Logistic function from Lakshay Garg

Percentage Accurate: 55.2% → 97.8%

Time: 5.0s

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: 55.2% 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: 97.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -1 \cdot 10^{+24}:\\ \;\;\;\;1\\ \mathbf{elif}\;-2 \cdot x \leq 2 \cdot 10^{-7}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -1e+24)
   1.0
   (if (<= (* -2.0 x) 2e-7)
     (+ x (* -0.3333333333333333 (pow x 3.0)))
     (+ (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -1e+24) {
		tmp = 1.0;
	} else if ((-2.0 * x) <= 2e-7) {
		tmp = x + (-0.3333333333333333 * pow(x, 3.0));
	} else {
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) + -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) <= (-1d+24)) then
        tmp = 1.0d0
    else if (((-2.0d0) * x) <= 2d-7) then
        tmp = x + ((-0.3333333333333333d0) * (x ** 3.0d0))
    else
        tmp = (2.0d0 / (1.0d0 + exp(((-2.0d0) * x)))) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -1e+24) {
		tmp = 1.0;
	} else if ((-2.0 * x) <= 2e-7) {
		tmp = x + (-0.3333333333333333 * Math.pow(x, 3.0));
	} else {
		tmp = (2.0 / (1.0 + Math.exp((-2.0 * x)))) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (-2.0 * x) <= -1e+24:
		tmp = 1.0
	elif (-2.0 * x) <= 2e-7:
		tmp = x + (-0.3333333333333333 * math.pow(x, 3.0))
	else:
		tmp = (2.0 / (1.0 + math.exp((-2.0 * x)))) + -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -1e+24)
		tmp = 1.0;
	elseif (Float64(-2.0 * x) <= 2e-7)
		tmp = Float64(x + Float64(-0.3333333333333333 * (x ^ 3.0)));
	else
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((-2.0 * x) <= -1e+24)
		tmp = 1.0;
	elseif ((-2.0 * x) <= 2e-7)
		tmp = x + (-0.3333333333333333 * (x ^ 3.0));
	else
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -1e+24], 1.0, If[LessEqual[N[(-2.0 * x), $MachinePrecision], 2e-7], N[(x + N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 -2 x) < -9.9999999999999998e23

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 4.9%

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

      1. +-commutative 4.9%

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

   4. Simplified 4.9%

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

      1. flip-- 4.6%

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

      2. div-inv 4.6%

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

      3. metadata-eval 4.6%

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

      4. difference-of-sqr-1 4.6%

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

      5. associate-+l+ 4.6%

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

      6. metadata-eval 4.6%

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

      7. associate--l+ 4.6%

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

      8. metadata-eval 4.6%

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

      9. +-rgt-identity 4.6%

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

      10. associate-+l+ 4.6%

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

      11. metadata-eval 4.6%

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

   6. Applied egg-rr 4.6%

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

   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 3.1%

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

      3. exp-lft-sqr 3.1%

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

      4. log-prod 3.1%

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

      5. rem-log-exp 3.1%

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

      6. rem-log-exp 18.8%

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

   9. Simplified 18.8%

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

   11. Applied egg-rr 100.0%

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

   if -9.9999999999999998e23 < (*.f64 -2 x) < 1.9999999999999999e-7

   1. Initial program 7.2%

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

   2. Taylor expanded in x around 0 100.0%

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

   if 1.9999999999999999e-7 < (*.f64 -2 x)

   1. Initial program 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -1 \cdot 10^{+24}:\\ \;\;\;\;1\\ \mathbf{elif}\;-2 \cdot x \leq 2 \cdot 10^{-7}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \end{array} \]

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 1.2:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.15)
   -1.0
   (if (<= x 1.2) (+ x (* -0.3333333333333333 (pow x 3.0))) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.15) {
		tmp = -1.0;
	} else if (x <= 1.2) {
		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 (x <= (-1.15d0)) then
        tmp = -1.0d0
    else if (x <= 1.2d0) 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 (x <= -1.15) {
		tmp = -1.0;
	} else if (x <= 1.2) {
		tmp = x + (-0.3333333333333333 * Math.pow(x, 3.0));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.15:
		tmp = -1.0
	elif x <= 1.2:
		tmp = x + (-0.3333333333333333 * math.pow(x, 3.0))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.15)
		tmp = -1.0;
	elseif (x <= 1.2)
		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 (x <= -1.15)
		tmp = -1.0;
	elseif (x <= 1.2)
		tmp = x + (-0.3333333333333333 * (x ^ 3.0));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.15], -1.0, If[LessEqual[x, 1.2], N[(x + N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if x < -1.1499999999999999

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 5.4%

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

      1. +-commutative 5.4%

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

   4. Simplified 5.4%

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

      1. flip-- 5.1%

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

      2. div-inv 5.1%

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

      3. metadata-eval 5.1%

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

      4. difference-of-sqr-1 5.1%

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

      5. associate-+l+ 5.1%

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

      6. metadata-eval 5.1%

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

      7. associate--l+ 5.1%

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

      8. metadata-eval 5.1%

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

      9. +-rgt-identity 5.1%

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

      10. associate-+l+ 5.1%

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

      11. metadata-eval 5.1%

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

   6. Applied egg-rr 5.1%

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

   7. Taylor expanded in x around 0 1.6%

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

      1. *-commutative 1.6%

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

      2. rem-log-exp 0.6%

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

      3. exp-lft-sqr 0.6%

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

      4. log-prod 0.6%

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

      5. rem-log-exp 0.6%

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

      6. rem-log-exp 1.6%

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

   9. Simplified 1.6%

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

   11. Applied egg-rr 100.0%

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

   if -1.1499999999999999 < x < 1.19999999999999996

   1. Initial program 7.9%

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

   2. Taylor expanded in x around 0 99.7%

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

   if 1.19999999999999996 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 4.9%

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

      1. +-commutative 4.9%

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

   4. Simplified 4.9%

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

      1. flip-- 4.6%

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

      2. div-inv 4.6%

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

      3. metadata-eval 4.6%

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

      4. difference-of-sqr-1 4.6%

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

      5. associate-+l+ 4.6%

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

      6. metadata-eval 4.6%

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

      7. associate--l+ 4.6%

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

      8. metadata-eval 4.6%

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

      9. +-rgt-identity 4.6%

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

      10. associate-+l+ 4.6%

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

      11. metadata-eval 4.6%

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

   6. Applied egg-rr 4.6%

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

   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 3.1%

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

      3. exp-lft-sqr 3.1%

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

      4. log-prod 3.1%

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

      5. rem-log-exp 3.1%

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

      6. rem-log-exp 18.8%

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

   9. Simplified 18.8%

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

   11. Applied egg-rr 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 1.2:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3: 99.3% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 1.35:\\ \;\;\;\;\left(x \cdot x + x \cdot 2\right) \cdot \left(0.5 + x \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.35)
   -1.0
   (if (<= x 1.35) (* (+ (* x x) (* x 2.0)) (+ 0.5 (* x -0.25))) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.35) {
		tmp = -1.0;
	} else if (x <= 1.35) {
		tmp = ((x * x) + (x * 2.0)) * (0.5 + (x * -0.25));
	} 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 <= (-1.35d0)) then
        tmp = -1.0d0
    else if (x <= 1.35d0) then
        tmp = ((x * x) + (x * 2.0d0)) * (0.5d0 + (x * (-0.25d0)))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.35) {
		tmp = -1.0;
	} else if (x <= 1.35) {
		tmp = ((x * x) + (x * 2.0)) * (0.5 + (x * -0.25));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.35:
		tmp = -1.0
	elif x <= 1.35:
		tmp = ((x * x) + (x * 2.0)) * (0.5 + (x * -0.25))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.35)
		tmp = -1.0;
	elseif (x <= 1.35)
		tmp = Float64(Float64(Float64(x * x) + Float64(x * 2.0)) * Float64(0.5 + Float64(x * -0.25)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.35)
		tmp = -1.0;
	elseif (x <= 1.35)
		tmp = ((x * x) + (x * 2.0)) * (0.5 + (x * -0.25));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.35], -1.0, If[LessEqual[x, 1.35], N[(N[(N[(x * x), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(x * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if x < -1.3500000000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 5.4%

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

      1. +-commutative 5.4%

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

   4. Simplified 5.4%

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

      1. flip-- 5.1%

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

      2. div-inv 5.1%

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

      3. metadata-eval 5.1%

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

      4. difference-of-sqr-1 5.1%

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

      5. associate-+l+ 5.1%

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

      6. metadata-eval 5.1%

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

      7. associate--l+ 5.1%

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

      8. metadata-eval 5.1%

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

      9. +-rgt-identity 5.1%

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

      10. associate-+l+ 5.1%

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

      11. metadata-eval 5.1%

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

   6. Applied egg-rr 5.1%

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

   7. Taylor expanded in x around 0 1.6%

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

      1. *-commutative 1.6%

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

      2. rem-log-exp 0.6%

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

      3. exp-lft-sqr 0.6%

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

      4. log-prod 0.6%

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

      5. rem-log-exp 0.6%

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

      6. rem-log-exp 1.6%

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

   9. Simplified 1.6%

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

   11. Applied egg-rr 100.0%

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

   if -1.3500000000000001 < x < 1.3500000000000001

   1. Initial program 7.9%

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

   2. Taylor expanded in x around 0 7.5%

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

      1. +-commutative 7.5%

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

   4. Simplified 7.5%

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

      1. flip-- 7.4%

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

      2. div-inv 7.4%

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

      3. metadata-eval 7.4%

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

      4. difference-of-sqr-1 7.5%

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

      5. associate-+l+ 7.5%

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

      6. metadata-eval 7.5%

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

      7. associate--l+ 99.5%

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

      8. metadata-eval 99.5%

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

      9. +-rgt-identity 99.5%

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

      10. associate-+l+ 99.5%

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

      11. metadata-eval 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in x around 0 99.5%

      \[\leadsto \left(\left(x + 2\right) \cdot x\right) \cdot \color{blue}{\left(0.5 + -0.25 \cdot x\right)} \]
   8. Step-by-step derivation

      1. *-commutative 99.5%

         \[\leadsto \left(\left(x + 2\right) \cdot x\right) \cdot \left(0.5 + \color{blue}{x \cdot -0.25}\right) \]

   9. Simplified 99.5%

      \[\leadsto \left(\left(x + 2\right) \cdot x\right) \cdot \color{blue}{\left(0.5 + x \cdot -0.25\right)} \]
   10. Step-by-step derivation

       1. *-commutative 99.5%

          \[\leadsto \color{blue}{\left(x \cdot \left(x + 2\right)\right)} \cdot \left(0.5 + x \cdot -0.25\right) \]

       2. distribute-lft-in 99.6%

          \[\leadsto \color{blue}{\left(x \cdot x + x \cdot 2\right)} \cdot \left(0.5 + x \cdot -0.25\right) \]

   11. Applied egg-rr 99.6%

       \[\leadsto \color{blue}{\left(x \cdot x + x \cdot 2\right)} \cdot \left(0.5 + x \cdot -0.25\right) \]

   if 1.3500000000000001 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 4.9%

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

      1. +-commutative 4.9%

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

   4. Simplified 4.9%

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

      1. flip-- 4.6%

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

      2. div-inv 4.6%

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

      3. metadata-eval 4.6%

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

      4. difference-of-sqr-1 4.6%

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

      5. associate-+l+ 4.6%

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

      6. metadata-eval 4.6%

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

      7. associate--l+ 4.6%

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

      8. metadata-eval 4.6%

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

      9. +-rgt-identity 4.6%

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

      10. associate-+l+ 4.6%

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

      11. metadata-eval 4.6%

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

   6. Applied egg-rr 4.6%

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

   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 3.1%

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

      3. exp-lft-sqr 3.1%

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

      4. log-prod 3.1%

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

      5. rem-log-exp 3.1%

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

      6. rem-log-exp 18.8%

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

   9. Simplified 18.8%

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

   11. Applied egg-rr 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 1.35:\\ \;\;\;\;\left(x \cdot x + x \cdot 2\right) \cdot \left(0.5 + x \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4: 99.3% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 1.35:\\ \;\;\;\;\left(0.5 + x \cdot -0.25\right) \cdot \left(x \cdot \left(x + 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.35)
   -1.0
   (if (<= x 1.35) (* (+ 0.5 (* x -0.25)) (* x (+ x 2.0))) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.35) {
		tmp = -1.0;
	} else if (x <= 1.35) {
		tmp = (0.5 + (x * -0.25)) * (x * (x + 2.0));
	} 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 <= (-1.35d0)) then
        tmp = -1.0d0
    else if (x <= 1.35d0) then
        tmp = (0.5d0 + (x * (-0.25d0))) * (x * (x + 2.0d0))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.35) {
		tmp = -1.0;
	} else if (x <= 1.35) {
		tmp = (0.5 + (x * -0.25)) * (x * (x + 2.0));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.35:
		tmp = -1.0
	elif x <= 1.35:
		tmp = (0.5 + (x * -0.25)) * (x * (x + 2.0))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.35)
		tmp = -1.0;
	elseif (x <= 1.35)
		tmp = Float64(Float64(0.5 + Float64(x * -0.25)) * Float64(x * Float64(x + 2.0)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.35)
		tmp = -1.0;
	elseif (x <= 1.35)
		tmp = (0.5 + (x * -0.25)) * (x * (x + 2.0));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.35], -1.0, If[LessEqual[x, 1.35], N[(N[(0.5 + N[(x * -0.25), $MachinePrecision]), $MachinePrecision] * N[(x * N[(x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if x < -1.3500000000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 5.4%

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

      1. +-commutative 5.4%

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

   4. Simplified 5.4%

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

      1. flip-- 5.1%

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

      2. div-inv 5.1%

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

      3. metadata-eval 5.1%

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

      4. difference-of-sqr-1 5.1%

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

      5. associate-+l+ 5.1%

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

      6. metadata-eval 5.1%

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

      7. associate--l+ 5.1%

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

      8. metadata-eval 5.1%

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

      9. +-rgt-identity 5.1%

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

      10. associate-+l+ 5.1%

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

      11. metadata-eval 5.1%

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

   6. Applied egg-rr 5.1%

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

   7. Taylor expanded in x around 0 1.6%

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

      1. *-commutative 1.6%

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

      2. rem-log-exp 0.6%

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

      3. exp-lft-sqr 0.6%

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

      4. log-prod 0.6%

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

      5. rem-log-exp 0.6%

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

      6. rem-log-exp 1.6%

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

   9. Simplified 1.6%

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

   11. Applied egg-rr 100.0%

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

   if -1.3500000000000001 < x < 1.3500000000000001

   1. Initial program 7.9%

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

   2. Taylor expanded in x around 0 7.5%

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

      1. +-commutative 7.5%

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

   4. Simplified 7.5%

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

      1. flip-- 7.4%

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

      2. div-inv 7.4%

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

      3. metadata-eval 7.4%

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

      4. difference-of-sqr-1 7.5%

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

      5. associate-+l+ 7.5%

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

      6. metadata-eval 7.5%

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

      7. associate--l+ 99.5%

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

      8. metadata-eval 99.5%

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

      9. +-rgt-identity 99.5%

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

      10. associate-+l+ 99.5%

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

      11. metadata-eval 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in x around 0 99.5%

      \[\leadsto \left(\left(x + 2\right) \cdot x\right) \cdot \color{blue}{\left(0.5 + -0.25 \cdot x\right)} \]
   8. Step-by-step derivation

      1. *-commutative 99.5%

         \[\leadsto \left(\left(x + 2\right) \cdot x\right) \cdot \left(0.5 + \color{blue}{x \cdot -0.25}\right) \]

   9. Simplified 99.5%

      \[\leadsto \left(\left(x + 2\right) \cdot x\right) \cdot \color{blue}{\left(0.5 + x \cdot -0.25\right)} \]

   if 1.3500000000000001 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 4.9%

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

      1. +-commutative 4.9%

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

   4. Simplified 4.9%

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

      1. flip-- 4.6%

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

      2. div-inv 4.6%

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

      3. metadata-eval 4.6%

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

      4. difference-of-sqr-1 4.6%

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

      5. associate-+l+ 4.6%

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

      6. metadata-eval 4.6%

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

      7. associate--l+ 4.6%

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

      8. metadata-eval 4.6%

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

      9. +-rgt-identity 4.6%

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

      10. associate-+l+ 4.6%

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

      11. metadata-eval 4.6%

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

   6. Applied egg-rr 4.6%

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

   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 3.1%

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

      3. exp-lft-sqr 3.1%

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

      4. log-prod 3.1%

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

      5. rem-log-exp 3.1%

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

      6. rem-log-exp 18.8%

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

   9. Simplified 18.8%

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

   11. Applied egg-rr 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 1.35:\\ \;\;\;\;\left(0.5 + x \cdot -0.25\right) \cdot \left(x \cdot \left(x + 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5: 99.3% accurate, 21.3× 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. Taylor expanded in x around 0 5.4%

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

      1. +-commutative 5.4%

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

   4. Simplified 5.4%

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

      1. flip-- 5.1%

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

      2. div-inv 5.1%

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

      3. metadata-eval 5.1%

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

      4. difference-of-sqr-1 5.1%

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

      5. associate-+l+ 5.1%

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

      6. metadata-eval 5.1%

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

      7. associate--l+ 5.1%

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

      8. metadata-eval 5.1%

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

      9. +-rgt-identity 5.1%

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

      10. associate-+l+ 5.1%

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

      11. metadata-eval 5.1%

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

   6. Applied egg-rr 5.1%

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

   7. Taylor expanded in x around 0 1.6%

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

      1. *-commutative 1.6%

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

      2. rem-log-exp 0.6%

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

      3. exp-lft-sqr 0.6%

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

      4. log-prod 0.6%

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

      5. rem-log-exp 0.6%

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

      6. rem-log-exp 1.6%

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

   9. Simplified 1.6%

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

   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. Taylor expanded in x around 0 99.5%

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

   if 1 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 4.9%

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

      1. +-commutative 4.9%

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

   4. Simplified 4.9%

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

      1. flip-- 4.6%

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

      2. div-inv 4.6%

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

      3. metadata-eval 4.6%

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

      4. difference-of-sqr-1 4.6%

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

      5. associate-+l+ 4.6%

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

      6. metadata-eval 4.6%

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

      7. associate--l+ 4.6%

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

      8. metadata-eval 4.6%

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

      9. +-rgt-identity 4.6%

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

      10. associate-+l+ 4.6%

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

      11. metadata-eval 4.6%

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

   6. Applied egg-rr 4.6%

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

   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 3.1%

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

      3. exp-lft-sqr 3.1%

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

      4. log-prod 3.1%

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

      5. rem-log-exp 3.1%

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

      6. rem-log-exp 18.8%

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

   9. Simplified 18.8%

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

   11. Applied egg-rr 100.0%

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

4. Final simplification 99.7%

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

Alternative 6: 53.4% accurate, 35.6× 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 49.5%

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

   2. Taylor expanded in x around 0 6.9%

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

      1. +-commutative 6.9%

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

   4. Simplified 6.9%

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

      1. flip-- 6.8%

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

      2. div-inv 6.8%

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

      3. metadata-eval 6.8%

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

      4. difference-of-sqr-1 6.8%

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

      5. associate-+l+ 6.8%

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

      6. metadata-eval 6.8%

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

      7. associate--l+ 57.1%

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

      8. metadata-eval 57.1%

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

      9. +-rgt-identity 57.1%

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

      10. associate-+l+ 57.2%

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

      11. metadata-eval 57.2%

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

   6. Applied egg-rr 57.2%

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

   7. Taylor expanded in x around 0 54.8%

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

      1. *-commutative 54.8%

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

      2. rem-log-exp 4.7%

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

      3. exp-lft-sqr 4.6%

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

      4. log-prod 4.6%

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

      5. rem-log-exp 11.6%

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

      6. rem-log-exp 54.8%

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

   9. Simplified 54.8%

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

   11. Applied egg-rr 47.7%

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

   if -4.999999999999985e-310 < x

   1. Initial program 47.5%

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

   2. Taylor expanded in x around 0 6.0%

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

      1. +-commutative 6.0%

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

   4. Simplified 6.0%

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

      1. flip-- 5.8%

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

      2. div-inv 5.8%

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

      3. metadata-eval 5.8%

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

      4. difference-of-sqr-1 5.9%

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

      5. associate-+l+ 5.9%

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

      6. metadata-eval 5.9%

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

      7. associate--l+ 58.4%

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

      8. metadata-eval 58.4%

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

      9. +-rgt-identity 58.4%

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

      10. associate-+l+ 58.4%

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

      11. metadata-eval 58.4%

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

   6. Applied egg-rr 58.4%

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

   7. Taylor expanded in x around 0 64.2%

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

      1. *-commutative 64.2%

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

      2. rem-log-exp 5.2%

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

      3. exp-lft-sqr 5.1%

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

      4. log-prod 5.2%

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

      5. rem-log-exp 12.3%

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

      6. rem-log-exp 64.2%

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

   9. Simplified 64.2%

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

   11. Applied egg-rr 46.4%

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

4. Final simplification 47.1%

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

Alternative 7: 27.4% 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 48.5%

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

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+ 57.7%

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

   8. metadata-eval 57.7%

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

   9. +-rgt-identity 57.7%

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

   10. associate-+l+ 57.8%

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

   11. metadata-eval 57.8%

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

6. Applied egg-rr 57.8%

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

7. Taylor expanded in x around 0 59.4%

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

   1. *-commutative 59.4%

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

   2. rem-log-exp 4.9%

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

   3. exp-lft-sqr 4.9%

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

   4. log-prod 4.9%

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

   5. rem-log-exp 11.9%

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

   6. rem-log-exp 59.4%

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

9. Simplified 59.4%

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

11. Applied egg-rr 25.2%

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

12. Final simplification 25.2%

    \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023199 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))