Logistic function from Lakshay Garg

Percentage Accurate: 54.6% → 99.6%

Time: 15.1s

Alternatives: 8

Speedup: 35.6×

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: 54.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{if}\;-2 \cdot x \leq -1000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;-2 \cdot x \leq 2 \cdot 10^{-7}:\\ \;\;\;\;-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \log \left(\sqrt{e^{t_0}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ -1.0 (/ 2.0 (+ 1.0 (exp (* -2.0 x)))))))
   (if (<= (* -2.0 x) -1000.0)
     t_0
     (if (<= (* -2.0 x) 2e-7)
       (+
        (* -0.3333333333333333 (pow x 3.0))
        (+ x (* 0.13333333333333333 (pow x 5.0))))
       (* 2.0 (log (sqrt (exp t_0))))))))

C

double code(double x, double y) {
	double t_0 = -1.0 + (2.0 / (1.0 + exp((-2.0 * x))));
	double tmp;
	if ((-2.0 * x) <= -1000.0) {
		tmp = t_0;
	} else if ((-2.0 * x) <= 2e-7) {
		tmp = (-0.3333333333333333 * pow(x, 3.0)) + (x + (0.13333333333333333 * pow(x, 5.0)));
	} else {
		tmp = 2.0 * log(sqrt(exp(t_0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-1.0d0) + (2.0d0 / (1.0d0 + exp(((-2.0d0) * x))))
    if (((-2.0d0) * x) <= (-1000.0d0)) then
        tmp = t_0
    else if (((-2.0d0) * x) <= 2d-7) then
        tmp = ((-0.3333333333333333d0) * (x ** 3.0d0)) + (x + (0.13333333333333333d0 * (x ** 5.0d0)))
    else
        tmp = 2.0d0 * log(sqrt(exp(t_0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = -1.0 + (2.0 / (1.0 + Math.exp((-2.0 * x))));
	double tmp;
	if ((-2.0 * x) <= -1000.0) {
		tmp = t_0;
	} else if ((-2.0 * x) <= 2e-7) {
		tmp = (-0.3333333333333333 * Math.pow(x, 3.0)) + (x + (0.13333333333333333 * Math.pow(x, 5.0)));
	} else {
		tmp = 2.0 * Math.log(Math.sqrt(Math.exp(t_0)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = -1.0 + (2.0 / (1.0 + math.exp((-2.0 * x))))
	tmp = 0
	if (-2.0 * x) <= -1000.0:
		tmp = t_0
	elif (-2.0 * x) <= 2e-7:
		tmp = (-0.3333333333333333 * math.pow(x, 3.0)) + (x + (0.13333333333333333 * math.pow(x, 5.0)))
	else:
		tmp = 2.0 * math.log(math.sqrt(math.exp(t_0)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(-1.0 + Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))))
	tmp = 0.0
	if (Float64(-2.0 * x) <= -1000.0)
		tmp = t_0;
	elseif (Float64(-2.0 * x) <= 2e-7)
		tmp = Float64(Float64(-0.3333333333333333 * (x ^ 3.0)) + Float64(x + Float64(0.13333333333333333 * (x ^ 5.0))));
	else
		tmp = Float64(2.0 * log(sqrt(exp(t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = -1.0 + (2.0 / (1.0 + exp((-2.0 * x))));
	tmp = 0.0;
	if ((-2.0 * x) <= -1000.0)
		tmp = t_0;
	elseif ((-2.0 * x) <= 2e-7)
		tmp = (-0.3333333333333333 * (x ^ 3.0)) + (x + (0.13333333333333333 * (x ^ 5.0)));
	else
		tmp = 2.0 * log(sqrt(exp(t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = 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], -1000.0], t$95$0, If[LessEqual[N[(-2.0 * x), $MachinePrecision], 2e-7], N[(N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x + N[(0.13333333333333333 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Log[N[Sqrt[N[Exp[t$95$0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 -2 x) < -1e3

   1. Initial program 100.0%

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

   if -1e3 < (*.f64 -2 x) < 1.9999999999999999e-7

   1. Initial program 7.1%

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

   2. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot {x}^{3} + \left(0.13333333333333333 \cdot {x}^{5} + x\right)} \]

   if 1.9999999999999999e-7 < (*.f64 -2 x)

   1. Initial program 99.9%

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

      1. add-log-exp 99.9%

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

      2. add-sqr-sqrt 99.9%

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

      3. log-prod 100.0%

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

      4. add-exp-log 100.0%

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

      5. expm1-def 100.0%

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

      6. log-div 99.9%

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

      7. log1p-udef 100.0%

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

      8. exp-prod 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. count-2 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 100.0%

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

      1. sub-neg 100.0%

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

      2. metadata-eval 100.0%

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

      3. exp-sum 99.9%

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

      4. exp-diff 99.9%

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

      5. log1p-def 100.0%

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

      6. exp-prod 100.0%

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

      7. div-exp 100.0%

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

      8. log1p-def 99.9%

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

      9. log-div 99.9%

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

      10. rem-exp-log 99.9%

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

   8. Simplified 99.9%

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

      1. pow-exp 100.0%

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

   10. Applied egg-rr 100.0%

       \[\leadsto 2 \cdot \log \left(\sqrt{e^{-1 + \frac{2}{1 + \color{blue}{e^{x \cdot -2}}}}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -1000:\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{elif}\;-2 \cdot x \leq 2 \cdot 10^{-7}:\\ \;\;\;\;-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \log \left(\sqrt{e^{-1 + \frac{2}{1 + e^{-2 \cdot x}}}}\right)\\ \end{array} \]

Alternative 2: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-2 \cdot x}\\ \mathbf{if}\;-2 \cdot x \leq -1000:\\ \;\;\;\;-1 + \frac{2}{1 + t_0}\\ \mathbf{elif}\;-2 \cdot x \leq 2 \cdot 10^{-7}:\\ \;\;\;\;-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(t_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (exp (* -2.0 x))))
   (if (<= (* -2.0 x) -1000.0)
     (+ -1.0 (/ 2.0 (+ 1.0 t_0)))
     (if (<= (* -2.0 x) 2e-7)
       (+
        (* -0.3333333333333333 (pow x 3.0))
        (+ x (* 0.13333333333333333 (pow x 5.0))))
       (expm1 (- (log 2.0) (log1p t_0)))))))

C

double code(double x, double y) {
	double t_0 = exp((-2.0 * x));
	double tmp;
	if ((-2.0 * x) <= -1000.0) {
		tmp = -1.0 + (2.0 / (1.0 + t_0));
	} else if ((-2.0 * x) <= 2e-7) {
		tmp = (-0.3333333333333333 * pow(x, 3.0)) + (x + (0.13333333333333333 * pow(x, 5.0)));
	} else {
		tmp = expm1((log(2.0) - log1p(t_0)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = Math.exp((-2.0 * x));
	double tmp;
	if ((-2.0 * x) <= -1000.0) {
		tmp = -1.0 + (2.0 / (1.0 + t_0));
	} else if ((-2.0 * x) <= 2e-7) {
		tmp = (-0.3333333333333333 * Math.pow(x, 3.0)) + (x + (0.13333333333333333 * Math.pow(x, 5.0)));
	} else {
		tmp = Math.expm1((Math.log(2.0) - Math.log1p(t_0)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.exp((-2.0 * x))
	tmp = 0
	if (-2.0 * x) <= -1000.0:
		tmp = -1.0 + (2.0 / (1.0 + t_0))
	elif (-2.0 * x) <= 2e-7:
		tmp = (-0.3333333333333333 * math.pow(x, 3.0)) + (x + (0.13333333333333333 * math.pow(x, 5.0)))
	else:
		tmp = math.expm1((math.log(2.0) - math.log1p(t_0)))
	return tmp

Julia

function code(x, y)
	t_0 = exp(Float64(-2.0 * x))
	tmp = 0.0
	if (Float64(-2.0 * x) <= -1000.0)
		tmp = Float64(-1.0 + Float64(2.0 / Float64(1.0 + t_0)));
	elseif (Float64(-2.0 * x) <= 2e-7)
		tmp = Float64(Float64(-0.3333333333333333 * (x ^ 3.0)) + Float64(x + Float64(0.13333333333333333 * (x ^ 5.0))));
	else
		tmp = expm1(Float64(log(2.0) - log1p(t_0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -1000.0], N[(-1.0 + N[(2.0 / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 2e-7], N[(N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x + N[(0.13333333333333333 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Exp[N[(N[Log[2.0], $MachinePrecision] - N[Log[1 + t$95$0], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 -2 x) < -1e3

   1. Initial program 100.0%

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

   if -1e3 < (*.f64 -2 x) < 1.9999999999999999e-7

   1. Initial program 7.1%

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

   2. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot {x}^{3} + \left(0.13333333333333333 \cdot {x}^{5} + x\right)} \]

   if 1.9999999999999999e-7 < (*.f64 -2 x)

   1. Initial program 99.9%

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

      1. add-log-exp 99.9%

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

      2. *-un-lft-identity 99.9%

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

      3. log-prod 99.9%

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

      4. metadata-eval 99.9%

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

      5. add-log-exp 99.9%

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

      6. add-exp-log 99.9%

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

      7. expm1-def 99.9%

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

      8. log-div 99.9%

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

      9. log1p-udef 100.0%

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

      10. exp-prod 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. +-lft-identity 100.0%

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

   5. Simplified 100.0%

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

      1. pow-exp 100.0%

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

      2. *-commutative 100.0%

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

   7. Applied egg-rr 100.0%

      \[\leadsto \mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(\color{blue}{e^{x \cdot -2}}\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -1000:\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{elif}\;-2 \cdot x \leq 2 \cdot 10^{-7}:\\ \;\;\;\;-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(e^{-2 \cdot x}\right)\right)\\ \end{array} \]

Alternative 3: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-2 \cdot x}\\ \mathbf{if}\;-2 \cdot x \leq -1000:\\ \;\;\;\;-1 + \frac{2}{1 + t_0}\\ \mathbf{elif}\;-2 \cdot x \leq 2 \cdot 10^{-7}:\\ \;\;\;\;-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{2}{e^{\mathsf{log1p}\left(t_0\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (exp (* -2.0 x))))
   (if (<= (* -2.0 x) -1000.0)
     (+ -1.0 (/ 2.0 (+ 1.0 t_0)))
     (if (<= (* -2.0 x) 2e-7)
       (+
        (* -0.3333333333333333 (pow x 3.0))
        (+ x (* 0.13333333333333333 (pow x 5.0))))
       (+ -1.0 (/ 2.0 (exp (log1p t_0))))))))

C

double code(double x, double y) {
	double t_0 = exp((-2.0 * x));
	double tmp;
	if ((-2.0 * x) <= -1000.0) {
		tmp = -1.0 + (2.0 / (1.0 + t_0));
	} else if ((-2.0 * x) <= 2e-7) {
		tmp = (-0.3333333333333333 * pow(x, 3.0)) + (x + (0.13333333333333333 * pow(x, 5.0)));
	} else {
		tmp = -1.0 + (2.0 / exp(log1p(t_0)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = Math.exp((-2.0 * x));
	double tmp;
	if ((-2.0 * x) <= -1000.0) {
		tmp = -1.0 + (2.0 / (1.0 + t_0));
	} else if ((-2.0 * x) <= 2e-7) {
		tmp = (-0.3333333333333333 * Math.pow(x, 3.0)) + (x + (0.13333333333333333 * Math.pow(x, 5.0)));
	} else {
		tmp = -1.0 + (2.0 / Math.exp(Math.log1p(t_0)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.exp((-2.0 * x))
	tmp = 0
	if (-2.0 * x) <= -1000.0:
		tmp = -1.0 + (2.0 / (1.0 + t_0))
	elif (-2.0 * x) <= 2e-7:
		tmp = (-0.3333333333333333 * math.pow(x, 3.0)) + (x + (0.13333333333333333 * math.pow(x, 5.0)))
	else:
		tmp = -1.0 + (2.0 / math.exp(math.log1p(t_0)))
	return tmp

Julia

function code(x, y)
	t_0 = exp(Float64(-2.0 * x))
	tmp = 0.0
	if (Float64(-2.0 * x) <= -1000.0)
		tmp = Float64(-1.0 + Float64(2.0 / Float64(1.0 + t_0)));
	elseif (Float64(-2.0 * x) <= 2e-7)
		tmp = Float64(Float64(-0.3333333333333333 * (x ^ 3.0)) + Float64(x + Float64(0.13333333333333333 * (x ^ 5.0))));
	else
		tmp = Float64(-1.0 + Float64(2.0 / exp(log1p(t_0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -1000.0], N[(-1.0 + N[(2.0 / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 2e-7], N[(N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x + N[(0.13333333333333333 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(2.0 / N[Exp[N[Log[1 + t$95$0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 -2 x) < -1e3

   1. Initial program 100.0%

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

   if -1e3 < (*.f64 -2 x) < 1.9999999999999999e-7

   1. Initial program 7.1%

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

   2. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot {x}^{3} + \left(0.13333333333333333 \cdot {x}^{5} + x\right)} \]

   if 1.9999999999999999e-7 < (*.f64 -2 x)

   1. Initial program 99.9%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. 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. log-div 99.9%

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

      3. log1p-udef 100.0%

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

      4. exp-prod 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. exp-diff 100.0%

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

      2. rem-exp-log 100.0%

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

   5. Simplified 100.0%

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

      1. pow-exp 100.0%

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

      2. *-commutative 100.0%

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

   7. Applied egg-rr 100.0%

      \[\leadsto \frac{2}{e^{\mathsf{log1p}\left(\color{blue}{e^{x \cdot -2}}\right)}} - 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -1000:\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{elif}\;-2 \cdot x \leq 2 \cdot 10^{-7}:\\ \;\;\;\;-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{2}{e^{\mathsf{log1p}\left(e^{-2 \cdot x}\right)}}\\ \end{array} \]

Alternative 4: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -1000 \lor \neg \left(-2 \cdot x \leq 2 \cdot 10^{-7}\right):\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* -2.0 x) -1000.0) (not (<= (* -2.0 x) 2e-7)))
   (+ -1.0 (/ 2.0 (+ 1.0 (exp (* -2.0 x)))))
   (+
    (* -0.3333333333333333 (pow x 3.0))
    (+ x (* 0.13333333333333333 (pow x 5.0))))))

C

double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -1000.0) || !((-2.0 * x) <= 2e-7)) {
		tmp = -1.0 + (2.0 / (1.0 + exp((-2.0 * x))));
	} else {
		tmp = (-0.3333333333333333 * pow(x, 3.0)) + (x + (0.13333333333333333 * pow(x, 5.0)));
	}
	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) <= (-1000.0d0)) .or. (.not. (((-2.0d0) * x) <= 2d-7))) then
        tmp = (-1.0d0) + (2.0d0 / (1.0d0 + exp(((-2.0d0) * x))))
    else
        tmp = ((-0.3333333333333333d0) * (x ** 3.0d0)) + (x + (0.13333333333333333d0 * (x ** 5.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -1000.0) || !((-2.0 * x) <= 2e-7)) {
		tmp = -1.0 + (2.0 / (1.0 + Math.exp((-2.0 * x))));
	} else {
		tmp = (-0.3333333333333333 * Math.pow(x, 3.0)) + (x + (0.13333333333333333 * Math.pow(x, 5.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((-2.0 * x) <= -1000.0) or not ((-2.0 * x) <= 2e-7):
		tmp = -1.0 + (2.0 / (1.0 + math.exp((-2.0 * x))))
	else:
		tmp = (-0.3333333333333333 * math.pow(x, 3.0)) + (x + (0.13333333333333333 * math.pow(x, 5.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((Float64(-2.0 * x) <= -1000.0) || !(Float64(-2.0 * x) <= 2e-7))
		tmp = Float64(-1.0 + Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))));
	else
		tmp = Float64(Float64(-0.3333333333333333 * (x ^ 3.0)) + Float64(x + Float64(0.13333333333333333 * (x ^ 5.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((-2.0 * x) <= -1000.0) || ~(((-2.0 * x) <= 2e-7)))
		tmp = -1.0 + (2.0 / (1.0 + exp((-2.0 * x))));
	else
		tmp = (-0.3333333333333333 * (x ^ 3.0)) + (x + (0.13333333333333333 * (x ^ 5.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[(-2.0 * x), $MachinePrecision], -1000.0], N[Not[LessEqual[N[(-2.0 * x), $MachinePrecision], 2e-7]], $MachinePrecision]], N[(-1.0 + N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x + N[(0.13333333333333333 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 -2 x) < -1e3 or 1.9999999999999999e-7 < (*.f64 -2 x)

   1. Initial program 100.0%

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

   if -1e3 < (*.f64 -2 x) < 1.9999999999999999e-7

   1. Initial program 7.1%

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

   2. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot {x}^{3} + \left(0.13333333333333333 \cdot {x}^{5} + x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -1000 \lor \neg \left(-2 \cdot x \leq 2 \cdot 10^{-7}\right):\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\\ \end{array} \]

Alternative 5: 99.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -1000 \lor \neg \left(-2 \cdot x \leq 2 \cdot 10^{-7}\right):\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* -2.0 x) -1000.0) (not (<= (* -2.0 x) 2e-7)))
   (+ -1.0 (/ 2.0 (+ 1.0 (exp (* -2.0 x)))))
   (+ x (* -0.3333333333333333 (pow x 3.0)))))

C

double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -1000.0) || !((-2.0 * x) <= 2e-7)) {
		tmp = -1.0 + (2.0 / (1.0 + exp((-2.0 * x))));
	} else {
		tmp = x + (-0.3333333333333333 * pow(x, 3.0));
	}
	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) <= (-1000.0d0)) .or. (.not. (((-2.0d0) * x) <= 2d-7))) then
        tmp = (-1.0d0) + (2.0d0 / (1.0d0 + exp(((-2.0d0) * x))))
    else
        tmp = x + ((-0.3333333333333333d0) * (x ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -1000.0) || !((-2.0 * x) <= 2e-7)) {
		tmp = -1.0 + (2.0 / (1.0 + Math.exp((-2.0 * x))));
	} else {
		tmp = x + (-0.3333333333333333 * Math.pow(x, 3.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((-2.0 * x) <= -1000.0) or not ((-2.0 * x) <= 2e-7):
		tmp = -1.0 + (2.0 / (1.0 + math.exp((-2.0 * x))))
	else:
		tmp = x + (-0.3333333333333333 * math.pow(x, 3.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((Float64(-2.0 * x) <= -1000.0) || !(Float64(-2.0 * x) <= 2e-7))
		tmp = Float64(-1.0 + Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))));
	else
		tmp = Float64(x + Float64(-0.3333333333333333 * (x ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((-2.0 * x) <= -1000.0) || ~(((-2.0 * x) <= 2e-7)))
		tmp = -1.0 + (2.0 / (1.0 + exp((-2.0 * x))));
	else
		tmp = x + (-0.3333333333333333 * (x ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[(-2.0 * x), $MachinePrecision], -1000.0], N[Not[LessEqual[N[(-2.0 * x), $MachinePrecision], 2e-7]], $MachinePrecision]], N[(-1.0 + N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 -2 x) < -1e3 or 1.9999999999999999e-7 < (*.f64 -2 x)

   1. Initial program 100.0%

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

   if -1e3 < (*.f64 -2 x) < 1.9999999999999999e-7

   1. Initial program 7.1%

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

   2. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot {x}^{3} + x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -1000 \lor \neg \left(-2 \cdot x \leq 2 \cdot 10^{-7}\right):\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \end{array} \]

Alternative 6: 78.5% accurate, 9.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.66)
		tmp = -1.0;
	else
		tmp = Float64(Float64(x + x) * Float64(1.0 / 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 + x) * (1.0 / (x + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.66], -1.0, N[(N[(x + x), $MachinePrecision] * N[(1.0 / 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. Taylor expanded in x around 0 96.6%

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

      1. *-commutative 96.6%

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

   4. Simplified 96.6%

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

   5. Taylor expanded in x around inf 98.8%

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

   if -0.660000000000000031 < x

   1. Initial program 35.3%

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

   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. div-inv 6.5%

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

      3. metadata-eval 6.5%

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

      4. difference-of-sqr-1 6.5%

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

      5. associate-+l+ 6.5%

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

      6. metadata-eval 6.5%

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

      7. associate--l+ 71.0%

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

      8. metadata-eval 71.0%

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

      9. +-rgt-identity 71.0%

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

      10. associate-+l+ 71.0%

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

      11. metadata-eval 71.0%

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

   6. Applied egg-rr 71.0%

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

   7. Taylor expanded in x around 0 74.4%

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

      1. *-commutative 74.4%

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

      2. rem-log-exp 5.6%

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

      3. exp-lft-sqr 5.6%

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

      4. log-prod 5.7%

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

      5. rem-log-exp 14.7%

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

      6. rem-log-exp 74.4%

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

   9. Simplified 74.4%

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

4. Final simplification 80.6%

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

Alternative 7: 75.7% accurate, 35.6× 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. Taylor expanded in x around 0 97.8%

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

      1. *-commutative 97.8%

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

   4. Simplified 97.8%

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

   5. Taylor expanded in x around inf 100.0%

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

   if -1 < x

   1. Initial program 35.6%

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

   2. Taylor expanded in x around 0 70.8%

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

4. Final simplification 78.1%

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

Alternative 8: 27.5% 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 51.7%

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

2. Taylor expanded in x around 0 28.4%

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

   1. *-commutative 28.4%

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

4. Simplified 28.4%

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

5. Taylor expanded in x around inf 27.5%

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

6. Final simplification 27.5%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023175 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))