Logistic function from Lakshay Garg

Percentage Accurate: 54.7% → 99.6%

Time: 12.3s

Alternatives: 6

Speedup: 7.8×

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: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -200:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x \cdot \left(1 + x \cdot \left({x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot -0.022222222222222223\right) - 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -200.0)
   (+ (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) -1.0)
   (expm1
    (*
     x
     (+
      1.0
      (*
       x
       (-
        (*
         (pow x 2.0)
         (+ 0.08333333333333333 (* (pow x 2.0) -0.022222222222222223)))
        0.5)))))))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -200.0) {
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	} else {
		tmp = expm1((x * (1.0 + (x * ((pow(x, 2.0) * (0.08333333333333333 + (pow(x, 2.0) * -0.022222222222222223))) - 0.5)))));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -200.0) {
		tmp = (2.0 / (1.0 + Math.exp((-2.0 * x)))) + -1.0;
	} else {
		tmp = Math.expm1((x * (1.0 + (x * ((Math.pow(x, 2.0) * (0.08333333333333333 + (Math.pow(x, 2.0) * -0.022222222222222223))) - 0.5)))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (-2.0 * x) <= -200.0:
		tmp = (2.0 / (1.0 + math.exp((-2.0 * x)))) + -1.0
	else:
		tmp = math.expm1((x * (1.0 + (x * ((math.pow(x, 2.0) * (0.08333333333333333 + (math.pow(x, 2.0) * -0.022222222222222223))) - 0.5)))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -200.0)
		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 * Float64(Float64((x ^ 2.0) * Float64(0.08333333333333333 + Float64((x ^ 2.0) * -0.022222222222222223))) - 0.5)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -200.0], 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 * N[(N[(N[Power[x, 2.0], $MachinePrecision] * N[(0.08333333333333333 + N[(N[Power[x, 2.0], $MachinePrecision] * -0.022222222222222223), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -200

   1. Initial program 100.0%

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

   if -200 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 37.9%

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

      1. add-exp-log 37.9%

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

      2. expm1-define 37.9%

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

      3. log-div 37.8%

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

      4. log1p-define 37.9%

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

      5. exp-prod 37.9%

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

   4. Applied egg-rr 37.9%

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

   5. Taylor expanded in x around 0 99.7%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{x \cdot \left(1 + x \cdot \left({x}^{2} \cdot \left(0.08333333333333333 + -0.022222222222222223 \cdot {x}^{2}\right) - 0.5\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -200:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x \cdot \left(1 + x \cdot \left({x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot -0.022222222222222223\right) - 0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.0004:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x \cdot \left(1 + x \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -0.0004)
   (+ (/ 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) <= -0.0004) {
		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) <= -0.0004) {
		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) <= -0.0004:
		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) <= -0.0004)
		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], -0.0004], 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) < -4.00000000000000019e-4

   1. Initial program 99.7%

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

   if -4.00000000000000019e-4 < (*.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-exp-log 37.7%

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

      2. expm1-define 37.7%

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

      3. log-div 37.6%

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

      4. log1p-define 37.6%

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

      5. exp-prod 37.6%

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

   4. Applied egg-rr 37.6%

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

   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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.0004:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x \cdot \left(1 + x \cdot -0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.95:\\ \;\;\;\;\mathsf{expm1}\left(x \cdot \left(1 + x \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x + 2}{x \cdot 2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.95)
   (expm1 (* x (+ 1.0 (* x -0.5))))
   (/ 1.0 (/ (+ x 2.0) (* x 2.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.95) {
		tmp = expm1((x * (1.0 + (x * -0.5))));
	} else {
		tmp = 1.0 / ((x + 2.0) / (x * 2.0));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.95) {
		tmp = Math.expm1((x * (1.0 + (x * -0.5))));
	} else {
		tmp = 1.0 / ((x + 2.0) / (x * 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.95:
		tmp = math.expm1((x * (1.0 + (x * -0.5))))
	else:
		tmp = 1.0 / ((x + 2.0) / (x * 2.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.95)
		tmp = expm1(Float64(x * Float64(1.0 + Float64(x * -0.5))));
	else
		tmp = Float64(1.0 / Float64(Float64(x + 2.0) / Float64(x * 2.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.95], N[(Exp[N[(x * N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision], N[(1.0 / N[(N[(x + 2.0), $MachinePrecision] / N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.94999999999999996

   1. Initial program 37.9%

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

      1. add-exp-log 37.9%

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

      2. expm1-define 37.9%

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

      3. log-div 37.8%

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

      4. log1p-define 37.9%

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

      5. exp-prod 37.9%

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

   4. Applied egg-rr 37.9%

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

   5. Taylor expanded in x around 0 99.3%

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

      1. *-commutative 99.3%

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

   7. Simplified 99.3%

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

   if 0.94999999999999996 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 5.3%

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

      1. +-commutative 5.3%

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

   5. Simplified 5.3%

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

      1. flip-- 5.0%

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

      2. clear-num 5.0%

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

      3. associate-+l+ 5.0%

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

      4. metadata-eval 5.0%

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

      5. metadata-eval 5.0%

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

      6. difference-of-sqr-1 5.0%

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

      7. associate-+l+ 5.0%

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

      8. metadata-eval 5.0%

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

      9. associate--l+ 5.0%

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

      10. metadata-eval 5.0%

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

      11. +-rgt-identity 5.0%

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

   7. Applied egg-rr 5.0%

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

   8. Taylor expanded in x around 0 18.8%

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

      1. *-commutative 18.8%

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

   10. Simplified 18.8%

       \[\leadsto \frac{1}{\frac{x + 2}{\color{blue}{x \cdot 2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.95:\\ \;\;\;\;\mathsf{expm1}\left(x \cdot \left(1 + x \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x + 2}{x \cdot 2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x + 2}{x \cdot 2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2e-20) (expm1 x) (/ 1.0 (/ (+ x 2.0) (* x 2.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2e-20) {
		tmp = expm1(x);
	} else {
		tmp = 1.0 / ((x + 2.0) / (x * 2.0));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 2e-20) {
		tmp = Math.expm1(x);
	} else {
		tmp = 1.0 / ((x + 2.0) / (x * 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 2e-20:
		tmp = math.expm1(x)
	else:
		tmp = 1.0 / ((x + 2.0) / (x * 2.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2e-20)
		tmp = expm1(x);
	else
		tmp = Float64(1.0 / Float64(Float64(x + 2.0) / Float64(x * 2.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2e-20], N[(Exp[x] - 1), $MachinePrecision], N[(1.0 / N[(N[(x + 2.0), $MachinePrecision] / N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.99999999999999989e-20

   1. Initial program 37.0%

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

      1. add-exp-log 37.0%

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

      2. expm1-define 37.0%

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

      3. log-div 37.0%

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

      4. log1p-define 37.1%

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

      5. exp-prod 37.1%

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

   4. Applied egg-rr 37.1%

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

   5. Taylor expanded in x around 0 99.7%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{x \cdot \left(1 + x \cdot \left({x}^{2} \cdot \left(0.08333333333333333 + -0.022222222222222223 \cdot {x}^{2}\right) - 0.5\right)\right)}\right) \]

   6. Taylor expanded in x around 0 99.2%

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

   if 1.99999999999999989e-20 < x

   1. Initial program 96.6%

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

   3. Taylor expanded in x around 0 10.5%

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

      1. +-commutative 10.5%

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

   5. Simplified 10.5%

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

      1. flip-- 10.1%

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

      2. clear-num 10.1%

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

      3. associate-+l+ 10.1%

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

      4. metadata-eval 10.1%

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

      5. metadata-eval 10.1%

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

      6. difference-of-sqr-1 10.2%

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

      7. associate-+l+ 10.2%

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

      8. metadata-eval 10.2%

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

      9. associate--l+ 12.4%

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

      10. metadata-eval 12.4%

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

      11. +-rgt-identity 12.4%

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

   7. Applied egg-rr 12.4%

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

   8. Taylor expanded in x around 0 22.5%

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

      1. *-commutative 22.5%

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

   10. Simplified 22.5%

       \[\leadsto \frac{1}{\frac{x + 2}{\color{blue}{x \cdot 2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x + 2}{x \cdot 2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 55.1% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.66:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x + 2}{x \cdot 2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.66) x (/ 1.0 (/ (+ x 2.0) (* x 2.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.66) {
		tmp = x;
	} else {
		tmp = 1.0 / ((x + 2.0) / (x * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.66d0) then
        tmp = x
    else
        tmp = 1.0d0 / ((x + 2.0d0) / (x * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.66) {
		tmp = x;
	} else {
		tmp = 1.0 / ((x + 2.0) / (x * 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.66:
		tmp = x
	else:
		tmp = 1.0 / ((x + 2.0) / (x * 2.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.66)
		tmp = x;
	else
		tmp = Float64(1.0 / Float64(Float64(x + 2.0) / Float64(x * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.66)
		tmp = x;
	else
		tmp = 1.0 / ((x + 2.0) / (x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.66], x, N[(1.0 / N[(N[(x + 2.0), $MachinePrecision] / N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.660000000000000031

   1. Initial program 37.9%

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

   3. Taylor expanded in x around 0 68.9%

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

   if 0.660000000000000031 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 5.3%

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

      1. +-commutative 5.3%

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

   5. Simplified 5.3%

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

      1. flip-- 5.0%

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

      2. clear-num 5.0%

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

      3. associate-+l+ 5.0%

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

      4. metadata-eval 5.0%

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

      5. metadata-eval 5.0%

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

      6. difference-of-sqr-1 5.0%

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

      7. associate-+l+ 5.0%

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

      8. metadata-eval 5.0%

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

      9. associate--l+ 5.0%

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

      10. metadata-eval 5.0%

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

      11. +-rgt-identity 5.0%

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

   7. Applied egg-rr 5.0%

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

   8. Taylor expanded in x around 0 18.8%

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

      1. *-commutative 18.8%

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

   10. Simplified 18.8%

       \[\leadsto \frac{1}{\frac{x + 2}{\color{blue}{x \cdot 2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.66:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x + 2}{x \cdot 2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.7% 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 51.5%

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

3. Taylor expanded in x around 0 55.0%

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

4. Final simplification 55.0%

   \[\leadsto x \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024078 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))