Logistic function from Lakshay Garg

Percentage Accurate: 54.7% → 99.7%

Time: 12.0s

Alternatives: 6

Speedup: 35.6×

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: 54.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -20:\\ \;\;\;\;\sqrt[3]{{\left(-1 + \frac{2}{1 + {\left(e^{x}\right)}^{-2}}\right)}^{3}}\\ \mathbf{elif}\;-2 \cdot x \leq 0.01:\\ \;\;\;\;x + \left(-0.3333333333333333 \cdot {x}^{3} + \left(-0.05396825396825397 \cdot {x}^{7} + 0.13333333333333333 \cdot {x}^{5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -20.0)
   (cbrt (pow (+ -1.0 (/ 2.0 (+ 1.0 (pow (exp x) -2.0)))) 3.0))
   (if (<= (* -2.0 x) 0.01)
     (+
      x
      (+
       (* -0.3333333333333333 (pow x 3.0))
       (+
        (* -0.05396825396825397 (pow x 7.0))
        (* 0.13333333333333333 (pow x 5.0)))))
     -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -20.0) {
		tmp = cbrt(pow((-1.0 + (2.0 / (1.0 + pow(exp(x), -2.0)))), 3.0));
	} else if ((-2.0 * x) <= 0.01) {
		tmp = x + ((-0.3333333333333333 * pow(x, 3.0)) + ((-0.05396825396825397 * pow(x, 7.0)) + (0.13333333333333333 * pow(x, 5.0))));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -20.0) {
		tmp = Math.cbrt(Math.pow((-1.0 + (2.0 / (1.0 + Math.pow(Math.exp(x), -2.0)))), 3.0));
	} else if ((-2.0 * x) <= 0.01) {
		tmp = x + ((-0.3333333333333333 * Math.pow(x, 3.0)) + ((-0.05396825396825397 * Math.pow(x, 7.0)) + (0.13333333333333333 * Math.pow(x, 5.0))));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -20.0)
		tmp = cbrt((Float64(-1.0 + Float64(2.0 / Float64(1.0 + (exp(x) ^ -2.0)))) ^ 3.0));
	elseif (Float64(-2.0 * x) <= 0.01)
		tmp = Float64(x + Float64(Float64(-0.3333333333333333 * (x ^ 3.0)) + Float64(Float64(-0.05396825396825397 * (x ^ 7.0)) + Float64(0.13333333333333333 * (x ^ 5.0)))));
	else
		tmp = -1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -20.0], N[Power[N[Power[N[(-1.0 + N[(2.0 / N[(1.0 + N[Power[N[Exp[x], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.01], N[(x + N[(N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.05396825396825397 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision] + N[(0.13333333333333333 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. exp-prod 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. pow-exp 100.0%

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

      2. *-commutative 100.0%

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

      3. metadata-eval 100.0%

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

      4. sub-neg 100.0%

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

      5. add-cbrt-cube 100.0%

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

      6. pow3 100.0%

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

      7. add-exp-log 100.0%

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

      8. expm1-def 100.0%

         \[\leadsto \sqrt[3]{{\color{blue}{\left(\mathsf{expm1}\left(\log \left(\frac{2}{1 + e^{-2 \cdot x}}\right)\right)\right)}}^{3}} \]

      9. log-div 100.0%

         \[\leadsto \sqrt[3]{{\left(\mathsf{expm1}\left(\color{blue}{\log 2 - \log \left(1 + e^{-2 \cdot x}\right)}\right)\right)}^{3}} \]

      10. log1p-udef 100.0%

          \[\leadsto \sqrt[3]{{\left(\mathsf{expm1}\left(\log 2 - \color{blue}{\mathsf{log1p}\left(e^{-2 \cdot x}\right)}\right)\right)}^{3}} \]

      11. *-commutative 100.0%

          \[\leadsto \sqrt[3]{{\left(\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(e^{\color{blue}{x \cdot -2}}\right)\right)\right)}^{3}} \]

      12. pow-exp 100.0%

          \[\leadsto \sqrt[3]{{\left(\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(\color{blue}{{\left(e^{x}\right)}^{-2}}\right)\right)\right)}^{3}} \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{x}\right)}^{-2}\right)\right)\right)}^{3}}} \]
   6. Step-by-step derivation

      1. pow-exp 100.0%

         \[\leadsto \sqrt[3]{{\left(\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(\color{blue}{e^{x \cdot -2}}\right)\right)\right)}^{3}} \]

   7. Applied egg-rr 100.0%

      \[\leadsto \sqrt[3]{{\left(\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(\color{blue}{e^{x \cdot -2}}\right)\right)\right)}^{3}} \]

   8. Taylor expanded in x around inf 100.0%

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

      1. sub-neg 100.0%

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

      2. metadata-eval 100.0%

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

      3. +-commutative 100.0%

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

      4. exp-diff 100.0%

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

      5. rem-exp-log 100.0%

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

      6. rem-exp-log 100.0%

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

      7. *-commutative 100.0%

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

      8. exp-prod 100.0%

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

   10. Simplified 100.0%

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

   if -20 < (*.f64 -2 x) < 0.0100000000000000002

   1. Initial program 8.2%

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

      1. sub-neg 8.2%

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

      2. *-commutative 8.2%

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

      3. exp-prod 8.2%

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

      4. metadata-eval 8.2%

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

   3. Simplified 8.2%

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

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{x + \left(-0.3333333333333333 \cdot {x}^{3} + \left(-0.05396825396825397 \cdot {x}^{7} + 0.13333333333333333 \cdot {x}^{5}\right)\right)} \]

   if 0.0100000000000000002 < (*.f64 -2 x)

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. exp-prod 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 96.7%

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

      1. *-commutative 96.7%

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

   6. Simplified 96.7%

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

   7. Taylor expanded in x around inf 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -20:\\ \;\;\;\;\sqrt[3]{{\left(-1 + \frac{2}{1 + {\left(e^{x}\right)}^{-2}}\right)}^{3}}\\ \mathbf{elif}\;-2 \cdot x \leq 0.01:\\ \;\;\;\;x + \left(-0.3333333333333333 \cdot {x}^{3} + \left(-0.05396825396825397 \cdot {x}^{7} + 0.13333333333333333 \cdot {x}^{5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 2: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -20:\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{elif}\;-2 \cdot x \leq 0.01:\\ \;\;\;\;x + \left(-0.3333333333333333 \cdot {x}^{3} + \left(-0.05396825396825397 \cdot {x}^{7} + 0.13333333333333333 \cdot {x}^{5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -20.0)
   (+ -1.0 (/ 2.0 (+ 1.0 (exp (* -2.0 x)))))
   (if (<= (* -2.0 x) 0.01)
     (+
      x
      (+
       (* -0.3333333333333333 (pow x 3.0))
       (+
        (* -0.05396825396825397 (pow x 7.0))
        (* 0.13333333333333333 (pow x 5.0)))))
     -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -20.0) {
		tmp = -1.0 + (2.0 / (1.0 + exp((-2.0 * x))));
	} else if ((-2.0 * x) <= 0.01) {
		tmp = x + ((-0.3333333333333333 * pow(x, 3.0)) + ((-0.05396825396825397 * pow(x, 7.0)) + (0.13333333333333333 * pow(x, 5.0))));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((-2.0d0) * x) <= (-20.0d0)) then
        tmp = (-1.0d0) + (2.0d0 / (1.0d0 + exp(((-2.0d0) * x))))
    else if (((-2.0d0) * x) <= 0.01d0) then
        tmp = x + (((-0.3333333333333333d0) * (x ** 3.0d0)) + (((-0.05396825396825397d0) * (x ** 7.0d0)) + (0.13333333333333333d0 * (x ** 5.0d0))))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -20.0) {
		tmp = -1.0 + (2.0 / (1.0 + Math.exp((-2.0 * x))));
	} else if ((-2.0 * x) <= 0.01) {
		tmp = x + ((-0.3333333333333333 * Math.pow(x, 3.0)) + ((-0.05396825396825397 * Math.pow(x, 7.0)) + (0.13333333333333333 * Math.pow(x, 5.0))));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (-2.0 * x) <= -20.0:
		tmp = -1.0 + (2.0 / (1.0 + math.exp((-2.0 * x))))
	elif (-2.0 * x) <= 0.01:
		tmp = x + ((-0.3333333333333333 * math.pow(x, 3.0)) + ((-0.05396825396825397 * math.pow(x, 7.0)) + (0.13333333333333333 * math.pow(x, 5.0))))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -20.0)
		tmp = Float64(-1.0 + Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))));
	elseif (Float64(-2.0 * x) <= 0.01)
		tmp = Float64(x + Float64(Float64(-0.3333333333333333 * (x ^ 3.0)) + Float64(Float64(-0.05396825396825397 * (x ^ 7.0)) + Float64(0.13333333333333333 * (x ^ 5.0)))));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((-2.0 * x) <= -20.0)
		tmp = -1.0 + (2.0 / (1.0 + exp((-2.0 * x))));
	elseif ((-2.0 * x) <= 0.01)
		tmp = x + ((-0.3333333333333333 * (x ^ 3.0)) + ((-0.05396825396825397 * (x ^ 7.0)) + (0.13333333333333333 * (x ^ 5.0))));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -20.0], N[(-1.0 + N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.01], N[(x + N[(N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.05396825396825397 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision] + N[(0.13333333333333333 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   if -20 < (*.f64 -2 x) < 0.0100000000000000002

   1. Initial program 8.2%

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

      1. sub-neg 8.2%

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

      2. *-commutative 8.2%

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

      3. exp-prod 8.2%

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

      4. metadata-eval 8.2%

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

   3. Simplified 8.2%

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

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{x + \left(-0.3333333333333333 \cdot {x}^{3} + \left(-0.05396825396825397 \cdot {x}^{7} + 0.13333333333333333 \cdot {x}^{5}\right)\right)} \]

   if 0.0100000000000000002 < (*.f64 -2 x)

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. exp-prod 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 96.7%

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

      1. *-commutative 96.7%

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

   6. Simplified 96.7%

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

   7. Taylor expanded in x around inf 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -20:\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{elif}\;-2 \cdot x \leq 0.01:\\ \;\;\;\;x + \left(-0.3333333333333333 \cdot {x}^{3} + \left(-0.05396825396825397 \cdot {x}^{7} + 0.13333333333333333 \cdot {x}^{5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 3: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -20:\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{elif}\;-2 \cdot x \leq 0.01:\\ \;\;\;\;x + \left(-0.3333333333333333 \cdot {x}^{3} + 0.13333333333333333 \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -20.0)
   (+ -1.0 (/ 2.0 (+ 1.0 (exp (* -2.0 x)))))
   (if (<= (* -2.0 x) 0.01)
     (+
      x
      (+
       (* -0.3333333333333333 (pow x 3.0))
       (* 0.13333333333333333 (pow x 5.0))))
     -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -20.0) {
		tmp = -1.0 + (2.0 / (1.0 + exp((-2.0 * x))));
	} else if ((-2.0 * x) <= 0.01) {
		tmp = x + ((-0.3333333333333333 * pow(x, 3.0)) + (0.13333333333333333 * pow(x, 5.0)));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((-2.0d0) * x) <= (-20.0d0)) then
        tmp = (-1.0d0) + (2.0d0 / (1.0d0 + exp(((-2.0d0) * x))))
    else if (((-2.0d0) * x) <= 0.01d0) then
        tmp = x + (((-0.3333333333333333d0) * (x ** 3.0d0)) + (0.13333333333333333d0 * (x ** 5.0d0)))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -20.0) {
		tmp = -1.0 + (2.0 / (1.0 + Math.exp((-2.0 * x))));
	} else if ((-2.0 * x) <= 0.01) {
		tmp = x + ((-0.3333333333333333 * Math.pow(x, 3.0)) + (0.13333333333333333 * Math.pow(x, 5.0)));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (-2.0 * x) <= -20.0:
		tmp = -1.0 + (2.0 / (1.0 + math.exp((-2.0 * x))))
	elif (-2.0 * x) <= 0.01:
		tmp = x + ((-0.3333333333333333 * math.pow(x, 3.0)) + (0.13333333333333333 * math.pow(x, 5.0)))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -20.0)
		tmp = Float64(-1.0 + Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))));
	elseif (Float64(-2.0 * x) <= 0.01)
		tmp = Float64(x + Float64(Float64(-0.3333333333333333 * (x ^ 3.0)) + Float64(0.13333333333333333 * (x ^ 5.0))));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((-2.0 * x) <= -20.0)
		tmp = -1.0 + (2.0 / (1.0 + exp((-2.0 * x))));
	elseif ((-2.0 * x) <= 0.01)
		tmp = x + ((-0.3333333333333333 * (x ^ 3.0)) + (0.13333333333333333 * (x ^ 5.0)));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -20.0], N[(-1.0 + N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.01], N[(x + N[(N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(0.13333333333333333 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   if -20 < (*.f64 -2 x) < 0.0100000000000000002

   1. Initial program 8.2%

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

      1. sub-neg 8.2%

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

      2. *-commutative 8.2%

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

      3. exp-prod 8.2%

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

      4. metadata-eval 8.2%

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

   3. Simplified 8.2%

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

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{x + \left(-0.3333333333333333 \cdot {x}^{3} + 0.13333333333333333 \cdot {x}^{5}\right)} \]

   if 0.0100000000000000002 < (*.f64 -2 x)

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. exp-prod 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 96.7%

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

      1. *-commutative 96.7%

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

   6. Simplified 96.7%

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

   7. Taylor expanded in x around inf 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -20:\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{elif}\;-2 \cdot x \leq 0.01:\\ \;\;\;\;x + \left(-0.3333333333333333 \cdot {x}^{3} + 0.13333333333333333 \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 4: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -20 \lor \neg \left(-2 \cdot x \leq 10^{-8}\right):\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* -2.0 x) -20.0) (not (<= (* -2.0 x) 1e-8)))
   (+ -1.0 (/ 2.0 (+ 1.0 (exp (* -2.0 x)))))
   (+ x (* -0.3333333333333333 (pow x 3.0)))))

C

double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -20.0) || !((-2.0 * x) <= 1e-8)) {
		tmp = -1.0 + (2.0 / (1.0 + exp((-2.0 * x))));
	} else {
		tmp = x + (-0.3333333333333333 * pow(x, 3.0));
	}
	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) <= 1d-8))) then
        tmp = (-1.0d0) + (2.0d0 / (1.0d0 + exp(((-2.0d0) * x))))
    else
        tmp = x + ((-0.3333333333333333d0) * (x ** 3.0d0))
    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) <= 1e-8)) {
		tmp = -1.0 + (2.0 / (1.0 + Math.exp((-2.0 * x))));
	} else {
		tmp = x + (-0.3333333333333333 * Math.pow(x, 3.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((-2.0 * x) <= -20.0) or not ((-2.0 * x) <= 1e-8):
		tmp = -1.0 + (2.0 / (1.0 + math.exp((-2.0 * x))))
	else:
		tmp = x + (-0.3333333333333333 * math.pow(x, 3.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((Float64(-2.0 * x) <= -20.0) || !(Float64(-2.0 * x) <= 1e-8))
		tmp = Float64(-1.0 + Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))));
	else
		tmp = Float64(x + Float64(-0.3333333333333333 * (x ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((-2.0 * x) <= -20.0) || ~(((-2.0 * x) <= 1e-8)))
		tmp = -1.0 + (2.0 / (1.0 + exp((-2.0 * x))));
	else
		tmp = x + (-0.3333333333333333 * (x ^ 3.0));
	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], 1e-8]], $MachinePrecision]], N[(-1.0 + N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 -2 x) < -20 or 1e-8 < (*.f64 -2 x)

   1. Initial program 99.9%

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

   if -20 < (*.f64 -2 x) < 1e-8

   1. Initial program 7.7%

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

      1. sub-neg 7.7%

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

      2. *-commutative 7.7%

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

      3. exp-prod 7.6%

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

      4. metadata-eval 7.6%

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

   3. Simplified 7.6%

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

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{x + -0.3333333333333333 \cdot {x}^{3}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -20 \lor \neg \left(-2 \cdot x \leq 10^{-8}\right):\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \end{array} \]

Alternative 5: 75.5% accurate, 35.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. exp-prod 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 96.7%

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

      1. *-commutative 96.7%

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

   6. Simplified 96.7%

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

   7. Taylor expanded in x around inf 100.0%

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

   if -1 < x

   1. Initial program 38.1%

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

      1. sub-neg 38.1%

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

      2. *-commutative 38.1%

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

      3. exp-prod 38.1%

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

      4. metadata-eval 38.1%

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

   3. Simplified 38.1%

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

   4. Taylor expanded in x around 0 68.9%

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

4. Final simplification 74.6%

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

Alternative 6: 27.5% 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 49.5%

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

   1. sub-neg 49.5%

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

   2. *-commutative 49.5%

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

   3. exp-prod 49.4%

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

   4. metadata-eval 49.4%

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

3. Simplified 49.4%

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

4. Taylor expanded in x around 0 22.3%

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

   1. *-commutative 22.3%

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

6. Simplified 22.3%

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

7. Taylor expanded in x around inf 20.9%

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

8. Final simplification 20.9%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023275 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))