Logistic function from Lakshay Garg

Percentage Accurate: 54.8% → 99.3%

Time: 7.9s

Alternatives: 6

Speedup: 17.5×

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: 54.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{if}\;-2 \cdot x \leq -40000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;-2 \cdot x \leq 0.02:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.05396825396825397, 0.13333333333333333\right), -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) -1.0)))
   (if (<= (* -2.0 x) -40000000.0)
     t_0
     (if (<= (* -2.0 x) 0.02)
       (fma
        (fma
         (* x x)
         (fma (* x x) -0.05396825396825397 0.13333333333333333)
         -0.3333333333333333)
        (* x (* x x))
        x)
       t_0))))

C

double code(double x, double y) {
	double t_0 = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	double tmp;
	if ((-2.0 * x) <= -40000000.0) {
		tmp = t_0;
	} else if ((-2.0 * x) <= 0.02) {
		tmp = fma(fma((x * x), fma((x * x), -0.05396825396825397, 0.13333333333333333), -0.3333333333333333), (x * (x * x)), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) + -1.0)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -40000000.0)
		tmp = t_0;
	elseif (Float64(-2.0 * x) <= 0.02)
		tmp = fma(fma(Float64(x * x), fma(Float64(x * x), -0.05396825396825397, 0.13333333333333333), -0.3333333333333333), Float64(x * Float64(x * x)), x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$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], -40000000.0], t$95$0, If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.02], 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], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -4e7 or 0.0200000000000000004 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 100.0%

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

   if -4e7 < (*.f64 #s(literal -2 binary64) x) < 0.0200000000000000004

   1. Initial program 8.5%

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

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.05396825396825397, 0.13333333333333333\right), -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -40000000:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.02:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.05396825396825397, 0.13333333333333333\right), -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.1% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.08 \cdot 10^{-8}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -1.3333333333333333, 2\right), -2\right), 2\right)} + -1\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2\right) \cdot \frac{1}{x + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.08e-8)
   (+ (/ 2.0 (fma x (fma x (fma x -1.3333333333333333 2.0) -2.0) 2.0)) -1.0)
   (* (* x 2.0) (/ 1.0 (+ x 2.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.08e-8) {
		tmp = (2.0 / fma(x, fma(x, fma(x, -1.3333333333333333, 2.0), -2.0), 2.0)) + -1.0;
	} else {
		tmp = (x * 2.0) * (1.0 / (x + 2.0));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.08e-8)
		tmp = Float64(Float64(2.0 / fma(x, fma(x, fma(x, -1.3333333333333333, 2.0), -2.0), 2.0)) + -1.0);
	else
		tmp = Float64(Float64(x * 2.0) * Float64(1.0 / Float64(x + 2.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.08e-8], N[(N[(2.0 / N[(x * N[(x * N[(x * -1.3333333333333333 + 2.0), $MachinePrecision] + -2.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(x * 2.0), $MachinePrecision] * N[(1.0 / N[(x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.0800000000000001e-8

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 98.2

         \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, -1.3333333333333333, 2\right)}, -2\right), 2\right)} - 1 \]

   5. Applied rewrites 98.2%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -1.3333333333333333, 2\right), -2\right), 2\right)}} - 1 \]

   if -1.0800000000000001e-8 < x

   1. Initial program 35.3%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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 6.6

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

   5. Applied rewrites 6.6%

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

      1. lift-+.f64 N/A

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

      2. flip-- N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. difference-of-sqr-1 N/A

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

      7. lift-+.f64 N/A

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

      8. associate--l+ N/A

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

      9. metadata-eval N/A

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

      10. +-rgt-identity N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. associate-+l+ N/A

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

      14. metadata-eval N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. associate-+l+ N/A

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

      19. metadata-eval N/A

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

      20. lower-+.f64 71.1

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

   7. Applied rewrites 71.1%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 74.7

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

   10. Applied rewrites 74.7%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.08 \cdot 10^{-8}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -1.3333333333333333, 2\right), -2\right), 2\right)} + -1\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2\right) \cdot \frac{1}{x + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.1% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-0.66d0)) then
        tmp = -1.0d0
    else
        tmp = (x * 2.0d0) * (1.0d0 / (x + 2.0d0))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := If[LessEqual[x, -0.66], -1.0, N[(N[(x * 2.0), $MachinePrecision] * N[(1.0 / N[(x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.660000000000000031

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. count-2 N/A

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

      8. lower-+.f64 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 99.8%

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

   if -0.660000000000000031 < x

   1. Initial program 35.8%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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 7.1

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

   5. Applied rewrites 7.1%

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

      1. lift-+.f64 N/A

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

      2. flip-- N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. difference-of-sqr-1 N/A

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

      7. lift-+.f64 N/A

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

      8. associate--l+ N/A

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

      9. metadata-eval N/A

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

      10. +-rgt-identity N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. associate-+l+ N/A

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

      14. metadata-eval N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. associate-+l+ N/A

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

      19. metadata-eval N/A

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

      20. lower-+.f64 70.9

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

   7. Applied rewrites 70.9%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 74.3

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

   10. Applied rewrites 74.3%

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

Alternative 4: 75.1% accurate, 17.5× 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. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. count-2 N/A

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

      8. lower-+.f64 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 99.8%

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

   if -1 < x

   1. Initial program 35.8%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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 7.1

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

   5. Applied rewrites 7.1%

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

      1. associate--l+ N/A

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

      2. metadata-eval N/A

         \[\leadsto x + \color{blue}{0} \]

      3. +-rgt-identity 71.0

         \[\leadsto \color{blue}{x} \]

   7. Applied rewrites 71.0%

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

Alternative 5: 28.9% accurate, 17.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.12 \cdot 10^{-154}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -1.12e-154) -1.0 0.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.12e-154) {
		tmp = -1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.12d-154)) then
        tmp = -1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.12e-154) {
		tmp = -1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.12e-154:
		tmp = -1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.12e-154)
		tmp = -1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.12e-154)
		tmp = -1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.12e-154], -1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < -1.12e-154

   1. Initial program 73.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. count-2 N/A

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

      8. lower-+.f64 72.2

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

   5. Applied rewrites 72.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 72.5%

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

   if -1.12e-154 < x

   1. Initial program 40.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 5.0%

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

      1. metadata-eval 5.0

         \[\leadsto \color{blue}{0} \]

   7. Applied rewrites 5.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 27.3% 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 53.9%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. +-commutative N/A

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

   6. lower-+.f64 N/A

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

   7. count-2 N/A

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

   8. lower-+.f64 32.0

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

5. Applied rewrites 32.0%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 30.3%

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

Reproduce

herbie shell --seed 2024214 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))