Logistic function from Lakshay Garg

Percentage Accurate: 54.0% → 99.6%

Time: 9.8s

Alternatives: 7

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: 54.0% 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, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.05:\\ \;\;\;\;\mathsf{expm1}\left({\left({\left({\left({\left(\log 2 - \mathsf{log1p}\left(e^{-2 \cdot x}\right)\right)}^{6}\right)}^{0.16666666666666666}\right)}^{3}\right)}^{0.3333333333333333}\right)\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-5}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -0.05)
   (expm1
    (pow
     (pow
      (pow
       (pow (- (log 2.0) (log1p (exp (* -2.0 x)))) 6.0)
       0.16666666666666666)
      3.0)
     0.3333333333333333))
   (if (<= (* -2.0 x) 5e-5) (+ x (* -0.3333333333333333 (pow x 3.0))) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.05) {
		tmp = expm1(pow(pow(pow(pow((log(2.0) - log1p(exp((-2.0 * x)))), 6.0), 0.16666666666666666), 3.0), 0.3333333333333333));
	} else if ((-2.0 * x) <= 5e-5) {
		tmp = x + (-0.3333333333333333 * pow(x, 3.0));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.05) {
		tmp = Math.expm1(Math.pow(Math.pow(Math.pow(Math.pow((Math.log(2.0) - Math.log1p(Math.exp((-2.0 * x)))), 6.0), 0.16666666666666666), 3.0), 0.3333333333333333));
	} else if ((-2.0 * x) <= 5e-5) {
		tmp = x + (-0.3333333333333333 * Math.pow(x, 3.0));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (-2.0 * x) <= -0.05:
		tmp = math.expm1(math.pow(math.pow(math.pow(math.pow((math.log(2.0) - math.log1p(math.exp((-2.0 * x)))), 6.0), 0.16666666666666666), 3.0), 0.3333333333333333))
	elif (-2.0 * x) <= 5e-5:
		tmp = x + (-0.3333333333333333 * math.pow(x, 3.0))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.05)
		tmp = expm1(((((Float64(log(2.0) - log1p(exp(Float64(-2.0 * x)))) ^ 6.0) ^ 0.16666666666666666) ^ 3.0) ^ 0.3333333333333333));
	elseif (Float64(-2.0 * x) <= 5e-5)
		tmp = Float64(x + Float64(-0.3333333333333333 * (x ^ 3.0)));
	else
		tmp = -1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.05], N[(Exp[N[Power[N[Power[N[Power[N[Power[N[(N[Log[2.0], $MachinePrecision] - N[Log[1 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 6.0], $MachinePrecision], 0.16666666666666666], $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]] - 1), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 5e-5], N[(x + N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 -2 x) < -0.050000000000000003

   1. Initial program 99.9%

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

      1. add-exp-log 99.9%

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

      2. expm1-def 99.9%

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

      3. log-div 99.9%

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

      4. log1p-udef 99.9%

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

      5. exp-prod 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. add-cbrt-cube 99.9%

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\sqrt[3]{\left(\left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right) \cdot \left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)\right) \cdot \left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)}}\right) \]

      2. pow1/3 99.9%

         \[\leadsto \mathsf{expm1}\left(\color{blue}{{\left(\left(\left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right) \cdot \left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)\right) \cdot \left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)\right)}^{0.3333333333333333}}\right) \]

      3. pow3 100.0%

         \[\leadsto \mathsf{expm1}\left({\color{blue}{\left({\left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)}^{3}\right)}}^{0.3333333333333333}\right) \]

      4. metadata-eval 100.0%

         \[\leadsto \mathsf{expm1}\left({\left({\left(\log \color{blue}{\left(1 + 1\right)} - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)}^{3}\right)}^{0.3333333333333333}\right) \]

      5. log1p-udef 100.0%

         \[\leadsto \mathsf{expm1}\left({\left({\left(\color{blue}{\mathsf{log1p}\left(1\right)} - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)}^{3}\right)}^{0.3333333333333333}\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{{\left({\left(\mathsf{log1p}\left(1\right) - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)}^{3}\right)}^{0.3333333333333333}}\right) \]
   7. Step-by-step derivation

      1. rem-cbrt-cube 99.9%

         \[\leadsto \mathsf{expm1}\left({\left({\color{blue}{\left(\sqrt[3]{{\left(\mathsf{log1p}\left(1\right) - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)}^{3}}\right)}}^{3}\right)}^{0.3333333333333333}\right) \]

      2. unpow1/3 100.0%

         \[\leadsto \mathsf{expm1}\left({\left({\color{blue}{\left({\left({\left(\mathsf{log1p}\left(1\right) - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)}^{3}\right)}^{0.3333333333333333}\right)}}^{3}\right)}^{0.3333333333333333}\right) \]

      3. sqr-pow 98.5%

         \[\leadsto \mathsf{expm1}\left({\left({\color{blue}{\left({\left({\left(\mathsf{log1p}\left(1\right) - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \cdot {\left({\left(\mathsf{log1p}\left(1\right) - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}\right)}}^{3}\right)}^{0.3333333333333333}\right) \]

      4. pow-prod-down 100.0%

         \[\leadsto \mathsf{expm1}\left({\left({\color{blue}{\left({\left({\left(\mathsf{log1p}\left(1\right) - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)}^{3} \cdot {\left(\mathsf{log1p}\left(1\right) - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}\right)}}^{3}\right)}^{0.3333333333333333}\right) \]

      5. pow-prod-up 100.0%

         \[\leadsto \mathsf{expm1}\left({\left({\left({\color{blue}{\left({\left(\mathsf{log1p}\left(1\right) - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)}^{\left(3 + 3\right)}\right)}}^{\left(\frac{0.3333333333333333}{2}\right)}\right)}^{3}\right)}^{0.3333333333333333}\right) \]

      6. log1p-udef 100.0%

         \[\leadsto \mathsf{expm1}\left({\left({\left({\left({\left(\color{blue}{\log \left(1 + 1\right)} - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)}^{\left(3 + 3\right)}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}\right)}^{3}\right)}^{0.3333333333333333}\right) \]

      7. metadata-eval 100.0%

         \[\leadsto \mathsf{expm1}\left({\left({\left({\left({\left(\log \color{blue}{2} - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)}^{\left(3 + 3\right)}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}\right)}^{3}\right)}^{0.3333333333333333}\right) \]

      8. pow-exp 100.0%

         \[\leadsto \mathsf{expm1}\left({\left({\left({\left({\left(\log 2 - \mathsf{log1p}\left(\color{blue}{e^{-2 \cdot x}}\right)\right)}^{\left(3 + 3\right)}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}\right)}^{3}\right)}^{0.3333333333333333}\right) \]

      9. rem-log-exp 100.0%

         \[\leadsto \mathsf{expm1}\left({\left({\left({\left({\left(\log 2 - \mathsf{log1p}\left(e^{\color{blue}{\log \left(e^{-2}\right)} \cdot x}\right)\right)}^{\left(3 + 3\right)}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}\right)}^{3}\right)}^{0.3333333333333333}\right) \]

      10. *-commutative 100.0%

          \[\leadsto \mathsf{expm1}\left({\left({\left({\left({\left(\log 2 - \mathsf{log1p}\left(e^{\color{blue}{x \cdot \log \left(e^{-2}\right)}}\right)\right)}^{\left(3 + 3\right)}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}\right)}^{3}\right)}^{0.3333333333333333}\right) \]

      11. rem-log-exp 100.0%

          \[\leadsto \mathsf{expm1}\left({\left({\left({\left({\left(\log 2 - \mathsf{log1p}\left(e^{x \cdot \color{blue}{-2}}\right)\right)}^{\left(3 + 3\right)}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}\right)}^{3}\right)}^{0.3333333333333333}\right) \]

      12. metadata-eval 100.0%

          \[\leadsto \mathsf{expm1}\left({\left({\left({\left({\left(\log 2 - \mathsf{log1p}\left(e^{x \cdot -2}\right)\right)}^{\color{blue}{6}}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}\right)}^{3}\right)}^{0.3333333333333333}\right) \]

      13. metadata-eval 100.0%

          \[\leadsto \mathsf{expm1}\left({\left({\left({\left({\left(\log 2 - \mathsf{log1p}\left(e^{x \cdot -2}\right)\right)}^{6}\right)}^{\color{blue}{0.16666666666666666}}\right)}^{3}\right)}^{0.3333333333333333}\right) \]

   8. Applied egg-rr 100.0%

      \[\leadsto \mathsf{expm1}\left({\left({\color{blue}{\left({\left({\left(\log 2 - \mathsf{log1p}\left(e^{x \cdot -2}\right)\right)}^{6}\right)}^{0.16666666666666666}\right)}}^{3}\right)}^{0.3333333333333333}\right) \]

   if -0.050000000000000003 < (*.f64 -2 x) < 5.00000000000000024e-5

   1. Initial program 7.0%

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

   3. Taylor expanded in x around 0 100.0%

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

   if 5.00000000000000024e-5 < (*.f64 -2 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 97.7%

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

      1. *-commutative 97.7%

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

   5. Simplified 97.7%

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

   6. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{-1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.05:\\ \;\;\;\;\mathsf{expm1}\left({\left({\left({\left({\left(\log 2 - \mathsf{log1p}\left(e^{-2 \cdot x}\right)\right)}^{6}\right)}^{0.16666666666666666}\right)}^{3}\right)}^{0.3333333333333333}\right)\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-5}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.05:\\ \;\;\;\;\mathsf{expm1}\left({\left({\left(\mathsf{log1p}\left(1\right) - \mathsf{log1p}\left(e^{-2 \cdot x}\right)\right)}^{3}\right)}^{0.3333333333333333}\right)\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-5}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -0.05)
   (expm1
    (pow
     (pow (- (log1p 1.0) (log1p (exp (* -2.0 x)))) 3.0)
     0.3333333333333333))
   (if (<= (* -2.0 x) 5e-5) (+ x (* -0.3333333333333333 (pow x 3.0))) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.05) {
		tmp = expm1(pow(pow((log1p(1.0) - log1p(exp((-2.0 * x)))), 3.0), 0.3333333333333333));
	} else if ((-2.0 * x) <= 5e-5) {
		tmp = x + (-0.3333333333333333 * pow(x, 3.0));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.05) {
		tmp = Math.expm1(Math.pow(Math.pow((Math.log1p(1.0) - Math.log1p(Math.exp((-2.0 * x)))), 3.0), 0.3333333333333333));
	} else if ((-2.0 * x) <= 5e-5) {
		tmp = x + (-0.3333333333333333 * Math.pow(x, 3.0));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (-2.0 * x) <= -0.05:
		tmp = math.expm1(math.pow(math.pow((math.log1p(1.0) - math.log1p(math.exp((-2.0 * x)))), 3.0), 0.3333333333333333))
	elif (-2.0 * x) <= 5e-5:
		tmp = x + (-0.3333333333333333 * math.pow(x, 3.0))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.05)
		tmp = expm1(((Float64(log1p(1.0) - log1p(exp(Float64(-2.0 * x)))) ^ 3.0) ^ 0.3333333333333333));
	elseif (Float64(-2.0 * x) <= 5e-5)
		tmp = Float64(x + Float64(-0.3333333333333333 * (x ^ 3.0)));
	else
		tmp = -1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.05], N[(Exp[N[Power[N[Power[N[(N[Log[1 + 1.0], $MachinePrecision] - N[Log[1 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]] - 1), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 5e-5], N[(x + N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 -2 x) < -0.050000000000000003

   1. Initial program 99.9%

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

      1. add-exp-log 99.9%

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

      2. expm1-def 99.9%

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

      3. log-div 99.9%

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

      4. log1p-udef 99.9%

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

      5. exp-prod 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. add-cbrt-cube 99.9%

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\sqrt[3]{\left(\left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right) \cdot \left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)\right) \cdot \left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)}}\right) \]

      2. pow1/3 99.9%

         \[\leadsto \mathsf{expm1}\left(\color{blue}{{\left(\left(\left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right) \cdot \left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)\right) \cdot \left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)\right)}^{0.3333333333333333}}\right) \]

      3. pow3 100.0%

         \[\leadsto \mathsf{expm1}\left({\color{blue}{\left({\left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)}^{3}\right)}}^{0.3333333333333333}\right) \]

      4. metadata-eval 100.0%

         \[\leadsto \mathsf{expm1}\left({\left({\left(\log \color{blue}{\left(1 + 1\right)} - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)}^{3}\right)}^{0.3333333333333333}\right) \]

      5. log1p-udef 100.0%

         \[\leadsto \mathsf{expm1}\left({\left({\left(\color{blue}{\mathsf{log1p}\left(1\right)} - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)}^{3}\right)}^{0.3333333333333333}\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{{\left({\left(\mathsf{log1p}\left(1\right) - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)}^{3}\right)}^{0.3333333333333333}}\right) \]

   7. Taylor expanded in x around inf 100.0%

      \[\leadsto \mathsf{expm1}\left({\left({\left(\mathsf{log1p}\left(1\right) - \mathsf{log1p}\left(\color{blue}{e^{-2 \cdot x}}\right)\right)}^{3}\right)}^{0.3333333333333333}\right) \]

   if -0.050000000000000003 < (*.f64 -2 x) < 5.00000000000000024e-5

   1. Initial program 7.0%

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

   3. Taylor expanded in x around 0 100.0%

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

   if 5.00000000000000024e-5 < (*.f64 -2 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 97.7%

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

      1. *-commutative 97.7%

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

   5. Simplified 97.7%

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

   6. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{-1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.05:\\ \;\;\;\;\mathsf{expm1}\left({\left({\left(\mathsf{log1p}\left(1\right) - \mathsf{log1p}\left(e^{-2 \cdot x}\right)\right)}^{3}\right)}^{0.3333333333333333}\right)\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-5}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.05:\\ \;\;\;\;\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(e^{-2 \cdot x}\right)\right)\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-5}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -0.05)
   (expm1 (- (log 2.0) (log1p (exp (* -2.0 x)))))
   (if (<= (* -2.0 x) 5e-5) (+ x (* -0.3333333333333333 (pow x 3.0))) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.05) {
		tmp = expm1((log(2.0) - log1p(exp((-2.0 * x)))));
	} else if ((-2.0 * x) <= 5e-5) {
		tmp = x + (-0.3333333333333333 * pow(x, 3.0));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.05) {
		tmp = Math.expm1((Math.log(2.0) - Math.log1p(Math.exp((-2.0 * x)))));
	} else if ((-2.0 * x) <= 5e-5) {
		tmp = x + (-0.3333333333333333 * Math.pow(x, 3.0));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (-2.0 * x) <= -0.05:
		tmp = math.expm1((math.log(2.0) - math.log1p(math.exp((-2.0 * x)))))
	elif (-2.0 * x) <= 5e-5:
		tmp = x + (-0.3333333333333333 * math.pow(x, 3.0))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.05)
		tmp = expm1(Float64(log(2.0) - log1p(exp(Float64(-2.0 * x)))));
	elseif (Float64(-2.0 * x) <= 5e-5)
		tmp = Float64(x + Float64(-0.3333333333333333 * (x ^ 3.0)));
	else
		tmp = -1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.05], N[(Exp[N[(N[Log[2.0], $MachinePrecision] - N[Log[1 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 5e-5], N[(x + N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 -2 x) < -0.050000000000000003

   1. Initial program 99.9%

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

      1. add-exp-log 99.9%

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

      2. expm1-def 99.9%

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

      3. log-div 99.9%

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

      4. log1p-udef 99.9%

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

      5. exp-prod 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf 99.9%

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

   if -0.050000000000000003 < (*.f64 -2 x) < 5.00000000000000024e-5

   1. Initial program 7.0%

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

   3. Taylor expanded in x around 0 100.0%

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

   if 5.00000000000000024e-5 < (*.f64 -2 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 97.7%

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

      1. *-commutative 97.7%

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

   5. Simplified 97.7%

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

   6. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{-1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.05:\\ \;\;\;\;\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(e^{-2 \cdot x}\right)\right)\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-5}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.05:\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-5}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -0.05)
   (+ -1.0 (/ 2.0 (+ 1.0 (exp (* -2.0 x)))))
   (if (<= (* -2.0 x) 5e-5) (+ x (* -0.3333333333333333 (pow x 3.0))) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.05) {
		tmp = -1.0 + (2.0 / (1.0 + exp((-2.0 * x))));
	} else if ((-2.0 * x) <= 5e-5) {
		tmp = x + (-0.3333333333333333 * pow(x, 3.0));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((-2.0d0) * x) <= (-0.05d0)) then
        tmp = (-1.0d0) + (2.0d0 / (1.0d0 + exp(((-2.0d0) * x))))
    else if (((-2.0d0) * x) <= 5d-5) then
        tmp = x + ((-0.3333333333333333d0) * (x ** 3.0d0))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.05) {
		tmp = -1.0 + (2.0 / (1.0 + Math.exp((-2.0 * x))));
	} else if ((-2.0 * x) <= 5e-5) {
		tmp = x + (-0.3333333333333333 * Math.pow(x, 3.0));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (-2.0 * x) <= -0.05:
		tmp = -1.0 + (2.0 / (1.0 + math.exp((-2.0 * x))))
	elif (-2.0 * x) <= 5e-5:
		tmp = x + (-0.3333333333333333 * math.pow(x, 3.0))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.05)
		tmp = Float64(-1.0 + Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))));
	elseif (Float64(-2.0 * x) <= 5e-5)
		tmp = Float64(x + Float64(-0.3333333333333333 * (x ^ 3.0)));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((-2.0 * x) <= -0.05)
		tmp = -1.0 + (2.0 / (1.0 + exp((-2.0 * x))));
	elseif ((-2.0 * x) <= 5e-5)
		tmp = x + (-0.3333333333333333 * (x ^ 3.0));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.05], N[(-1.0 + N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 5e-5], N[(x + N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 -2 x) < -0.050000000000000003

   1. Initial program 99.9%

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

   if -0.050000000000000003 < (*.f64 -2 x) < 5.00000000000000024e-5

   1. Initial program 7.0%

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

   3. Taylor expanded in x around 0 100.0%

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

   if 5.00000000000000024e-5 < (*.f64 -2 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 97.7%

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

      1. *-commutative 97.7%

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

   5. Simplified 97.7%

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

   6. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{-1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.05:\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-5}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.7% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.66:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x + 2}}{\frac{0.5}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.66) -1.0 (/ (/ 1.0 (+ x 2.0)) (/ 0.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.66) {
		tmp = -1.0;
	} else {
		tmp = (1.0 / (x + 2.0)) / (0.5 / 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 <= (-0.66d0)) then
        tmp = -1.0d0
    else
        tmp = (1.0d0 / (x + 2.0d0)) / (0.5d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.66) {
		tmp = -1.0;
	} else {
		tmp = (1.0 / (x + 2.0)) / (0.5 / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.66:
		tmp = -1.0
	else:
		tmp = (1.0 / (x + 2.0)) / (0.5 / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.66)
		tmp = -1.0;
	else
		tmp = Float64(Float64(1.0 / Float64(x + 2.0)) / Float64(0.5 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.66)
		tmp = -1.0;
	else
		tmp = (1.0 / (x + 2.0)) / (0.5 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.66], -1.0, N[(N[(1.0 / N[(x + 2.0), $MachinePrecision]), $MachinePrecision] / N[(0.5 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.660000000000000031

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 97.7%

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

      1. *-commutative 97.7%

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

   5. Simplified 97.7%

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

   6. Taylor expanded in x around inf 100.0%

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

   if -0.660000000000000031 < x

   1. Initial program 39.1%

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

   3. Taylor expanded in x around 0 6.7%

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

      1. +-commutative 6.7%

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

   5. Simplified 6.7%

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

      1. flip-- 6.6%

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

      2. div-inv 6.6%

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

      3. metadata-eval 6.6%

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

      4. difference-of-sqr-1 6.6%

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

      5. associate-+l+ 6.6%

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

      6. metadata-eval 6.6%

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

      7. associate--l+ 67.3%

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

      8. metadata-eval 67.3%

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

      9. +-rgt-identity 67.3%

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

      10. associate-+l+ 67.3%

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

      11. metadata-eval 67.3%

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

   7. Applied egg-rr 67.3%

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

      1. div-inv 67.3%

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

      2. clear-num 67.2%

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

      3. div-inv 67.2%

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

      4. associate-/r* 67.2%

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

      5. +-commutative 67.2%

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

      6. *-commutative 67.2%

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

      7. associate-/r* 67.2%

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

      8. +-commutative 67.2%

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

   9. Applied egg-rr 67.2%

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

   10. Taylor expanded in x around 0 71.2%

       \[\leadsto \frac{\frac{1}{2 + x}}{\color{blue}{\frac{0.5}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.66:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x + 2}}{\frac{0.5}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.1% 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 97.7%

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

      1. *-commutative 97.7%

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

   5. Simplified 97.7%

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

   6. Taylor expanded in x around inf 100.0%

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

   if -1 < x

   1. Initial program 39.1%

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

   3. Taylor expanded in x around 0 67.5%

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

4. Final simplification 75.7%

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

Alternative 7: 27.3% accurate, 109.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 -1.0)

C

double code(double x, double y) {
	return -1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = -1.0d0
end function

Java

public static double code(double x, double y) {
	return -1.0;
}

Python

def code(x, y):
	return -1.0

Julia

function code(x, y)
	return -1.0
end

MATLAB

function tmp = code(x, y)
	tmp = -1.0;
end

Wolfram

code[x_, y_] := -1.0

Derivation

1. Initial program 54.6%

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

3. Taylor expanded in x around 0 28.6%

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

   1. *-commutative 28.6%

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

5. Simplified 28.6%

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

6. Taylor expanded in x around inf 27.6%

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

7. Final simplification 27.6%

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

Reproduce

herbie shell --seed 2024029 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))