Logistic function from Lakshay Garg

Percentage Accurate: 54.1% → 99.8%

Time: 9.2s

Alternatives: 5

Speedup: 10.2×

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: 54.1% 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, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-2 \cdot x}\\ t_1 := t\_0 + 1\\ t_2 := {\left(\mathsf{fma}\left(t\_0, 0.5, 0.5\right)\right)}^{-2}\\ t_3 := -1 + \frac{2}{t\_1}\\ \mathbf{if}\;-2 \cdot x \leq -1:\\ \;\;\;\;\mathsf{fma}\left(\frac{t\_2}{-1 + t\_2}, t\_3, \frac{1}{-1 + \frac{-2}{t\_1}}\right)\\ \mathbf{elif}\;-2 \cdot x \leq 0.0002:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (exp (* -2.0 x)))
        (t_1 (+ t_0 1.0))
        (t_2 (pow (fma t_0 0.5 0.5) -2.0))
        (t_3 (+ -1.0 (/ 2.0 t_1))))
   (if (<= (* -2.0 x) -1.0)
     (fma (/ t_2 (+ -1.0 t_2)) t_3 (/ 1.0 (+ -1.0 (/ -2.0 t_1))))
     (if (<= (* -2.0 x) 0.0002)
       (fma -0.3333333333333333 (* x (* x x)) x)
       t_3))))

C

double code(double x, double y) {
	double t_0 = exp((-2.0 * x));
	double t_1 = t_0 + 1.0;
	double t_2 = pow(fma(t_0, 0.5, 0.5), -2.0);
	double t_3 = -1.0 + (2.0 / t_1);
	double tmp;
	if ((-2.0 * x) <= -1.0) {
		tmp = fma((t_2 / (-1.0 + t_2)), t_3, (1.0 / (-1.0 + (-2.0 / t_1))));
	} else if ((-2.0 * x) <= 0.0002) {
		tmp = fma(-0.3333333333333333, (x * (x * x)), x);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = exp(Float64(-2.0 * x))
	t_1 = Float64(t_0 + 1.0)
	t_2 = fma(t_0, 0.5, 0.5) ^ -2.0
	t_3 = Float64(-1.0 + Float64(2.0 / t_1))
	tmp = 0.0
	if (Float64(-2.0 * x) <= -1.0)
		tmp = fma(Float64(t_2 / Float64(-1.0 + t_2)), t_3, Float64(1.0 / Float64(-1.0 + Float64(-2.0 / t_1))));
	elseif (Float64(-2.0 * x) <= 0.0002)
		tmp = fma(-0.3333333333333333, Float64(x * Float64(x * x)), x);
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(t$95$0 * 0.5 + 0.5), $MachinePrecision], -2.0], $MachinePrecision]}, Block[{t$95$3 = N[(-1.0 + N[(2.0 / t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -1.0], N[(N[(t$95$2 / N[(-1.0 + t$95$2), $MachinePrecision]), $MachinePrecision] * t$95$3 + N[(1.0 / N[(-1.0 + N[(-2.0 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.0002], N[(-0.3333333333333333 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -1

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(\left(1 + e^{-2 \cdot x}\right) \cdot 0.5\right)}^{-2}}{\mathsf{fma}\left(4, {\left(1 + e^{-2 \cdot x}\right)}^{-2}, -1\right)}, -1 - \frac{-2}{1 + e^{-2 \cdot x}}, -\frac{1}{1 - \frac{-2}{1 + e^{-2 \cdot x}}}\right)} \]

   6. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(\mathsf{fma}\left(e^{-2 \cdot x}, 0.5, 0.5\right)\right)}^{-2}}{-1 + {\left(\mathsf{fma}\left(e^{-2 \cdot x}, 0.5, 0.5\right)\right)}^{-2}}, -1 + \frac{2}{1 + e^{-2 \cdot x}}, \frac{1}{\frac{-2}{1 + e^{-2 \cdot x}} + -1}\right)} \]

   if -1 < (*.f64 #s(literal -2 binary64) x) < 2.0000000000000001e-4

   1. Initial program 7.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right)} \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

   if 2.0000000000000001e-4 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 100.0%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Add Preprocessing
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -1:\\ \;\;\;\;\mathsf{fma}\left(\frac{{\left(\mathsf{fma}\left(e^{-2 \cdot x}, 0.5, 0.5\right)\right)}^{-2}}{-1 + {\left(\mathsf{fma}\left(e^{-2 \cdot x}, 0.5, 0.5\right)\right)}^{-2}}, -1 + \frac{2}{e^{-2 \cdot x} + 1}, \frac{1}{-1 + \frac{-2}{e^{-2 \cdot x} + 1}}\right)\\ \mathbf{elif}\;-2 \cdot x \leq 0.0002:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{2}{e^{-2 \cdot x} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2}{e^{-2 \cdot x} + 1} \leq 0.05:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x + 2}{x \cdot 2}}\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((2.0d0 / (exp(((-2.0d0) * x)) + 1.0d0)) <= 0.05d0) then
        tmp = -1.0d0
    else
        tmp = 1.0d0 / ((x + 2.0d0) / (x * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((2.0 / (Math.exp((-2.0 * x)) + 1.0)) <= 0.05) {
		tmp = -1.0;
	} else {
		tmp = 1.0 / ((x + 2.0) / (x * 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (2.0 / (math.exp((-2.0 * x)) + 1.0)) <= 0.05:
		tmp = -1.0
	else:
		tmp = 1.0 / ((x + 2.0) / (x * 2.0))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((2.0 / (exp((-2.0 * x)) + 1.0)) <= 0.05)
		tmp = -1.0;
	else
		tmp = 1.0 / ((x + 2.0) / (x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (*.f64 #s(literal -2 binary64) x)))) < 0.050000000000000003

   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 97.4

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

   5. Applied rewrites 97.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 99.0%

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

   if 0.050000000000000003 < (/.f64 #s(literal 2 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (*.f64 #s(literal -2 binary64) x))))

   1. Initial program 40.2%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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 6.7

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

   5. Applied rewrites 6.7%

      \[\leadsto \color{blue}{\left(x + 1\right)} - 1 \]
   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. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. lower-+.f64 N/A

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

      10. metadata-eval N/A

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

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

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

      12. lift-+.f64 N/A

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

      13. associate--l+ N/A

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

      14. metadata-eval N/A

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

      15. +-rgt-identity N/A

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

      16. lower-*.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. associate-+l+ N/A

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

      19. metadata-eval N/A

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

      20. lower-+.f64 66.1

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

   7. Applied rewrites 66.1%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 70.4

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

   10. Applied rewrites 70.4%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{e^{-2 \cdot x} + 1} \leq 0.05:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x + 2}{x \cdot 2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 + \frac{2}{e^{-2 \cdot x} + 1}\\ \mathbf{if}\;-2 \cdot x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;-2 \cdot x \leq 0.0002:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ -1.0 (/ 2.0 (+ (exp (* -2.0 x)) 1.0)))))
   (if (<= (* -2.0 x) -1.0)
     t_0
     (if (<= (* -2.0 x) 0.0002)
       (fma -0.3333333333333333 (* x (* x x)) x)
       t_0))))

C

double code(double x, double y) {
	double t_0 = -1.0 + (2.0 / (exp((-2.0 * x)) + 1.0));
	double tmp;
	if ((-2.0 * x) <= -1.0) {
		tmp = t_0;
	} else if ((-2.0 * x) <= 0.0002) {
		tmp = fma(-0.3333333333333333, (x * (x * x)), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(-1.0 + Float64(2.0 / Float64(exp(Float64(-2.0 * x)) + 1.0)))
	tmp = 0.0
	if (Float64(-2.0 * x) <= -1.0)
		tmp = t_0;
	elseif (Float64(-2.0 * x) <= 0.0002)
		tmp = fma(-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[(-1.0 + N[(2.0 / N[(N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -1.0], t$95$0, If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.0002], N[(-0.3333333333333333 * 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) < -1 or 2.0000000000000001e-4 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 100.0%

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

   if -1 < (*.f64 #s(literal -2 binary64) x) < 2.0000000000000001e-4

   1. Initial program 7.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right)} \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -1:\\ \;\;\;\;-1 + \frac{2}{e^{-2 \cdot x} + 1}\\ \mathbf{elif}\;-2 \cdot x \leq 0.0002:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{2}{e^{-2 \cdot x} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.8% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq 0.0002:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* -2.0 x) 0.0002) x -1.0))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= 0.0002) {
		tmp = x;
	} 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) <= 0.0002d0) then
        tmp = x
    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) <= 0.0002) {
		tmp = x;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (-2.0 * x) <= 0.0002:
		tmp = x
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= 0.0002)
		tmp = x;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((-2.0 * x) <= 0.0002)
		tmp = x;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.0002], x, -1.0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal -2 binary64) x) < 2.0000000000000001e-4

   1. Initial program 40.2%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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 6.7

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

   5. Applied rewrites 6.7%

      \[\leadsto \color{blue}{\left(x + 1\right)} - 1 \]
   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 66.4

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

   7. Applied rewrites 66.4%

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

   if 2.0000000000000001e-4 < (*.f64 #s(literal -2 binary64) x)

   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 97.4

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

   5. Applied rewrites 97.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 99.0%

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

Alternative 5: 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 57.7%

   \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
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 32.3

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

5. Applied rewrites 32.3%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 31.2%

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

Reproduce

herbie shell --seed 2024216 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))