   if -20 < (*.f64 -2 x) < 2e-16

   1. Initial program 7.1%

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

   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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -20 \lor \neg \left(-2 \cdot x \leq 2 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\\ \end{array} \]

Alternative 2: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.002 \lor \neg \left(-2 \cdot x \leq 2 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* -2.0 x) -0.002) (not (<= (* -2.0 x) 2e-16)))
   (+ (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) -1.0)
   x))

C

double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -0.002) || !((-2.0 * x) <= 2e-16)) {
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) + -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 ((((-2.0d0) * x) <= (-0.002d0)) .or. (.not. (((-2.0d0) * x) <= 2d-16))) then
        tmp = (2.0d0 / (1.0d0 + exp(((-2.0d0) * x)))) + (-1.0d0)
    else
        tmp = x
    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) <= 2e-16)) {
		tmp = (2.0 / (1.0 + Math.exp((-2.0 * x)))) + -1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((-2.0 * x) <= -0.002) or not ((-2.0 * x) <= 2e-16):
		tmp = (2.0 / (1.0 + math.exp((-2.0 * x)))) + -1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((Float64(-2.0 * x) <= -0.002) || !(Float64(-2.0 * x) <= 2e-16))
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) + -1.0);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((-2.0 * x) <= -0.002) || ~(((-2.0 * x) <= 2e-16)))
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	else
		tmp = x;
	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], 2e-16]], $MachinePrecision]], N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if (*.f64 -2 x) < -2e-3 or 2e-16 < (*.f64 -2 x)

   1. Initial program 99.8%

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

   if -2e-3 < (*.f64 -2 x) < 2e-16

   1. Initial program 5.9%

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

   2. Taylor expanded in x around 0 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.002 \lor \neg \left(-2 \cdot x \leq 2 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3: 79.1% accurate, 9.9× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.68) {
		tmp = -1.0;
	} else {
		tmp = (x + x) * (1.0 / (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 + x) * (1.0d0 / (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 + x) * (1.0 / (x + 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.68:
		tmp = -1.0
	else:
		tmp = (x + x) * (1.0 / (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 + x) * Float64(1.0 / 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 + x) * (1.0 / (x + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.68], -1.0, N[(N[(x + x), $MachinePrecision] * N[(1.0 / 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 98.9%

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

      1. *-commutative 98.9%

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

   4. Simplified 98.9%

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

   5. Taylor expanded in x around inf 100.0%

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

   if -0.680000000000000049 < x

   1. Initial program 36.2%

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

   2. Taylor expanded in x around 0 6.3%

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

      1. +-commutative 6.3%

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

   4. Simplified 6.3%

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

      1. flip-- 6.2%

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

      2. div-inv 6.2%

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

      3. metadata-eval 6.2%

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

      4. difference-of-sqr-1 6.2%

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

      5. associate-+l+ 6.2%

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

      6. metadata-eval 6.2%

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

      7. associate--l+ 69.8%

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

      8. metadata-eval 69.8%

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

      9. +-rgt-identity 69.8%

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

      10. associate-+l+ 69.8%

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

      11. metadata-eval 69.8%

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

   6. Applied egg-rr 69.8%

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

   7. Taylor expanded in x around 0 73.8%

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

      1. *-commutative 73.8%

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

      2. rem-log-exp 5.6%

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

      3. exp-lft-sqr 5.6%

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

      4. log-prod 5.6%

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

      5. rem-log-exp 14.4%

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

      6. rem-log-exp 73.8%

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

   9. Simplified 73.8%

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

4. Final simplification 79.4%

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

Alternative 4: 79.6% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 2.5:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;2 - \frac{4}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0) -1.0 (if (<= x 2.5) 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 <= 2.5) {
		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 <= 2.5d0) 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 <= 2.5) {
		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 <= 2.5:
		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 <= 2.5)
		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 <= 2.5)
		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, 2.5], 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 98.9%

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

      1. *-commutative 98.9%

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

   4. Simplified 98.9%

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

   5. Taylor expanded in x around inf 100.0%

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

   if -1 < x < 2.5

   1. Initial program 7.7%

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

   2. Taylor expanded in x around 0 98.7%

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

   if 2.5 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 5.2%

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

      1. +-commutative 5.2%

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

   4. Simplified 5.2%

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

      1. flip-- 4.8%

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

      2. div-inv 4.8%

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

      3. metadata-eval 4.8%

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

      4. difference-of-sqr-1 4.8%

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

      5. associate-+l+ 4.8%

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

      6. metadata-eval 4.8%

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

      7. associate--l+ 4.8%

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

      8. metadata-eval 4.8%

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

      9. +-rgt-identity 4.8%

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

      10. associate-+l+ 4.8%

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

      11. metadata-eval 4.8%

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

   6. Applied egg-rr 4.8%

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

   7. Taylor expanded in x around 0 18.8%

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

      1. *-commutative 18.8%

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

      2. rem-log-exp 3.4%

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

      3. exp-lft-sqr 3.4%

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

      4. log-prod 3.4%

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

      5. rem-log-exp 3.4%

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

      6. rem-log-exp 18.8%

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

   9. Simplified 18.8%

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

   10. Taylor expanded in x around inf 18.8%

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

       1. associate-*r/ 18.8%

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

       2. metadata-eval 18.8%

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

   12. Simplified 18.8%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 2.5:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;2 - \frac{4}{x}\\ \end{array} \]

Alternative 5: 79.6% 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 98.9%

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

      1. *-commutative 98.9%

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

   4. Simplified 98.9%

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

   5. Taylor expanded in x around inf 100.0%

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

   if -1 < x < 2

   1. Initial program 7.7%

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

   2. Taylor expanded in x around 0 98.7%

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

   if 2 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 5.2%

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

      1. +-commutative 5.2%

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

   4. Simplified 5.2%

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

      1. flip-- 4.8%

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

      2. div-inv 4.8%

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

      3. metadata-eval 4.8%

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

      4. difference-of-sqr-1 4.8%

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

      5. associate-+l+ 4.8%

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

      6. metadata-eval 4.8%

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

      7. associate--l+ 4.8%

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

      8. metadata-eval 4.8%

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

      9. +-rgt-identity 4.8%

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

      10. associate-+l+ 4.8%

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

      11. metadata-eval 4.8%

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

   6. Applied egg-rr 4.8%

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

   7. Taylor expanded in x around 0 18.8%

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

      1. *-commutative 18.8%

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

      2. rem-log-exp 3.4%

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

      3. exp-lft-sqr 3.4%

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

      4. log-prod 3.4%

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

      5. rem-log-exp 3.4%

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

      6. rem-log-exp 18.8%

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

   9. Simplified 18.8%

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

   10. Taylor expanded in x around inf 18.8%

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

4. Final simplification 79.6%

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

Alternative 6: 32.8% accurate, 35.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{-308}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 1.1e-308) -1.0 2.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.1e-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 <= 1.1d-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 <= 1.1e-308) {
		tmp = -1.0;
	} else {
		tmp = 2.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.1e-308:
		tmp = -1.0
	else:
		tmp = 2.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.1e-308)
		tmp = -1.0;
	else
		tmp = 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.1e-308)
		tmp = -1.0;
	else
		tmp = 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.1e-308], -1.0, 2.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001e-308

   1. Initial program 47.8%

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

   2. Taylor expanded in x around 0 46.7%

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

      1. *-commutative 46.7%

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

   4. Simplified 46.7%

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

   5. Taylor expanded in x around inf 46.8%

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

   if 1.1000000000000001e-308 < x

   1. Initial program 51.9%

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

   2. Taylor expanded in x around 0 6.5%

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

      1. +-commutative 6.5%

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

   4. Simplified 6.5%

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

      1. flip-- 6.3%

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

      2. div-inv 6.3%

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

      3. metadata-eval 6.3%

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

      4. difference-of-sqr-1 6.3%

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

      5. associate-+l+ 6.3%

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

      6. metadata-eval 6.3%

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

      7. associate--l+ 54.1%

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

      8. metadata-eval 54.1%

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

      9. +-rgt-identity 54.1%

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

      10. associate-+l+ 54.1%

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

      11. metadata-eval 54.1%

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

   6. Applied egg-rr 54.1%

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

   7. Taylor expanded in x around 0 60.3%

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

      1. *-commutative 60.3%

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

      2. rem-log-exp 5.4%

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

      3. exp-lft-sqr 5.4%

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

      4. log-prod 5.4%

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

      5. rem-log-exp 12.0%

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

      6. rem-log-exp 60.3%

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

   9. Simplified 60.3%

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

   10. Taylor expanded in x around inf 11.7%

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

4. Final simplification 28.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{-308}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]

Alternative 7: 27.9% 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 49.9%

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

2. Taylor expanded in x around 0 25.1%

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

   1. *-commutative 25.1%

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

4. Simplified 25.1%

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

5. Taylor expanded in x around inf 23.9%

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

6. Final simplification 23.9%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023182 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))