Logistic function from Lakshay Garg

Percentage Accurate: 54.3% → 100.0%

Time: 15.5s

Alternatives: 6

Speedup: 18.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.05 \lor \neg \left(-2 \cdot x \leq 0.01\right):\\ \;\;\;\;\mathsf{expm1}\left(-\log \left(\left(1 + e^{-2 \cdot x}\right) \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(-0.3333333333333333 \cdot {x}^{3} + \left(-0.05396825396825397 \cdot {x}^{7} + 0.13333333333333333 \cdot {x}^{5}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* -2.0 x) -0.05) (not (<= (* -2.0 x) 0.01)))
   (expm1 (- (log (* (+ 1.0 (exp (* -2.0 x))) 0.5))))
   (+
    x
    (+
     (* -0.3333333333333333 (pow x 3.0))
     (+
      (* -0.05396825396825397 (pow x 7.0))
      (* 0.13333333333333333 (pow x 5.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -0.05) || !((-2.0 * x) <= 0.01)) {
		tmp = expm1(-log(((1.0 + exp((-2.0 * x))) * 0.5)));
	} else {
		tmp = x + ((-0.3333333333333333 * pow(x, 3.0)) + ((-0.05396825396825397 * pow(x, 7.0)) + (0.13333333333333333 * pow(x, 5.0))));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -0.05) || !((-2.0 * x) <= 0.01)) {
		tmp = Math.expm1(-Math.log(((1.0 + Math.exp((-2.0 * x))) * 0.5)));
	} else {
		tmp = x + ((-0.3333333333333333 * Math.pow(x, 3.0)) + ((-0.05396825396825397 * Math.pow(x, 7.0)) + (0.13333333333333333 * Math.pow(x, 5.0))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((-2.0 * x) <= -0.05) or not ((-2.0 * x) <= 0.01):
		tmp = math.expm1(-math.log(((1.0 + math.exp((-2.0 * x))) * 0.5)))
	else:
		tmp = x + ((-0.3333333333333333 * math.pow(x, 3.0)) + ((-0.05396825396825397 * math.pow(x, 7.0)) + (0.13333333333333333 * math.pow(x, 5.0))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((Float64(-2.0 * x) <= -0.05) || !(Float64(-2.0 * x) <= 0.01))
		tmp = expm1(Float64(-log(Float64(Float64(1.0 + exp(Float64(-2.0 * x))) * 0.5))));
	else
		tmp = Float64(x + Float64(Float64(-0.3333333333333333 * (x ^ 3.0)) + Float64(Float64(-0.05396825396825397 * (x ^ 7.0)) + Float64(0.13333333333333333 * (x ^ 5.0)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.05], N[Not[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.01]], $MachinePrecision]], N[(Exp[(-N[Log[N[(N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision])] - 1), $MachinePrecision], 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]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 -2 x) < -0.050000000000000003 or 0.0100000000000000002 < (*.f64 -2 x)

   1. Initial program 99.9%

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

      1. add-exp-log 99.9%

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

      2. expm1-def 99.9%

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

      3. log-div 100.0%

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

      4. log1p-udef 99.9%

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

      5. exp-prod 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. exp-prod 99.9%

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

      2. *-commutative 99.9%

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

      3. exp-prod 99.9%

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

   6. Simplified 99.9%

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

      1. log1p-udef 100.0%

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

      2. pow-exp 100.0%

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

      3. *-commutative 100.0%

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

      4. pow-exp 100.0%

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

      5. diff-log 99.9%

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

      6. clear-num 99.9%

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

      7. log-rec 100.0%

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

      8. div-inv 100.0%

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

      9. metadata-eval 100.0%

         \[\leadsto \mathsf{expm1}\left(-\log \left(\left(1 + {\left(e^{-2}\right)}^{x}\right) \cdot \color{blue}{0.5}\right)\right) \]

   8. Applied egg-rr 100.0%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{-\log \left(\left(1 + {\left(e^{-2}\right)}^{x}\right) \cdot 0.5\right)}\right) \]
   9. Step-by-step derivation

      1. pow-exp 100.0%

         \[\leadsto \mathsf{expm1}\left(-\log \left(\left(1 + \color{blue}{e^{-2 \cdot x}}\right) \cdot 0.5\right)\right) \]

      2. *-commutative 100.0%

         \[\leadsto \mathsf{expm1}\left(-\log \left(\left(1 + e^{\color{blue}{x \cdot -2}}\right) \cdot 0.5\right)\right) \]

   10. Applied egg-rr 100.0%

       \[\leadsto \mathsf{expm1}\left(-\log \left(\left(1 + \color{blue}{e^{x \cdot -2}}\right) \cdot 0.5\right)\right) \]

   if -0.050000000000000003 < (*.f64 -2 x) < 0.0100000000000000002

   1. Initial program 8.4%

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

   3. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{x + \left(-0.3333333333333333 \cdot {x}^{3} + \left(-0.05396825396825397 \cdot {x}^{7} + 0.13333333333333333 \cdot {x}^{5}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.05 \lor \neg \left(-2 \cdot x \leq 0.01\right):\\ \;\;\;\;\mathsf{expm1}\left(-\log \left(\left(1 + e^{-2 \cdot x}\right) \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(-0.3333333333333333 \cdot {x}^{3} + \left(-0.05396825396825397 \cdot {x}^{7} + 0.13333333333333333 \cdot {x}^{5}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 100.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.05 \lor \neg \left(-2 \cdot x \leq 0.01\right):\\ \;\;\;\;\mathsf{expm1}\left(-\log \left(\left(1 + e^{-2 \cdot x}\right) \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(-0.3333333333333333 \cdot {x}^{3} + 0.13333333333333333 \cdot {x}^{5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* -2.0 x) -0.05) (not (<= (* -2.0 x) 0.01)))
   (expm1 (- (log (* (+ 1.0 (exp (* -2.0 x))) 0.5))))
   (+
    x
    (+
     (* -0.3333333333333333 (pow x 3.0))
     (* 0.13333333333333333 (pow x 5.0))))))

C

double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -0.05) || !((-2.0 * x) <= 0.01)) {
		tmp = expm1(-log(((1.0 + exp((-2.0 * x))) * 0.5)));
	} else {
		tmp = x + ((-0.3333333333333333 * pow(x, 3.0)) + (0.13333333333333333 * pow(x, 5.0)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -0.05) || !((-2.0 * x) <= 0.01)) {
		tmp = Math.expm1(-Math.log(((1.0 + Math.exp((-2.0 * x))) * 0.5)));
	} else {
		tmp = x + ((-0.3333333333333333 * Math.pow(x, 3.0)) + (0.13333333333333333 * Math.pow(x, 5.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((-2.0 * x) <= -0.05) or not ((-2.0 * x) <= 0.01):
		tmp = math.expm1(-math.log(((1.0 + math.exp((-2.0 * x))) * 0.5)))
	else:
		tmp = x + ((-0.3333333333333333 * math.pow(x, 3.0)) + (0.13333333333333333 * math.pow(x, 5.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((Float64(-2.0 * x) <= -0.05) || !(Float64(-2.0 * x) <= 0.01))
		tmp = expm1(Float64(-log(Float64(Float64(1.0 + exp(Float64(-2.0 * x))) * 0.5))));
	else
		tmp = Float64(x + Float64(Float64(-0.3333333333333333 * (x ^ 3.0)) + Float64(0.13333333333333333 * (x ^ 5.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.05], N[Not[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.01]], $MachinePrecision]], N[(Exp[(-N[Log[N[(N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision])] - 1), $MachinePrecision], 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]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 -2 x) < -0.050000000000000003 or 0.0100000000000000002 < (*.f64 -2 x)

   1. Initial program 99.9%

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

      1. add-exp-log 99.9%

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

      2. expm1-def 99.9%

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

      3. log-div 100.0%

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

      4. log1p-udef 99.9%

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

      5. exp-prod 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. exp-prod 99.9%

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

      2. *-commutative 99.9%

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

      3. exp-prod 99.9%

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

   6. Simplified 99.9%

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

      1. log1p-udef 100.0%

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

      2. pow-exp 100.0%

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

      3. *-commutative 100.0%

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

      4. pow-exp 100.0%

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

      5. diff-log 99.9%

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

      6. clear-num 99.9%

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

      7. log-rec 100.0%

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

      8. div-inv 100.0%

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

      9. metadata-eval 100.0%

         \[\leadsto \mathsf{expm1}\left(-\log \left(\left(1 + {\left(e^{-2}\right)}^{x}\right) \cdot \color{blue}{0.5}\right)\right) \]

   8. Applied egg-rr 100.0%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{-\log \left(\left(1 + {\left(e^{-2}\right)}^{x}\right) \cdot 0.5\right)}\right) \]
   9. Step-by-step derivation

      1. pow-exp 100.0%

         \[\leadsto \mathsf{expm1}\left(-\log \left(\left(1 + \color{blue}{e^{-2 \cdot x}}\right) \cdot 0.5\right)\right) \]

      2. *-commutative 100.0%

         \[\leadsto \mathsf{expm1}\left(-\log \left(\left(1 + e^{\color{blue}{x \cdot -2}}\right) \cdot 0.5\right)\right) \]

   10. Applied egg-rr 100.0%

       \[\leadsto \mathsf{expm1}\left(-\log \left(\left(1 + \color{blue}{e^{x \cdot -2}}\right) \cdot 0.5\right)\right) \]

   if -0.050000000000000003 < (*.f64 -2 x) < 0.0100000000000000002

   1. Initial program 8.4%

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

   3. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{x + \left(-0.3333333333333333 \cdot {x}^{3} + 0.13333333333333333 \cdot {x}^{5}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.05 \lor \neg \left(-2 \cdot x \leq 0.01\right):\\ \;\;\;\;\mathsf{expm1}\left(-\log \left(\left(1 + e^{-2 \cdot x}\right) \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(-0.3333333333333333 \cdot {x}^{3} + 0.13333333333333333 \cdot {x}^{5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 100.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.05 \lor \neg \left(-2 \cdot x \leq 0.01\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{else}:\\ \;\;\;\;x + \left(-0.3333333333333333 \cdot {x}^{3} + 0.13333333333333333 \cdot {x}^{5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* -2.0 x) -0.05) (not (<= (* -2.0 x) 0.01)))
   (+ (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) -1.0)
   (+
    x
    (+
     (* -0.3333333333333333 (pow x 3.0))
     (* 0.13333333333333333 (pow x 5.0))))))

C

double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -0.05) || !((-2.0 * x) <= 0.01)) {
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	} else {
		tmp = x + ((-0.3333333333333333 * pow(x, 3.0)) + (0.13333333333333333 * pow(x, 5.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((((-2.0d0) * x) <= (-0.05d0)) .or. (.not. (((-2.0d0) * x) <= 0.01d0))) then
        tmp = (2.0d0 / (1.0d0 + exp(((-2.0d0) * x)))) + (-1.0d0)
    else
        tmp = x + (((-0.3333333333333333d0) * (x ** 3.0d0)) + (0.13333333333333333d0 * (x ** 5.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -0.05) || !((-2.0 * x) <= 0.01)) {
		tmp = (2.0 / (1.0 + Math.exp((-2.0 * x)))) + -1.0;
	} else {
		tmp = x + ((-0.3333333333333333 * Math.pow(x, 3.0)) + (0.13333333333333333 * Math.pow(x, 5.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((-2.0 * x) <= -0.05) or not ((-2.0 * x) <= 0.01):
		tmp = (2.0 / (1.0 + math.exp((-2.0 * x)))) + -1.0
	else:
		tmp = x + ((-0.3333333333333333 * math.pow(x, 3.0)) + (0.13333333333333333 * math.pow(x, 5.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((Float64(-2.0 * x) <= -0.05) || !(Float64(-2.0 * x) <= 0.01))
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) + -1.0);
	else
		tmp = Float64(x + Float64(Float64(-0.3333333333333333 * (x ^ 3.0)) + Float64(0.13333333333333333 * (x ^ 5.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((-2.0 * x) <= -0.05) || ~(((-2.0 * x) <= 0.01)))
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	else
		tmp = x + ((-0.3333333333333333 * (x ^ 3.0)) + (0.13333333333333333 * (x ^ 5.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.05], N[Not[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.01]], $MachinePrecision]], N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], 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]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 -2 x) < -0.050000000000000003 or 0.0100000000000000002 < (*.f64 -2 x)

   1. Initial program 99.9%

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

   if -0.050000000000000003 < (*.f64 -2 x) < 0.0100000000000000002

   1. Initial program 8.4%

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

   3. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{x + \left(-0.3333333333333333 \cdot {x}^{3} + 0.13333333333333333 \cdot {x}^{5}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.05 \lor \neg \left(-2 \cdot x \leq 0.01\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{else}:\\ \;\;\;\;x + \left(-0.3333333333333333 \cdot {x}^{3} + 0.13333333333333333 \cdot {x}^{5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.05 \lor \neg \left(-2 \cdot x \leq 0.001\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{else}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* -2.0 x) -0.05) (not (<= (* -2.0 x) 0.001)))
   (+ (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) -1.0)
   (+ x (* -0.3333333333333333 (pow x 3.0)))))

C

double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -0.05) || !((-2.0 * x) <= 0.001)) {
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	} 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) <= (-0.05d0)) .or. (.not. (((-2.0d0) * x) <= 0.001d0))) then
        tmp = (2.0d0 / (1.0d0 + exp(((-2.0d0) * x)))) + (-1.0d0)
    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) <= -0.05) || !((-2.0 * x) <= 0.001)) {
		tmp = (2.0 / (1.0 + Math.exp((-2.0 * x)))) + -1.0;
	} else {
		tmp = x + (-0.3333333333333333 * Math.pow(x, 3.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((-2.0 * x) <= -0.05) or not ((-2.0 * x) <= 0.001):
		tmp = (2.0 / (1.0 + math.exp((-2.0 * x)))) + -1.0
	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) <= -0.05) || !(Float64(-2.0 * x) <= 0.001))
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) + -1.0);
	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) <= -0.05) || ~(((-2.0 * x) <= 0.001)))
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	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], -0.05], N[Not[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.001]], $MachinePrecision]], N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $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) < -0.050000000000000003 or 1e-3 < (*.f64 -2 x)

   1. Initial program 99.8%

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

   if -0.050000000000000003 < (*.f64 -2 x) < 1e-3

   1. Initial program 7.8%

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

   3. Taylor expanded in x around 0 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.05 \lor \neg \left(-2 \cdot x \leq 0.001\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{else}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.5% accurate, 18.1× 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. Add Preprocessing

   3. Taylor expanded in x around 0 98.0%

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

      1. *-commutative 98.0%

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

   5. Simplified 98.0%

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

   6. Taylor expanded in x around inf 100.0%

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

   if -1 < x

   1. Initial program 32.7%

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

   3. Taylor expanded in x around 0 74.6%

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

4. Final simplification 80.9%

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

Alternative 6: 27.2% 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. Add Preprocessing

3. Taylor expanded in x around 0 29.1%

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

   1. *-commutative 29.1%

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

5. Simplified 29.1%

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

6. Taylor expanded in x around inf 27.5%

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

7. Final simplification 27.5%

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

Reproduce

herbie shell --seed 2024040 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))