Logistic function from Lakshay Garg

Percentage Accurate: 54.7% → 99.7%

Time: 11.9s

Alternatives: 8

Speedup: 9.9×

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: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{if}\;-2 \cdot x \leq -0.2:\\ \;\;\;\;\sqrt[3]{t\_0} \cdot \sqrt[3]{{t\_0}^{2}}\\ \mathbf{elif}\;-2 \cdot x \leq 0.002:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.13333333333333333 - 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) -1.0)))
   (if (<= (* -2.0 x) -0.2)
     (* (cbrt t_0) (cbrt (pow t_0 2.0)))
     (if (<= (* -2.0 x) 0.002)
       (*
        x
        (+
         1.0
         (* (* x x) (- (* (* x x) 0.13333333333333333) 0.3333333333333333))))
       -1.0))))

C

double code(double x, double y) {
	double t_0 = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	double tmp;
	if ((-2.0 * x) <= -0.2) {
		tmp = cbrt(t_0) * cbrt(pow(t_0, 2.0));
	} else if ((-2.0 * x) <= 0.002) {
		tmp = x * (1.0 + ((x * x) * (((x * x) * 0.13333333333333333) - 0.3333333333333333)));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = (2.0 / (1.0 + Math.exp((-2.0 * x)))) + -1.0;
	double tmp;
	if ((-2.0 * x) <= -0.2) {
		tmp = Math.cbrt(t_0) * Math.cbrt(Math.pow(t_0, 2.0));
	} else if ((-2.0 * x) <= 0.002) {
		tmp = x * (1.0 + ((x * x) * (((x * x) * 0.13333333333333333) - 0.3333333333333333)));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) + -1.0)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.2)
		tmp = Float64(cbrt(t_0) * cbrt((t_0 ^ 2.0)));
	elseif (Float64(-2.0 * x) <= 0.002)
		tmp = Float64(x * Float64(1.0 + Float64(Float64(x * x) * Float64(Float64(Float64(x * x) * 0.13333333333333333) - 0.3333333333333333))));
	else
		tmp = -1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.2], N[(N[Power[t$95$0, 1/3], $MachinePrecision] * N[Power[N[Power[t$95$0, 2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.002], N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -0.20000000000000001

   1. Initial program 100.0%

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

      1. add-cbrt-cube 100.0%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)}} \]

      2. pow1/3 100.0%

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

      3. associate-*l* 100.0%

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

      4. unpow-prod-down 100.0%

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

      5. pow1/3 100.0%

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

      6. sub-neg 100.0%

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

      7. exp-prod 100.0%

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

      8. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{2}{1 + {\left(e^{-2}\right)}^{x}} + -1} \cdot {\left({\left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}} + -1\right)}^{2}\right)}^{0.3333333333333333}} \]
   5. Step-by-step derivation

      1. unpow1/3 100.0%

         \[\leadsto \sqrt[3]{\frac{2}{1 + {\left(e^{-2}\right)}^{x}} + -1} \cdot \color{blue}{\sqrt[3]{{\left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}} + -1\right)}^{2}}} \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{2}{1 + {\left(e^{-2}\right)}^{x}} + -1} \cdot \sqrt[3]{{\left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}} + -1\right)}^{2}}} \]

   7. Taylor expanded in x around inf 100.0%

      \[\leadsto \sqrt[3]{\frac{2}{1 + {\left(e^{-2}\right)}^{x}} + -1} \cdot \sqrt[3]{{\left(\color{blue}{\frac{2}{1 + e^{-2 \cdot x}}} + -1\right)}^{2}} \]

   8. Taylor expanded in x around inf 100.0%

      \[\leadsto \sqrt[3]{\frac{2}{\color{blue}{1 + e^{-2 \cdot x}}} + -1} \cdot \sqrt[3]{{\left(\frac{2}{1 + e^{-2 \cdot x}} + -1\right)}^{2}} \]

   if -0.20000000000000001 < (*.f64 #s(literal -2 binary64) x) < 2e-3

   1. Initial program 8.4%

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

   3. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.13333333333333333 \cdot {x}^{2} - 0.3333333333333333\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.13333333333333333 \cdot \color{blue}{\left(x \cdot x\right)} - 0.3333333333333333\right)\right) \]

   5. Applied egg-rr 100.0%

      \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.13333333333333333 \cdot \color{blue}{\left(x \cdot x\right)} - 0.3333333333333333\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.13333333333333333 \cdot \color{blue}{\left(x \cdot x\right)} - 0.3333333333333333\right)\right) \]

   7. Applied egg-rr 100.0%

      \[\leadsto x \cdot \left(1 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.13333333333333333 \cdot \left(x \cdot x\right) - 0.3333333333333333\right)\right) \]

   if 2e-3 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.2%

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

   4. Taylor expanded in x around inf 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.2:\\ \;\;\;\;\sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} + -1} \cdot \sqrt[3]{{\left(\frac{2}{1 + e^{-2 \cdot x}} + -1\right)}^{2}}\\ \mathbf{elif}\;-2 \cdot x \leq 0.002:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.13333333333333333 - 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.2:\\ \;\;\;\;\log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} + -1}\right)\\ \mathbf{elif}\;-2 \cdot x \leq 0.002:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.13333333333333333 - 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -0.2)
   (log (exp (+ (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) -1.0)))
   (if (<= (* -2.0 x) 0.002)
     (*
      x
      (+
       1.0
       (* (* x x) (- (* (* x x) 0.13333333333333333) 0.3333333333333333))))
     -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.2) {
		tmp = log(exp(((2.0 / (1.0 + exp((-2.0 * x)))) + -1.0)));
	} else if ((-2.0 * x) <= 0.002) {
		tmp = x * (1.0 + ((x * x) * (((x * x) * 0.13333333333333333) - 0.3333333333333333)));
	} 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) <= (-0.2d0)) then
        tmp = log(exp(((2.0d0 / (1.0d0 + exp(((-2.0d0) * x)))) + (-1.0d0))))
    else if (((-2.0d0) * x) <= 0.002d0) then
        tmp = x * (1.0d0 + ((x * x) * (((x * x) * 0.13333333333333333d0) - 0.3333333333333333d0)))
    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) <= -0.2) {
		tmp = Math.log(Math.exp(((2.0 / (1.0 + Math.exp((-2.0 * x)))) + -1.0)));
	} else if ((-2.0 * x) <= 0.002) {
		tmp = x * (1.0 + ((x * x) * (((x * x) * 0.13333333333333333) - 0.3333333333333333)));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (-2.0 * x) <= -0.2:
		tmp = math.log(math.exp(((2.0 / (1.0 + math.exp((-2.0 * x)))) + -1.0)))
	elif (-2.0 * x) <= 0.002:
		tmp = x * (1.0 + ((x * x) * (((x * x) * 0.13333333333333333) - 0.3333333333333333)))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.2)
		tmp = log(exp(Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) + -1.0)));
	elseif (Float64(-2.0 * x) <= 0.002)
		tmp = Float64(x * Float64(1.0 + Float64(Float64(x * x) * Float64(Float64(Float64(x * x) * 0.13333333333333333) - 0.3333333333333333))));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((-2.0 * x) <= -0.2)
		tmp = log(exp(((2.0 / (1.0 + exp((-2.0 * x)))) + -1.0)));
	elseif ((-2.0 * x) <= 0.002)
		tmp = x * (1.0 + ((x * x) * (((x * x) * 0.13333333333333333) - 0.3333333333333333)));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.2], N[Log[N[Exp[N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.002], N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -0.20000000000000001

   1. Initial program 100.0%

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

      1. add-log-exp 100.0%

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

      2. sub-neg 100.0%

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

      3. exp-prod 100.0%

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

      4. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

   if -0.20000000000000001 < (*.f64 #s(literal -2 binary64) x) < 2e-3

   1. Initial program 8.4%

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

   3. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.13333333333333333 \cdot {x}^{2} - 0.3333333333333333\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.13333333333333333 \cdot \color{blue}{\left(x \cdot x\right)} - 0.3333333333333333\right)\right) \]

   5. Applied egg-rr 100.0%

      \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.13333333333333333 \cdot \color{blue}{\left(x \cdot x\right)} - 0.3333333333333333\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.13333333333333333 \cdot \color{blue}{\left(x \cdot x\right)} - 0.3333333333333333\right)\right) \]

   7. Applied egg-rr 100.0%

      \[\leadsto x \cdot \left(1 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.13333333333333333 \cdot \left(x \cdot x\right) - 0.3333333333333333\right)\right) \]

   if 2e-3 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.2%

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

   4. Taylor expanded in x around inf 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.2:\\ \;\;\;\;\log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} + -1}\right)\\ \mathbf{elif}\;-2 \cdot x \leq 0.002:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.13333333333333333 - 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.2:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.002:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.13333333333333333 - 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -0.2)
   (+ (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) -1.0)
   (if (<= (* -2.0 x) 0.002)
     (*
      x
      (+
       1.0
       (* (* x x) (- (* (* x x) 0.13333333333333333) 0.3333333333333333))))
     -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.2) {
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	} else if ((-2.0 * x) <= 0.002) {
		tmp = x * (1.0 + ((x * x) * (((x * x) * 0.13333333333333333) - 0.3333333333333333)));
	} 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) <= (-0.2d0)) then
        tmp = (2.0d0 / (1.0d0 + exp(((-2.0d0) * x)))) + (-1.0d0)
    else if (((-2.0d0) * x) <= 0.002d0) then
        tmp = x * (1.0d0 + ((x * x) * (((x * x) * 0.13333333333333333d0) - 0.3333333333333333d0)))
    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) <= -0.2) {
		tmp = (2.0 / (1.0 + Math.exp((-2.0 * x)))) + -1.0;
	} else if ((-2.0 * x) <= 0.002) {
		tmp = x * (1.0 + ((x * x) * (((x * x) * 0.13333333333333333) - 0.3333333333333333)));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (-2.0 * x) <= -0.2:
		tmp = (2.0 / (1.0 + math.exp((-2.0 * x)))) + -1.0
	elif (-2.0 * x) <= 0.002:
		tmp = x * (1.0 + ((x * x) * (((x * x) * 0.13333333333333333) - 0.3333333333333333)))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.2)
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) + -1.0);
	elseif (Float64(-2.0 * x) <= 0.002)
		tmp = Float64(x * Float64(1.0 + Float64(Float64(x * x) * Float64(Float64(Float64(x * x) * 0.13333333333333333) - 0.3333333333333333))));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((-2.0 * x) <= -0.2)
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	elseif ((-2.0 * x) <= 0.002)
		tmp = x * (1.0 + ((x * x) * (((x * x) * 0.13333333333333333) - 0.3333333333333333)));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.2], N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.002], N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -0.20000000000000001

   1. Initial program 100.0%

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

   if -0.20000000000000001 < (*.f64 #s(literal -2 binary64) x) < 2e-3

   1. Initial program 8.4%

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

   3. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.13333333333333333 \cdot {x}^{2} - 0.3333333333333333\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.13333333333333333 \cdot \color{blue}{\left(x \cdot x\right)} - 0.3333333333333333\right)\right) \]

   5. Applied egg-rr 100.0%

      \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.13333333333333333 \cdot \color{blue}{\left(x \cdot x\right)} - 0.3333333333333333\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.13333333333333333 \cdot \color{blue}{\left(x \cdot x\right)} - 0.3333333333333333\right)\right) \]

   7. Applied egg-rr 100.0%

      \[\leadsto x \cdot \left(1 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.13333333333333333 \cdot \left(x \cdot x\right) - 0.3333333333333333\right)\right) \]

   if 2e-3 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.2%

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

   4. Taylor expanded in x around inf 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.2:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.002:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.13333333333333333 - 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.3% accurate, 7.3× 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. Add Preprocessing

   3. Taylor expanded in x around 0 99.2%

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

   4. Taylor expanded in x around inf 100.0%

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

   if -1 < x < 2.5

   1. Initial program 9.1%

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

   3. Taylor expanded in x around 0 99.0%

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

   if 2.5 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 5.5%

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

      1. +-commutative 5.5%

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

   5. Simplified 5.5%

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

      1. flip-- 5.2%

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

      2. clear-num 5.2%

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

      3. associate-+l+ 5.2%

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

      4. metadata-eval 5.2%

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

      5. metadata-eval 5.2%

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

      6. difference-of-sqr-1 5.2%

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

      7. associate-+l+ 5.2%

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

      8. metadata-eval 5.2%

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

      9. associate--l+ 5.2%

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

      10. metadata-eval 5.2%

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

      11. +-rgt-identity 5.2%

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

   7. Applied egg-rr 5.2%

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

   8. Taylor expanded in x around 0 18.5%

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

   9. Taylor expanded in x around inf 18.8%

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

       1. associate-*r/ 18.8%

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

       2. metadata-eval 18.8%

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

   11. Simplified 18.8%

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

Alternative 5: 78.7% accurate, 9.1× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.66) {
		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 <= (-0.66d0)) 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 <= -0.66) {
		tmp = -1.0;
	} else {
		tmp = x * (2.0 / (x + 2.0));
	}
	return tmp;
}

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -0.660000000000000031

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.2%

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

   4. Taylor expanded in x around inf 100.0%

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

   if -0.660000000000000031 < x

   1. Initial program 36.4%

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

   3. Taylor expanded in x around 0 7.4%

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

      1. +-commutative 7.4%

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

   5. Simplified 7.4%

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

      1. flip-- 7.3%

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

      2. clear-num 7.3%

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

      3. associate-+l+ 7.3%

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

      4. metadata-eval 7.3%

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

      5. metadata-eval 7.3%

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

      6. difference-of-sqr-1 7.3%

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

      7. associate-+l+ 7.3%

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

      8. metadata-eval 7.3%

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

      9. associate--l+ 70.6%

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

      10. metadata-eval 70.6%

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

      11. +-rgt-identity 70.6%

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

   7. Applied egg-rr 70.6%

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

   8. Taylor expanded in x around 0 73.7%

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

      1. clear-num 73.9%

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

      2. *-commutative 73.9%

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

      3. *-un-lft-identity 73.9%

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

      4. times-frac 74.0%

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

   10. Applied egg-rr 74.0%

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

4. Final simplification 81.1%

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

Alternative 6: 79.3% accurate, 9.9× 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. Add Preprocessing

   3. Taylor expanded in x around 0 99.2%

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

   4. Taylor expanded in x around inf 100.0%

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

   if -1 < x < 2

   1. Initial program 9.1%

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

   3. Taylor expanded in x around 0 99.0%

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

   if 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 5.5%

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

      1. +-commutative 5.5%

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

   5. Simplified 5.5%

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

      1. flip-- 5.2%

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

      2. clear-num 5.2%

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

      3. associate-+l+ 5.2%

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

      4. metadata-eval 5.2%

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

      5. metadata-eval 5.2%

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

      6. difference-of-sqr-1 5.2%

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

      7. associate-+l+ 5.2%

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

      8. metadata-eval 5.2%

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

      9. associate--l+ 5.2%

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

      10. metadata-eval 5.2%

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

      11. +-rgt-identity 5.2%

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

   7. Applied egg-rr 5.2%

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

   8. Taylor expanded in x around 0 18.5%

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

   9. Taylor expanded in x around inf 18.8%

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

Alternative 7: 32.8% accurate, 18.1× 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 57.9%

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

   3. Taylor expanded in x around 0 57.2%

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

   4. Taylor expanded in x around inf 56.3%

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

   if 1.1000000000000001e-308 < x

   1. Initial program 49.6%

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

   3. Taylor expanded in x around 0 7.1%

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

      1. +-commutative 7.1%

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

   5. Simplified 7.1%

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

      1. flip-- 6.8%

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

      2. clear-num 6.8%

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

      3. associate-+l+ 6.8%

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

      4. metadata-eval 6.8%

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

      5. metadata-eval 6.8%

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

      6. difference-of-sqr-1 6.9%

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

      7. associate-+l+ 6.9%

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

      8. metadata-eval 6.9%

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

      9. associate--l+ 57.1%

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

      10. metadata-eval 57.1%

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

      11. +-rgt-identity 57.1%

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

   7. Applied egg-rr 57.1%

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

   8. Taylor expanded in x around 0 62.3%

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

   9. Taylor expanded in x around inf 11.1%

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

Alternative 8: 27.8% 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 53.8%

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

3. Taylor expanded in x around 0 31.6%

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

4. Taylor expanded in x around inf 29.6%

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

Reproduce

herbie shell --seed 2024149 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))