Logistic function from Lakshay Garg

Percentage Accurate: 54.4% → 99.0%

Time: 17.6s

Alternatives: 9

Speedup: 21.3×

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: 54.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.001:\\ \;\;\;\;2 \cdot \log \left(\sqrt{e^{-1 + \frac{2}{1 + e^{-2 \cdot x}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -0.001)
   (* 2.0 (log (sqrt (exp (+ -1.0 (/ 2.0 (+ 1.0 (exp (* -2.0 x)))))))))
   (expm1 x)))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.001) {
		tmp = 2.0 * log(sqrt(exp((-1.0 + (2.0 / (1.0 + exp((-2.0 * x))))))));
	} else {
		tmp = expm1(x);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.001) {
		tmp = 2.0 * Math.log(Math.sqrt(Math.exp((-1.0 + (2.0 / (1.0 + Math.exp((-2.0 * x))))))));
	} else {
		tmp = Math.expm1(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (-2.0 * x) <= -0.001:
		tmp = 2.0 * math.log(math.sqrt(math.exp((-1.0 + (2.0 / (1.0 + math.exp((-2.0 * x))))))))
	else:
		tmp = math.expm1(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.001)
		tmp = Float64(2.0 * log(sqrt(exp(Float64(-1.0 + Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))))))));
	else
		tmp = expm1(x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.001], N[(2.0 * N[Log[N[Sqrt[N[Exp[N[(-1.0 + N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(Exp[x] - 1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 -2 x) < -1e-3

   1. Initial program 99.7%

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

      1. add-log-exp 99.7%

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

      2. *-un-lft-identity 99.7%

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

      3. log-prod 99.7%

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

      4. metadata-eval 99.7%

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

      5. add-log-exp 99.7%

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

      6. add-exp-log 99.7%

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

      7. expm1-def 99.7%

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

      8. log-div 99.7%

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

      9. log1p-udef 99.7%

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

      10. exp-prod 99.7%

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

   3. Applied egg-rr 99.7%

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

      1. +-lft-identity 99.7%

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

   5. Simplified 99.7%

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

      1. add-log-exp 99.7%

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

      2. add-sqr-sqrt 99.7%

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

      3. log-prod 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. count-2 99.8%

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

      2. +-commutative 99.8%

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

      3. +-commutative 99.8%

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

   9. Simplified 99.8%

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

   10. Taylor expanded in x around inf 99.8%

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

   if -1e-3 < (*.f64 -2 x)

   1. Initial program 42.9%

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

      1. add-log-exp 42.9%

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

      2. *-un-lft-identity 42.9%

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

      3. log-prod 42.9%

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

      4. metadata-eval 42.9%

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

      5. add-log-exp 42.9%

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

      6. add-exp-log 42.9%

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

      7. expm1-def 42.9%

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

      8. log-div 42.9%

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

      9. log1p-udef 42.9%

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

      10. exp-prod 42.9%

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

   3. Applied egg-rr 42.9%

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

      1. +-lft-identity 42.9%

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

   5. Simplified 42.9%

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

   6. Taylor expanded in x around 0 99.3%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.001:\\ \;\;\;\;2 \cdot \log \left(\sqrt{e^{-1 + \frac{2}{1 + e^{-2 \cdot x}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \]

Alternative 2: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.001:\\ \;\;\;\;\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(e^{-2 \cdot x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -0.001)
   (expm1 (- (log 2.0) (log1p (exp (* -2.0 x)))))
   (expm1 x)))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.001) {
		tmp = expm1((log(2.0) - log1p(exp((-2.0 * x)))));
	} else {
		tmp = expm1(x);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.001) {
		tmp = Math.expm1((Math.log(2.0) - Math.log1p(Math.exp((-2.0 * x)))));
	} else {
		tmp = Math.expm1(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (-2.0 * x) <= -0.001:
		tmp = math.expm1((math.log(2.0) - math.log1p(math.exp((-2.0 * x)))))
	else:
		tmp = math.expm1(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.001)
		tmp = expm1(Float64(log(2.0) - log1p(exp(Float64(-2.0 * x)))));
	else
		tmp = expm1(x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.001], 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[x] - 1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 -2 x) < -1e-3

   1. Initial program 99.7%

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

      1. add-log-exp 99.7%

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

      2. *-un-lft-identity 99.7%

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

      3. log-prod 99.7%

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

      4. metadata-eval 99.7%

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

      5. add-log-exp 99.7%

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

      6. add-exp-log 99.7%

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

      7. expm1-def 99.7%

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

      8. log-div 99.7%

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

      9. log1p-udef 99.7%

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

      10. exp-prod 99.7%

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

   3. Applied egg-rr 99.7%

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

      1. +-lft-identity 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in x around inf 99.7%

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

   if -1e-3 < (*.f64 -2 x)

   1. Initial program 42.9%

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

      1. add-log-exp 42.9%

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

      2. *-un-lft-identity 42.9%

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

      3. log-prod 42.9%

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

      4. metadata-eval 42.9%

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

      5. add-log-exp 42.9%

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

      6. add-exp-log 42.9%

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

      7. expm1-def 42.9%

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

      8. log-div 42.9%

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

      9. log1p-udef 42.9%

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

      10. exp-prod 42.9%

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

   3. Applied egg-rr 42.9%

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

      1. +-lft-identity 42.9%

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

   5. Simplified 42.9%

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

   6. Taylor expanded in x around 0 99.3%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.001:\\ \;\;\;\;\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(e^{-2 \cdot x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \]

Alternative 3: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.001:\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -0.001)
   (+ -1.0 (/ 2.0 (+ 1.0 (exp (* -2.0 x)))))
   (expm1 x)))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.001) {
		tmp = -1.0 + (2.0 / (1.0 + exp((-2.0 * x))));
	} else {
		tmp = expm1(x);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.001) {
		tmp = -1.0 + (2.0 / (1.0 + Math.exp((-2.0 * x))));
	} else {
		tmp = Math.expm1(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (-2.0 * x) <= -0.001:
		tmp = -1.0 + (2.0 / (1.0 + math.exp((-2.0 * x))))
	else:
		tmp = math.expm1(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.001)
		tmp = Float64(-1.0 + Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))));
	else
		tmp = expm1(x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.001], N[(-1.0 + N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Exp[x] - 1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 -2 x) < -1e-3

   1. Initial program 99.7%

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

   if -1e-3 < (*.f64 -2 x)

   1. Initial program 42.9%

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

      1. add-log-exp 42.9%

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

      2. *-un-lft-identity 42.9%

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

      3. log-prod 42.9%

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

      4. metadata-eval 42.9%

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

      5. add-log-exp 42.9%

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

      6. add-exp-log 42.9%

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

      7. expm1-def 42.9%

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

      8. log-div 42.9%

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

      9. log1p-udef 42.9%

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

      10. exp-prod 42.9%

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

   3. Applied egg-rr 42.9%

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

      1. +-lft-identity 42.9%

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

   5. Simplified 42.9%

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

   6. Taylor expanded in x around 0 99.3%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.001:\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \]

Alternative 4: 79.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{x + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4e-14) (expm1 x) (* x (/ 2.0 (+ x 2.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4e-14) {
		tmp = expm1(x);
	} else {
		tmp = x * (2.0 / (x + 2.0));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -4e-14) {
		tmp = Math.expm1(x);
	} else {
		tmp = x * (2.0 / (x + 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4e-14:
		tmp = math.expm1(x)
	else:
		tmp = x * (2.0 / (x + 2.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4e-14)
		tmp = expm1(x);
	else
		tmp = Float64(x * Float64(2.0 / Float64(x + 2.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4e-14], N[(Exp[x] - 1), $MachinePrecision], N[(x * N[(2.0 / N[(x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4e-14

   1. Initial program 98.4%

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

      1. add-log-exp 98.4%

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

      2. *-un-lft-identity 98.4%

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

      3. log-prod 98.4%

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

      4. metadata-eval 98.4%

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

      5. add-log-exp 98.4%

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

      6. add-exp-log 98.4%

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

      7. expm1-def 98.4%

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

      8. log-div 98.4%

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

      9. log1p-udef 98.5%

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

      10. exp-prod 98.5%

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

   3. Applied egg-rr 98.5%

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

      1. +-lft-identity 98.5%

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

   5. Simplified 98.5%

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

   6. Taylor expanded in x around 0 98.4%

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

   if -4e-14 < x

   1. Initial program 41.8%

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

   2. Taylor expanded in x around 0 6.6%

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

      1. +-commutative 6.6%

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

   4. Simplified 6.6%

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

      1. flip-- 6.5%

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

      2. clear-num 6.5%

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

      3. associate-+l+ 6.5%

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

      4. metadata-eval 6.5%

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

      5. metadata-eval 6.5%

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

      6. difference-of-sqr-1 6.5%

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

      7. associate-+l+ 6.5%

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

      8. metadata-eval 6.5%

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

      9. associate--l+ 64.4%

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

      10. metadata-eval 64.4%

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

      11. +-rgt-identity 64.4%

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

   6. Applied egg-rr 64.4%

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

   7. Taylor expanded in x around 0 69.1%

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

      1. *-commutative 69.1%

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

   9. Simplified 69.1%

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

       1. expm1-log1p-u 69.1%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{x + 2}{x \cdot 2}}\right)\right)} \]

       2. expm1-udef 11.3%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\frac{x + 2}{x \cdot 2}}\right)} - 1} \]

       3. associate-/r/ 11.3%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{x + 2} \cdot \left(x \cdot 2\right)}\right)} - 1 \]

       4. +-commutative 11.3%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{2 + x}} \cdot \left(x \cdot 2\right)\right)} - 1 \]

       5. *-commutative 11.3%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{2 + x} \cdot \color{blue}{\left(2 \cdot x\right)}\right)} - 1 \]

   11. Applied egg-rr 11.3%

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

       1. expm1-def 69.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{2 + x} \cdot \left(2 \cdot x\right)\right)\right)} \]

       2. expm1-log1p 69.3%

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

       3. associate-*r* 69.3%

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

       4. *-commutative 69.3%

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

       5. associate-*l/ 69.3%

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

       6. metadata-eval 69.3%

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

       7. +-commutative 69.3%

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

   13. Simplified 69.3%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{x + 2}\\ \end{array} \]

Alternative 5: 79.7% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 2.5:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;2 - \frac{4}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0) -1.0 (if (<= x 2.5) x (- 2.0 (/ 4.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = -1.0;
	} else if (x <= 2.5) {
		tmp = x;
	} else {
		tmp = 2.0 - (4.0 / 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 <= 2.5d0) then
        tmp = x
    else
        tmp = 2.0d0 - (4.0d0 / 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 <= 2.5) {
		tmp = x;
	} else {
		tmp = 2.0 - (4.0 / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = -1.0
	elif x <= 2.5:
		tmp = x
	else:
		tmp = 2.0 - (4.0 / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = -1.0;
	elseif (x <= 2.5)
		tmp = x;
	else
		tmp = Float64(2.0 - Float64(4.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = -1.0;
	elseif (x <= 2.5)
		tmp = x;
	else
		tmp = 2.0 - (4.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], -1.0, If[LessEqual[x, 2.5], x, N[(2.0 - N[(4.0 / x), $MachinePrecision]), $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. Taylor expanded in x around 0 98.7%

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

      1. *-commutative 98.7%

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

   4. Simplified 98.7%

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

   5. Taylor expanded in x around inf 100.0%

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

   if -1 < x < 2.5

   1. Initial program 9.3%

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

   2. Taylor expanded in x around 0 98.5%

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

   if 2.5 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 5.8%

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

      1. +-commutative 5.8%

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

   4. Simplified 5.8%

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

      1. flip-- 5.5%

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

      2. clear-num 5.5%

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

      3. associate-+l+ 5.5%

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

      4. metadata-eval 5.5%

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

      5. metadata-eval 5.5%

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

      6. difference-of-sqr-1 5.5%

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

      7. associate-+l+ 5.5%

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

      8. metadata-eval 5.5%

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

      9. associate--l+ 5.5%

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

      10. metadata-eval 5.5%

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

      11. +-rgt-identity 5.5%

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

   6. Applied egg-rr 5.5%

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

   7. Taylor expanded in x around 0 18.8%

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

      1. *-commutative 18.8%

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

   9. Simplified 18.8%

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

   10. Taylor expanded in x around inf 18.8%

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

       1. associate-*r/ 18.8%

          \[\leadsto 2 - \color{blue}{\frac{4 \cdot 1}{x}} \]

       2. metadata-eval 18.8%

          \[\leadsto 2 - \frac{\color{blue}{4}}{x} \]

   12. Simplified 18.8%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 2.5:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;2 - \frac{4}{x}\\ \end{array} \]

Alternative 6: 79.1% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.68:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{x + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.68) -1.0 (* x (/ 2.0 (+ x 2.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.68) {
		tmp = -1.0;
	} else {
		tmp = 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 <= (-0.68d0)) then
        tmp = -1.0d0
    else
        tmp = 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 <= -0.68) {
		tmp = -1.0;
	} else {
		tmp = x * (2.0 / (x + 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.68:
		tmp = -1.0
	else:
		tmp = x * (2.0 / (x + 2.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.68)
		tmp = -1.0;
	else
		tmp = Float64(x * Float64(2.0 / Float64(x + 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.68)
		tmp = -1.0;
	else
		tmp = x * (2.0 / (x + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.68], -1.0, N[(x * N[(2.0 / N[(x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.680000000000000049

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.7%

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

      1. *-commutative 98.7%

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

   4. Simplified 98.7%

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

   5. Taylor expanded in x around inf 100.0%

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

   if -0.680000000000000049 < x

   1. Initial program 42.1%

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

   2. Taylor expanded in x around 0 7.2%

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

      1. +-commutative 7.2%

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

   4. Simplified 7.2%

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

      1. flip-- 7.1%

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

      2. clear-num 7.1%

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

      3. associate-+l+ 7.1%

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

      4. metadata-eval 7.1%

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

      5. metadata-eval 7.1%

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

      6. difference-of-sqr-1 7.1%

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

      7. associate-+l+ 7.1%

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

      8. metadata-eval 7.1%

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

      9. associate--l+ 64.6%

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

      10. metadata-eval 64.6%

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

      11. +-rgt-identity 64.6%

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

   6. Applied egg-rr 64.6%

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

   7. Taylor expanded in x around 0 68.9%

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

      1. *-commutative 68.9%

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

   9. Simplified 68.9%

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

       1. expm1-log1p-u 68.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{x + 2}{x \cdot 2}}\right)\right)} \]

       2. expm1-udef 11.8%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\frac{x + 2}{x \cdot 2}}\right)} - 1} \]

       3. associate-/r/ 11.8%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{x + 2} \cdot \left(x \cdot 2\right)}\right)} - 1 \]

       4. +-commutative 11.8%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{2 + x}} \cdot \left(x \cdot 2\right)\right)} - 1 \]

       5. *-commutative 11.8%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{2 + x} \cdot \color{blue}{\left(2 \cdot x\right)}\right)} - 1 \]

   11. Applied egg-rr 11.8%

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

       1. expm1-def 69.1%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{2 + x} \cdot \left(2 \cdot x\right)\right)\right)} \]

       2. expm1-log1p 69.1%

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

       3. associate-*r* 69.1%

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

       4. *-commutative 69.1%

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

       5. associate-*l/ 69.1%

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

       6. metadata-eval 69.1%

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

       7. +-commutative 69.1%

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

   13. Simplified 69.1%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.68:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{x + 2}\\ \end{array} \]

Alternative 7: 79.7% accurate, 21.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -1.0) -1.0 (if (<= x 2.0) x 2.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = -1.0;
	} else if (x <= 2.0) {
		tmp = x;
	} else {
		tmp = 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 <= (-1.0d0)) then
        tmp = -1.0d0
    else if (x <= 2.0d0) then
        tmp = x
    else
        tmp = 2.0d0
    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 <= 2.0) {
		tmp = x;
	} else {
		tmp = 2.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = -1.0
	elif x <= 2.0:
		tmp = x
	else:
		tmp = 2.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = -1.0;
	elseif (x <= 2.0)
		tmp = x;
	else
		tmp = 2.0;
	end
	return tmp
end

MATLAB

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

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], -1.0, If[LessEqual[x, 2.0], x, 2.0]]

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. Taylor expanded in x around 0 98.7%

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

      1. *-commutative 98.7%

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

   4. Simplified 98.7%

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

   5. Taylor expanded in x around inf 100.0%

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

   if -1 < x < 2

   1. Initial program 9.3%

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

   2. Taylor expanded in x around 0 98.5%

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

   if 2 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 5.8%

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

      1. +-commutative 5.8%

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

   4. Simplified 5.8%

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

      1. flip-- 5.5%

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

      2. clear-num 5.5%

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

      3. associate-+l+ 5.5%

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

      4. metadata-eval 5.5%

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

      5. metadata-eval 5.5%

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

      6. difference-of-sqr-1 5.5%

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

      7. associate-+l+ 5.5%

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

      8. metadata-eval 5.5%

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

      9. associate--l+ 5.5%

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

      10. metadata-eval 5.5%

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

      11. +-rgt-identity 5.5%

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

   6. Applied egg-rr 5.5%

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

   7. Taylor expanded in x around 0 18.8%

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

      1. *-commutative 18.8%

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

   9. Simplified 18.8%

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

   10. Taylor expanded in x around inf 18.7%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]

Alternative 8: 33.0% accurate, 35.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{-308}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 1.1e-308) -1.0 2.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.1e-308) {
		tmp = -1.0;
	} else {
		tmp = 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 <= 1.1d-308) then
        tmp = -1.0d0
    else
        tmp = 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.1e-308) {
		tmp = -1.0;
	} else {
		tmp = 2.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.1e-308:
		tmp = -1.0
	else:
		tmp = 2.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.1e-308)
		tmp = -1.0;
	else
		tmp = 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.1e-308)
		tmp = -1.0;
	else
		tmp = 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.1e-308], -1.0, 2.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001e-308

   1. Initial program 60.2%

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

   2. Taylor expanded in x around 0 58.9%

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

      1. *-commutative 58.9%

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

   4. Simplified 58.9%

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

   5. Taylor expanded in x around inf 58.2%

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

   if 1.1000000000000001e-308 < x

   1. Initial program 56.2%

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

   2. Taylor expanded in x around 0 6.5%

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

      1. +-commutative 6.5%

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

   4. Simplified 6.5%

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

      1. flip-- 6.4%

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

      2. clear-num 6.4%

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

      3. associate-+l+ 6.4%

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

      4. metadata-eval 6.4%

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

      5. metadata-eval 6.4%

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

      6. difference-of-sqr-1 6.4%

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

      7. associate-+l+ 6.4%

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

      8. metadata-eval 6.4%

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

      9. associate--l+ 49.9%

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

      10. metadata-eval 49.9%

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

      11. +-rgt-identity 49.9%

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

   6. Applied egg-rr 49.9%

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

   7. Taylor expanded in x around 0 56.6%

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

      1. *-commutative 56.6%

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

   9. Simplified 56.6%

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

   10. Taylor expanded in x around inf 12.3%

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

4. Final simplification 35.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{-308}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]

Alternative 9: 28.1% 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 58.2%

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

2. Taylor expanded in x around 0 31.3%

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

   1. *-commutative 31.3%

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

4. Simplified 31.3%

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

5. Taylor expanded in x around inf 29.9%

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

6. Final simplification 29.9%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023187 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))