Logistic function from Lakshay Garg

Percentage Accurate: 54.4% → 99.8%

Time: 7.6s

Alternatives: 12

Speedup: 2.7×

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: 54.4% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{x \cdot -2} + 1\\ \mathbf{if}\;x \cdot -2 \leq -20:\\ \;\;\;\;\frac{2}{t\_0} - 1\\ \mathbf{elif}\;x \cdot -2 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left({x}^{5}, 0.13333333333333333, \mathsf{fma}\left({x}^{3}, -0.3333333333333333, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(-1 - \frac{-2}{t\_0}\right)}^{-1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (exp (* x -2.0)) 1.0)))
   (if (<= (* x -2.0) -20.0)
     (- (/ 2.0 t_0) 1.0)
     (if (<= (* x -2.0) 5e-5)
       (fma
        (pow x 5.0)
        0.13333333333333333
        (fma (pow x 3.0) -0.3333333333333333 x))
       (/ 1.0 (pow (- -1.0 (/ -2.0 t_0)) -1.0))))))

C

double code(double x, double y) {
	double t_0 = exp((x * -2.0)) + 1.0;
	double tmp;
	if ((x * -2.0) <= -20.0) {
		tmp = (2.0 / t_0) - 1.0;
	} else if ((x * -2.0) <= 5e-5) {
		tmp = fma(pow(x, 5.0), 0.13333333333333333, fma(pow(x, 3.0), -0.3333333333333333, x));
	} else {
		tmp = 1.0 / pow((-1.0 - (-2.0 / t_0)), -1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(exp(Float64(x * -2.0)) + 1.0)
	tmp = 0.0
	if (Float64(x * -2.0) <= -20.0)
		tmp = Float64(Float64(2.0 / t_0) - 1.0);
	elseif (Float64(x * -2.0) <= 5e-5)
		tmp = fma((x ^ 5.0), 0.13333333333333333, fma((x ^ 3.0), -0.3333333333333333, x));
	else
		tmp = Float64(1.0 / (Float64(-1.0 - Float64(-2.0 / t_0)) ^ -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Exp[N[(x * -2.0), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[N[(x * -2.0), $MachinePrecision], -20.0], N[(N[(2.0 / t$95$0), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[N[(x * -2.0), $MachinePrecision], 5e-5], N[(N[Power[x, 5.0], $MachinePrecision] * 0.13333333333333333 + N[(N[Power[x, 3.0], $MachinePrecision] * -0.3333333333333333 + x), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Power[N[(-1.0 - N[(-2.0 / t$95$0), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -20

   1. Initial program 100.0%

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

   if -20 < (*.f64 #s(literal -2 binary64) x) < 5.00000000000000024e-5

   1. Initial program 9.4%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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. +-commutative N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. associate-+l+ N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{5}, 0.13333333333333333, \mathsf{fma}\left({x}^{3}, -0.3333333333333333, x\right)\right)} \]

   if 5.00000000000000024e-5 < (*.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. flip3-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-- N/A

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

      7. lift--.f64 N/A

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

      8. inv-pow N/A

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

      9. metadata-eval N/A

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

   4. Applied rewrites 100.0%

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

      1. lift-pow.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. pow-exp N/A

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

      4. *-commutative N/A

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

      5. lower-exp.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 100.0

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

   6. Applied rewrites 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot -2 \leq -20:\\ \;\;\;\;\frac{2}{e^{x \cdot -2} + 1} - 1\\ \mathbf{elif}\;x \cdot -2 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left({x}^{5}, 0.13333333333333333, \mathsf{fma}\left({x}^{3}, -0.3333333333333333, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(-1 - \frac{-2}{e^{x \cdot -2} + 1}\right)}^{-1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{e^{x \cdot -2} + 1} - 1\\ \mathbf{if}\;x \cdot -2 \leq -20:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \cdot -2 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left({x}^{5}, 0.13333333333333333, \mathsf{fma}\left({x}^{3}, -0.3333333333333333, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (/ 2.0 (+ (exp (* x -2.0)) 1.0)) 1.0)))
   (if (<= (* x -2.0) -20.0)
     t_0
     (if (<= (* x -2.0) 5e-5)
       (fma
        (pow x 5.0)
        0.13333333333333333
        (fma (pow x 3.0) -0.3333333333333333 x))
       t_0))))

C

double code(double x, double y) {
	double t_0 = (2.0 / (exp((x * -2.0)) + 1.0)) - 1.0;
	double tmp;
	if ((x * -2.0) <= -20.0) {
		tmp = t_0;
	} else if ((x * -2.0) <= 5e-5) {
		tmp = fma(pow(x, 5.0), 0.13333333333333333, fma(pow(x, 3.0), -0.3333333333333333, x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(2.0 / Float64(exp(Float64(x * -2.0)) + 1.0)) - 1.0)
	tmp = 0.0
	if (Float64(x * -2.0) <= -20.0)
		tmp = t_0;
	elseif (Float64(x * -2.0) <= 5e-5)
		tmp = fma((x ^ 5.0), 0.13333333333333333, fma((x ^ 3.0), -0.3333333333333333, x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(2.0 / N[(N[Exp[N[(x * -2.0), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[N[(x * -2.0), $MachinePrecision], -20.0], t$95$0, If[LessEqual[N[(x * -2.0), $MachinePrecision], 5e-5], N[(N[Power[x, 5.0], $MachinePrecision] * 0.13333333333333333 + N[(N[Power[x, 3.0], $MachinePrecision] * -0.3333333333333333 + x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -20 or 5.00000000000000024e-5 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 100.0%

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

   if -20 < (*.f64 #s(literal -2 binary64) x) < 5.00000000000000024e-5

   1. Initial program 9.4%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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. +-commutative N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. associate-+l+ N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{5}, 0.13333333333333333, \mathsf{fma}\left({x}^{3}, -0.3333333333333333, x\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot -2 \leq -20:\\ \;\;\;\;\frac{2}{e^{x \cdot -2} + 1} - 1\\ \mathbf{elif}\;x \cdot -2 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left({x}^{5}, 0.13333333333333333, \mathsf{fma}\left({x}^{3}, -0.3333333333333333, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{e^{x \cdot -2} + 1} - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2}{e^{x \cdot -2} + 1} \leq 0:\\ \;\;\;\;\frac{2}{\left(2 \cdot x\right) \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ 2.0 (+ (exp (* x -2.0)) 1.0)) 0.0)
   (- (/ 2.0 (* (* 2.0 x) x)) 1.0)
   (/ 1.0 (/ 1.0 x))))

C

double code(double x, double y) {
	double tmp;
	if ((2.0 / (exp((x * -2.0)) + 1.0)) <= 0.0) {
		tmp = (2.0 / ((2.0 * x) * x)) - 1.0;
	} else {
		tmp = 1.0 / (1.0 / x);
	}
	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((x * (-2.0d0))) + 1.0d0)) <= 0.0d0) then
        tmp = (2.0d0 / ((2.0d0 * x) * x)) - 1.0d0
    else
        tmp = 1.0d0 / (1.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((2.0 / (Math.exp((x * -2.0)) + 1.0)) <= 0.0) {
		tmp = (2.0 / ((2.0 * x) * x)) - 1.0;
	} else {
		tmp = 1.0 / (1.0 / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (2.0 / (math.exp((x * -2.0)) + 1.0)) <= 0.0:
		tmp = (2.0 / ((2.0 * x) * x)) - 1.0
	else:
		tmp = 1.0 / (1.0 / x)
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := If[LessEqual[N[(2.0 / N[(N[Exp[N[(x * -2.0), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(2.0 / N[(N[(2.0 * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(1.0 / N[(1.0 / x), $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)))) < 0.0

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 100.0%

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

   if 0.0 < (/.f64 #s(literal 2 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (*.f64 #s(literal -2 binary64) x))))

   1. Initial program 39.6%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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. flip3-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-- N/A

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

      7. lift--.f64 N/A

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

      8. inv-pow N/A

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

      9. metadata-eval N/A

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

   4. Applied rewrites 39.6%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 67.7

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]

   7. Applied rewrites 67.7%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{e^{x \cdot -2} + 1} \leq 0:\\ \;\;\;\;\frac{2}{\left(2 \cdot x\right) \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2}{e^{x \cdot -2} + 1} \leq 0:\\ \;\;\;\;\frac{2}{x \cdot -2} - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ 2.0 (+ (exp (* x -2.0)) 1.0)) 0.0)
   (- (/ 2.0 (* x -2.0)) 1.0)
   (/ 1.0 (/ 1.0 x))))

C

double code(double x, double y) {
	double tmp;
	if ((2.0 / (exp((x * -2.0)) + 1.0)) <= 0.0) {
		tmp = (2.0 / (x * -2.0)) - 1.0;
	} else {
		tmp = 1.0 / (1.0 / x);
	}
	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((x * (-2.0d0))) + 1.0d0)) <= 0.0d0) then
        tmp = (2.0d0 / (x * (-2.0d0))) - 1.0d0
    else
        tmp = 1.0d0 / (1.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((2.0 / (Math.exp((x * -2.0)) + 1.0)) <= 0.0) {
		tmp = (2.0 / (x * -2.0)) - 1.0;
	} else {
		tmp = 1.0 / (1.0 / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (2.0 / (math.exp((x * -2.0)) + 1.0)) <= 0.0:
		tmp = (2.0 / (x * -2.0)) - 1.0
	else:
		tmp = 1.0 / (1.0 / x)
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := If[LessEqual[N[(2.0 / N[(N[Exp[N[(x * -2.0), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(2.0 / N[(x * -2.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(1.0 / N[(1.0 / x), $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)))) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 99.8%

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

   if 0.0 < (/.f64 #s(literal 2 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (*.f64 #s(literal -2 binary64) x))))

   1. Initial program 39.6%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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. flip3-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-- N/A

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

      7. lift--.f64 N/A

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

      8. inv-pow N/A

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

      9. metadata-eval N/A

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

   4. Applied rewrites 39.6%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 67.7

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]

   7. Applied rewrites 67.7%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{e^{x \cdot -2} + 1} \leq 0:\\ \;\;\;\;\frac{2}{x \cdot -2} - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{e^{x \cdot -2} + 1} - 1\\ \mathbf{if}\;x \cdot -2 \leq -0.002:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \cdot -2 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left({x}^{3}, -0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (/ 2.0 (+ (exp (* x -2.0)) 1.0)) 1.0)))
   (if (<= (* x -2.0) -0.002)
     t_0
     (if (<= (* x -2.0) 5e-5) (fma (pow x 3.0) -0.3333333333333333 x) t_0))))

C

double code(double x, double y) {
	double t_0 = (2.0 / (exp((x * -2.0)) + 1.0)) - 1.0;
	double tmp;
	if ((x * -2.0) <= -0.002) {
		tmp = t_0;
	} else if ((x * -2.0) <= 5e-5) {
		tmp = fma(pow(x, 3.0), -0.3333333333333333, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(2.0 / Float64(exp(Float64(x * -2.0)) + 1.0)) - 1.0)
	tmp = 0.0
	if (Float64(x * -2.0) <= -0.002)
		tmp = t_0;
	elseif (Float64(x * -2.0) <= 5e-5)
		tmp = fma((x ^ 3.0), -0.3333333333333333, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(2.0 / N[(N[Exp[N[(x * -2.0), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[N[(x * -2.0), $MachinePrecision], -0.002], t$95$0, If[LessEqual[N[(x * -2.0), $MachinePrecision], 5e-5], N[(N[Power[x, 3.0], $MachinePrecision] * -0.3333333333333333 + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -2e-3 or 5.00000000000000024e-5 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 99.9%

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

   if -2e-3 < (*.f64 #s(literal -2 binary64) x) < 5.00000000000000024e-5

   1. Initial program 8.8%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. pow-plus N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 100.0

          \[\leadsto \mathsf{fma}\left({x}^{\color{blue}{3}}, -0.3333333333333333, x\right) \]

   5. Applied rewrites 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot -2 \leq -0.002:\\ \;\;\;\;\frac{2}{e^{x \cdot -2} + 1} - 1\\ \mathbf{elif}\;x \cdot -2 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left({x}^{3}, -0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{e^{x \cdot -2} + 1} - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x \cdot -2} \leq 1.00002:\\ \;\;\;\;\frac{1}{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.3333333333333333, x, 2\right), x, -2\right), x, 2\right)} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (exp (* x -2.0)) 1.00002)
   (/ 1.0 (/ 1.0 x))
   (- (/ 2.0 (fma (fma (fma -1.3333333333333333 x 2.0) x -2.0) x 2.0)) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (exp((x * -2.0)) <= 1.00002) {
		tmp = 1.0 / (1.0 / x);
	} else {
		tmp = (2.0 / fma(fma(fma(-1.3333333333333333, x, 2.0), x, -2.0), x, 2.0)) - 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (exp(Float64(x * -2.0)) <= 1.00002)
		tmp = Float64(1.0 / Float64(1.0 / x));
	else
		tmp = Float64(Float64(2.0 / fma(fma(fma(-1.3333333333333333, x, 2.0), x, -2.0), x, 2.0)) - 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Exp[N[(x * -2.0), $MachinePrecision]], $MachinePrecision], 1.00002], N[(1.0 / N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(N[(-1.3333333333333333 * x + 2.0), $MachinePrecision] * x + -2.0), $MachinePrecision] * x + 2.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 (*.f64 #s(literal -2 binary64) x)) < 1.00001999999999991

   1. Initial program 39.1%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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. flip3-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-- N/A

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

      7. lift--.f64 N/A

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

      8. inv-pow N/A

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

      9. metadata-eval N/A

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

   4. Applied rewrites 39.1%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 67.9

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]

   7. Applied rewrites 67.9%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]

   if 1.00001999999999991 < (exp.f64 (*.f64 #s(literal -2 binary64) x))

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 98.5

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

   5. Applied rewrites 98.5%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.3333333333333333, x, 2\right), x, -2\right), x, 2\right)}} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x \cdot -2} \leq 1.00002:\\ \;\;\;\;\frac{1}{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.3333333333333333, x, 2\right), x, -2\right), x, 2\right)} - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot -2 \leq 0.5:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x}{\mathsf{fma}\left(x \cdot x, 0.3333333333333333, 1\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot x\right) \cdot x} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x -2.0) 0.5)
   (/ 1.0 (/ 1.0 (/ x (fma (* x x) 0.3333333333333333 1.0))))
   (- (/ 2.0 (* (* 2.0 x) x)) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((x * -2.0) <= 0.5) {
		tmp = 1.0 / (1.0 / (x / fma((x * x), 0.3333333333333333, 1.0)));
	} else {
		tmp = (2.0 / ((2.0 * x) * x)) - 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * -2.0) <= 0.5)
		tmp = Float64(1.0 / Float64(1.0 / Float64(x / fma(Float64(x * x), 0.3333333333333333, 1.0))));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(2.0 * x) * x)) - 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * -2.0), $MachinePrecision], 0.5], N[(1.0 / N[(1.0 / N[(x / N[(N[(x * x), $MachinePrecision] * 0.3333333333333333 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(2.0 * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal -2 binary64) x) < 0.5

   1. Initial program 39.6%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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. flip3-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-- N/A

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

      7. lift--.f64 N/A

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

      8. inv-pow N/A

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

      9. metadata-eval N/A

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

   4. Applied rewrites 39.6%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{3} + \frac{-1}{45} \cdot {x}^{2}\right)}{x}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{3} + \frac{-1}{45} \cdot {x}^{2}\right)}{x}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{{x}^{2} \cdot \left(\frac{1}{3} + \frac{-1}{45} \cdot {x}^{2}\right) + 1}}{x}} \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(\frac{1}{3} + \frac{-1}{45} \cdot {x}^{2}\right) \cdot {x}^{2}} + 1}{x}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{fma}\left(\frac{1}{3} + \frac{-1}{45} \cdot {x}^{2}, {x}^{2}, 1\right)}}{x}} \]

      5. +-commutative N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{45} \cdot {x}^{2} + \frac{1}{3}}, {x}^{2}, 1\right)}{x}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{45}, {x}^{2}, \frac{1}{3}\right)}, {x}^{2}, 1\right)}{x}} \]

      7. unpow2 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{45}, \color{blue}{x \cdot x}, \frac{1}{3}\right), {x}^{2}, 1\right)}{x}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{45}, \color{blue}{x \cdot x}, \frac{1}{3}\right), {x}^{2}, 1\right)}{x}} \]

      9. unpow2 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{45}, x \cdot x, \frac{1}{3}\right), \color{blue}{x \cdot x}, 1\right)}{x}} \]

      10. lower-*.f64 67.7

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, x \cdot x, 0.3333333333333333\right), \color{blue}{x \cdot x}, 1\right)}{x}} \]

   7. Applied rewrites 67.7%

      \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, x \cdot x, 0.3333333333333333\right), x \cdot x, 1\right)}{x}}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\frac{1}{3}, x \cdot x, 1\right)}{x}} \]
   9. Step-by-step derivation

   10. Applied rewrites 68.3%

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(0.3333333333333333, x \cdot x, 1\right)}{x}} \]
   11. Step-by-step derivation

   12. Applied rewrites 68.3%

       \[\leadsto \frac{1}{\frac{1}{\color{blue}{\frac{x}{\mathsf{fma}\left(x \cdot x, 0.3333333333333333, 1\right)}}}} \]

   if 0.5 < (*.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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 100.0%

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

4. Final simplification 75.9%

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

Alternative 8: 74.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot -2 \leq 0.5:\\ \;\;\;\;\frac{1}{\frac{-1}{x} \cdot \left(-\mathsf{fma}\left(x \cdot x, 0.3333333333333333, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot x\right) \cdot x} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x -2.0) 0.5)
   (/ 1.0 (* (/ -1.0 x) (- (fma (* x x) 0.3333333333333333 1.0))))
   (- (/ 2.0 (* (* 2.0 x) x)) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((x * -2.0) <= 0.5) {
		tmp = 1.0 / ((-1.0 / x) * -fma((x * x), 0.3333333333333333, 1.0));
	} else {
		tmp = (2.0 / ((2.0 * x) * x)) - 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * -2.0) <= 0.5)
		tmp = Float64(1.0 / Float64(Float64(-1.0 / x) * Float64(-fma(Float64(x * x), 0.3333333333333333, 1.0))));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(2.0 * x) * x)) - 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * -2.0), $MachinePrecision], 0.5], N[(1.0 / N[(N[(-1.0 / x), $MachinePrecision] * (-N[(N[(x * x), $MachinePrecision] * 0.3333333333333333 + 1.0), $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(2.0 * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal -2 binary64) x) < 0.5

   1. Initial program 39.6%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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. flip3-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-- N/A

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

      7. lift--.f64 N/A

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

      8. inv-pow N/A

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

      9. metadata-eval N/A

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

   4. Applied rewrites 39.6%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{3} + \frac{-1}{45} \cdot {x}^{2}\right)}{x}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{3} + \frac{-1}{45} \cdot {x}^{2}\right)}{x}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{{x}^{2} \cdot \left(\frac{1}{3} + \frac{-1}{45} \cdot {x}^{2}\right) + 1}}{x}} \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(\frac{1}{3} + \frac{-1}{45} \cdot {x}^{2}\right) \cdot {x}^{2}} + 1}{x}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{fma}\left(\frac{1}{3} + \frac{-1}{45} \cdot {x}^{2}, {x}^{2}, 1\right)}}{x}} \]

      5. +-commutative N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{45} \cdot {x}^{2} + \frac{1}{3}}, {x}^{2}, 1\right)}{x}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{45}, {x}^{2}, \frac{1}{3}\right)}, {x}^{2}, 1\right)}{x}} \]

      7. unpow2 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{45}, \color{blue}{x \cdot x}, \frac{1}{3}\right), {x}^{2}, 1\right)}{x}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{45}, \color{blue}{x \cdot x}, \frac{1}{3}\right), {x}^{2}, 1\right)}{x}} \]

      9. unpow2 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{45}, x \cdot x, \frac{1}{3}\right), \color{blue}{x \cdot x}, 1\right)}{x}} \]

      10. lower-*.f64 67.7

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, x \cdot x, 0.3333333333333333\right), \color{blue}{x \cdot x}, 1\right)}{x}} \]

   7. Applied rewrites 67.7%

      \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, x \cdot x, 0.3333333333333333\right), x \cdot x, 1\right)}{x}}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\frac{1}{3}, x \cdot x, 1\right)}{x}} \]
   9. Step-by-step derivation

   10. Applied rewrites 68.3%

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(0.3333333333333333, x \cdot x, 1\right)}{x}} \]
   11. Step-by-step derivation

   12. Applied rewrites 68.3%

       \[\leadsto \frac{1}{\left(-\mathsf{fma}\left(x \cdot x, 0.3333333333333333, 1\right)\right) \cdot \color{blue}{\frac{-1}{x}}} \]

   if 0.5 < (*.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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 100.0%

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

4. Final simplification 75.9%

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

Alternative 9: 74.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot -2 \leq 0.5:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(0.3333333333333333, x \cdot x, 1\right)}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot x\right) \cdot x} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x -2.0) 0.5)
   (/ 1.0 (/ (fma 0.3333333333333333 (* x x) 1.0) x))
   (- (/ 2.0 (* (* 2.0 x) x)) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((x * -2.0) <= 0.5) {
		tmp = 1.0 / (fma(0.3333333333333333, (x * x), 1.0) / x);
	} else {
		tmp = (2.0 / ((2.0 * x) * x)) - 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * -2.0) <= 0.5)
		tmp = Float64(1.0 / Float64(fma(0.3333333333333333, Float64(x * x), 1.0) / x));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(2.0 * x) * x)) - 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * -2.0), $MachinePrecision], 0.5], N[(1.0 / N[(N[(0.3333333333333333 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(2.0 * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal -2 binary64) x) < 0.5

   1. Initial program 39.6%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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. flip3-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-- N/A

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

      7. lift--.f64 N/A

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

      8. inv-pow N/A

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

      9. metadata-eval N/A

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

   4. Applied rewrites 39.6%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1 + \frac{1}{3} \cdot {x}^{2}}{x}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\frac{1}{3} \cdot {x}^{2} + 1}}{x}} \]

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 68.3

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(0.3333333333333333, \color{blue}{x \cdot x}, 1\right)}{x}} \]

   7. Applied rewrites 68.3%

      \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, x \cdot x, 1\right)}{x}}} \]

   if 0.5 < (*.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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 100.0%

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

4. Final simplification 75.9%

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

Alternative 10: 29.0% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot -2 \leq 0.5:\\ \;\;\;\;\left(1 + x\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot -2} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x -2.0) 0.5) (- (+ 1.0 x) 1.0) (- (/ 2.0 (* x -2.0)) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((x * -2.0) <= 0.5) {
		tmp = (1.0 + x) - 1.0;
	} else {
		tmp = (2.0 / (x * -2.0)) - 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 * (-2.0d0)) <= 0.5d0) then
        tmp = (1.0d0 + x) - 1.0d0
    else
        tmp = (2.0d0 / (x * (-2.0d0))) - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x * -2.0) <= 0.5) {
		tmp = (1.0 + x) - 1.0;
	} else {
		tmp = (2.0 / (x * -2.0)) - 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x * -2.0) <= 0.5:
		tmp = (1.0 + x) - 1.0
	else:
		tmp = (2.0 / (x * -2.0)) - 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * -2.0) <= 0.5)
		tmp = Float64(Float64(1.0 + x) - 1.0);
	else
		tmp = Float64(Float64(2.0 / Float64(x * -2.0)) - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x * -2.0) <= 0.5)
		tmp = (1.0 + x) - 1.0;
	else
		tmp = (2.0 / (x * -2.0)) - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * -2.0), $MachinePrecision], 0.5], N[(N[(1.0 + x), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(2.0 / N[(x * -2.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal -2 binary64) x) < 0.5

   1. Initial program 39.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-+.f64 8.1

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

   5. Applied rewrites 8.1%

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

   if 0.5 < (*.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

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

      1. +-commutative N/A

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

      2. lower-fma.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 99.8%

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

4. Final simplification 30.0%

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

Alternative 11: 6.6% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \left(1 + x\right) - 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- (+ 1.0 x) 1.0))

C

double code(double x, double y) {
	return (1.0 + x) - 1.0;
}

Fortran

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

Java

public static double code(double x, double y) {
	return (1.0 + x) - 1.0;
}

Python

def code(x, y):
	return (1.0 + x) - 1.0

Julia

function code(x, y)
	return Float64(Float64(1.0 + x) - 1.0)
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 + x) - 1.0;
end

Wolfram

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

Derivation

1. Initial program 54.0%

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

3. Taylor expanded in x around 0

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

   1. lower-+.f64 7.4

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

5. Applied rewrites 7.4%

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

Alternative 12: 4.2% accurate, 30.8× speedup

Math

\[\begin{array}{l} \\ 1 - 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- 1.0 1.0))

C

double code(double x, double y) {
	return 1.0 - 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 - 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0 - 1.0;
}

Python

def code(x, y):
	return 1.0 - 1.0

Julia

function code(x, y)
	return Float64(1.0 - 1.0)
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 - 1.0;
end

Wolfram

code[x_, y_] := N[(1.0 - 1.0), $MachinePrecision]

Derivation

1. Initial program 54.0%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 4.2%

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

Reproduce

herbie shell --seed 2024331 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))