Logistic function from Lakshay Garg

Percentage Accurate: 53.8% → 99.8%

Time: 16.1s

Alternatives: 11

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: 53.8% 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: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(e^{-2}\right)}^{x} + 1\\ \mathbf{if}\;-2 \cdot x \leq -0.05:\\ \;\;\;\;\frac{\frac{8}{{t_0}^{3}} + -1}{4 \cdot {t_0}^{-2} + \left(1 + \frac{2}{t_0}\right)}\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-10}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(e^{-2 \cdot x}\right)\right)}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (pow (exp -2.0) x) 1.0)))
   (if (<= (* -2.0 x) -0.05)
     (/
      (+ (/ 8.0 (pow t_0 3.0)) -1.0)
      (+ (* 4.0 (pow t_0 -2.0)) (+ 1.0 (/ 2.0 t_0))))
     (if (<= (* -2.0 x) 5e-10)
       (+ x (* -0.3333333333333333 (pow x 3.0)))
       (pow (cbrt (expm1 (- (log 2.0) (log1p (exp (* -2.0 x)))))) 3.0)))))

C

double code(double x, double y) {
	double t_0 = pow(exp(-2.0), x) + 1.0;
	double tmp;
	if ((-2.0 * x) <= -0.05) {
		tmp = ((8.0 / pow(t_0, 3.0)) + -1.0) / ((4.0 * pow(t_0, -2.0)) + (1.0 + (2.0 / t_0)));
	} else if ((-2.0 * x) <= 5e-10) {
		tmp = x + (-0.3333333333333333 * pow(x, 3.0));
	} else {
		tmp = pow(cbrt(expm1((log(2.0) - log1p(exp((-2.0 * x)))))), 3.0);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = Math.pow(Math.exp(-2.0), x) + 1.0;
	double tmp;
	if ((-2.0 * x) <= -0.05) {
		tmp = ((8.0 / Math.pow(t_0, 3.0)) + -1.0) / ((4.0 * Math.pow(t_0, -2.0)) + (1.0 + (2.0 / t_0)));
	} else if ((-2.0 * x) <= 5e-10) {
		tmp = x + (-0.3333333333333333 * Math.pow(x, 3.0));
	} else {
		tmp = Math.pow(Math.cbrt(Math.expm1((Math.log(2.0) - Math.log1p(Math.exp((-2.0 * x)))))), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64((exp(-2.0) ^ x) + 1.0)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.05)
		tmp = Float64(Float64(Float64(8.0 / (t_0 ^ 3.0)) + -1.0) / Float64(Float64(4.0 * (t_0 ^ -2.0)) + Float64(1.0 + Float64(2.0 / t_0))));
	elseif (Float64(-2.0 * x) <= 5e-10)
		tmp = Float64(x + Float64(-0.3333333333333333 * (x ^ 3.0)));
	else
		tmp = cbrt(expm1(Float64(log(2.0) - log1p(exp(Float64(-2.0 * x)))))) ^ 3.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Power[N[Exp[-2.0], $MachinePrecision], x], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.05], N[(N[(N[(8.0 / N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / N[(N[(4.0 * N[Power[t$95$0, -2.0], $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(2.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 5e-10], N[(x + N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(Exp[N[(N[Log[2.0], $MachinePrecision] - N[Log[1 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]]]]

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. Step-by-step derivation

      1. add-cube-cbrt 99.9%

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

      2. pow3 99.9%

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

      3. add-exp-log 99.9%

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

      4. expm1-def 99.9%

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

      5. log-div 99.9%

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

      6. log1p-udef 99.9%

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

      7. exp-prod 99.9%

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

   3. Applied egg-rr 99.9%

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

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\frac{8}{{\left({\left(e^{-2}\right)}^{x} + 1\right)}^{3}} + -1}{4 \cdot {\left({\left(e^{-2}\right)}^{x} + 1\right)}^{-2} + \left(1 + \frac{2}{{\left(e^{-2}\right)}^{x} + 1}\right)}} \]

   if -0.050000000000000003 < (*.f64 -2 x) < 5.00000000000000031e-10

   1. Initial program 6.6%

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

   2. Taylor expanded in x around 0 100.0%

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

   if 5.00000000000000031e-10 < (*.f64 -2 x)

   1. Initial program 99.9%

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

      1. add-cube-cbrt 99.9%

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

      2. pow3 99.9%

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

      3. add-exp-log 99.9%

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

      4. expm1-def 99.9%

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

      5. log-div 99.9%

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

      6. log1p-udef 100.0%

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

      7. exp-prod 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around inf 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.05:\\ \;\;\;\;\frac{\frac{8}{{\left({\left(e^{-2}\right)}^{x} + 1\right)}^{3}} + -1}{4 \cdot {\left({\left(e^{-2}\right)}^{x} + 1\right)}^{-2} + \left(1 + \frac{2}{{\left(e^{-2}\right)}^{x} + 1}\right)}\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-10}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(e^{-2 \cdot x}\right)\right)}\right)}^{3}\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-2 \cdot x}\\ t_1 := 1 + t_0\\ t_2 := 1 + \frac{2}{t_1}\\ \mathbf{if}\;-2 \cdot x \leq -0.05:\\ \;\;\;\;\frac{4}{{t_1}^{2} \cdot t_2} + \frac{-1}{t_2}\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-10}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(t_0\right)\right)}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (exp (* -2.0 x))) (t_1 (+ 1.0 t_0)) (t_2 (+ 1.0 (/ 2.0 t_1))))
   (if (<= (* -2.0 x) -0.05)
     (+ (/ 4.0 (* (pow t_1 2.0) t_2)) (/ -1.0 t_2))
     (if (<= (* -2.0 x) 5e-10)
       (+ x (* -0.3333333333333333 (pow x 3.0)))
       (pow (cbrt (expm1 (- (log 2.0) (log1p t_0)))) 3.0)))))

C

double code(double x, double y) {
	double t_0 = exp((-2.0 * x));
	double t_1 = 1.0 + t_0;
	double t_2 = 1.0 + (2.0 / t_1);
	double tmp;
	if ((-2.0 * x) <= -0.05) {
		tmp = (4.0 / (pow(t_1, 2.0) * t_2)) + (-1.0 / t_2);
	} else if ((-2.0 * x) <= 5e-10) {
		tmp = x + (-0.3333333333333333 * pow(x, 3.0));
	} else {
		tmp = pow(cbrt(expm1((log(2.0) - log1p(t_0)))), 3.0);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = Math.exp((-2.0 * x));
	double t_1 = 1.0 + t_0;
	double t_2 = 1.0 + (2.0 / t_1);
	double tmp;
	if ((-2.0 * x) <= -0.05) {
		tmp = (4.0 / (Math.pow(t_1, 2.0) * t_2)) + (-1.0 / t_2);
	} else if ((-2.0 * x) <= 5e-10) {
		tmp = x + (-0.3333333333333333 * Math.pow(x, 3.0));
	} else {
		tmp = Math.pow(Math.cbrt(Math.expm1((Math.log(2.0) - Math.log1p(t_0)))), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = exp(Float64(-2.0 * x))
	t_1 = Float64(1.0 + t_0)
	t_2 = Float64(1.0 + Float64(2.0 / t_1))
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.05)
		tmp = Float64(Float64(4.0 / Float64((t_1 ^ 2.0) * t_2)) + Float64(-1.0 / t_2));
	elseif (Float64(-2.0 * x) <= 5e-10)
		tmp = Float64(x + Float64(-0.3333333333333333 * (x ^ 3.0)));
	else
		tmp = cbrt(expm1(Float64(log(2.0) - log1p(t_0)))) ^ 3.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(2.0 / t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.05], N[(N[(4.0 / N[(N[Power[t$95$1, 2.0], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 5e-10], N[(x + N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(Exp[N[(N[Log[2.0], $MachinePrecision] - N[Log[1 + t$95$0], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]]]]]]

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. Step-by-step derivation

      1. add-cube-cbrt 99.9%

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

      2. pow3 99.9%

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

      3. add-exp-log 99.9%

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

      4. expm1-def 99.9%

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

      5. log-div 99.9%

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

      6. log1p-udef 99.9%

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

      7. exp-prod 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in x around inf 99.9%

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

      1. rem-cube-cbrt 99.9%

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

      2. log1p-udef 99.9%

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

      3. pow-exp 99.9%

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

      4. +-commutative 99.9%

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

      5. diff-log 99.9%

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

      6. expm1-def 99.9%

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

      7. add-exp-log 99.9%

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

      8. sub-neg 99.9%

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

      9. metadata-eval 99.9%

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

      10. flip-+ 99.9%

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

      11. metadata-eval 99.9%

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

      12. div-sub 99.9%

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

   6. Applied egg-rr 100.0%

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

      1. sub-neg 100.0%

         \[\leadsto \color{blue}{\frac{\frac{4}{{\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{2}}}{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}} + \left(-\frac{1}{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}\right)} \]

      2. associate-/l/ 100.0%

         \[\leadsto \color{blue}{\frac{4}{\left(1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right) \cdot {\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{2}}} + \left(-\frac{1}{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}\right) \]

      3. *-commutative 100.0%

         \[\leadsto \frac{4}{\color{blue}{{\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{2} \cdot \left(1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)}} + \left(-\frac{1}{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}\right) \]

      4. exp-prod 100.0%

         \[\leadsto \frac{4}{{\left(1 + \color{blue}{e^{-2 \cdot x}}\right)}^{2} \cdot \left(1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)} + \left(-\frac{1}{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}\right) \]

      5. *-commutative 100.0%

         \[\leadsto \frac{4}{{\left(1 + e^{\color{blue}{x \cdot -2}}\right)}^{2} \cdot \left(1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)} + \left(-\frac{1}{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}\right) \]

      6. exp-prod 100.0%

         \[\leadsto \frac{4}{{\left(1 + e^{x \cdot -2}\right)}^{2} \cdot \left(1 + \frac{2}{1 + \color{blue}{e^{-2 \cdot x}}}\right)} + \left(-\frac{1}{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}\right) \]

      7. *-commutative 100.0%

         \[\leadsto \frac{4}{{\left(1 + e^{x \cdot -2}\right)}^{2} \cdot \left(1 + \frac{2}{1 + e^{\color{blue}{x \cdot -2}}}\right)} + \left(-\frac{1}{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}\right) \]

      8. distribute-neg-frac 100.0%

         \[\leadsto \frac{4}{{\left(1 + e^{x \cdot -2}\right)}^{2} \cdot \left(1 + \frac{2}{1 + e^{x \cdot -2}}\right)} + \color{blue}{\frac{-1}{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}} \]

      9. metadata-eval 100.0%

         \[\leadsto \frac{4}{{\left(1 + e^{x \cdot -2}\right)}^{2} \cdot \left(1 + \frac{2}{1 + e^{x \cdot -2}}\right)} + \frac{\color{blue}{-1}}{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}} \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{4}{{\left(1 + e^{x \cdot -2}\right)}^{2} \cdot \left(1 + \frac{2}{1 + e^{x \cdot -2}}\right)} + \frac{-1}{1 + \frac{2}{1 + e^{x \cdot -2}}}} \]

   if -0.050000000000000003 < (*.f64 -2 x) < 5.00000000000000031e-10

   1. Initial program 6.6%

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

   2. Taylor expanded in x around 0 100.0%

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

   if 5.00000000000000031e-10 < (*.f64 -2 x)

   1. Initial program 99.9%

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

      1. add-cube-cbrt 99.9%

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

      2. pow3 99.9%

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

      3. add-exp-log 99.9%

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

      4. expm1-def 99.9%

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

      5. log-div 99.9%

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

      6. log1p-udef 100.0%

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

      7. exp-prod 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around inf 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.05:\\ \;\;\;\;\frac{4}{{\left(1 + e^{-2 \cdot x}\right)}^{2} \cdot \left(1 + \frac{2}{1 + e^{-2 \cdot x}}\right)} + \frac{-1}{1 + \frac{2}{1 + e^{-2 \cdot x}}}\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-10}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(e^{-2 \cdot x}\right)\right)}\right)}^{3}\\ \end{array} \]

Alternative 3: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-2 \cdot x}\\ t_1 := 1 + t_0\\ t_2 := 1 + \frac{2}{t_1}\\ \mathbf{if}\;-2 \cdot x \leq -0.05:\\ \;\;\;\;\frac{4}{{t_1}^{2} \cdot t_2} + \frac{-1}{t_2}\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-10}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(t_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (exp (* -2.0 x))) (t_1 (+ 1.0 t_0)) (t_2 (+ 1.0 (/ 2.0 t_1))))
   (if (<= (* -2.0 x) -0.05)
     (+ (/ 4.0 (* (pow t_1 2.0) t_2)) (/ -1.0 t_2))
     (if (<= (* -2.0 x) 5e-10)
       (+ x (* -0.3333333333333333 (pow x 3.0)))
       (expm1 (- (log 2.0) (log1p t_0)))))))

C

double code(double x, double y) {
	double t_0 = exp((-2.0 * x));
	double t_1 = 1.0 + t_0;
	double t_2 = 1.0 + (2.0 / t_1);
	double tmp;
	if ((-2.0 * x) <= -0.05) {
		tmp = (4.0 / (pow(t_1, 2.0) * t_2)) + (-1.0 / t_2);
	} else if ((-2.0 * x) <= 5e-10) {
		tmp = x + (-0.3333333333333333 * pow(x, 3.0));
	} 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 t_1 = 1.0 + t_0;
	double t_2 = 1.0 + (2.0 / t_1);
	double tmp;
	if ((-2.0 * x) <= -0.05) {
		tmp = (4.0 / (Math.pow(t_1, 2.0) * t_2)) + (-1.0 / t_2);
	} else if ((-2.0 * x) <= 5e-10) {
		tmp = x + (-0.3333333333333333 * Math.pow(x, 3.0));
	} 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))
	t_1 = 1.0 + t_0
	t_2 = 1.0 + (2.0 / t_1)
	tmp = 0
	if (-2.0 * x) <= -0.05:
		tmp = (4.0 / (math.pow(t_1, 2.0) * t_2)) + (-1.0 / t_2)
	elif (-2.0 * x) <= 5e-10:
		tmp = x + (-0.3333333333333333 * math.pow(x, 3.0))
	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))
	t_1 = Float64(1.0 + t_0)
	t_2 = Float64(1.0 + Float64(2.0 / t_1))
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.05)
		tmp = Float64(Float64(4.0 / Float64((t_1 ^ 2.0) * t_2)) + Float64(-1.0 / t_2));
	elseif (Float64(-2.0 * x) <= 5e-10)
		tmp = Float64(x + Float64(-0.3333333333333333 * (x ^ 3.0)));
	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]}, Block[{t$95$1 = N[(1.0 + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(2.0 / t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.05], N[(N[(4.0 / N[(N[Power[t$95$1, 2.0], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 5e-10], N[(x + N[(-0.3333333333333333 * N[Power[x, 3.0], $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) < -0.050000000000000003

   1. Initial program 99.9%

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

      1. add-cube-cbrt 99.9%

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

      2. pow3 99.9%

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

      3. add-exp-log 99.9%

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

      4. expm1-def 99.9%

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

      5. log-div 99.9%

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

      6. log1p-udef 99.9%

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

      7. exp-prod 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in x around inf 99.9%

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

      1. rem-cube-cbrt 99.9%

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

      2. log1p-udef 99.9%

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

      3. pow-exp 99.9%

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

      4. +-commutative 99.9%

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

      5. diff-log 99.9%

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

      6. expm1-def 99.9%

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

      7. add-exp-log 99.9%

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

      8. sub-neg 99.9%

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

      9. metadata-eval 99.9%

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

      10. flip-+ 99.9%

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

      11. metadata-eval 99.9%

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

      12. div-sub 99.9%

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

   6. Applied egg-rr 100.0%

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

      1. sub-neg 100.0%

         \[\leadsto \color{blue}{\frac{\frac{4}{{\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{2}}}{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}} + \left(-\frac{1}{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}\right)} \]

      2. associate-/l/ 100.0%

         \[\leadsto \color{blue}{\frac{4}{\left(1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right) \cdot {\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{2}}} + \left(-\frac{1}{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}\right) \]

      3. *-commutative 100.0%

         \[\leadsto \frac{4}{\color{blue}{{\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{2} \cdot \left(1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)}} + \left(-\frac{1}{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}\right) \]

      4. exp-prod 100.0%

         \[\leadsto \frac{4}{{\left(1 + \color{blue}{e^{-2 \cdot x}}\right)}^{2} \cdot \left(1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)} + \left(-\frac{1}{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}\right) \]

      5. *-commutative 100.0%

         \[\leadsto \frac{4}{{\left(1 + e^{\color{blue}{x \cdot -2}}\right)}^{2} \cdot \left(1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)} + \left(-\frac{1}{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}\right) \]

      6. exp-prod 100.0%

         \[\leadsto \frac{4}{{\left(1 + e^{x \cdot -2}\right)}^{2} \cdot \left(1 + \frac{2}{1 + \color{blue}{e^{-2 \cdot x}}}\right)} + \left(-\frac{1}{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}\right) \]

      7. *-commutative 100.0%

         \[\leadsto \frac{4}{{\left(1 + e^{x \cdot -2}\right)}^{2} \cdot \left(1 + \frac{2}{1 + e^{\color{blue}{x \cdot -2}}}\right)} + \left(-\frac{1}{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}\right) \]

      8. distribute-neg-frac 100.0%

         \[\leadsto \frac{4}{{\left(1 + e^{x \cdot -2}\right)}^{2} \cdot \left(1 + \frac{2}{1 + e^{x \cdot -2}}\right)} + \color{blue}{\frac{-1}{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}} \]

      9. metadata-eval 100.0%

         \[\leadsto \frac{4}{{\left(1 + e^{x \cdot -2}\right)}^{2} \cdot \left(1 + \frac{2}{1 + e^{x \cdot -2}}\right)} + \frac{\color{blue}{-1}}{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}} \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{4}{{\left(1 + e^{x \cdot -2}\right)}^{2} \cdot \left(1 + \frac{2}{1 + e^{x \cdot -2}}\right)} + \frac{-1}{1 + \frac{2}{1 + e^{x \cdot -2}}}} \]

   if -0.050000000000000003 < (*.f64 -2 x) < 5.00000000000000031e-10

   1. Initial program 6.6%

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

   2. Taylor expanded in x around 0 100.0%

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

   if 5.00000000000000031e-10 < (*.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 99.9%

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

      10. exp-prod 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. +-lft-identity 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around inf 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.05:\\ \;\;\;\;\frac{4}{{\left(1 + e^{-2 \cdot x}\right)}^{2} \cdot \left(1 + \frac{2}{1 + e^{-2 \cdot x}}\right)} + \frac{-1}{1 + \frac{2}{1 + e^{-2 \cdot x}}}\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-10}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(e^{-2 \cdot x}\right)\right)\\ \end{array} \]

Alternative 4: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-2 \cdot x}\\ t_1 := 1 + t_0\\ \mathbf{if}\;-2 \cdot x \leq -0.05:\\ \;\;\;\;\frac{-1 + \frac{4}{{t_1}^{2}}}{1 + \frac{2}{t_1}}\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-10}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(t_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (exp (* -2.0 x))) (t_1 (+ 1.0 t_0)))
   (if (<= (* -2.0 x) -0.05)
     (/ (+ -1.0 (/ 4.0 (pow t_1 2.0))) (+ 1.0 (/ 2.0 t_1)))
     (if (<= (* -2.0 x) 5e-10)
       (+ x (* -0.3333333333333333 (pow x 3.0)))
       (expm1 (- (log 2.0) (log1p t_0)))))))

C

double code(double x, double y) {
	double t_0 = exp((-2.0 * x));
	double t_1 = 1.0 + t_0;
	double tmp;
	if ((-2.0 * x) <= -0.05) {
		tmp = (-1.0 + (4.0 / pow(t_1, 2.0))) / (1.0 + (2.0 / t_1));
	} else if ((-2.0 * x) <= 5e-10) {
		tmp = x + (-0.3333333333333333 * pow(x, 3.0));
	} 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 t_1 = 1.0 + t_0;
	double tmp;
	if ((-2.0 * x) <= -0.05) {
		tmp = (-1.0 + (4.0 / Math.pow(t_1, 2.0))) / (1.0 + (2.0 / t_1));
	} else if ((-2.0 * x) <= 5e-10) {
		tmp = x + (-0.3333333333333333 * Math.pow(x, 3.0));
	} 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))
	t_1 = 1.0 + t_0
	tmp = 0
	if (-2.0 * x) <= -0.05:
		tmp = (-1.0 + (4.0 / math.pow(t_1, 2.0))) / (1.0 + (2.0 / t_1))
	elif (-2.0 * x) <= 5e-10:
		tmp = x + (-0.3333333333333333 * math.pow(x, 3.0))
	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))
	t_1 = Float64(1.0 + t_0)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.05)
		tmp = Float64(Float64(-1.0 + Float64(4.0 / (t_1 ^ 2.0))) / Float64(1.0 + Float64(2.0 / t_1)));
	elseif (Float64(-2.0 * x) <= 5e-10)
		tmp = Float64(x + Float64(-0.3333333333333333 * (x ^ 3.0)));
	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]}, Block[{t$95$1 = N[(1.0 + t$95$0), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.05], N[(N[(-1.0 + N[(4.0 / N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 5e-10], N[(x + N[(-0.3333333333333333 * N[Power[x, 3.0], $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) < -0.050000000000000003

   1. Initial program 99.9%

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

      1. add-cube-cbrt 99.9%

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

      2. pow3 99.9%

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

      3. add-exp-log 99.9%

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

      4. expm1-def 99.9%

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

      5. log-div 99.9%

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

      6. log1p-udef 99.9%

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

      7. exp-prod 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in x around inf 99.9%

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

      1. rem-cube-cbrt 99.9%

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

      2. log1p-udef 99.9%

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

      3. pow-exp 99.9%

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

      4. +-commutative 99.9%

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

      5. diff-log 99.9%

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

      6. expm1-def 99.9%

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

      7. add-exp-log 99.9%

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

      8. sub-neg 99.9%

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

      9. metadata-eval 99.9%

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

      10. flip-+ 99.9%

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

      11. div-inv 99.9%

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

   6. Applied egg-rr 100.0%

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

      1. associate-*r/ 100.0%

         \[\leadsto \color{blue}{\frac{\left(\frac{4}{{\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{2}} + -1\right) \cdot 1}{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}} \]

      2. *-rgt-identity 100.0%

         \[\leadsto \frac{\color{blue}{\frac{4}{{\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{2}} + -1}}{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}} \]

      3. +-commutative 100.0%

         \[\leadsto \frac{\color{blue}{-1 + \frac{4}{{\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{2}}}}{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}} \]

      4. exp-prod 100.0%

         \[\leadsto \frac{-1 + \frac{4}{{\left(1 + \color{blue}{e^{-2 \cdot x}}\right)}^{2}}}{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}} \]

      5. *-commutative 100.0%

         \[\leadsto \frac{-1 + \frac{4}{{\left(1 + e^{\color{blue}{x \cdot -2}}\right)}^{2}}}{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}} \]

      6. exp-prod 100.0%

         \[\leadsto \frac{-1 + \frac{4}{{\left(1 + e^{x \cdot -2}\right)}^{2}}}{1 + \frac{2}{1 + \color{blue}{e^{-2 \cdot x}}}} \]

      7. *-commutative 100.0%

         \[\leadsto \frac{-1 + \frac{4}{{\left(1 + e^{x \cdot -2}\right)}^{2}}}{1 + \frac{2}{1 + e^{\color{blue}{x \cdot -2}}}} \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{-1 + \frac{4}{{\left(1 + e^{x \cdot -2}\right)}^{2}}}{1 + \frac{2}{1 + e^{x \cdot -2}}}} \]

   if -0.050000000000000003 < (*.f64 -2 x) < 5.00000000000000031e-10

   1. Initial program 6.6%

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

   2. Taylor expanded in x around 0 100.0%

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

   if 5.00000000000000031e-10 < (*.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 99.9%

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

      10. exp-prod 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. +-lft-identity 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around inf 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.05:\\ \;\;\;\;\frac{-1 + \frac{4}{{\left(1 + e^{-2 \cdot x}\right)}^{2}}}{1 + \frac{2}{1 + e^{-2 \cdot x}}}\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-10}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(e^{-2 \cdot x}\right)\right)\\ \end{array} \]

Alternative 5: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-2 \cdot x}\\ t_1 := 1 + t_0\\ \mathbf{if}\;-2 \cdot x \leq -0.05:\\ \;\;\;\;\frac{-1 + \frac{4}{{t_1}^{2}}}{1 + \frac{2}{t_1}}\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-10}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{2}{e^{\mathsf{log1p}\left(t_0\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (exp (* -2.0 x))) (t_1 (+ 1.0 t_0)))
   (if (<= (* -2.0 x) -0.05)
     (/ (+ -1.0 (/ 4.0 (pow t_1 2.0))) (+ 1.0 (/ 2.0 t_1)))
     (if (<= (* -2.0 x) 5e-10)
       (+ x (* -0.3333333333333333 (pow x 3.0)))
       (+ -1.0 (/ 2.0 (exp (log1p t_0))))))))

C

double code(double x, double y) {
	double t_0 = exp((-2.0 * x));
	double t_1 = 1.0 + t_0;
	double tmp;
	if ((-2.0 * x) <= -0.05) {
		tmp = (-1.0 + (4.0 / pow(t_1, 2.0))) / (1.0 + (2.0 / t_1));
	} else if ((-2.0 * x) <= 5e-10) {
		tmp = x + (-0.3333333333333333 * pow(x, 3.0));
	} 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 t_1 = 1.0 + t_0;
	double tmp;
	if ((-2.0 * x) <= -0.05) {
		tmp = (-1.0 + (4.0 / Math.pow(t_1, 2.0))) / (1.0 + (2.0 / t_1));
	} else if ((-2.0 * x) <= 5e-10) {
		tmp = x + (-0.3333333333333333 * Math.pow(x, 3.0));
	} 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))
	t_1 = 1.0 + t_0
	tmp = 0
	if (-2.0 * x) <= -0.05:
		tmp = (-1.0 + (4.0 / math.pow(t_1, 2.0))) / (1.0 + (2.0 / t_1))
	elif (-2.0 * x) <= 5e-10:
		tmp = x + (-0.3333333333333333 * math.pow(x, 3.0))
	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))
	t_1 = Float64(1.0 + t_0)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.05)
		tmp = Float64(Float64(-1.0 + Float64(4.0 / (t_1 ^ 2.0))) / Float64(1.0 + Float64(2.0 / t_1)));
	elseif (Float64(-2.0 * x) <= 5e-10)
		tmp = Float64(x + Float64(-0.3333333333333333 * (x ^ 3.0)));
	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]}, Block[{t$95$1 = N[(1.0 + t$95$0), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.05], N[(N[(-1.0 + N[(4.0 / N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 5e-10], N[(x + N[(-0.3333333333333333 * N[Power[x, 3.0], $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) < -0.050000000000000003

   1. Initial program 99.9%

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

      1. add-cube-cbrt 99.9%

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

      2. pow3 99.9%

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

      3. add-exp-log 99.9%

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

      4. expm1-def 99.9%

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

      5. log-div 99.9%

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

      6. log1p-udef 99.9%

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

      7. exp-prod 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in x around inf 99.9%

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

      1. rem-cube-cbrt 99.9%

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

      2. log1p-udef 99.9%

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

      3. pow-exp 99.9%

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

      4. +-commutative 99.9%

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

      5. diff-log 99.9%

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

      6. expm1-def 99.9%

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

      7. add-exp-log 99.9%

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

      8. sub-neg 99.9%

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

      9. metadata-eval 99.9%

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

      10. flip-+ 99.9%

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

      11. div-inv 99.9%

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

   6. Applied egg-rr 100.0%

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

      1. associate-*r/ 100.0%

         \[\leadsto \color{blue}{\frac{\left(\frac{4}{{\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{2}} + -1\right) \cdot 1}{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}} \]

      2. *-rgt-identity 100.0%

         \[\leadsto \frac{\color{blue}{\frac{4}{{\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{2}} + -1}}{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}} \]

      3. +-commutative 100.0%

         \[\leadsto \frac{\color{blue}{-1 + \frac{4}{{\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{2}}}}{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}} \]

      4. exp-prod 100.0%

         \[\leadsto \frac{-1 + \frac{4}{{\left(1 + \color{blue}{e^{-2 \cdot x}}\right)}^{2}}}{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}} \]

      5. *-commutative 100.0%

         \[\leadsto \frac{-1 + \frac{4}{{\left(1 + e^{\color{blue}{x \cdot -2}}\right)}^{2}}}{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}} \]

      6. exp-prod 100.0%

         \[\leadsto \frac{-1 + \frac{4}{{\left(1 + e^{x \cdot -2}\right)}^{2}}}{1 + \frac{2}{1 + \color{blue}{e^{-2 \cdot x}}}} \]

      7. *-commutative 100.0%

         \[\leadsto \frac{-1 + \frac{4}{{\left(1 + e^{x \cdot -2}\right)}^{2}}}{1 + \frac{2}{1 + e^{\color{blue}{x \cdot -2}}}} \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{-1 + \frac{4}{{\left(1 + e^{x \cdot -2}\right)}^{2}}}{1 + \frac{2}{1 + e^{x \cdot -2}}}} \]

   if -0.050000000000000003 < (*.f64 -2 x) < 5.00000000000000031e-10

   1. Initial program 6.6%

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

   2. Taylor expanded in x around 0 100.0%

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

   if 5.00000000000000031e-10 < (*.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 99.9%

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

      4. exp-prod 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. exp-diff 99.9%

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

      2. rem-exp-log 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around inf 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.05:\\ \;\;\;\;\frac{-1 + \frac{4}{{\left(1 + e^{-2 \cdot x}\right)}^{2}}}{1 + \frac{2}{1 + e^{-2 \cdot x}}}\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-10}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{2}{e^{\mathsf{log1p}\left(e^{-2 \cdot x}\right)}}\\ \end{array} \]

Alternative 6: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-2 \cdot x}\\ \mathbf{if}\;-2 \cdot x \leq -0.05:\\ \;\;\;\;-1 + \frac{2}{1 + t_0}\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-10}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{2}{e^{\mathsf{log1p}\left(t_0\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (exp (* -2.0 x))))
   (if (<= (* -2.0 x) -0.05)
     (+ -1.0 (/ 2.0 (+ 1.0 t_0)))
     (if (<= (* -2.0 x) 5e-10)
       (+ x (* -0.3333333333333333 (pow x 3.0)))
       (+ -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) <= -0.05) {
		tmp = -1.0 + (2.0 / (1.0 + t_0));
	} else if ((-2.0 * x) <= 5e-10) {
		tmp = x + (-0.3333333333333333 * pow(x, 3.0));
	} 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) <= -0.05) {
		tmp = -1.0 + (2.0 / (1.0 + t_0));
	} else if ((-2.0 * x) <= 5e-10) {
		tmp = x + (-0.3333333333333333 * Math.pow(x, 3.0));
	} 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) <= -0.05:
		tmp = -1.0 + (2.0 / (1.0 + t_0))
	elif (-2.0 * x) <= 5e-10:
		tmp = x + (-0.3333333333333333 * math.pow(x, 3.0))
	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) <= -0.05)
		tmp = Float64(-1.0 + Float64(2.0 / Float64(1.0 + t_0)));
	elseif (Float64(-2.0 * x) <= 5e-10)
		tmp = Float64(x + Float64(-0.3333333333333333 * (x ^ 3.0)));
	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], -0.05], N[(-1.0 + N[(2.0 / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 5e-10], N[(x + N[(-0.3333333333333333 * N[Power[x, 3.0], $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) < -0.050000000000000003

   1. Initial program 99.9%

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

   if -0.050000000000000003 < (*.f64 -2 x) < 5.00000000000000031e-10

   1. Initial program 6.6%

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

   2. Taylor expanded in x around 0 100.0%

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

   if 5.00000000000000031e-10 < (*.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 99.9%

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

      4. exp-prod 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. exp-diff 99.9%

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

      2. rem-exp-log 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around inf 99.9%

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

4. Final simplification 99.9%

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

Alternative 7: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-2 \cdot x}\\ \mathbf{if}\;-2 \cdot x \leq -0.05:\\ \;\;\;\;{\left({\left(-1 + \frac{2}{1 + t_0}\right)}^{0.3333333333333333}\right)}^{3}\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-10}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{2}{e^{\mathsf{log1p}\left(t_0\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (exp (* -2.0 x))))
   (if (<= (* -2.0 x) -0.05)
     (pow (pow (+ -1.0 (/ 2.0 (+ 1.0 t_0))) 0.3333333333333333) 3.0)
     (if (<= (* -2.0 x) 5e-10)
       (+ x (* -0.3333333333333333 (pow x 3.0)))
       (+ -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) <= -0.05) {
		tmp = pow(pow((-1.0 + (2.0 / (1.0 + t_0))), 0.3333333333333333), 3.0);
	} else if ((-2.0 * x) <= 5e-10) {
		tmp = x + (-0.3333333333333333 * pow(x, 3.0));
	} 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) <= -0.05) {
		tmp = Math.pow(Math.pow((-1.0 + (2.0 / (1.0 + t_0))), 0.3333333333333333), 3.0);
	} else if ((-2.0 * x) <= 5e-10) {
		tmp = x + (-0.3333333333333333 * Math.pow(x, 3.0));
	} 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) <= -0.05:
		tmp = math.pow(math.pow((-1.0 + (2.0 / (1.0 + t_0))), 0.3333333333333333), 3.0)
	elif (-2.0 * x) <= 5e-10:
		tmp = x + (-0.3333333333333333 * math.pow(x, 3.0))
	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) <= -0.05)
		tmp = (Float64(-1.0 + Float64(2.0 / Float64(1.0 + t_0))) ^ 0.3333333333333333) ^ 3.0;
	elseif (Float64(-2.0 * x) <= 5e-10)
		tmp = Float64(x + Float64(-0.3333333333333333 * (x ^ 3.0)));
	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], -0.05], N[Power[N[Power[N[(-1.0 + N[(2.0 / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.3333333333333333], $MachinePrecision], 3.0], $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 5e-10], N[(x + N[(-0.3333333333333333 * N[Power[x, 3.0], $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) < -0.050000000000000003

   1. Initial program 99.9%

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

      1. add-cube-cbrt 99.9%

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

      2. pow3 99.9%

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

      3. add-exp-log 99.9%

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

      4. expm1-def 99.9%

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

      5. log-div 99.9%

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

      6. log1p-udef 99.9%

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

      7. exp-prod 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. pow1/3 100.0%

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

      2. log1p-udef 100.0%

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

      3. diff-log 99.9%

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

      4. add-exp-log 99.9%

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

      5. log1p-udef 99.9%

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

      6. expm1-def 99.9%

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

      7. add-exp-log 99.9%

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

      8. sub-neg 99.9%

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

      9. log1p-udef 99.9%

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

      10. add-exp-log 99.9%

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

      11. +-commutative 99.9%

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

      12. metadata-eval 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around inf 99.9%

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

   if -0.050000000000000003 < (*.f64 -2 x) < 5.00000000000000031e-10

   1. Initial program 6.6%

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

   2. Taylor expanded in x around 0 100.0%

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

   if 5.00000000000000031e-10 < (*.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 99.9%

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

      4. exp-prod 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. exp-diff 99.9%

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

      2. rem-exp-log 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around inf 99.9%

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

4. Final simplification 99.9%

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

Alternative 8: 99.8% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* -2.0 x) -0.05) (not (<= (* -2.0 x) 5e-10)))
   (+ -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) <= -0.05) || !((-2.0 * x) <= 5e-10)) {
		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) <= (-0.05d0)) .or. (.not. (((-2.0d0) * x) <= 5d-10))) 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) <= -0.05) || !((-2.0 * x) <= 5e-10)) {
		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) <= -0.05) or not ((-2.0 * x) <= 5e-10):
		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) <= -0.05) || !(Float64(-2.0 * x) <= 5e-10))
		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) <= -0.05) || ~(((-2.0 * x) <= 5e-10)))
		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], -0.05], N[Not[LessEqual[N[(-2.0 * x), $MachinePrecision], 5e-10]], $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) < -0.050000000000000003 or 5.00000000000000031e-10 < (*.f64 -2 x)

   1. Initial program 99.9%

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

   if -0.050000000000000003 < (*.f64 -2 x) < 5.00000000000000031e-10

   1. Initial program 6.6%

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

   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 99.9%

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

Alternative 9: 79.1% accurate, 9.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-0.68d0)) then
        tmp = -1.0d0
    else
        tmp = 1.0d0 / ((x + 2.0d0) / (x * 2.0d0))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -0.680000000000000049

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 95.9%

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

      1. *-commutative 95.9%

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

   4. Simplified 95.9%

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

   5. Taylor expanded in x around inf 100.0%

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

   if -0.680000000000000049 < x

   1. Initial program 40.6%

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

   2. Taylor expanded in x around 0 6.4%

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

      1. +-commutative 6.4%

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

   4. Simplified 6.4%

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

      1. flip-- 6.3%

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

      2. clear-num 6.3%

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

      3. associate-+l+ 6.3%

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

      4. metadata-eval 6.3%

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

      5. metadata-eval 6.3%

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

      6. difference-of-sqr-1 6.3%

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

      7. associate-+l+ 6.3%

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

      8. metadata-eval 6.3%

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

      9. associate--l+ 65.3%

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

      10. metadata-eval 65.3%

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

      11. +-rgt-identity 65.3%

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

   6. Applied egg-rr 65.3%

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

   7. Taylor expanded in x around 0 69.8%

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

      1. *-commutative 69.8%

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

   9. Simplified 69.8%

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

4. Final simplification 78.0%

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

Alternative 10: 76.6% 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 95.9%

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

      1. *-commutative 95.9%

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

   4. Simplified 95.9%

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

   5. Taylor expanded in x around inf 100.0%

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

   if -1 < x

   1. Initial program 40.6%

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

   2. Taylor expanded in x around 0 65.6%

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

4. Final simplification 75.0%

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

Alternative 11: 27.6% 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 56.9%

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

2. Taylor expanded in x around 0 29.7%

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

   1. *-commutative 29.7%

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

4. Simplified 29.7%

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

5. Taylor expanded in x around inf 29.5%

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

6. Final simplification 29.5%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023217 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))