Logistic function from Lakshay Garg

Percentage Accurate: 53.9% → 99.4%

Time: 18.3s

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: 53.9% 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: 99.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -5000:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x \cdot \left(1 + x \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -5000.0)
   (+ (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) -1.0)
   (expm1 (* x (+ 1.0 (* x -0.5))))))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -5000.0) {
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) + -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) <= -5000.0) {
		tmp = (2.0 / (1.0 + Math.exp((-2.0 * x)))) + -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) <= -5000.0:
		tmp = (2.0 / (1.0 + math.exp((-2.0 * x)))) + -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) <= -5000.0)
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) + -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], -5000.0], N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $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) < -5e3

   1. Initial program 100.0%

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

   if -5e3 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 37.7%

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

      1. add-log-exp 37.7%

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

      2. add-exp-log 37.7%

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

      3. expm1-define 37.7%

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

      4. log-div 37.7%

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

      5. log1p-define 37.7%

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

      6. exp-prod 37.7%

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

   4. Applied egg-rr 37.7%

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

   5. Taylor expanded in x around inf 37.7%

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

      1. expm1-define 37.7%

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

      2. log1p-define 37.7%

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

      3. *-commutative 37.7%

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

   7. Simplified 37.7%

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

   8. Taylor expanded in x around 0 98.6%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -5000:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x \cdot \left(1 + x \cdot -0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 1.5:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.15)
   -1.0
   (if (<= x 1.5) (+ x (* -0.3333333333333333 (pow x 3.0))) (log x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.15) {
		tmp = -1.0;
	} else if (x <= 1.5) {
		tmp = x + (-0.3333333333333333 * pow(x, 3.0));
	} else {
		tmp = log(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 <= (-1.15d0)) then
        tmp = -1.0d0
    else if (x <= 1.5d0) then
        tmp = x + ((-0.3333333333333333d0) * (x ** 3.0d0))
    else
        tmp = log(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.15) {
		tmp = -1.0;
	} else if (x <= 1.5) {
		tmp = x + (-0.3333333333333333 * Math.pow(x, 3.0));
	} else {
		tmp = Math.log(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.15:
		tmp = -1.0
	elif x <= 1.5:
		tmp = x + (-0.3333333333333333 * math.pow(x, 3.0))
	else:
		tmp = math.log(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.15)
		tmp = -1.0;
	elseif (x <= 1.5)
		tmp = Float64(x + Float64(-0.3333333333333333 * (x ^ 3.0)));
	else
		tmp = log(x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.15)
		tmp = -1.0;
	elseif (x <= 1.5)
		tmp = x + (-0.3333333333333333 * (x ^ 3.0));
	else
		tmp = log(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.15], -1.0, If[LessEqual[x, 1.5], N[(x + N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[x], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.1499999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 94.0%

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

   4. Taylor expanded in x around inf 98.5%

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

   if -1.1499999999999999 < x < 1.5

   1. Initial program 8.0%

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

   3. Taylor expanded in x around 0 98.8%

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

      1. distribute-rgt-in 98.8%

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

      2. *-lft-identity 98.8%

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

      3. associate-*l* 98.8%

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

      4. pow-plus 98.8%

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

      5. metadata-eval 98.8%

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

   5. Simplified 98.8%

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

   if 1.5 < 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-exp-log 100.0%

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

      3. expm1-define 100.0%

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

      4. log-div 100.0%

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

      5. log1p-define 100.0%

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

      6. exp-prod 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. log1p-define 100.0%

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

      2. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0 14.2%

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

      1. +-commutative 14.2%

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

   10. Simplified 14.2%

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

   11. Taylor expanded in x around inf 14.2%

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

       1. mul-1-neg 14.2%

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

       2. log-rec 14.2%

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

       3. remove-double-neg 14.2%

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

   13. Simplified 14.2%

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

Alternative 3: 78.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.55:\\ \;\;\;\;\mathsf{expm1}\left(x \cdot \left(1 + x \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.55) (expm1 (* x (+ 1.0 (* x -0.5)))) (log x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.55) {
		tmp = expm1((x * (1.0 + (x * -0.5))));
	} else {
		tmp = log(x);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.55) {
		tmp = Math.expm1((x * (1.0 + (x * -0.5))));
	} else {
		tmp = Math.log(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.55:
		tmp = math.expm1((x * (1.0 + (x * -0.5))))
	else:
		tmp = math.log(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.55)
		tmp = expm1(Float64(x * Float64(1.0 + Float64(x * -0.5))));
	else
		tmp = log(x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.55], N[(Exp[N[(x * N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision], N[Log[x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.55000000000000004

   1. Initial program 37.7%

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

      1. add-log-exp 37.7%

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

      2. add-exp-log 37.7%

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

      3. expm1-define 37.7%

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

      4. log-div 37.7%

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

      5. log1p-define 37.7%

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

      6. exp-prod 37.7%

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

   4. Applied egg-rr 37.7%

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

   5. Taylor expanded in x around inf 37.7%

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

      1. expm1-define 37.7%

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

      2. log1p-define 37.7%

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

      3. *-commutative 37.7%

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

   7. Simplified 37.7%

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

   8. Taylor expanded in x around 0 98.6%

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

   if 1.55000000000000004 < 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-exp-log 100.0%

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

      3. expm1-define 100.0%

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

      4. log-div 100.0%

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

      5. log1p-define 100.0%

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

      6. exp-prod 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. log1p-define 100.0%

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

      2. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0 14.2%

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

      1. +-commutative 14.2%

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

   10. Simplified 14.2%

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

   11. Taylor expanded in x around inf 14.2%

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

       1. mul-1-neg 14.2%

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

       2. log-rec 14.2%

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

       3. remove-double-neg 14.2%

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

   13. Simplified 14.2%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.55:\\ \;\;\;\;\mathsf{expm1}\left(x \cdot \left(1 + x \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 1.65:\\ \;\;\;\;\frac{x \cdot \left(x + 2\right)}{x + 2}\\ \mathbf{else}:\\ \;\;\;\;\log x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0) -1.0 (if (<= x 1.65) (/ (* x (+ x 2.0)) (+ x 2.0)) (log x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = -1.0;
	} else if (x <= 1.65) {
		tmp = (x * (x + 2.0)) / (x + 2.0);
	} else {
		tmp = log(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 if (x <= 1.65d0) then
        tmp = (x * (x + 2.0d0)) / (x + 2.0d0)
    else
        tmp = log(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 if (x <= 1.65) {
		tmp = (x * (x + 2.0)) / (x + 2.0);
	} else {
		tmp = Math.log(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = -1.0
	elif x <= 1.65:
		tmp = (x * (x + 2.0)) / (x + 2.0)
	else:
		tmp = math.log(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = -1.0;
	elseif (x <= 1.65)
		tmp = Float64(Float64(x * Float64(x + 2.0)) / Float64(x + 2.0));
	else
		tmp = log(x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = -1.0;
	elseif (x <= 1.65)
		tmp = (x * (x + 2.0)) / (x + 2.0);
	else
		tmp = log(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], -1.0, If[LessEqual[x, 1.65], N[(N[(x * N[(x + 2.0), $MachinePrecision]), $MachinePrecision] / N[(x + 2.0), $MachinePrecision]), $MachinePrecision], N[Log[x], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 94.0%

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

   4. Taylor expanded in x around inf 98.5%

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

   if -1 < x < 1.6499999999999999

   1. Initial program 8.0%

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

   3. Taylor expanded in x around 0 6.6%

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

      1. +-commutative 6.6%

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

   5. Simplified 6.6%

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

      1. flip-- 6.6%

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

      2. metadata-eval 6.6%

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

      3. difference-of-sqr-1 6.6%

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

      4. associate-+l+ 6.6%

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

      5. metadata-eval 6.6%

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

      6. associate--l+ 98.5%

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

      7. metadata-eval 98.5%

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

      8. +-rgt-identity 98.5%

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

      9. associate-+l+ 98.5%

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

      10. metadata-eval 98.5%

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

   7. Applied egg-rr 98.5%

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

   if 1.6499999999999999 < 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-exp-log 100.0%

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

      3. expm1-define 100.0%

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

      4. log-div 100.0%

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

      5. log1p-define 100.0%

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

      6. exp-prod 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. log1p-define 100.0%

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

      2. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0 14.2%

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

      1. +-commutative 14.2%

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

   10. Simplified 14.2%

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

   11. Taylor expanded in x around inf 14.2%

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

       1. mul-1-neg 14.2%

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

       2. log-rec 14.2%

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

       3. remove-double-neg 14.2%

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

   13. Simplified 14.2%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 1.65:\\ \;\;\;\;\frac{x \cdot \left(x + 2\right)}{x + 2}\\ \mathbf{else}:\\ \;\;\;\;\log x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.9% 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 94.0%

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

   4. Taylor expanded in x around inf 98.5%

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

   if -1 < x

   1. Initial program 39.6%

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

   3. Taylor expanded in x around 0 66.6%

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

Alternative 6: 27.1% 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 54.0%

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

3. Taylor expanded in x around 0 26.0%

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

4. Taylor expanded in x around inf 25.9%

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

Reproduce

herbie shell --seed 2024096 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))