Logistic function from Lakshay Garg

Percentage Accurate: 54.3% → 99.7%

Time: 8.3s

Alternatives: 6

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: 54.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + {\left(e^{-2}\right)}^{x}\\ \mathbf{if}\;-2 \cdot x \leq -5:\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-6}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{\mathsf{fma}\left(4, {t_0}^{-2}, -1\right)}{1 + \frac{2}{t_0}}}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (pow (exp -2.0) x))))
   (if (<= (* -2.0 x) -5.0)
     (+ -1.0 (/ 2.0 (+ 1.0 (exp (* -2.0 x)))))
     (if (<= (* -2.0 x) 5e-6)
       (+ x (* -0.3333333333333333 (pow x 3.0)))
       (pow
        (cbrt (/ (fma 4.0 (pow t_0 -2.0) -1.0) (+ 1.0 (/ 2.0 t_0))))
        3.0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 + pow(exp(-2.0), x);
	double tmp;
	if ((-2.0 * x) <= -5.0) {
		tmp = -1.0 + (2.0 / (1.0 + exp((-2.0 * x))));
	} else if ((-2.0 * x) <= 5e-6) {
		tmp = x + (-0.3333333333333333 * pow(x, 3.0));
	} else {
		tmp = pow(cbrt((fma(4.0, pow(t_0, -2.0), -1.0) / (1.0 + (2.0 / t_0)))), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 + (exp(-2.0) ^ x))
	tmp = 0.0
	if (Float64(-2.0 * x) <= -5.0)
		tmp = Float64(-1.0 + Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))));
	elseif (Float64(-2.0 * x) <= 5e-6)
		tmp = Float64(x + Float64(-0.3333333333333333 * (x ^ 3.0)));
	else
		tmp = cbrt(Float64(fma(4.0, (t_0 ^ -2.0), -1.0) / Float64(1.0 + Float64(2.0 / t_0)))) ^ 3.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[Power[N[Exp[-2.0], $MachinePrecision], x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -5.0], 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], 5e-6], N[(x + N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(N[(4.0 * N[Power[t$95$0, -2.0], $MachinePrecision] + -1.0), $MachinePrecision] / N[(1.0 + N[(2.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   if -5 < (*.f64 -2 x) < 5.00000000000000041e-6

   1. Initial program 6.8%

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

   2. Taylor expanded in x around 0 100.0%

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

   if 5.00000000000000041e-6 < (*.f64 -2 x)

   1. Initial program 99.9%

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

      1. flip-- 99.9%

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

      2. div-inv 99.9%

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

   3. Applied egg-rr 100.0%

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

      1. associate-*r/ 100.0%

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

      2. *-rgt-identity 100.0%

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

   5. Simplified 100.0%

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

      1. pow-exp 100.0%

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

      2. *-commutative 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. pow-exp 100.0%

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

      2. *-commutative 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. add-cube-cbrt 100.0%

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

       2. pow3 100.0%

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

       3. *-commutative 100.0%

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

       4. pow-exp 100.0%

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

       5. *-commutative 100.0%

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

       6. pow-exp 100.0%

          \[\leadsto {\left(\sqrt[3]{\frac{\mathsf{fma}\left(4, {\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{-2}, -1\right)}{1 + \frac{2}{1 + \color{blue}{{\left(e^{-2}\right)}^{x}}}}}\right)}^{3} \]

   11. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{\mathsf{fma}\left(4, {\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{-2}, -1\right)}{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}}\right)}^{3}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -5:\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-6}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{\mathsf{fma}\left(4, {\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{-2}, -1\right)}{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}}\right)}^{3}\\ \end{array} \]

Alternative 2: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + e^{-2 \cdot x}\\ t_1 := \frac{2}{t_0}\\ \mathbf{if}\;-2 \cdot x \leq -5:\\ \;\;\;\;-1 + t_1\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-6}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{\mathsf{fma}\left(4, {t_0}^{-2}, -1\right)}\right)}{1 + t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (exp (* -2.0 x)))) (t_1 (/ 2.0 t_0)))
   (if (<= (* -2.0 x) -5.0)
     (+ -1.0 t_1)
     (if (<= (* -2.0 x) 5e-6)
       (+ x (* -0.3333333333333333 (pow x 3.0)))
       (/ (log (exp (fma 4.0 (pow t_0 -2.0) -1.0))) (+ 1.0 t_1))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + exp((-2.0 * x));
	double t_1 = 2.0 / t_0;
	double tmp;
	if ((-2.0 * x) <= -5.0) {
		tmp = -1.0 + t_1;
	} else if ((-2.0 * x) <= 5e-6) {
		tmp = x + (-0.3333333333333333 * pow(x, 3.0));
	} else {
		tmp = log(exp(fma(4.0, pow(t_0, -2.0), -1.0))) / (1.0 + t_1);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 + exp(Float64(-2.0 * x)))
	t_1 = Float64(2.0 / t_0)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -5.0)
		tmp = Float64(-1.0 + t_1);
	elseif (Float64(-2.0 * x) <= 5e-6)
		tmp = Float64(x + Float64(-0.3333333333333333 * (x ^ 3.0)));
	else
		tmp = Float64(log(exp(fma(4.0, (t_0 ^ -2.0), -1.0))) / Float64(1.0 + t_1));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / t$95$0), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -5.0], N[(-1.0 + t$95$1), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 5e-6], N[(x + N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Log[N[Exp[N[(4.0 * N[Power[t$95$0, -2.0], $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   if -5 < (*.f64 -2 x) < 5.00000000000000041e-6

   1. Initial program 6.8%

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

   2. Taylor expanded in x around 0 100.0%

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

   if 5.00000000000000041e-6 < (*.f64 -2 x)

   1. Initial program 99.9%

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

      1. flip-- 99.9%

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

      2. div-inv 99.9%

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

   3. Applied egg-rr 100.0%

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

      1. associate-*r/ 100.0%

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

      2. *-rgt-identity 100.0%

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

   5. Simplified 100.0%

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

      1. pow-exp 100.0%

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

      2. *-commutative 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. add-log-exp 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. pow-exp 100.0%

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

       2. *-commutative 100.0%

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

   11. Applied egg-rr 100.0%

       \[\leadsto \frac{\log \left(e^{\mathsf{fma}\left(4, {\left(1 + \color{blue}{e^{x \cdot -2}}\right)}^{-2}, -1\right)}\right)}{1 + \frac{2}{1 + e^{x \cdot -2}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -5:\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-6}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{\mathsf{fma}\left(4, {\left(1 + e^{-2 \cdot x}\right)}^{-2}, -1\right)}\right)}{1 + \frac{2}{1 + e^{-2 \cdot x}}}\\ \end{array} \]

Alternative 3: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + e^{-2 \cdot x}\\ t_1 := \frac{2}{t_0}\\ \mathbf{if}\;-2 \cdot x \leq -5:\\ \;\;\;\;-1 + t_1\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-6}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(4, {t_0}^{-2}, -1\right)}{1 + t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (exp (* -2.0 x)))) (t_1 (/ 2.0 t_0)))
   (if (<= (* -2.0 x) -5.0)
     (+ -1.0 t_1)
     (if (<= (* -2.0 x) 5e-6)
       (+ x (* -0.3333333333333333 (pow x 3.0)))
       (/ (fma 4.0 (pow t_0 -2.0) -1.0) (+ 1.0 t_1))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + exp((-2.0 * x));
	double t_1 = 2.0 / t_0;
	double tmp;
	if ((-2.0 * x) <= -5.0) {
		tmp = -1.0 + t_1;
	} else if ((-2.0 * x) <= 5e-6) {
		tmp = x + (-0.3333333333333333 * pow(x, 3.0));
	} else {
		tmp = fma(4.0, pow(t_0, -2.0), -1.0) / (1.0 + t_1);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 + exp(Float64(-2.0 * x)))
	t_1 = Float64(2.0 / t_0)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -5.0)
		tmp = Float64(-1.0 + t_1);
	elseif (Float64(-2.0 * x) <= 5e-6)
		tmp = Float64(x + Float64(-0.3333333333333333 * (x ^ 3.0)));
	else
		tmp = Float64(fma(4.0, (t_0 ^ -2.0), -1.0) / Float64(1.0 + t_1));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / t$95$0), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -5.0], N[(-1.0 + t$95$1), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 5e-6], N[(x + N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(4.0 * N[Power[t$95$0, -2.0], $MachinePrecision] + -1.0), $MachinePrecision] / N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   if -5 < (*.f64 -2 x) < 5.00000000000000041e-6

   1. Initial program 6.8%

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

   2. Taylor expanded in x around 0 100.0%

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

   if 5.00000000000000041e-6 < (*.f64 -2 x)

   1. Initial program 99.9%

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

      1. flip-- 99.9%

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

      2. div-inv 99.9%

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

   3. Applied egg-rr 100.0%

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

      1. associate-*r/ 100.0%

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

      2. *-rgt-identity 100.0%

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

   5. Simplified 100.0%

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

      1. pow-exp 100.0%

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

      2. *-commutative 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. pow-exp 100.0%

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

      2. *-commutative 100.0%

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

   9. Applied egg-rr 100.0%

      \[\leadsto \frac{\mathsf{fma}\left(4, {\left(1 + \color{blue}{e^{x \cdot -2}}\right)}^{-2}, -1\right)}{1 + \frac{2}{1 + e^{x \cdot -2}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -5:\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-6}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(4, {\left(1 + e^{-2 \cdot x}\right)}^{-2}, -1\right)}{1 + \frac{2}{1 + e^{-2 \cdot x}}}\\ \end{array} \]

Alternative 4: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -5 \lor \neg \left(-2 \cdot x \leq 5 \cdot 10^{-6}\right):\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* -2.0 x) -5.0) (not (<= (* -2.0 x) 5e-6)))
   (+ -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) <= -5.0) || !((-2.0 * x) <= 5e-6)) {
		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) <= (-5.0d0)) .or. (.not. (((-2.0d0) * x) <= 5d-6))) 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) <= -5.0) || !((-2.0 * x) <= 5e-6)) {
		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) <= -5.0) or not ((-2.0 * x) <= 5e-6):
		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) <= -5.0) || !(Float64(-2.0 * x) <= 5e-6))
		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) <= -5.0) || ~(((-2.0 * x) <= 5e-6)))
		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], -5.0], N[Not[LessEqual[N[(-2.0 * x), $MachinePrecision], 5e-6]], $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) < -5 or 5.00000000000000041e-6 < (*.f64 -2 x)

   1. Initial program 100.0%

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

   if -5 < (*.f64 -2 x) < 5.00000000000000041e-6

   1. Initial program 6.8%

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

   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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -5 \lor \neg \left(-2 \cdot x \leq 5 \cdot 10^{-6}\right):\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \end{array} \]

Alternative 5: 75.1% 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 98.8%

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

      1. *-commutative 98.8%

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

   4. Simplified 98.8%

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

   5. Taylor expanded in x around inf 100.0%

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

   if -1 < x

   1. Initial program 36.7%

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

   2. Taylor expanded in x around 0 69.7%

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

4. Final simplification 77.8%

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

Alternative 6: 26.5% 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 53.7%

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

2. Taylor expanded in x around 0 30.3%

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

   1. *-commutative 30.3%

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

4. Simplified 30.3%

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

5. Taylor expanded in x around inf 29.4%

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

6. Final simplification 29.4%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023229 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))