Logistic function from Lakshay Garg

Percentage Accurate: 54.7% → 99.8%

Time: 14.0s

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: 54.7% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.0005:\\ \;\;\;\;\frac{1 + \frac{-4}{{\left(\mathsf{hypot}\left(1, e^{-x}\right)\right)}^{4}}}{-1 + \frac{2}{-1 - e^{-2 \cdot x}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x \cdot \left(1 + x \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -0.0005)
   (/
    (+ 1.0 (/ -4.0 (pow (hypot 1.0 (exp (- x))) 4.0)))
    (+ -1.0 (/ 2.0 (- -1.0 (exp (* -2.0 x))))))
   (expm1 (* x (+ 1.0 (* x -0.5))))))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.0005) {
		tmp = (1.0 + (-4.0 / pow(hypot(1.0, exp(-x)), 4.0))) / (-1.0 + (2.0 / (-1.0 - 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) <= -0.0005) {
		tmp = (1.0 + (-4.0 / Math.pow(Math.hypot(1.0, Math.exp(-x)), 4.0))) / (-1.0 + (2.0 / (-1.0 - 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) <= -0.0005:
		tmp = (1.0 + (-4.0 / math.pow(math.hypot(1.0, math.exp(-x)), 4.0))) / (-1.0 + (2.0 / (-1.0 - 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) <= -0.0005)
		tmp = Float64(Float64(1.0 + Float64(-4.0 / (hypot(1.0, exp(Float64(-x))) ^ 4.0))) / Float64(-1.0 + Float64(2.0 / Float64(-1.0 - 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], -0.0005], N[(N[(1.0 + N[(-4.0 / N[Power[N[Sqrt[1.0 ^ 2 + N[Exp[(-x)], $MachinePrecision] ^ 2], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + N[(2.0 / N[(-1.0 - N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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) < -5.0000000000000001e-4

   1. Initial program 99.7%

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

      1. flip-- 99.7%

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

      2. div-inv 99.7%

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

      3. metadata-eval 99.7%

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

      4. sub-neg 99.7%

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

      5. frac-times 99.8%

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

      6. metadata-eval 99.8%

         \[\leadsto \left(\frac{\color{blue}{4}}{\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)} + \left(-1\right)\right) \cdot \frac{1}{\frac{2}{1 + e^{-2 \cdot x}} + 1} \]

      7. pow2 99.8%

         \[\leadsto \left(\frac{4}{\color{blue}{{\left(1 + e^{-2 \cdot x}\right)}^{2}}} + \left(-1\right)\right) \cdot \frac{1}{\frac{2}{1 + e^{-2 \cdot x}} + 1} \]

      8. exp-prod 99.8%

         \[\leadsto \left(\frac{4}{{\left(1 + \color{blue}{{\left(e^{-2}\right)}^{x}}\right)}^{2}} + \left(-1\right)\right) \cdot \frac{1}{\frac{2}{1 + e^{-2 \cdot x}} + 1} \]

      9. metadata-eval 99.8%

         \[\leadsto \left(\frac{4}{{\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{2}} + \color{blue}{-1}\right) \cdot \frac{1}{\frac{2}{1 + e^{-2 \cdot x}} + 1} \]

      10. +-commutative 99.8%

          \[\leadsto \left(\frac{4}{{\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{2}} + -1\right) \cdot \frac{1}{\color{blue}{1 + \frac{2}{1 + e^{-2 \cdot x}}}} \]

      11. exp-prod 99.8%

          \[\leadsto \left(\frac{4}{{\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{2}} + -1\right) \cdot \frac{1}{1 + \frac{2}{1 + \color{blue}{{\left(e^{-2}\right)}^{x}}}} \]

   4. Applied egg-rr 99.8%

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

      1. associate-*r/ 99.8%

         \[\leadsto \color{blue}{\frac{\left(\frac{4}{{\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{2}} + -1\right) \cdot 1}{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}} \]

      2. *-rgt-identity 99.8%

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

      3. remove-double-neg 99.8%

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

      4. distribute-frac-neg2 99.8%

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

      5. distribute-frac-neg 99.8%

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

   6. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{1 + \frac{-4}{{\left(\mathsf{hypot}\left(1, e^{-x}\right)\right)}^{4}}}{-1 + \frac{2}{-1 - {\left(e^{-2}\right)}^{x}}}} \]

   7. Taylor expanded in x around inf 99.8%

      \[\leadsto \frac{1 + \frac{-4}{{\left(\mathsf{hypot}\left(1, e^{-x}\right)\right)}^{4}}}{-1 + \frac{2}{-1 - \color{blue}{e^{-2 \cdot x}}}} \]

   if -5.0000000000000001e-4 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 39.8%

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

      1. add-exp-log 39.8%

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

      2. expm1-define 39.8%

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

      3. log-div 39.8%

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

      4. log1p-define 39.8%

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

      5. exp-prod 39.8%

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

   4. Applied egg-rr 39.8%

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

   5. Taylor expanded in x around 0 99.5%

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

      1. *-commutative 99.5%

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

   7. Simplified 99.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.0005:\\ \;\;\;\;\frac{1 + \frac{-4}{{\left(\mathsf{hypot}\left(1, e^{-x}\right)\right)}^{4}}}{-1 + \frac{2}{-1 - e^{-2 \cdot x}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x \cdot \left(1 + x \cdot -0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.0005:\\ \;\;\;\;\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(e^{-2 \cdot x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x \cdot \left(1 + x \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -0.0005)
   (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) <= -0.0005) {
		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) <= -0.0005) {
		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) <= -0.0005:
		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) <= -0.0005)
		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], -0.0005], 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) < -5.0000000000000001e-4

   1. Initial program 99.7%

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

      1. add-exp-log 99.7%

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

      2. expm1-define 99.7%

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

      3. log-div 99.7%

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

      4. log1p-define 99.7%

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

      5. exp-prod 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around inf 99.7%

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

   if -5.0000000000000001e-4 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 39.8%

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

      1. add-exp-log 39.8%

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

      2. expm1-define 39.8%

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

      3. log-div 39.8%

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

      4. log1p-define 39.8%

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

      5. exp-prod 39.8%

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

   4. Applied egg-rr 39.8%

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

   5. Taylor expanded in x around 0 99.5%

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

      1. *-commutative 99.5%

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

   7. Simplified 99.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.0005:\\ \;\;\;\;\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(e^{-2 \cdot x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x \cdot \left(1 + x \cdot -0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.0005:\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x \cdot \left(1 + x \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -0.0005)
   (+ -1.0 (/ 2.0 (+ 1.0 (exp (* -2.0 x)))))
   (expm1 (* x (+ 1.0 (* x -0.5))))))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.0005) {
		tmp = -1.0 + (2.0 / (1.0 + 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) <= -0.0005) {
		tmp = -1.0 + (2.0 / (1.0 + 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) <= -0.0005:
		tmp = -1.0 + (2.0 / (1.0 + 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) <= -0.0005)
		tmp = Float64(-1.0 + Float64(2.0 / Float64(1.0 + 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], -0.0005], N[(-1.0 + N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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) < -5.0000000000000001e-4

   1. Initial program 99.7%

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

   if -5.0000000000000001e-4 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 39.8%

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

      1. add-exp-log 39.8%

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

      2. expm1-define 39.8%

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

      3. log-div 39.8%

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

      4. log1p-define 39.8%

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

      5. exp-prod 39.8%

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

   4. Applied egg-rr 39.8%

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

   5. Taylor expanded in x around 0 99.5%

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

      1. *-commutative 99.5%

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

   7. Simplified 99.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.0005:\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x \cdot \left(1 + x \cdot -0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{expm1}\left(x \cdot \left(1 + x \cdot -0.5\right)\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (expm1 (* x (+ 1.0 (* x -0.5)))))

C

double code(double x, double y) {
	return expm1((x * (1.0 + (x * -0.5))));
}

Java

public static double code(double x, double y) {
	return Math.expm1((x * (1.0 + (x * -0.5))));
}

Python

def code(x, y):
	return math.expm1((x * (1.0 + (x * -0.5))))

Julia

function code(x, y)
	return expm1(Float64(x * Float64(1.0 + Float64(x * -0.5))))
end

Wolfram

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

Derivation

1. Initial program 58.3%

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

   1. add-exp-log 58.3%

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

   2. expm1-define 58.3%

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

   3. log-div 58.3%

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

   4. log1p-define 58.3%

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

   5. exp-prod 58.3%

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

4. Applied egg-rr 58.3%

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

5. Taylor expanded in x around 0 69.8%

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

   1. *-commutative 69.8%

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

7. Simplified 69.8%

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

8. Final simplification 69.8%

   \[\leadsto \mathsf{expm1}\left(x \cdot \left(1 + x \cdot -0.5\right)\right) \]
9. Add Preprocessing

Alternative 5: 76.4% 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 97.0%

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

      1. *-commutative 97.0%

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

   5. Simplified 97.0%

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

   6. Taylor expanded in x around inf 98.7%

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

   if -1 < x

   1. Initial program 45.0%

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

   3. Taylor expanded in x around 0 61.6%

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

4. Final simplification 70.6%

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

Alternative 6: 28.4% 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 58.3%

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

3. Taylor expanded in x around 0 27.3%

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

   1. *-commutative 27.3%

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

5. Simplified 27.3%

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

6. Taylor expanded in x around inf 26.1%

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

7. Final simplification 26.1%

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

Reproduce

herbie shell --seed 2024073 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))