Logistic function from Lakshay Garg

Percentage Accurate: 54.7% → 99.8%

Time: 11.2s

Alternatives: 8

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: 54.7% 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: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.02:\\ \;\;\;\;\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(e^{-2 \cdot x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x \cdot \left(1 + x \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -0.02)
   (expm1 (- (log 2.0) (log1p (exp (* -2.0 x)))))
   (expm1 (* x (+ 1.0 (* x -0.5))))))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.02) {
		tmp = expm1((log(2.0) - log1p(exp((-2.0 * x)))));
	} else {
		tmp = expm1((x * (1.0 + (x * -0.5))));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.02) {
		tmp = Math.expm1((Math.log(2.0) - Math.log1p(Math.exp((-2.0 * x)))));
	} else {
		tmp = Math.expm1((x * (1.0 + (x * -0.5))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (-2.0 * x) <= -0.02:
		tmp = math.expm1((math.log(2.0) - math.log1p(math.exp((-2.0 * x)))))
	else:
		tmp = math.expm1((x * (1.0 + (x * -0.5))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.02)
		tmp = expm1(Float64(log(2.0) - log1p(exp(Float64(-2.0 * x)))));
	else
		tmp = expm1(Float64(x * Float64(1.0 + Float64(x * -0.5))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.02], N[(Exp[N[(N[Log[2.0], $MachinePrecision] - N[Log[1 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision], N[(Exp[N[(x * N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -0.0200000000000000004

   1. Initial program 99.8%

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

      1. add-exp-log 99.8%

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

      2. expm1-define 99.8%

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

      3. log-div 99.9%

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

      4. log1p-define 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. Taylor expanded in x around inf 99.9%

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

   if -0.0200000000000000004 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 45.0%

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

      1. add-exp-log 45.0%

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

      2. expm1-define 45.0%

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

      3. log-div 45.0%

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

      4. log1p-define 45.0%

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

      5. exp-prod 45.0%

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

   4. Applied egg-rr 45.0%

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

   5. Taylor expanded in x around 0 99.4%

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

      1. *-commutative 99.4%

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

   7. Simplified 99.4%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{x \cdot \left(1 + x \cdot -0.5\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.02:\\ \;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} + -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x \cdot \left(1 + x \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -0.02)
   (+ (/ 2.0 (+ (exp (* -2.0 x)) 1.0)) -1.0)
   (expm1 (* x (+ 1.0 (* x -0.5))))))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.02) {
		tmp = (2.0 / (exp((-2.0 * x)) + 1.0)) + -1.0;
	} else {
		tmp = expm1((x * (1.0 + (x * -0.5))));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.02) {
		tmp = (2.0 / (Math.exp((-2.0 * x)) + 1.0)) + -1.0;
	} else {
		tmp = Math.expm1((x * (1.0 + (x * -0.5))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (-2.0 * x) <= -0.02:
		tmp = (2.0 / (math.exp((-2.0 * x)) + 1.0)) + -1.0
	else:
		tmp = math.expm1((x * (1.0 + (x * -0.5))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.02)
		tmp = Float64(Float64(2.0 / Float64(exp(Float64(-2.0 * x)) + 1.0)) + -1.0);
	else
		tmp = expm1(Float64(x * Float64(1.0 + Float64(x * -0.5))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.02], N[(N[(2.0 / N[(N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(Exp[N[(x * N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -0.0200000000000000004

   1. Initial program 99.8%

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

   if -0.0200000000000000004 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 45.0%

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

      1. add-exp-log 45.0%

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

      2. expm1-define 45.0%

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

      3. log-div 45.0%

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

      4. log1p-define 45.0%

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

      5. exp-prod 45.0%

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

   4. Applied egg-rr 45.0%

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

   5. Taylor expanded in x around 0 99.4%

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

      1. *-commutative 99.4%

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

   7. Simplified 99.4%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{x \cdot \left(1 + x \cdot -0.5\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.02:\\ \;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} + -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x \cdot \left(1 + x \cdot -0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.6:\\ \;\;\;\;\mathsf{expm1}\left(x \cdot \left(1 + x \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \mathsf{log1p}\left(x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.6) (expm1 (* x (+ 1.0 (* x -0.5)))) (* 2.0 (log1p (* x 0.5)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.6) {
		tmp = expm1((x * (1.0 + (x * -0.5))));
	} else {
		tmp = 2.0 * log1p((x * 0.5));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.6) {
		tmp = Math.expm1((x * (1.0 + (x * -0.5))));
	} else {
		tmp = 2.0 * Math.log1p((x * 0.5));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.6:
		tmp = math.expm1((x * (1.0 + (x * -0.5))))
	else:
		tmp = 2.0 * math.log1p((x * 0.5))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.6)
		tmp = expm1(Float64(x * Float64(1.0 + Float64(x * -0.5))));
	else
		tmp = Float64(2.0 * log1p(Float64(x * 0.5)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.6], N[(Exp[N[(x * N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision], N[(2.0 * N[Log[1 + N[(x * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.599999999999999978

   1. Initial program 45.5%

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

      1. add-exp-log 45.5%

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

      2. expm1-define 45.5%

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

      3. log-div 45.5%

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

      4. log1p-define 45.5%

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

      5. exp-prod 45.5%

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

   4. Applied egg-rr 45.5%

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

   5. Taylor expanded in x around 0 98.9%

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

      1. *-commutative 98.9%

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

   7. Simplified 98.9%

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

   if 0.599999999999999978 < x

   1. Initial program 100.0%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Add Preprocessing
   3. 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. add-sqr-sqrt 100.0%

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

      3. log-prod 100.0%

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

      4. sub-neg 100.0%

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

      5. exp-prod 100.0%

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

      6. metadata-eval 100.0%

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

      7. sub-neg 100.0%

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

      8. exp-prod 100.0%

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

      9. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. count-2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 13.9%

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

      1. *-commutative 13.9%

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

   9. Simplified 13.9%

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

       1. pow1 13.9%

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

       2. log1p-define 13.9%

          \[\leadsto {\left(2 \cdot \color{blue}{\mathsf{log1p}\left(x \cdot 0.5\right)}\right)}^{1} \]

   11. Applied egg-rr 13.9%

       \[\leadsto \color{blue}{{\left(2 \cdot \mathsf{log1p}\left(x \cdot 0.5\right)\right)}^{1}} \]
   12. Step-by-step derivation

       1. unpow1 13.9%

          \[\leadsto \color{blue}{2 \cdot \mathsf{log1p}\left(x \cdot 0.5\right)} \]

   13. Simplified 13.9%

       \[\leadsto \color{blue}{2 \cdot \mathsf{log1p}\left(x \cdot 0.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 77.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-8}:\\ \;\;\;\;\frac{2}{1 + \left(1 + x \cdot \left(x \cdot \left(2 + x \cdot -1.3333333333333333\right) - 2\right)\right)} + -1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \mathsf{log1p}\left(x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.1e-8)
   (+
    (/
     2.0
     (+ 1.0 (+ 1.0 (* x (- (* x (+ 2.0 (* x -1.3333333333333333))) 2.0)))))
    -1.0)
   (* 2.0 (log1p (* x 0.5)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.1e-8) {
		tmp = (2.0 / (1.0 + (1.0 + (x * ((x * (2.0 + (x * -1.3333333333333333))) - 2.0))))) + -1.0;
	} else {
		tmp = 2.0 * log1p((x * 0.5));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.1e-8) {
		tmp = (2.0 / (1.0 + (1.0 + (x * ((x * (2.0 + (x * -1.3333333333333333))) - 2.0))))) + -1.0;
	} else {
		tmp = 2.0 * Math.log1p((x * 0.5));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.1e-8:
		tmp = (2.0 / (1.0 + (1.0 + (x * ((x * (2.0 + (x * -1.3333333333333333))) - 2.0))))) + -1.0
	else:
		tmp = 2.0 * math.log1p((x * 0.5))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.1e-8)
		tmp = Float64(Float64(2.0 / Float64(1.0 + Float64(1.0 + Float64(x * Float64(Float64(x * Float64(2.0 + Float64(x * -1.3333333333333333))) - 2.0))))) + -1.0);
	else
		tmp = Float64(2.0 * log1p(Float64(x * 0.5)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.1e-8], N[(N[(2.0 / N[(1.0 + N[(1.0 + N[(x * N[(N[(x * N[(2.0 + N[(x * -1.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(2.0 * N[Log[1 + N[(x * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.09999999999999994e-8

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 98.4%

      \[\leadsto \frac{2}{1 + \color{blue}{\left(1 + x \cdot \left(x \cdot \left(2 + -1.3333333333333333 \cdot x\right) - 2\right)\right)}} - 1 \]

   if -2.09999999999999994e-8 < x

   1. Initial program 38.8%

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

      1. add-log-exp 38.8%

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

      2. add-sqr-sqrt 38.8%

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

      3. log-prod 38.8%

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

      4. sub-neg 38.8%

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

      5. exp-prod 38.8%

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

      6. metadata-eval 38.8%

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

      7. sub-neg 38.8%

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

      8. exp-prod 38.8%

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

      9. metadata-eval 38.8%

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

   4. Applied egg-rr 38.8%

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

      1. count-2 38.8%

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

   6. Simplified 38.8%

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

   7. Taylor expanded in x around 0 9.0%

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

      1. *-commutative 9.0%

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

   9. Simplified 9.0%

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

       1. pow1 9.0%

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

       2. log1p-define 69.9%

          \[\leadsto {\left(2 \cdot \color{blue}{\mathsf{log1p}\left(x \cdot 0.5\right)}\right)}^{1} \]

   11. Applied egg-rr 69.9%

       \[\leadsto \color{blue}{{\left(2 \cdot \mathsf{log1p}\left(x \cdot 0.5\right)\right)}^{1}} \]
   12. Step-by-step derivation

       1. unpow1 69.9%

          \[\leadsto \color{blue}{2 \cdot \mathsf{log1p}\left(x \cdot 0.5\right)} \]

   13. Simplified 69.9%

       \[\leadsto \color{blue}{2 \cdot \mathsf{log1p}\left(x \cdot 0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-8}:\\ \;\;\;\;\frac{2}{1 + \left(1 + x \cdot \left(x \cdot \left(2 + x \cdot -1.3333333333333333\right) - 2\right)\right)} + -1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \mathsf{log1p}\left(x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.6% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{1 + \left(1 + x \cdot \left(x \cdot \left(2 + x \cdot -1.3333333333333333\right) - 2\right)\right)} + -1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -7e-6)
   (+
    (/
     2.0
     (+ 1.0 (+ 1.0 (* x (- (* x (+ 2.0 (* x -1.3333333333333333))) 2.0)))))
    -1.0)
   x))

C

double code(double x, double y) {
	double tmp;
	if (x <= -7e-6) {
		tmp = (2.0 / (1.0 + (1.0 + (x * ((x * (2.0 + (x * -1.3333333333333333))) - 2.0))))) + -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 <= (-7d-6)) then
        tmp = (2.0d0 / (1.0d0 + (1.0d0 + (x * ((x * (2.0d0 + (x * (-1.3333333333333333d0)))) - 2.0d0))))) + (-1.0d0)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -7e-6) {
		tmp = (2.0 / (1.0 + (1.0 + (x * ((x * (2.0 + (x * -1.3333333333333333))) - 2.0))))) + -1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -7e-6:
		tmp = (2.0 / (1.0 + (1.0 + (x * ((x * (2.0 + (x * -1.3333333333333333))) - 2.0))))) + -1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -7e-6)
		tmp = Float64(Float64(2.0 / Float64(1.0 + Float64(1.0 + Float64(x * Float64(Float64(x * Float64(2.0 + Float64(x * -1.3333333333333333))) - 2.0))))) + -1.0);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -7e-6)
		tmp = (2.0 / (1.0 + (1.0 + (x * ((x * (2.0 + (x * -1.3333333333333333))) - 2.0))))) + -1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -7e-6], N[(N[(2.0 / N[(1.0 + N[(1.0 + N[(x * N[(N[(x * N[(2.0 + N[(x * -1.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if x < -6.99999999999999989e-6

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 98.4%

      \[\leadsto \frac{2}{1 + \color{blue}{\left(1 + x \cdot \left(x \cdot \left(2 + -1.3333333333333333 \cdot x\right) - 2\right)\right)}} - 1 \]

   if -6.99999999999999989e-6 < x

   1. Initial program 38.8%

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

   3. Taylor expanded in x around 0 67.4%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{1 + \left(1 + x \cdot \left(x \cdot \left(2 + x \cdot -1.3333333333333333\right) - 2\right)\right)} + -1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{2 + x \cdot \left(x \cdot \left(2 + x \cdot -1.3333333333333333\right) - 2\right)} + -1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -6e-6)
   (+
    (/ 2.0 (+ 2.0 (* x (- (* x (+ 2.0 (* x -1.3333333333333333))) 2.0))))
    -1.0)
   x))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6e-6) {
		tmp = (2.0 / (2.0 + (x * ((x * (2.0 + (x * -1.3333333333333333))) - 2.0)))) + -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 <= (-6d-6)) then
        tmp = (2.0d0 / (2.0d0 + (x * ((x * (2.0d0 + (x * (-1.3333333333333333d0)))) - 2.0d0)))) + (-1.0d0)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -6e-6) {
		tmp = (2.0 / (2.0 + (x * ((x * (2.0 + (x * -1.3333333333333333))) - 2.0)))) + -1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -6e-6:
		tmp = (2.0 / (2.0 + (x * ((x * (2.0 + (x * -1.3333333333333333))) - 2.0)))) + -1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6e-6)
		tmp = Float64(Float64(2.0 / Float64(2.0 + Float64(x * Float64(Float64(x * Float64(2.0 + Float64(x * -1.3333333333333333))) - 2.0)))) + -1.0);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -6e-6)
		tmp = (2.0 / (2.0 + (x * ((x * (2.0 + (x * -1.3333333333333333))) - 2.0)))) + -1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6e-6], N[(N[(2.0 / N[(2.0 + N[(x * N[(N[(x * N[(2.0 + N[(x * -1.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if x < -6.0000000000000002e-6

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 98.4%

      \[\leadsto \frac{2}{\color{blue}{2 + x \cdot \left(x \cdot \left(2 + -1.3333333333333333 \cdot x\right) - 2\right)}} - 1 \]

   if -6.0000000000000002e-6 < x

   1. Initial program 38.8%

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

   3. Taylor expanded in x around 0 67.4%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{2 + x \cdot \left(x \cdot \left(2 + x \cdot -1.3333333333333333\right) - 2\right)} + -1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.7% 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 98.6%

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

      1. *-commutative 98.6%

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

   5. Simplified 98.6%

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

   6. Taylor expanded in x around inf 99.1%

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

   if -1 < x

   1. Initial program 39.3%

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

   3. Taylor expanded in x around 0 67.3%

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

Alternative 8: 27.6% 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 58.0%

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

3. Taylor expanded in x around 0 34.0%

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

   1. *-commutative 34.0%

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

5. Simplified 34.0%

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

6. Taylor expanded in x around inf 32.7%

   \[\leadsto \color{blue}{-1} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024141 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))