Logistic function from Lakshay Garg

Percentage Accurate: 53.0% → 99.0%

Time: 10.0s

Alternatives: 6

Speedup: 109.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.0% 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: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.5:\\ \;\;\;\;\mathsf{expm1}\left(\log 2 - \log \left(1 + e^{-2 \cdot x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -0.5)
   (expm1 (- (log 2.0) (log (+ 1.0 (exp (* -2.0 x))))))
   (expm1 x)))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.5) {
		tmp = expm1((log(2.0) - log((1.0 + exp((-2.0 * x))))));
	} else {
		tmp = expm1(x);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.5) {
		tmp = Math.expm1((Math.log(2.0) - Math.log((1.0 + Math.exp((-2.0 * x))))));
	} else {
		tmp = Math.expm1(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (-2.0 * x) <= -0.5:
		tmp = math.expm1((math.log(2.0) - math.log((1.0 + math.exp((-2.0 * x))))))
	else:
		tmp = math.expm1(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.5)
		tmp = expm1(Float64(log(2.0) - log(Float64(1.0 + exp(Float64(-2.0 * x))))));
	else
		tmp = expm1(x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.5], N[(Exp[N[(N[Log[2.0], $MachinePrecision] - N[Log[N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision], N[(Exp[x] - 1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 -2 x) < -0.5

   1. Initial program 100.0%

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

      1. add-log-exp 100.0%

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

      2. *-un-lft-identity 100.0%

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

      3. log-prod 100.0%

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

      4. metadata-eval 100.0%

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

      5. add-log-exp 100.0%

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

      6. add-exp-log 100.0%

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

      7. expm1-def 100.0%

         \[\leadsto 0 + \color{blue}{\mathsf{expm1}\left(\log \left(\frac{2}{1 + e^{-2 \cdot x}}\right)\right)} \]

      8. log-div 100.0%

         \[\leadsto 0 + \mathsf{expm1}\left(\color{blue}{\log 2 - \log \left(1 + e^{-2 \cdot x}\right)}\right) \]

      9. log1p-udef 100.0%

         \[\leadsto 0 + \mathsf{expm1}\left(\log 2 - \color{blue}{\mathsf{log1p}\left(e^{-2 \cdot x}\right)}\right) \]

      10. exp-prod 100.0%

          \[\leadsto 0 + \mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(\color{blue}{{\left(e^{-2}\right)}^{x}}\right)\right) \]

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0 + \mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)} \]
   4. Step-by-step derivation

      1. +-lft-identity 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 100.0%

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

   if -0.5 < (*.f64 -2 x)

   1. Initial program 37.6%

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

      1. add-log-exp 37.6%

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

      2. *-un-lft-identity 37.6%

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

      3. log-prod 37.6%

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

      4. metadata-eval 37.6%

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

      5. add-log-exp 37.6%

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

      6. add-exp-log 37.6%

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

      7. expm1-def 37.6%

         \[\leadsto 0 + \color{blue}{\mathsf{expm1}\left(\log \left(\frac{2}{1 + e^{-2 \cdot x}}\right)\right)} \]

      8. log-div 37.6%

         \[\leadsto 0 + \mathsf{expm1}\left(\color{blue}{\log 2 - \log \left(1 + e^{-2 \cdot x}\right)}\right) \]

      9. log1p-udef 37.7%

         \[\leadsto 0 + \mathsf{expm1}\left(\log 2 - \color{blue}{\mathsf{log1p}\left(e^{-2 \cdot x}\right)}\right) \]

      10. exp-prod 37.7%

          \[\leadsto 0 + \mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(\color{blue}{{\left(e^{-2}\right)}^{x}}\right)\right) \]

   3. Applied egg-rr 37.7%

      \[\leadsto \color{blue}{0 + \mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)} \]
   4. Step-by-step derivation

      1. +-lft-identity 37.7%

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

   5. Simplified 37.7%

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

   6. Taylor expanded in x around 0 99.7%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{x}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.5:\\ \;\;\;\;\mathsf{expm1}\left(\log 2 - \log \left(1 + e^{-2 \cdot x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \]

Alternative 2: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.5:\\ \;\;\;\;\sqrt[3]{8 \cdot \frac{1}{{\left(1 + e^{-2 \cdot x}\right)}^{3}}} + -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -0.5)
   (+ (cbrt (* 8.0 (/ 1.0 (pow (+ 1.0 (exp (* -2.0 x))) 3.0)))) -1.0)
   (expm1 x)))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.5) {
		tmp = cbrt((8.0 * (1.0 / pow((1.0 + exp((-2.0 * x))), 3.0)))) + -1.0;
	} else {
		tmp = expm1(x);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.5) {
		tmp = Math.cbrt((8.0 * (1.0 / Math.pow((1.0 + Math.exp((-2.0 * x))), 3.0)))) + -1.0;
	} else {
		tmp = Math.expm1(x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.5)
		tmp = Float64(cbrt(Float64(8.0 * Float64(1.0 / (Float64(1.0 + exp(Float64(-2.0 * x))) ^ 3.0)))) + -1.0);
	else
		tmp = expm1(x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.5], N[(N[Power[N[(8.0 * N[(1.0 / N[Power[N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + -1.0), $MachinePrecision], N[(Exp[x] - 1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 -2 x) < -0.5

   1. Initial program 100.0%

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

      1. add-cbrt-cube 100.0%

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

      2. pow1/3 100.0%

         \[\leadsto \color{blue}{{\left(\left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right) \cdot \frac{2}{1 + e^{-2 \cdot x}}\right)}^{0.3333333333333333}} - 1 \]

      3. pow3 100.0%

         \[\leadsto {\color{blue}{\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)}}^{0.3333333333333333} - 1 \]

      4. div-inv 100.0%

         \[\leadsto {\left({\color{blue}{\left(2 \cdot \frac{1}{1 + e^{-2 \cdot x}}\right)}}^{3}\right)}^{0.3333333333333333} - 1 \]

      5. unpow-prod-down 100.0%

         \[\leadsto {\color{blue}{\left({2}^{3} \cdot {\left(\frac{1}{1 + e^{-2 \cdot x}}\right)}^{3}\right)}}^{0.3333333333333333} - 1 \]

      6. metadata-eval 100.0%

         \[\leadsto {\left(\color{blue}{8} \cdot {\left(\frac{1}{1 + e^{-2 \cdot x}}\right)}^{3}\right)}^{0.3333333333333333} - 1 \]

      7. inv-pow 100.0%

         \[\leadsto {\left(8 \cdot {\color{blue}{\left({\left(1 + e^{-2 \cdot x}\right)}^{-1}\right)}}^{3}\right)}^{0.3333333333333333} - 1 \]

      8. metadata-eval 100.0%

         \[\leadsto {\left(8 \cdot {\left({\left(1 + e^{-2 \cdot x}\right)}^{\color{blue}{\left(-1\right)}}\right)}^{3}\right)}^{0.3333333333333333} - 1 \]

      9. pow-pow 100.0%

         \[\leadsto {\left(8 \cdot \color{blue}{{\left(1 + e^{-2 \cdot x}\right)}^{\left(\left(-1\right) \cdot 3\right)}}\right)}^{0.3333333333333333} - 1 \]

      10. exp-prod 100.0%

          \[\leadsto {\left(8 \cdot {\left(1 + \color{blue}{{\left(e^{-2}\right)}^{x}}\right)}^{\left(\left(-1\right) \cdot 3\right)}\right)}^{0.3333333333333333} - 1 \]

      11. metadata-eval 100.0%

          \[\leadsto {\left(8 \cdot {\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{\left(\color{blue}{-1} \cdot 3\right)}\right)}^{0.3333333333333333} - 1 \]

      12. metadata-eval 100.0%

          \[\leadsto {\left(8 \cdot {\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{\color{blue}{-3}}\right)}^{0.3333333333333333} - 1 \]

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{{\left(8 \cdot {\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{-3}\right)}^{0.3333333333333333}} - 1 \]
   4. Step-by-step derivation

      1. unpow1/3 100.0%

         \[\leadsto \color{blue}{\sqrt[3]{8 \cdot {\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{-3}}} - 1 \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\sqrt[3]{8 \cdot {\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{-3}}} - 1 \]

   6. Taylor expanded in x around inf 100.0%

      \[\leadsto \sqrt[3]{8 \cdot \color{blue}{\frac{1}{{\left(1 + e^{-2 \cdot x}\right)}^{3}}}} - 1 \]

   if -0.5 < (*.f64 -2 x)

   1. Initial program 37.6%

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

      1. add-log-exp 37.6%

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

      2. *-un-lft-identity 37.6%

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

      3. log-prod 37.6%

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

      4. metadata-eval 37.6%

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

      5. add-log-exp 37.6%

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

      6. add-exp-log 37.6%

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

      7. expm1-def 37.6%

         \[\leadsto 0 + \color{blue}{\mathsf{expm1}\left(\log \left(\frac{2}{1 + e^{-2 \cdot x}}\right)\right)} \]

      8. log-div 37.6%

         \[\leadsto 0 + \mathsf{expm1}\left(\color{blue}{\log 2 - \log \left(1 + e^{-2 \cdot x}\right)}\right) \]

      9. log1p-udef 37.7%

         \[\leadsto 0 + \mathsf{expm1}\left(\log 2 - \color{blue}{\mathsf{log1p}\left(e^{-2 \cdot x}\right)}\right) \]

      10. exp-prod 37.7%

          \[\leadsto 0 + \mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(\color{blue}{{\left(e^{-2}\right)}^{x}}\right)\right) \]

   3. Applied egg-rr 37.7%

      \[\leadsto \color{blue}{0 + \mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)} \]
   4. Step-by-step derivation

      1. +-lft-identity 37.7%

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

   5. Simplified 37.7%

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

   6. Taylor expanded in x around 0 99.7%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{x}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.5:\\ \;\;\;\;\sqrt[3]{8 \cdot \frac{1}{{\left(1 + e^{-2 \cdot x}\right)}^{3}}} + -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \]

Alternative 3: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.5:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -0.5) (+ (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) -1.0) (expm1 x)))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.5) {
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	} else {
		tmp = expm1(x);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.5) {
		tmp = (2.0 / (1.0 + Math.exp((-2.0 * x)))) + -1.0;
	} else {
		tmp = Math.expm1(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (-2.0 * x) <= -0.5:
		tmp = (2.0 / (1.0 + math.exp((-2.0 * x)))) + -1.0
	else:
		tmp = math.expm1(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.5)
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) + -1.0);
	else
		tmp = expm1(x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.5], N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(Exp[x] - 1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 -2 x) < -0.5

   1. Initial program 100.0%

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

   if -0.5 < (*.f64 -2 x)

   1. Initial program 37.6%

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

      1. add-log-exp 37.6%

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

      2. *-un-lft-identity 37.6%

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

      3. log-prod 37.6%

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

      4. metadata-eval 37.6%

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

      5. add-log-exp 37.6%

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

      6. add-exp-log 37.6%

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

      7. expm1-def 37.6%

         \[\leadsto 0 + \color{blue}{\mathsf{expm1}\left(\log \left(\frac{2}{1 + e^{-2 \cdot x}}\right)\right)} \]

      8. log-div 37.6%

         \[\leadsto 0 + \mathsf{expm1}\left(\color{blue}{\log 2 - \log \left(1 + e^{-2 \cdot x}\right)}\right) \]

      9. log1p-udef 37.7%

         \[\leadsto 0 + \mathsf{expm1}\left(\log 2 - \color{blue}{\mathsf{log1p}\left(e^{-2 \cdot x}\right)}\right) \]

      10. exp-prod 37.7%

          \[\leadsto 0 + \mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(\color{blue}{{\left(e^{-2}\right)}^{x}}\right)\right) \]

   3. Applied egg-rr 37.7%

      \[\leadsto \color{blue}{0 + \mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)} \]
   4. Step-by-step derivation

      1. +-lft-identity 37.7%

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

   5. Simplified 37.7%

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

   6. Taylor expanded in x around 0 99.7%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{x}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.5:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \]

Alternative 4: 79.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-45}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-0.5}{x} \cdot \left(\left(-x\right) - 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2e-45) (expm1 x) (/ 1.0 (* (/ -0.5 x) (- (- x) 2.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2e-45) {
		tmp = expm1(x);
	} else {
		tmp = 1.0 / ((-0.5 / x) * (-x - 2.0));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 2e-45) {
		tmp = Math.expm1(x);
	} else {
		tmp = 1.0 / ((-0.5 / x) * (-x - 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 2e-45:
		tmp = math.expm1(x)
	else:
		tmp = 1.0 / ((-0.5 / x) * (-x - 2.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2e-45)
		tmp = expm1(x);
	else
		tmp = Float64(1.0 / Float64(Float64(-0.5 / x) * Float64(Float64(-x) - 2.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2e-45], N[(Exp[x] - 1), $MachinePrecision], N[(1.0 / N[(N[(-0.5 / x), $MachinePrecision] * N[((-x) - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.99999999999999997e-45

   1. Initial program 38.6%

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

      1. add-log-exp 38.6%

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

      2. *-un-lft-identity 38.6%

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

      3. log-prod 38.6%

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

      4. metadata-eval 38.6%

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

      5. add-log-exp 38.6%

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

      6. add-exp-log 38.6%

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

      7. expm1-def 38.6%

         \[\leadsto 0 + \color{blue}{\mathsf{expm1}\left(\log \left(\frac{2}{1 + e^{-2 \cdot x}}\right)\right)} \]

      8. log-div 38.6%

         \[\leadsto 0 + \mathsf{expm1}\left(\color{blue}{\log 2 - \log \left(1 + e^{-2 \cdot x}\right)}\right) \]

      9. log1p-udef 38.7%

         \[\leadsto 0 + \mathsf{expm1}\left(\log 2 - \color{blue}{\mathsf{log1p}\left(e^{-2 \cdot x}\right)}\right) \]

      10. exp-prod 38.7%

          \[\leadsto 0 + \mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(\color{blue}{{\left(e^{-2}\right)}^{x}}\right)\right) \]

   3. Applied egg-rr 38.7%

      \[\leadsto \color{blue}{0 + \mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)} \]
   4. Step-by-step derivation

      1. +-lft-identity 38.7%

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

   5. Simplified 38.7%

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

   6. Taylor expanded in x around 0 99.7%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{x}\right) \]

   if 1.99999999999999997e-45 < x

   1. Initial program 93.8%

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

   2. Taylor expanded in x around 0 5.1%

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

      1. +-commutative 5.1%

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

   4. Simplified 5.1%

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

      1. flip-- 4.8%

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

      2. clear-num 4.8%

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

      3. associate-+l+ 4.8%

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

      4. metadata-eval 4.8%

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

      5. metadata-eval 4.8%

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

      6. difference-of-sqr-1 4.8%

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

      7. associate-+l+ 4.8%

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

      8. metadata-eval 4.8%

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

      9. associate--l+ 10.9%

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

      10. metadata-eval 10.9%

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

      11. +-rgt-identity 10.9%

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

   6. Applied egg-rr 10.9%

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

      1. clear-num 10.9%

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

      2. frac-2neg 10.9%

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

      3. associate-/r/ 10.9%

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

      4. *-commutative 10.9%

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

      5. distribute-rgt-neg-in 10.9%

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

   8. Applied egg-rr 10.9%

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

   9. Taylor expanded in x around 0 23.9%

      \[\leadsto \frac{1}{\color{blue}{\frac{-0.5}{x}} \cdot \left(-\left(x + 2\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-45}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-0.5}{x} \cdot \left(\left(-x\right) - 2\right)}\\ \end{array} \]

Alternative 5: 55.1% accurate, 10.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{-0.5}{x} \cdot \left(\left(-x\right) - 2\right)} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ 1.0 (* (/ -0.5 x) (- (- x) 2.0))))

C

double code(double x, double y) {
	return 1.0 / ((-0.5 / x) * (-x - 2.0));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 / (((-0.5d0) / x) * (-x - 2.0d0))
end function

Java

public static double code(double x, double y) {
	return 1.0 / ((-0.5 / x) * (-x - 2.0));
}

Python

def code(x, y):
	return 1.0 / ((-0.5 / x) * (-x - 2.0))

Julia

function code(x, y)
	return Float64(1.0 / Float64(Float64(-0.5 / x) * Float64(Float64(-x) - 2.0)))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 / ((-0.5 / x) * (-x - 2.0));
end

Wolfram

code[x_, y_] := N[(1.0 / N[(N[(-0.5 / x), $MachinePrecision] * N[((-x) - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.6%

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

2. Taylor expanded in x around 0 5.7%

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

   1. +-commutative 5.7%

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

4. Simplified 5.7%

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

   1. flip-- 5.5%

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

   2. clear-num 5.5%

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

   3. associate-+l+ 5.5%

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

   4. metadata-eval 5.5%

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

   5. metadata-eval 5.5%

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

   6. difference-of-sqr-1 5.5%

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

   7. associate-+l+ 5.5%

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

   8. metadata-eval 5.5%

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

   9. associate--l+ 49.6%

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

   10. metadata-eval 49.6%

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

   11. +-rgt-identity 49.6%

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

6. Applied egg-rr 49.6%

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

   1. clear-num 49.6%

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

   2. frac-2neg 49.6%

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

   3. associate-/r/ 49.6%

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

   4. *-commutative 49.6%

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

   5. distribute-rgt-neg-in 49.6%

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

8. Applied egg-rr 49.6%

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

9. Taylor expanded in x around 0 52.7%

   \[\leadsto \frac{1}{\color{blue}{\frac{-0.5}{x}} \cdot \left(-\left(x + 2\right)\right)} \]

10. Final simplification 52.7%

    \[\leadsto \frac{1}{\frac{-0.5}{x} \cdot \left(\left(-x\right) - 2\right)} \]

Alternative 6: 53.4% accurate, 109.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 55.6%

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

2. Taylor expanded in x around 0 50.0%

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

3. Final simplification 50.0%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023318 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))