Logistic function from Lakshay Garg

Percentage Accurate: 53.7% → 99.8%

Time: 10.6s

Alternatives: 8

Speedup: 3.2×

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: 53.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -10:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{\mathsf{expm1}\left(x \cdot -4\right)}, \mathsf{expm1}\left(-2 \cdot x\right), -1\right)\\ \mathbf{elif}\;-2 \cdot x \leq 0.05:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.05396825396825397, 0.13333333333333333\right), -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -10.0)
   (fma (/ 2.0 (expm1 (* x -4.0))) (expm1 (* -2.0 x)) -1.0)
   (if (<= (* -2.0 x) 0.05)
     (fma
      (fma
       (* x x)
       (fma (* x x) -0.05396825396825397 0.13333333333333333)
       -0.3333333333333333)
      (* x (* x x))
      x)
     -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -10.0) {
		tmp = fma((2.0 / expm1((x * -4.0))), expm1((-2.0 * x)), -1.0);
	} else if ((-2.0 * x) <= 0.05) {
		tmp = fma(fma((x * x), fma((x * x), -0.05396825396825397, 0.13333333333333333), -0.3333333333333333), (x * (x * x)), x);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -10.0)
		tmp = fma(Float64(2.0 / expm1(Float64(x * -4.0))), expm1(Float64(-2.0 * x)), -1.0);
	elseif (Float64(-2.0 * x) <= 0.05)
		tmp = fma(fma(Float64(x * x), fma(Float64(x * x), -0.05396825396825397, 0.13333333333333333), -0.3333333333333333), Float64(x * Float64(x * x)), x);
	else
		tmp = -1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -10.0], 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], 0.05], N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.05396825396825397 + 0.13333333333333333), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -10

   1. Initial program 99.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-+ N/A

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

      7. associate-/r/ N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

   if -10 < (*.f64 #s(literal -2 binary64) x) < 0.050000000000000003

   1. Initial program 8.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right) + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)\right) \cdot x + 1 \cdot x} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)\right)} + 1 \cdot x \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)} + 1 \cdot x \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right) \cdot \left(x \cdot {x}^{2}\right)} + 1 \cdot x \]

      6. *-lft-identity N/A

         \[\leadsto \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right) \cdot \left(x \cdot {x}^{2}\right) + \color{blue}{x} \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}, x \cdot {x}^{2}, x\right)} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.05396825396825397, 0.13333333333333333\right), -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)} \]

   if 0.050000000000000003 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 100.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-+ N/A

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

      7. associate-/r/ N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1} \]
   6. Step-by-step derivation

   7. Applied rewrites 100.0%

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

Alternative 2: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -10:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.05:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.05396825396825397, 0.13333333333333333\right), -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -10.0)
   (+ (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) -1.0)
   (if (<= (* -2.0 x) 0.05)
     (fma
      (fma
       (* x x)
       (fma (* x x) -0.05396825396825397 0.13333333333333333)
       -0.3333333333333333)
      (* x (* x x))
      x)
     -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -10.0) {
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	} else if ((-2.0 * x) <= 0.05) {
		tmp = fma(fma((x * x), fma((x * x), -0.05396825396825397, 0.13333333333333333), -0.3333333333333333), (x * (x * x)), x);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -10.0)
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) + -1.0);
	elseif (Float64(-2.0 * x) <= 0.05)
		tmp = fma(fma(Float64(x * x), fma(Float64(x * x), -0.05396825396825397, 0.13333333333333333), -0.3333333333333333), Float64(x * Float64(x * x)), x);
	else
		tmp = -1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -10.0], N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.05], N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.05396825396825397 + 0.13333333333333333), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -10

   1. Initial program 99.9%

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

   if -10 < (*.f64 #s(literal -2 binary64) x) < 0.050000000000000003

   1. Initial program 8.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right) + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)\right) \cdot x + 1 \cdot x} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)\right)} + 1 \cdot x \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)} + 1 \cdot x \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right) \cdot \left(x \cdot {x}^{2}\right)} + 1 \cdot x \]

      6. *-lft-identity N/A

         \[\leadsto \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right) \cdot \left(x \cdot {x}^{2}\right) + \color{blue}{x} \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}, x \cdot {x}^{2}, x\right)} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.05396825396825397, 0.13333333333333333\right), -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)} \]

   if 0.050000000000000003 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 100.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-+ N/A

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

      7. associate-/r/ N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1} \]
   6. Step-by-step derivation

   7. Applied rewrites 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -10:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.05:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.05396825396825397, 0.13333333333333333\right), -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -10:\\ \;\;\;\;\frac{2}{1} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.05:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.05396825396825397, 0.13333333333333333\right), -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -10.0)
   (+ (/ 2.0 1.0) -1.0)
   (if (<= (* -2.0 x) 0.05)
     (fma
      (fma
       (* x x)
       (fma (* x x) -0.05396825396825397 0.13333333333333333)
       -0.3333333333333333)
      (* x (* x x))
      x)
     -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -10.0) {
		tmp = (2.0 / 1.0) + -1.0;
	} else if ((-2.0 * x) <= 0.05) {
		tmp = fma(fma((x * x), fma((x * x), -0.05396825396825397, 0.13333333333333333), -0.3333333333333333), (x * (x * x)), x);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -10.0)
		tmp = Float64(Float64(2.0 / 1.0) + -1.0);
	elseif (Float64(-2.0 * x) <= 0.05)
		tmp = fma(fma(Float64(x * x), fma(Float64(x * x), -0.05396825396825397, 0.13333333333333333), -0.3333333333333333), Float64(x * Float64(x * x)), x);
	else
		tmp = -1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -10.0], N[(N[(2.0 / 1.0), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.05], N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.05396825396825397 + 0.13333333333333333), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -10

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. lower--.f64 N/A

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

      4. count-2 N/A

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

      5. lower-+.f64 1.6

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

   5. Applied rewrites 1.6%

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

   7. Applied rewrites 96.7%

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

   8. Taylor expanded in x around inf

      \[\leadsto \frac{2}{1} - 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 98.6%

       \[\leadsto \frac{2}{1} - 1 \]

   if -10 < (*.f64 #s(literal -2 binary64) x) < 0.050000000000000003

   1. Initial program 8.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right) + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)\right) \cdot x + 1 \cdot x} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)\right)} + 1 \cdot x \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)} + 1 \cdot x \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right) \cdot \left(x \cdot {x}^{2}\right)} + 1 \cdot x \]

      6. *-lft-identity N/A

         \[\leadsto \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right) \cdot \left(x \cdot {x}^{2}\right) + \color{blue}{x} \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}, x \cdot {x}^{2}, x\right)} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.05396825396825397, 0.13333333333333333\right), -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)} \]

   if 0.050000000000000003 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 100.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-+ N/A

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

      7. associate-/r/ N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1} \]
   6. Step-by-step derivation

   7. Applied rewrites 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -10:\\ \;\;\;\;\frac{2}{1} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.05:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.05396825396825397, 0.13333333333333333\right), -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -10:\\ \;\;\;\;\frac{2}{1} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.05:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -10.0)
   (+ (/ 2.0 1.0) -1.0)
   (if (<= (* -2.0 x) 0.05)
     (fma
      (fma (* x x) 0.13333333333333333 -0.3333333333333333)
      (* x (* x x))
      x)
     -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -10.0) {
		tmp = (2.0 / 1.0) + -1.0;
	} else if ((-2.0 * x) <= 0.05) {
		tmp = fma(fma((x * x), 0.13333333333333333, -0.3333333333333333), (x * (x * x)), x);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -10.0)
		tmp = Float64(Float64(2.0 / 1.0) + -1.0);
	elseif (Float64(-2.0 * x) <= 0.05)
		tmp = fma(fma(Float64(x * x), 0.13333333333333333, -0.3333333333333333), Float64(x * Float64(x * x)), x);
	else
		tmp = -1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -10.0], N[(N[(2.0 / 1.0), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.05], N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333 + -0.3333333333333333), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -10

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. lower--.f64 N/A

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

      4. count-2 N/A

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

      5. lower-+.f64 1.6

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

   5. Applied rewrites 1.6%

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

   7. Applied rewrites 96.7%

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

   8. Taylor expanded in x around inf

      \[\leadsto \frac{2}{1} - 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 98.6%

       \[\leadsto \frac{2}{1} - 1 \]

   if -10 < (*.f64 #s(literal -2 binary64) x) < 0.050000000000000003

   1. Initial program 8.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \color{blue}{1 \cdot x + \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right) \cdot x} \]

      2. *-lft-identity N/A

         \[\leadsto \color{blue}{x} + \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right) \cdot x \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right) \cdot x + x} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)} + x \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) \cdot \left(x \cdot {x}^{2}\right)} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x \cdot {x}^{2}, x\right)} \]

      8. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{2}{15} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, x \cdot {x}^{2}, x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \frac{2}{15}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), x \cdot {x}^{2}, x\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{2} \cdot \frac{2}{15} + \color{blue}{\frac{-1}{3}}, x \cdot {x}^{2}, x\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{2}{15}, \frac{-1}{3}\right)}, x \cdot {x}^{2}, x\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{2}{15}, \frac{-1}{3}\right), x \cdot {x}^{2}, x\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{2}{15}, \frac{-1}{3}\right), x \cdot {x}^{2}, x\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{2}{15}, \frac{-1}{3}\right), \color{blue}{x \cdot {x}^{2}}, x\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{2}{15}, \frac{-1}{3}\right), x \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \]

      16. lower-*.f64 99.9

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right), x \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)} \]

   if 0.050000000000000003 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 100.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-+ N/A

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

      7. associate-/r/ N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1} \]
   6. Step-by-step derivation

   7. Applied rewrites 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -10:\\ \;\;\;\;\frac{2}{1} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.05:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -10:\\ \;\;\;\;\frac{2}{1} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.05:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -10.0)
   (+ (/ 2.0 1.0) -1.0)
   (if (<= (* -2.0 x) 0.05) (fma -0.3333333333333333 (* x (* x x)) x) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -10.0) {
		tmp = (2.0 / 1.0) + -1.0;
	} else if ((-2.0 * x) <= 0.05) {
		tmp = fma(-0.3333333333333333, (x * (x * x)), x);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -10.0)
		tmp = Float64(Float64(2.0 / 1.0) + -1.0);
	elseif (Float64(-2.0 * x) <= 0.05)
		tmp = fma(-0.3333333333333333, Float64(x * Float64(x * x)), x);
	else
		tmp = -1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -10.0], N[(N[(2.0 / 1.0), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.05], N[(-0.3333333333333333 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -10

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. lower--.f64 N/A

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

      4. count-2 N/A

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

      5. lower-+.f64 1.6

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

   5. Applied rewrites 1.6%

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

   7. Applied rewrites 96.7%

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

   8. Taylor expanded in x around inf

      \[\leadsto \frac{2}{1} - 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 98.6%

       \[\leadsto \frac{2}{1} - 1 \]

   if -10 < (*.f64 #s(literal -2 binary64) x) < 0.050000000000000003

   1. Initial program 8.1%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{3}, x \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \]

      10. lower-*.f64 99.7

          \[\leadsto \mathsf{fma}\left(-0.3333333333333333, x \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.3333333333333333, x \cdot \left(x \cdot x\right), x\right)} \]

   if 0.050000000000000003 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 100.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-+ N/A

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

      7. associate-/r/ N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1} \]
   6. Step-by-step derivation

   7. Applied rewrites 100.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -10:\\ \;\;\;\;\frac{2}{1} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.05:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.3% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{1} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5e-310) -1.0 (+ (/ 2.0 1.0) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5e-310) {
		tmp = -1.0;
	} else {
		tmp = (2.0 / 1.0) + -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 (x <= (-5d-310)) then
        tmp = -1.0d0
    else
        tmp = (2.0d0 / 1.0d0) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -5e-310) {
		tmp = -1.0;
	} else {
		tmp = (2.0 / 1.0) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5e-310:
		tmp = -1.0
	else:
		tmp = (2.0 / 1.0) + -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = -1.0;
	else
		tmp = Float64(Float64(2.0 / 1.0) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5e-310)
		tmp = -1.0;
	else
		tmp = (2.0 / 1.0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5e-310], -1.0, N[(N[(2.0 / 1.0), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 52.6%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-+ N/A

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

      7. associate-/r/ N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 3.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1} \]
   6. Step-by-step derivation

   7. Applied rewrites 51.5%

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

   if -4.999999999999985e-310 < x

   1. Initial program 59.2%

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

   3. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. lower--.f64 N/A

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

      4. count-2 N/A

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

      5. lower-+.f64 4.6

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

   5. Applied rewrites 4.6%

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

   7. Applied rewrites 56.2%

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

   8. Taylor expanded in x around inf

      \[\leadsto \frac{2}{1} - 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 56.8%

       \[\leadsto \frac{2}{1} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{1} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 29.6% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq 5 \cdot 10^{-161}:\\ \;\;\;\;\left(x + 1\right) + -1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) 5e-161) (+ (+ x 1.0) -1.0) -1.0))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= 5e-161) {
		tmp = (x + 1.0) + -1.0;
	} 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) <= 5d-161) then
        tmp = (x + 1.0d0) + (-1.0d0)
    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) <= 5e-161) {
		tmp = (x + 1.0) + -1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (-2.0 * x) <= 5e-161:
		tmp = (x + 1.0) + -1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= 5e-161)
		tmp = Float64(Float64(x + 1.0) + -1.0);
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((-2.0 * x) <= 5e-161)
		tmp = (x + 1.0) + -1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], 5e-161], N[(N[(x + 1.0), $MachinePrecision] + -1.0), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal -2 binary64) x) < 4.9999999999999999e-161

   1. Initial program 48.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 6.8

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

   5. Applied rewrites 6.8%

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

   if 4.9999999999999999e-161 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 69.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-+ N/A

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

      7. associate-/r/ N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 3.2%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1} \]
   6. Step-by-step derivation

   7. Applied rewrites 68.6%

      \[\leadsto \color{blue}{-1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq 5 \cdot 10^{-161}:\\ \;\;\;\;\left(x + 1\right) + -1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 27.0% accurate, 123.0× 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 55.8%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-/.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. flip-+ N/A

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

   7. associate-/r/ N/A

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

   8. lower-fma.f64 N/A

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

4. Applied rewrites 30.7%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{-1} \]
6. Step-by-step derivation

7. Applied rewrites 26.9%

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

Reproduce

herbie shell --seed 2024223 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))