Logistic function from Lakshay Garg

Percentage Accurate: 53.9% → 99.4%

Time: 5.1s

Alternatives: 4

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: 53.9% 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: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -1:\\ \;\;\;\;\mathsf{fma}\left(\frac{-2}{\mathsf{expm1}\left(x \cdot -4\right)}, -\mathsf{expm1}\left(-2 \cdot x\right), -1\right)\\ \mathbf{elif}\;-2 \cdot x \leq 2 \cdot 10^{-6}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -1.0)
   (fma (/ -2.0 (expm1 (* x -4.0))) (- (expm1 (* -2.0 x))) -1.0)
   (if (<= (* -2.0 x) 2e-6) (+ x (* -0.3333333333333333 (pow x 3.0))) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -1.0) {
		tmp = fma((-2.0 / expm1((x * -4.0))), -expm1((-2.0 * x)), -1.0);
	} else if ((-2.0 * x) <= 2e-6) {
		tmp = x + (-0.3333333333333333 * pow(x, 3.0));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -1.0)
		tmp = fma(Float64(-2.0 / expm1(Float64(x * -4.0))), Float64(-expm1(Float64(-2.0 * x))), -1.0);
	elseif (Float64(-2.0 * x) <= 2e-6)
		tmp = Float64(x + Float64(-0.3333333333333333 * (x ^ 3.0)));
	else
		tmp = -1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -1.0], N[(N[(-2.0 / N[(Exp[N[(x * -4.0), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] * (-N[(Exp[N[(-2.0 * x), $MachinePrecision]] - 1), $MachinePrecision]) + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 2e-6], N[(x + N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

      1. flip-+ 100.0%

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

      2. associate-/r/ 100.0%

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

      3. fma-neg 100.0%

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

      4. metadata-eval 100.0%

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

      5. pow2 100.0%

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

      6. *-commutative 100.0%

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

      7. exp-prod 100.0%

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

      8. pow-pow 100.0%

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

      9. metadata-eval 100.0%

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

      10. exp-prod 100.0%

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

      11. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. exp-prod 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. neg-sub0 100.0%

         \[\leadsto \mathsf{fma}\left(\frac{2}{1 - {\left(e^{x}\right)}^{-4}}, \color{blue}{\left(0 - {\left(e^{x}\right)}^{-2}\right)} + 1, -1\right) \]

      6. associate-+l- 100.0%

         \[\leadsto \mathsf{fma}\left(\frac{2}{1 - {\left(e^{x}\right)}^{-4}}, \color{blue}{0 - \left({\left(e^{x}\right)}^{-2} - 1\right)}, -1\right) \]

      7. sub0-neg 100.0%

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

      8. exp-prod 100.0%

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

      9. *-commutative 100.0%

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

      10. expm1-def 100.0%

          \[\leadsto \mathsf{fma}\left(\frac{2}{1 - {\left(e^{x}\right)}^{-4}}, -\color{blue}{\mathsf{expm1}\left(-2 \cdot x\right)}, -1\right) \]

      11. *-commutative 100.0%

          \[\leadsto \mathsf{fma}\left(\frac{2}{1 - {\left(e^{x}\right)}^{-4}}, -\mathsf{expm1}\left(\color{blue}{x \cdot -2}\right), -1\right) \]

   6. Simplified 100.0%

      \[\leadsto \mathsf{fma}\left(\frac{2}{1 - {\left(e^{x}\right)}^{-4}}, \color{blue}{-\mathsf{expm1}\left(x \cdot -2\right)}, -1\right) \]
   7. Step-by-step derivation

      1. sub-neg 100.0%

         \[\leadsto \mathsf{fma}\left(\frac{2}{\color{blue}{1 + \left(-{\left(e^{x}\right)}^{-4}\right)}}, -\mathsf{expm1}\left(x \cdot -2\right), -1\right) \]

   8. Applied egg-rr 100.0%

      \[\leadsto \mathsf{fma}\left(\frac{2}{\color{blue}{1 + \left(-{\left(e^{x}\right)}^{-4}\right)}}, -\mathsf{expm1}\left(x \cdot -2\right), -1\right) \]
   9. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \mathsf{fma}\left(\frac{2}{\color{blue}{\left(-{\left(e^{x}\right)}^{-4}\right) + 1}}, -\mathsf{expm1}\left(x \cdot -2\right), -1\right) \]

      2. neg-sub0 100.0%

         \[\leadsto \mathsf{fma}\left(\frac{2}{\color{blue}{\left(0 - {\left(e^{x}\right)}^{-4}\right)} + 1}, -\mathsf{expm1}\left(x \cdot -2\right), -1\right) \]

      3. associate-+l- 100.0%

         \[\leadsto \mathsf{fma}\left(\frac{2}{\color{blue}{0 - \left({\left(e^{x}\right)}^{-4} - 1\right)}}, -\mathsf{expm1}\left(x \cdot -2\right), -1\right) \]

      4. sub-neg 100.0%

         \[\leadsto \mathsf{fma}\left(\frac{2}{0 - \color{blue}{\left({\left(e^{x}\right)}^{-4} + \left(-1\right)\right)}}, -\mathsf{expm1}\left(x \cdot -2\right), -1\right) \]

      5. remove-double-neg 100.0%

         \[\leadsto \mathsf{fma}\left(\frac{2}{0 - \left(\color{blue}{\left(-\left(-{\left(e^{x}\right)}^{-4}\right)\right)} + \left(-1\right)\right)}, -\mathsf{expm1}\left(x \cdot -2\right), -1\right) \]

      6. distribute-neg-in 100.0%

         \[\leadsto \mathsf{fma}\left(\frac{2}{0 - \color{blue}{\left(-\left(\left(-{\left(e^{x}\right)}^{-4}\right) + 1\right)\right)}}, -\mathsf{expm1}\left(x \cdot -2\right), -1\right) \]

      7. +-commutative 100.0%

         \[\leadsto \mathsf{fma}\left(\frac{2}{0 - \left(-\color{blue}{\left(1 + \left(-{\left(e^{x}\right)}^{-4}\right)\right)}\right)}, -\mathsf{expm1}\left(x \cdot -2\right), -1\right) \]

      8. sub-neg 100.0%

         \[\leadsto \mathsf{fma}\left(\frac{2}{0 - \left(-\color{blue}{\left(1 - {\left(e^{x}\right)}^{-4}\right)}\right)}, -\mathsf{expm1}\left(x \cdot -2\right), -1\right) \]

      9. sub0-neg 100.0%

         \[\leadsto \mathsf{fma}\left(\frac{2}{\color{blue}{-\left(-\left(1 - {\left(e^{x}\right)}^{-4}\right)\right)}}, -\mathsf{expm1}\left(x \cdot -2\right), -1\right) \]

      10. sub-neg 100.0%

          \[\leadsto \mathsf{fma}\left(\frac{2}{-\left(-\color{blue}{\left(1 + \left(-{\left(e^{x}\right)}^{-4}\right)\right)}\right)}, -\mathsf{expm1}\left(x \cdot -2\right), -1\right) \]

      11. +-commutative 100.0%

          \[\leadsto \mathsf{fma}\left(\frac{2}{-\left(-\color{blue}{\left(\left(-{\left(e^{x}\right)}^{-4}\right) + 1\right)}\right)}, -\mathsf{expm1}\left(x \cdot -2\right), -1\right) \]

      12. distribute-neg-in 100.0%

          \[\leadsto \mathsf{fma}\left(\frac{2}{-\color{blue}{\left(\left(-\left(-{\left(e^{x}\right)}^{-4}\right)\right) + \left(-1\right)\right)}}, -\mathsf{expm1}\left(x \cdot -2\right), -1\right) \]

      13. remove-double-neg 100.0%

          \[\leadsto \mathsf{fma}\left(\frac{2}{-\left(\color{blue}{{\left(e^{x}\right)}^{-4}} + \left(-1\right)\right)}, -\mathsf{expm1}\left(x \cdot -2\right), -1\right) \]

      14. sub-neg 100.0%

          \[\leadsto \mathsf{fma}\left(\frac{2}{-\color{blue}{\left({\left(e^{x}\right)}^{-4} - 1\right)}}, -\mathsf{expm1}\left(x \cdot -2\right), -1\right) \]

      15. exp-prod 100.0%

          \[\leadsto \mathsf{fma}\left(\frac{2}{-\left(\color{blue}{e^{x \cdot -4}} - 1\right)}, -\mathsf{expm1}\left(x \cdot -2\right), -1\right) \]

      16. *-commutative 100.0%

          \[\leadsto \mathsf{fma}\left(\frac{2}{-\left(e^{\color{blue}{-4 \cdot x}} - 1\right)}, -\mathsf{expm1}\left(x \cdot -2\right), -1\right) \]

      17. expm1-def 100.0%

          \[\leadsto \mathsf{fma}\left(\frac{2}{-\color{blue}{\mathsf{expm1}\left(-4 \cdot x\right)}}, -\mathsf{expm1}\left(x \cdot -2\right), -1\right) \]

      18. *-commutative 100.0%

          \[\leadsto \mathsf{fma}\left(\frac{2}{-\mathsf{expm1}\left(\color{blue}{x \cdot -4}\right)}, -\mathsf{expm1}\left(x \cdot -2\right), -1\right) \]

   10. Simplified 100.0%

       \[\leadsto \mathsf{fma}\left(\frac{2}{\color{blue}{-\mathsf{expm1}\left(x \cdot -4\right)}}, -\mathsf{expm1}\left(x \cdot -2\right), -1\right) \]

   11. Taylor expanded in x around inf 100.0%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-2}{e^{-4 \cdot x} - 1}}, -\mathsf{expm1}\left(x \cdot -2\right), -1\right) \]
   12. Step-by-step derivation

       1. expm1-def 100.0%

          \[\leadsto \mathsf{fma}\left(\frac{-2}{\color{blue}{\mathsf{expm1}\left(-4 \cdot x\right)}}, -\mathsf{expm1}\left(x \cdot -2\right), -1\right) \]

       2. *-commutative 100.0%

          \[\leadsto \mathsf{fma}\left(\frac{-2}{\mathsf{expm1}\left(\color{blue}{x \cdot -4}\right)}, -\mathsf{expm1}\left(x \cdot -2\right), -1\right) \]

   13. Simplified 100.0%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-2}{\mathsf{expm1}\left(x \cdot -4\right)}}, -\mathsf{expm1}\left(x \cdot -2\right), -1\right) \]

   if -1 < (*.f64 -2 x) < 1.99999999999999991e-6

   1. Initial program 8.6%

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

   2. Taylor expanded in x around 0 100.0%

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

   if 1.99999999999999991e-6 < (*.f64 -2 x)

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 97.4%

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

      1. *-commutative 97.4%

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

   4. Simplified 97.4%

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

   5. Taylor expanded in x around inf 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -1:\\ \;\;\;\;\mathsf{fma}\left(\frac{-2}{\mathsf{expm1}\left(x \cdot -4\right)}, -\mathsf{expm1}\left(-2 \cdot x\right), -1\right)\\ \mathbf{elif}\;-2 \cdot x \leq 2 \cdot 10^{-6}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 2: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -1:\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{elif}\;-2 \cdot x \leq 2 \cdot 10^{-6}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -1.0)
   (+ -1.0 (/ 2.0 (+ 1.0 (exp (* -2.0 x)))))
   (if (<= (* -2.0 x) 2e-6) (+ x (* -0.3333333333333333 (pow x 3.0))) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -1.0) {
		tmp = -1.0 + (2.0 / (1.0 + exp((-2.0 * x))));
	} else if ((-2.0 * x) <= 2e-6) {
		tmp = x + (-0.3333333333333333 * pow(x, 3.0));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((-2.0d0) * x) <= (-1.0d0)) then
        tmp = (-1.0d0) + (2.0d0 / (1.0d0 + exp(((-2.0d0) * x))))
    else if (((-2.0d0) * x) <= 2d-6) then
        tmp = x + ((-0.3333333333333333d0) * (x ** 3.0d0))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -1.0) {
		tmp = -1.0 + (2.0 / (1.0 + Math.exp((-2.0 * x))));
	} else if ((-2.0 * x) <= 2e-6) {
		tmp = x + (-0.3333333333333333 * Math.pow(x, 3.0));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (-2.0 * x) <= -1.0:
		tmp = -1.0 + (2.0 / (1.0 + math.exp((-2.0 * x))))
	elif (-2.0 * x) <= 2e-6:
		tmp = x + (-0.3333333333333333 * math.pow(x, 3.0))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -1.0)
		tmp = Float64(-1.0 + Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))));
	elseif (Float64(-2.0 * x) <= 2e-6)
		tmp = Float64(x + Float64(-0.3333333333333333 * (x ^ 3.0)));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((-2.0 * x) <= -1.0)
		tmp = -1.0 + (2.0 / (1.0 + exp((-2.0 * x))));
	elseif ((-2.0 * x) <= 2e-6)
		tmp = x + (-0.3333333333333333 * (x ^ 3.0));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -1.0], 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], 2e-6], N[(x + N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   if -1 < (*.f64 -2 x) < 1.99999999999999991e-6

   1. Initial program 8.6%

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

   2. Taylor expanded in x around 0 100.0%

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

   if 1.99999999999999991e-6 < (*.f64 -2 x)

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 97.4%

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

      1. *-commutative 97.4%

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

   4. Simplified 97.4%

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

   5. Taylor expanded in x around inf 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -1:\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{elif}\;-2 \cdot x \leq 2 \cdot 10^{-6}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 3: 76.7% accurate, 35.6× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (if (<= x -1.0) -1.0 x))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = -1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

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

Java

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

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = -1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = -1.0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = -1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], -1.0, x]

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 97.4%

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

      1. *-commutative 97.4%

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

   4. Simplified 97.4%

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

   5. Taylor expanded in x around inf 100.0%

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

   if -1 < x

   1. Initial program 40.2%

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

   2. Taylor expanded in x around 0 67.0%

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

4. Final simplification 73.6%

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

Alternative 4: 27.9% 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 52.1%

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

2. Taylor expanded in x around 0 24.1%

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

   1. *-commutative 24.1%

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

4. Simplified 24.1%

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

5. Taylor expanded in x around inf 22.3%

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

6. Final simplification 22.3%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023189 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))