Logistic function from Lakshay Garg

Percentage Accurate: 54.3% → 99.2%

Time: 9.6s

Alternatives: 20

Speedup: 2.2×

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: 54.3% 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: 99.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;-2 \cdot x \leq -100000:\\ \;\;\;\;\frac{2}{2 - \frac{x + x}{x + x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, t\_0, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{64 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot t\_0\right)\right)} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (if (<= (* -2.0 x) -100000.0)
     (+ (/ 2.0 (- 2.0 (/ (+ x x) (+ x x)))) -1.0)
     (if (<= (* -2.0 x) 4e-5)
       (fma -0.3333333333333333 t_0 x)
       (+ (/ 2.0 (* 64.0 (* (* x x) (* x t_0)))) -1.0)))))

C

double code(double x, double y) {
	double t_0 = x * (x * x);
	double tmp;
	if ((-2.0 * x) <= -100000.0) {
		tmp = (2.0 / (2.0 - ((x + x) / (x + x)))) + -1.0;
	} else if ((-2.0 * x) <= 4e-5) {
		tmp = fma(-0.3333333333333333, t_0, x);
	} else {
		tmp = (2.0 / (64.0 * ((x * x) * (x * t_0)))) + -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (Float64(-2.0 * x) <= -100000.0)
		tmp = Float64(Float64(2.0 / Float64(2.0 - Float64(Float64(x + x) / Float64(x + x)))) + -1.0);
	elseif (Float64(-2.0 * x) <= 4e-5)
		tmp = fma(-0.3333333333333333, t_0, x);
	else
		tmp = Float64(Float64(2.0 / Float64(64.0 * Float64(Float64(x * x) * Float64(x * t_0)))) + -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -100000.0], N[(N[(2.0 / N[(2.0 - N[(N[(x + x), $MachinePrecision] / N[(x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 4e-5], N[(-0.3333333333333333 * t$95$0 + x), $MachinePrecision], N[(N[(2.0 / N[(64.0 * N[(N[(x * x), $MachinePrecision] * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -1e5

   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. metadata-eval N/A

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

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

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

      3. lower--.f64 N/A

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

      4. count-2 N/A

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

      5. lower-+.f64 1.6

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

   5. Applied rewrites 1.6%

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

   7. Applied rewrites 100.0%

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

   if -1e5 < (*.f64 #s(literal -2 binary64) x) < 4.00000000000000033e-5

   1. Initial program 6.8%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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. 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 4.00000000000000033e-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. metadata-eval N/A

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

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

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

      3. lower--.f64 N/A

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

      4. count-2 N/A

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

      5. lower-+.f64 97.5

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

   5. Applied rewrites 97.5%

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

   7. Applied rewrites 98.6%

      \[\leadsto \frac{2}{\mathsf{fma}\left(x + x, \color{blue}{x}, 2\right)} - 1 \]
   8. Step-by-step derivation

   9. Applied rewrites 100.0%

      \[\leadsto \frac{2}{\mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right), \color{blue}{64}, 2\right)} - 1 \]

   10. Taylor expanded in x around inf

       \[\leadsto \frac{2}{64 \cdot \color{blue}{{x}^{6}}} - 1 \]
   11. Step-by-step derivation

   12. Applied rewrites 100.0%

       \[\leadsto \frac{2}{64 \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}} - 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -100000:\\ \;\;\;\;\frac{2}{2 - \frac{x + x}{x + x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{64 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 75.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} \leq 0:\\ \;\;\;\;\frac{2}{\left(x + x\right) \cdot t\_0} + -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, t\_0, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (if (<= (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 0.0)
     (+ (/ 2.0 (* (+ x x) t_0)) -1.0)
     (fma -0.3333333333333333 t_0 x))))

C

double code(double x, double y) {
	double t_0 = x * (x * x);
	double tmp;
	if ((2.0 / (1.0 + exp((-2.0 * x)))) <= 0.0) {
		tmp = (2.0 / ((x + x) * t_0)) + -1.0;
	} else {
		tmp = fma(-0.3333333333333333, t_0, x);
	}
	return tmp;
}

Julia

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

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(2.0 / N[(N[(x + x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(-0.3333333333333333 * t$95$0 + x), $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. metadata-eval N/A

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

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

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

      3. lower--.f64 N/A

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

      4. count-2 N/A

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

      5. lower-+.f64 97.5

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

   5. Applied rewrites 97.5%

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

   7. Applied rewrites 99.3%

      \[\leadsto \frac{2}{\mathsf{fma}\left(8, \color{blue}{x \cdot \left(x \cdot x\right)}, 2\right)} - 1 \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{2}{8 \cdot \color{blue}{{x}^{3}}} - 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 99.3%

       \[\leadsto \frac{2}{x \cdot \color{blue}{\left(8 \cdot \left(x \cdot x\right)\right)}} - 1 \]

   12. Applied rewrites 99.6%

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

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

   1. Initial program 38.0%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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 66.8

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

   5. Applied rewrites 66.8%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} \leq 0:\\ \;\;\;\;\frac{2}{\left(x + x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)} + -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, x \cdot \left(x \cdot x\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} \leq 0:\\ \;\;\;\;\frac{2}{t\_0 \cdot 16} + -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, t\_0, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (if (<= (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 0.0)
     (+ (/ 2.0 (* t_0 16.0)) -1.0)
     (fma -0.3333333333333333 t_0 x))))

C

double code(double x, double y) {
	double t_0 = x * (x * x);
	double tmp;
	if ((2.0 / (1.0 + exp((-2.0 * x)))) <= 0.0) {
		tmp = (2.0 / (t_0 * 16.0)) + -1.0;
	} else {
		tmp = fma(-0.3333333333333333, t_0, x);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) <= 0.0)
		tmp = Float64(Float64(2.0 / Float64(t_0 * 16.0)) + -1.0);
	else
		tmp = fma(-0.3333333333333333, t_0, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(2.0 / N[(t$95$0 * 16.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(-0.3333333333333333 * t$95$0 + x), $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. metadata-eval N/A

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

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

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

      3. lower--.f64 N/A

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

      4. count-2 N/A

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

      5. lower-+.f64 97.5

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

   5. Applied rewrites 97.5%

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

   7. Applied rewrites 99.3%

      \[\leadsto \frac{2}{\mathsf{fma}\left(8, \color{blue}{x \cdot \left(x \cdot x\right)}, 2\right)} - 1 \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{2}{8 \cdot \color{blue}{{x}^{3}}} - 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 99.3%

       \[\leadsto \frac{2}{x \cdot \color{blue}{\left(8 \cdot \left(x \cdot x\right)\right)}} - 1 \]
   11. Step-by-step derivation

   12. Applied rewrites 99.3%

       \[\leadsto \frac{2}{\left(x \cdot \left(x \cdot x\right)\right) \cdot 16} - 1 \]

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

   1. Initial program 38.0%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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 66.8

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

   5. Applied rewrites 66.8%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} \leq 0:\\ \;\;\;\;\frac{2}{\left(x \cdot \left(x \cdot x\right)\right) \cdot 16} + -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, x \cdot \left(x \cdot x\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} \leq 0:\\ \;\;\;\;\frac{2}{x \cdot \left(\left(x \cdot x\right) \cdot 8\right)} + -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, x \cdot \left(x \cdot x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 0.0)
   (+ (/ 2.0 (* x (* (* x x) 8.0))) -1.0)
   (fma -0.3333333333333333 (* x (* x x)) x)))

C

double code(double x, double y) {
	double tmp;
	if ((2.0 / (1.0 + exp((-2.0 * x)))) <= 0.0) {
		tmp = (2.0 / (x * ((x * x) * 8.0))) + -1.0;
	} else {
		tmp = fma(-0.3333333333333333, (x * (x * x)), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) <= 0.0)
		tmp = Float64(Float64(2.0 / Float64(x * Float64(Float64(x * x) * 8.0))) + -1.0);
	else
		tmp = fma(-0.3333333333333333, Float64(x * Float64(x * x)), x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(2.0 / N[(x * N[(N[(x * x), $MachinePrecision] * 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(-0.3333333333333333 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $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. metadata-eval N/A

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

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

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

      3. lower--.f64 N/A

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

      4. count-2 N/A

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

      5. lower-+.f64 97.5

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

   5. Applied rewrites 97.5%

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

   7. Applied rewrites 99.3%

      \[\leadsto \frac{2}{\mathsf{fma}\left(8, \color{blue}{x \cdot \left(x \cdot x\right)}, 2\right)} - 1 \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{2}{8 \cdot \color{blue}{{x}^{3}}} - 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 99.3%

       \[\leadsto \frac{2}{x \cdot \color{blue}{\left(8 \cdot \left(x \cdot x\right)\right)}} - 1 \]

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

   1. Initial program 38.0%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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 66.8

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

   5. Applied rewrites 66.8%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} \leq 0:\\ \;\;\;\;\frac{2}{x \cdot \left(\left(x \cdot x\right) \cdot 8\right)} + -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, x \cdot \left(x \cdot x\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} \leq 0:\\ \;\;\;\;\frac{2}{\left(x + x\right) \cdot \left(x \cdot x\right)} + -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, x \cdot \left(x \cdot x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 0.0)
   (+ (/ 2.0 (* (+ x x) (* x x))) -1.0)
   (fma -0.3333333333333333 (* x (* x x)) x)))

C

double code(double x, double y) {
	double tmp;
	if ((2.0 / (1.0 + exp((-2.0 * x)))) <= 0.0) {
		tmp = (2.0 / ((x + x) * (x * x))) + -1.0;
	} else {
		tmp = fma(-0.3333333333333333, (x * (x * x)), x);
	}
	return tmp;
}

Julia

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

Wolfram

code[x_, y_] := If[LessEqual[N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(2.0 / N[(N[(x + x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(-0.3333333333333333 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $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. metadata-eval N/A

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

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

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

      3. lower--.f64 N/A

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

      4. count-2 N/A

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

      5. lower-+.f64 97.5

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

   5. Applied rewrites 97.5%

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

   7. Applied rewrites 99.3%

      \[\leadsto \frac{2}{\mathsf{fma}\left(8, \color{blue}{x \cdot \left(x \cdot x\right)}, 2\right)} - 1 \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{2}{8 \cdot \color{blue}{{x}^{3}}} - 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 99.3%

       \[\leadsto \frac{2}{x \cdot \color{blue}{\left(8 \cdot \left(x \cdot x\right)\right)}} - 1 \]

   12. Applied rewrites 99.2%

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

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

   1. Initial program 38.0%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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 66.8

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

   5. Applied rewrites 66.8%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} \leq 0:\\ \;\;\;\;\frac{2}{\left(x + x\right) \cdot \left(x \cdot x\right)} + -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, x \cdot \left(x \cdot x\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} \leq 0:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(x + x, x, 2\right)} + -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, x \cdot \left(x \cdot x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 0.0)
   (+ (/ 2.0 (fma (+ x x) x 2.0)) -1.0)
   (fma -0.3333333333333333 (* x (* x x)) x)))

C

double code(double x, double y) {
	double tmp;
	if ((2.0 / (1.0 + exp((-2.0 * x)))) <= 0.0) {
		tmp = (2.0 / fma((x + x), x, 2.0)) + -1.0;
	} else {
		tmp = fma(-0.3333333333333333, (x * (x * x)), x);
	}
	return tmp;
}

Julia

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

Wolfram

code[x_, y_] := If[LessEqual[N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(2.0 / N[(N[(x + x), $MachinePrecision] * x + 2.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(-0.3333333333333333 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $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. metadata-eval N/A

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

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

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

      3. lower--.f64 N/A

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

      4. count-2 N/A

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

      5. lower-+.f64 97.5

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

   5. Applied rewrites 97.5%

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

   7. Applied rewrites 98.6%

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

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

   1. Initial program 38.0%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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 66.8

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

   5. Applied rewrites 66.8%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} \leq 0:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(x + x, x, 2\right)} + -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, x \cdot \left(x \cdot x\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} \leq 0:\\ \;\;\;\;\frac{2}{x \cdot \left(x + x\right)} + -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, x \cdot \left(x \cdot x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 0.0)
   (+ (/ 2.0 (* x (+ x x))) -1.0)
   (fma -0.3333333333333333 (* x (* x x)) x)))

C

double code(double x, double y) {
	double tmp;
	if ((2.0 / (1.0 + exp((-2.0 * x)))) <= 0.0) {
		tmp = (2.0 / (x * (x + x))) + -1.0;
	} else {
		tmp = fma(-0.3333333333333333, (x * (x * x)), x);
	}
	return tmp;
}

Julia

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

Wolfram

code[x_, y_] := If[LessEqual[N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(2.0 / N[(x * N[(x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(-0.3333333333333333 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $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. metadata-eval N/A

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

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

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

      3. lower--.f64 N/A

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

      4. count-2 N/A

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

      5. lower-+.f64 97.5

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

   5. Applied rewrites 97.5%

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

   7. Applied rewrites 99.3%

      \[\leadsto \frac{2}{\mathsf{fma}\left(8, \color{blue}{x \cdot \left(x \cdot x\right)}, 2\right)} - 1 \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{2}{8 \cdot \color{blue}{{x}^{3}}} - 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 99.3%

       \[\leadsto \frac{2}{x \cdot \color{blue}{\left(8 \cdot \left(x \cdot x\right)\right)}} - 1 \]
   11. Step-by-step derivation

   12. Applied rewrites 98.6%

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

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

   1. Initial program 38.0%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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 66.8

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

   5. Applied rewrites 66.8%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} \leq 0:\\ \;\;\;\;\frac{2}{x \cdot \left(x + x\right)} + -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, x \cdot \left(x \cdot x\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;e^{-2 \cdot x} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, t\_0, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{64 \cdot \left(x \cdot t\_0\right)} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (if (<= (exp (* -2.0 x)) 2.0)
     (fma -0.3333333333333333 t_0 x)
     (+ (/ 2.0 (* 64.0 (* x t_0))) -1.0))))

C

double code(double x, double y) {
	double t_0 = x * (x * x);
	double tmp;
	if (exp((-2.0 * x)) <= 2.0) {
		tmp = fma(-0.3333333333333333, t_0, x);
	} else {
		tmp = (2.0 / (64.0 * (x * t_0))) + -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (exp(Float64(-2.0 * x)) <= 2.0)
		tmp = fma(-0.3333333333333333, t_0, x);
	else
		tmp = Float64(Float64(2.0 / Float64(64.0 * Float64(x * t_0))) + -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision], 2.0], N[(-0.3333333333333333 * t$95$0 + x), $MachinePrecision], N[(N[(2.0 / N[(64.0 * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 38.0%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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 66.8

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

   5. Applied rewrites 66.8%

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

   if 2 < (exp.f64 (*.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. metadata-eval N/A

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

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

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

      3. lower--.f64 N/A

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

      4. count-2 N/A

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

      5. lower-+.f64 97.5

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

   5. Applied rewrites 97.5%

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

   7. Applied rewrites 99.3%

      \[\leadsto \frac{2}{\mathsf{fma}\left(8, \color{blue}{x \cdot \left(x \cdot x\right)}, 2\right)} - 1 \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{2}{8 \cdot \color{blue}{{x}^{3}}} - 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 99.3%

       \[\leadsto \frac{2}{x \cdot \color{blue}{\left(8 \cdot \left(x \cdot x\right)\right)}} - 1 \]

   12. Applied rewrites 99.7%

       \[\leadsto \frac{2}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot 64} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-2 \cdot x} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{64 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} \leq 0:\\ \;\;\;\;\frac{2}{x + x} + -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, x \cdot \left(x \cdot x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 0.0)
   (+ (/ 2.0 (+ x x)) -1.0)
   (fma -0.3333333333333333 (* x (* x x)) x)))

C

double code(double x, double y) {
	double tmp;
	if ((2.0 / (1.0 + exp((-2.0 * x)))) <= 0.0) {
		tmp = (2.0 / (x + x)) + -1.0;
	} else {
		tmp = fma(-0.3333333333333333, (x * (x * x)), x);
	}
	return tmp;
}

Julia

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

Wolfram

code[x_, y_] := If[LessEqual[N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(2.0 / N[(x + x), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(-0.3333333333333333 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $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. metadata-eval N/A

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

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

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

      3. lower--.f64 N/A

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

      4. count-2 N/A

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

      5. lower-+.f64 97.5

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

   5. Applied rewrites 97.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 97.5%

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

   10. Applied rewrites 97.6%

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

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

   1. Initial program 38.0%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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 66.8

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

   5. Applied rewrites 66.8%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} \leq 0:\\ \;\;\;\;\frac{2}{x + x} + -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, x \cdot \left(x \cdot x\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{-2 \cdot x} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(x \cdot x, 16, 2\right)} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (exp (* -2.0 x)) 2.0)
   (fma -0.3333333333333333 (* x (* x x)) x)
   (+ (/ 2.0 (fma (* x x) 16.0 2.0)) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (exp((-2.0 * x)) <= 2.0) {
		tmp = fma(-0.3333333333333333, (x * (x * x)), x);
	} else {
		tmp = (2.0 / fma((x * x), 16.0, 2.0)) + -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (exp(Float64(-2.0 * x)) <= 2.0)
		tmp = fma(-0.3333333333333333, Float64(x * Float64(x * x)), x);
	else
		tmp = Float64(Float64(2.0 / fma(Float64(x * x), 16.0, 2.0)) + -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision], 2.0], N[(-0.3333333333333333 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(2.0 / N[(N[(x * x), $MachinePrecision] * 16.0 + 2.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 38.0%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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 66.8

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

   5. Applied rewrites 66.8%

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

   if 2 < (exp.f64 (*.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. metadata-eval N/A

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

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

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

      3. lower--.f64 N/A

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

      4. count-2 N/A

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

      5. lower-+.f64 97.5

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

   5. Applied rewrites 97.5%

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

   7. Applied rewrites 98.6%

      \[\leadsto \frac{2}{\mathsf{fma}\left(x + x, \color{blue}{x}, 2\right)} - 1 \]
   8. Step-by-step derivation

   9. Applied rewrites 100.0%

      \[\leadsto \frac{2}{\mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right), \color{blue}{64}, 2\right)} - 1 \]

   11. Applied rewrites 98.8%

       \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \color{blue}{16}, 2\right)} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-2 \cdot x} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(x \cdot x, 16, 2\right)} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{-2 \cdot x} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(x \cdot x\right) \cdot 16} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (exp (* -2.0 x)) 2.0)
   (fma -0.3333333333333333 (* x (* x x)) x)
   (+ (/ 2.0 (* (* x x) 16.0)) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (exp((-2.0 * x)) <= 2.0) {
		tmp = fma(-0.3333333333333333, (x * (x * x)), x);
	} else {
		tmp = (2.0 / ((x * x) * 16.0)) + -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (exp(Float64(-2.0 * x)) <= 2.0)
		tmp = fma(-0.3333333333333333, Float64(x * Float64(x * x)), x);
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(x * x) * 16.0)) + -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision], 2.0], N[(-0.3333333333333333 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(2.0 / N[(N[(x * x), $MachinePrecision] * 16.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 38.0%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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 66.8

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

   5. Applied rewrites 66.8%

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

   if 2 < (exp.f64 (*.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. metadata-eval N/A

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

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

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

      3. lower--.f64 N/A

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

      4. count-2 N/A

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

      5. lower-+.f64 97.5

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

   5. Applied rewrites 97.5%

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

   7. Applied rewrites 99.3%

      \[\leadsto \frac{2}{\mathsf{fma}\left(8, \color{blue}{x \cdot \left(x \cdot x\right)}, 2\right)} - 1 \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{2}{8 \cdot \color{blue}{{x}^{3}}} - 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 99.3%

       \[\leadsto \frac{2}{x \cdot \color{blue}{\left(8 \cdot \left(x \cdot x\right)\right)}} - 1 \]

   12. Applied rewrites 98.8%

       \[\leadsto \frac{2}{\left(x \cdot x\right) \cdot 16} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-2 \cdot x} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(x \cdot x\right) \cdot 16} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 75.2% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (exp (* -2.0 x)) 2.0)
   (fma -0.3333333333333333 (* x (* x x)) x)
   (+ (/ 2.0 (* x 4.0)) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (exp((-2.0 * x)) <= 2.0) {
		tmp = fma(-0.3333333333333333, (x * (x * x)), x);
	} else {
		tmp = (2.0 / (x * 4.0)) + -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (exp(Float64(-2.0 * x)) <= 2.0)
		tmp = fma(-0.3333333333333333, Float64(x * Float64(x * x)), x);
	else
		tmp = Float64(Float64(2.0 / Float64(x * 4.0)) + -1.0);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 38.0%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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 66.8

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

   5. Applied rewrites 66.8%

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

   if 2 < (exp.f64 (*.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. metadata-eval N/A

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

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

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

      3. lower--.f64 N/A

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

      4. count-2 N/A

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

      5. lower-+.f64 97.5

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

   5. Applied rewrites 97.5%

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

   7. Applied rewrites 99.3%

      \[\leadsto \frac{2}{\mathsf{fma}\left(8, \color{blue}{x \cdot \left(x \cdot x\right)}, 2\right)} - 1 \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{2}{8 \cdot \color{blue}{{x}^{3}}} - 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 99.3%

       \[\leadsto \frac{2}{x \cdot \color{blue}{\left(8 \cdot \left(x \cdot x\right)\right)}} - 1 \]
   11. Step-by-step derivation

   12. Applied rewrites 97.7%

       \[\leadsto \frac{2}{4 \cdot x} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-2 \cdot x} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot 4} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 99.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;-2 \cdot x \leq -100000:\\ \;\;\;\;\frac{2}{2 - \frac{x + x}{x + x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, t\_0, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_0 \cdot \left(\left(x + x\right) \cdot \left(x \cdot 16\right)\right)} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (if (<= (* -2.0 x) -100000.0)
     (+ (/ 2.0 (- 2.0 (/ (+ x x) (+ x x)))) -1.0)
     (if (<= (* -2.0 x) 4e-5)
       (fma -0.3333333333333333 t_0 x)
       (+ (/ 2.0 (* t_0 (* (+ x x) (* x 16.0)))) -1.0)))))

C

double code(double x, double y) {
	double t_0 = x * (x * x);
	double tmp;
	if ((-2.0 * x) <= -100000.0) {
		tmp = (2.0 / (2.0 - ((x + x) / (x + x)))) + -1.0;
	} else if ((-2.0 * x) <= 4e-5) {
		tmp = fma(-0.3333333333333333, t_0, x);
	} else {
		tmp = (2.0 / (t_0 * ((x + x) * (x * 16.0)))) + -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (Float64(-2.0 * x) <= -100000.0)
		tmp = Float64(Float64(2.0 / Float64(2.0 - Float64(Float64(x + x) / Float64(x + x)))) + -1.0);
	elseif (Float64(-2.0 * x) <= 4e-5)
		tmp = fma(-0.3333333333333333, t_0, x);
	else
		tmp = Float64(Float64(2.0 / Float64(t_0 * Float64(Float64(x + x) * Float64(x * 16.0)))) + -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -100000.0], N[(N[(2.0 / N[(2.0 - N[(N[(x + x), $MachinePrecision] / N[(x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 4e-5], N[(-0.3333333333333333 * t$95$0 + x), $MachinePrecision], N[(N[(2.0 / N[(t$95$0 * N[(N[(x + x), $MachinePrecision] * N[(x * 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -1e5

   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. metadata-eval N/A

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

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

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

      3. lower--.f64 N/A

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

      4. count-2 N/A

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

      5. lower-+.f64 1.6

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

   5. Applied rewrites 1.6%

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

   7. Applied rewrites 100.0%

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

   if -1e5 < (*.f64 #s(literal -2 binary64) x) < 4.00000000000000033e-5

   1. Initial program 6.8%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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. 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 4.00000000000000033e-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. metadata-eval N/A

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

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

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

      3. lower--.f64 N/A

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

      4. count-2 N/A

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

      5. lower-+.f64 97.5

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

   5. Applied rewrites 97.5%

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

   7. Applied rewrites 99.3%

      \[\leadsto \frac{2}{\mathsf{fma}\left(8, \color{blue}{x \cdot \left(x \cdot x\right)}, 2\right)} - 1 \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{2}{8 \cdot \color{blue}{{x}^{3}}} - 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 99.3%

       \[\leadsto \frac{2}{x \cdot \color{blue}{\left(8 \cdot \left(x \cdot x\right)\right)}} - 1 \]

   12. Applied rewrites 99.9%

       \[\leadsto \frac{2}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot 16\right) \cdot \color{blue}{\left(x + x\right)}\right)} - 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -100000:\\ \;\;\;\;\frac{2}{2 - \frac{x + x}{x + x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x + x\right) \cdot \left(x \cdot 16\right)\right)} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 99.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -100000:\\ \;\;\;\;\frac{2}{2 - \frac{x + x}{x + x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(16 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), x, 2\right)} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -100000.0)
   (+ (/ 2.0 (- 2.0 (/ (+ x x) (+ x x)))) -1.0)
   (if (<= (* -2.0 x) 4e-5)
     (fma -0.3333333333333333 (* x (* x x)) x)
     (+ (/ 2.0 (fma (* 16.0 (* (* x x) (* x x))) x 2.0)) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -100000.0) {
		tmp = (2.0 / (2.0 - ((x + x) / (x + x)))) + -1.0;
	} else if ((-2.0 * x) <= 4e-5) {
		tmp = fma(-0.3333333333333333, (x * (x * x)), x);
	} else {
		tmp = (2.0 / fma((16.0 * ((x * x) * (x * x))), x, 2.0)) + -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -100000.0)
		tmp = Float64(Float64(2.0 / Float64(2.0 - Float64(Float64(x + x) / Float64(x + x)))) + -1.0);
	elseif (Float64(-2.0 * x) <= 4e-5)
		tmp = fma(-0.3333333333333333, Float64(x * Float64(x * x)), x);
	else
		tmp = Float64(Float64(2.0 / fma(Float64(16.0 * Float64(Float64(x * x) * Float64(x * x))), x, 2.0)) + -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -100000.0], N[(N[(2.0 / N[(2.0 - N[(N[(x + x), $MachinePrecision] / N[(x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 4e-5], N[(-0.3333333333333333 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(2.0 / N[(N[(16.0 * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x + 2.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -1e5

   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. metadata-eval N/A

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

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

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

      3. lower--.f64 N/A

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

      4. count-2 N/A

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

      5. lower-+.f64 1.6

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

   5. Applied rewrites 1.6%

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

   7. Applied rewrites 100.0%

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

   if -1e5 < (*.f64 #s(literal -2 binary64) x) < 4.00000000000000033e-5

   1. Initial program 6.8%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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. 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 4.00000000000000033e-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. metadata-eval N/A

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

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

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

      3. lower--.f64 N/A

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

      4. count-2 N/A

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

      5. lower-+.f64 97.5

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

   5. Applied rewrites 97.5%

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

   7. Applied rewrites 98.6%

      \[\leadsto \frac{2}{\mathsf{fma}\left(x + x, \color{blue}{x}, 2\right)} - 1 \]
   8. Step-by-step derivation

   9. Applied rewrites 99.9%

      \[\leadsto \frac{2}{\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 16, x, 2\right)} - 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -100000:\\ \;\;\;\;\frac{2}{2 - \frac{x + x}{x + x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(16 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), x, 2\right)} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 99.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;-2 \cdot x \leq -100000:\\ \;\;\;\;\frac{2}{2 - \frac{x + x}{x + x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, t\_0, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\left(x + x\right) \cdot t\_0, x, 2\right)} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (if (<= (* -2.0 x) -100000.0)
     (+ (/ 2.0 (- 2.0 (/ (+ x x) (+ x x)))) -1.0)
     (if (<= (* -2.0 x) 4e-5)
       (fma -0.3333333333333333 t_0 x)
       (+ (/ 2.0 (fma (* (+ x x) t_0) x 2.0)) -1.0)))))

C

double code(double x, double y) {
	double t_0 = x * (x * x);
	double tmp;
	if ((-2.0 * x) <= -100000.0) {
		tmp = (2.0 / (2.0 - ((x + x) / (x + x)))) + -1.0;
	} else if ((-2.0 * x) <= 4e-5) {
		tmp = fma(-0.3333333333333333, t_0, x);
	} else {
		tmp = (2.0 / fma(((x + x) * t_0), x, 2.0)) + -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (Float64(-2.0 * x) <= -100000.0)
		tmp = Float64(Float64(2.0 / Float64(2.0 - Float64(Float64(x + x) / Float64(x + x)))) + -1.0);
	elseif (Float64(-2.0 * x) <= 4e-5)
		tmp = fma(-0.3333333333333333, t_0, x);
	else
		tmp = Float64(Float64(2.0 / fma(Float64(Float64(x + x) * t_0), x, 2.0)) + -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -100000.0], N[(N[(2.0 / N[(2.0 - N[(N[(x + x), $MachinePrecision] / N[(x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 4e-5], N[(-0.3333333333333333 * t$95$0 + x), $MachinePrecision], N[(N[(2.0 / N[(N[(N[(x + x), $MachinePrecision] * t$95$0), $MachinePrecision] * x + 2.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -1e5

   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. metadata-eval N/A

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

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

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

      3. lower--.f64 N/A

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

      4. count-2 N/A

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

      5. lower-+.f64 1.6

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

   5. Applied rewrites 1.6%

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

   7. Applied rewrites 100.0%

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

   if -1e5 < (*.f64 #s(literal -2 binary64) x) < 4.00000000000000033e-5

   1. Initial program 6.8%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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. 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 4.00000000000000033e-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. metadata-eval N/A

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

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

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

      3. lower--.f64 N/A

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

      4. count-2 N/A

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

      5. lower-+.f64 97.5

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

   5. Applied rewrites 97.5%

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

   7. Applied rewrites 98.6%

      \[\leadsto \frac{2}{\mathsf{fma}\left(x + x, \color{blue}{x}, 2\right)} - 1 \]
   8. Step-by-step derivation

   9. Applied rewrites 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -100000:\\ \;\;\;\;\frac{2}{2 - \frac{x + x}{x + x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\left(x + x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right), x, 2\right)} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 99.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -100000:\\ \;\;\;\;\frac{2}{2 - \frac{x + x}{x + x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(x + x\right) \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 64\right)\right)} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -100000.0)
   (+ (/ 2.0 (- 2.0 (/ (+ x x) (+ x x)))) -1.0)
   (if (<= (* -2.0 x) 4e-5)
     (fma -0.3333333333333333 (* x (* x x)) x)
     (+ (/ 2.0 (* (+ x x) (* x (* (* x x) 64.0)))) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -100000.0) {
		tmp = (2.0 / (2.0 - ((x + x) / (x + x)))) + -1.0;
	} else if ((-2.0 * x) <= 4e-5) {
		tmp = fma(-0.3333333333333333, (x * (x * x)), x);
	} else {
		tmp = (2.0 / ((x + x) * (x * ((x * x) * 64.0)))) + -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -100000.0)
		tmp = Float64(Float64(2.0 / Float64(2.0 - Float64(Float64(x + x) / Float64(x + x)))) + -1.0);
	elseif (Float64(-2.0 * x) <= 4e-5)
		tmp = fma(-0.3333333333333333, Float64(x * Float64(x * x)), x);
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(x + x) * Float64(x * Float64(Float64(x * x) * 64.0)))) + -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -100000.0], N[(N[(2.0 / N[(2.0 - N[(N[(x + x), $MachinePrecision] / N[(x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 4e-5], N[(-0.3333333333333333 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(2.0 / N[(N[(x + x), $MachinePrecision] * N[(x * N[(N[(x * x), $MachinePrecision] * 64.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -1e5

   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. metadata-eval N/A

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

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

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

      3. lower--.f64 N/A

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

      4. count-2 N/A

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

      5. lower-+.f64 1.6

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

   5. Applied rewrites 1.6%

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

   7. Applied rewrites 100.0%

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

   if -1e5 < (*.f64 #s(literal -2 binary64) x) < 4.00000000000000033e-5

   1. Initial program 6.8%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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. 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 4.00000000000000033e-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. metadata-eval N/A

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

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

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

      3. lower--.f64 N/A

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

      4. count-2 N/A

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

      5. lower-+.f64 97.5

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

   5. Applied rewrites 97.5%

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

   7. Applied rewrites 99.3%

      \[\leadsto \frac{2}{\mathsf{fma}\left(8, \color{blue}{x \cdot \left(x \cdot x\right)}, 2\right)} - 1 \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{2}{8 \cdot \color{blue}{{x}^{3}}} - 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 99.3%

       \[\leadsto \frac{2}{x \cdot \color{blue}{\left(8 \cdot \left(x \cdot x\right)\right)}} - 1 \]

   12. Applied rewrites 99.7%

       \[\leadsto \frac{2}{\left(x + x\right) \cdot \left(x \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot 64\right)}\right)} - 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -100000:\\ \;\;\;\;\frac{2}{2 - \frac{x + x}{x + x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(x + x\right) \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 64\right)\right)} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 99.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;-2 \cdot x \leq -100000:\\ \;\;\;\;\frac{2}{2 - \frac{x + x}{x + x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, t\_0, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{64 \cdot \left(x \cdot t\_0\right)} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (if (<= (* -2.0 x) -100000.0)
     (+ (/ 2.0 (- 2.0 (/ (+ x x) (+ x x)))) -1.0)
     (if (<= (* -2.0 x) 4e-5)
       (fma -0.3333333333333333 t_0 x)
       (+ (/ 2.0 (* 64.0 (* x t_0))) -1.0)))))

C

double code(double x, double y) {
	double t_0 = x * (x * x);
	double tmp;
	if ((-2.0 * x) <= -100000.0) {
		tmp = (2.0 / (2.0 - ((x + x) / (x + x)))) + -1.0;
	} else if ((-2.0 * x) <= 4e-5) {
		tmp = fma(-0.3333333333333333, t_0, x);
	} else {
		tmp = (2.0 / (64.0 * (x * t_0))) + -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (Float64(-2.0 * x) <= -100000.0)
		tmp = Float64(Float64(2.0 / Float64(2.0 - Float64(Float64(x + x) / Float64(x + x)))) + -1.0);
	elseif (Float64(-2.0 * x) <= 4e-5)
		tmp = fma(-0.3333333333333333, t_0, x);
	else
		tmp = Float64(Float64(2.0 / Float64(64.0 * Float64(x * t_0))) + -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -100000.0], N[(N[(2.0 / N[(2.0 - N[(N[(x + x), $MachinePrecision] / N[(x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 4e-5], N[(-0.3333333333333333 * t$95$0 + x), $MachinePrecision], N[(N[(2.0 / N[(64.0 * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -1e5

   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. metadata-eval N/A

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

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

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

      3. lower--.f64 N/A

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

      4. count-2 N/A

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

      5. lower-+.f64 1.6

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

   5. Applied rewrites 1.6%

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

   7. Applied rewrites 100.0%

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

   if -1e5 < (*.f64 #s(literal -2 binary64) x) < 4.00000000000000033e-5

   1. Initial program 6.8%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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. 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 4.00000000000000033e-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. metadata-eval N/A

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

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

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

      3. lower--.f64 N/A

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

      4. count-2 N/A

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

      5. lower-+.f64 97.5

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

   5. Applied rewrites 97.5%

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

   7. Applied rewrites 99.3%

      \[\leadsto \frac{2}{\mathsf{fma}\left(8, \color{blue}{x \cdot \left(x \cdot x\right)}, 2\right)} - 1 \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{2}{8 \cdot \color{blue}{{x}^{3}}} - 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 99.3%

       \[\leadsto \frac{2}{x \cdot \color{blue}{\left(8 \cdot \left(x \cdot x\right)\right)}} - 1 \]

   12. Applied rewrites 99.7%

       \[\leadsto \frac{2}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot 64} - 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -100000:\\ \;\;\;\;\frac{2}{2 - \frac{x + x}{x + x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{64 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 50.0% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.3333333333333333, x \cdot \left(x \cdot x\right), x\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (fma -0.3333333333333333 (* x (* x x)) x))

C

double code(double x, double y) {
	return fma(-0.3333333333333333, (x * (x * x)), x);
}

Julia

function code(x, y)
	return fma(-0.3333333333333333, Float64(x * Float64(x * x)), x)
end

Wolfram

code[x_, y_] := N[(-0.3333333333333333 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 55.2%

   \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
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 48.5

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

5. Applied rewrites 48.5%

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

Alternative 19: 6.5% accurate, 17.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.2%

   \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
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 6.0

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

5. Applied rewrites 6.0%

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

6. Final simplification 6.0%

   \[\leadsto \left(x + 1\right) + -1 \]
7. Add Preprocessing

Alternative 20: 4.3% 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 55.2%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 4.3%

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

6. Final simplification 4.3%

   \[\leadsto 1 + -1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024237 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))