Logistic function from Lakshay Garg

Percentage Accurate: 54.1% → 99.5%

Time: 8.4s

Alternatives: 10

Speedup: 5.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + {\left(e^{x}\right)}^{-2}\\ \mathbf{if}\;-2 \cdot x \leq -0.1:\\ \;\;\;\;\frac{\mathsf{fma}\left({t\_0}^{-6}, 64, -1\right)}{\mathsf{fma}\left({t\_0}^{-4}, 16, \mathsf{fma}\left(4, {t\_0}^{-2}, 1\right)\right) \cdot \left(\frac{2}{t\_0} - -1\right)}\\ \mathbf{elif}\;-2 \cdot x \leq 4 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, -0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (pow (exp x) -2.0))))
   (if (<= (* -2.0 x) -0.1)
     (/
      (fma (pow t_0 -6.0) 64.0 -1.0)
      (*
       (fma (pow t_0 -4.0) 16.0 (fma 4.0 (pow t_0 -2.0) 1.0))
       (- (/ 2.0 t_0) -1.0)))
     (if (<= (* -2.0 x) 4e-17)
       (fma (* (* x x) x) -0.3333333333333333 x)
       (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 + pow(exp(x), -2.0);
	double tmp;
	if ((-2.0 * x) <= -0.1) {
		tmp = fma(pow(t_0, -6.0), 64.0, -1.0) / (fma(pow(t_0, -4.0), 16.0, fma(4.0, pow(t_0, -2.0), 1.0)) * ((2.0 / t_0) - -1.0));
	} else if ((-2.0 * x) <= 4e-17) {
		tmp = fma(((x * x) * x), -0.3333333333333333, x);
	} else {
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) - 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 + (exp(x) ^ -2.0))
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.1)
		tmp = Float64(fma((t_0 ^ -6.0), 64.0, -1.0) / Float64(fma((t_0 ^ -4.0), 16.0, fma(4.0, (t_0 ^ -2.0), 1.0)) * Float64(Float64(2.0 / t_0) - -1.0)));
	elseif (Float64(-2.0 * x) <= 4e-17)
		tmp = fma(Float64(Float64(x * x) * x), -0.3333333333333333, x);
	else
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) - 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[Power[N[Exp[x], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.1], N[(N[(N[Power[t$95$0, -6.0], $MachinePrecision] * 64.0 + -1.0), $MachinePrecision] / N[(N[(N[Power[t$95$0, -4.0], $MachinePrecision] * 16.0 + N[(4.0 * N[Power[t$95$0, -2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / t$95$0), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 4e-17], N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * -0.3333333333333333 + x), $MachinePrecision], N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -0.10000000000000001

   1. Initial program 99.9%

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(64, {\left({\left(e^{x}\right)}^{-2} + 1\right)}^{-6}, -1\right)}{\left(\frac{2}{{\left(e^{x}\right)}^{-2} + 1} - -1\right) \cdot \left(\mathsf{fma}\left({\left({\left(e^{x}\right)}^{-2} + 1\right)}^{-2}, 4, 1\right) + {\left(0.5 \cdot \left({\left(e^{x}\right)}^{-2} + 1\right)\right)}^{-4}\right)}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{64 \cdot {\left({\left(e^{x}\right)}^{-2} + 1\right)}^{-6} + -1}}{\left(\frac{2}{{\left(e^{x}\right)}^{-2} + 1} - -1\right) \cdot \left(\mathsf{fma}\left({\left({\left(e^{x}\right)}^{-2} + 1\right)}^{-2}, 4, 1\right) + {\left(\frac{1}{2} \cdot \left({\left(e^{x}\right)}^{-2} + 1\right)\right)}^{-4}\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{\left({\left(e^{x}\right)}^{-2} + 1\right)}^{-6} \cdot 64} + -1}{\left(\frac{2}{{\left(e^{x}\right)}^{-2} + 1} - -1\right) \cdot \left(\mathsf{fma}\left({\left({\left(e^{x}\right)}^{-2} + 1\right)}^{-2}, 4, 1\right) + {\left(\frac{1}{2} \cdot \left({\left(e^{x}\right)}^{-2} + 1\right)\right)}^{-4}\right)} \]

      3. lower-fma.f64 100.0

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\left({\left(e^{x}\right)}^{-2} + 1\right)}^{-6}, 64, -1\right)}}{\left(\frac{2}{{\left(e^{x}\right)}^{-2} + 1} - -1\right) \cdot \left(\mathsf{fma}\left({\left({\left(e^{x}\right)}^{-2} + 1\right)}^{-2}, 4, 1\right) + {\left(0.5 \cdot \left({\left(e^{x}\right)}^{-2} + 1\right)\right)}^{-4}\right)} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\left({\left(e^{x}\right)}^{-2} + 1\right)}}^{-6}, 64, -1\right)}{\left(\frac{2}{{\left(e^{x}\right)}^{-2} + 1} - -1\right) \cdot \left(\mathsf{fma}\left({\left({\left(e^{x}\right)}^{-2} + 1\right)}^{-2}, 4, 1\right) + {\left(\frac{1}{2} \cdot \left({\left(e^{x}\right)}^{-2} + 1\right)\right)}^{-4}\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\left(1 + {\left(e^{x}\right)}^{-2}\right)}}^{-6}, 64, -1\right)}{\left(\frac{2}{{\left(e^{x}\right)}^{-2} + 1} - -1\right) \cdot \left(\mathsf{fma}\left({\left({\left(e^{x}\right)}^{-2} + 1\right)}^{-2}, 4, 1\right) + {\left(\frac{1}{2} \cdot \left({\left(e^{x}\right)}^{-2} + 1\right)\right)}^{-4}\right)} \]

      6. lower-+.f64 100.0

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\left(1 + {\left(e^{x}\right)}^{-2}\right)}}^{-6}, 64, -1\right)}{\left(\frac{2}{{\left(e^{x}\right)}^{-2} + 1} - -1\right) \cdot \left(\mathsf{fma}\left({\left({\left(e^{x}\right)}^{-2} + 1\right)}^{-2}, 4, 1\right) + {\left(0.5 \cdot \left({\left(e^{x}\right)}^{-2} + 1\right)\right)}^{-4}\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\left(1 + {\left(e^{x}\right)}^{-2}\right)}^{-6}, 64, -1\right)}{\color{blue}{\left(\frac{2}{{\left(e^{x}\right)}^{-2} + 1} - -1\right) \cdot \left(\mathsf{fma}\left({\left({\left(e^{x}\right)}^{-2} + 1\right)}^{-2}, 4, 1\right) + {\left(\frac{1}{2} \cdot \left({\left(e^{x}\right)}^{-2} + 1\right)\right)}^{-4}\right)}} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\left(1 + {\left(e^{x}\right)}^{-2}\right)}^{-6}, 64, -1\right)}{\color{blue}{\left(\mathsf{fma}\left({\left({\left(e^{x}\right)}^{-2} + 1\right)}^{-2}, 4, 1\right) + {\left(\frac{1}{2} \cdot \left({\left(e^{x}\right)}^{-2} + 1\right)\right)}^{-4}\right) \cdot \left(\frac{2}{{\left(e^{x}\right)}^{-2} + 1} - -1\right)}} \]

      9. lower-*.f64 100.0

         \[\leadsto \frac{\mathsf{fma}\left({\left(1 + {\left(e^{x}\right)}^{-2}\right)}^{-6}, 64, -1\right)}{\color{blue}{\left(\mathsf{fma}\left({\left({\left(e^{x}\right)}^{-2} + 1\right)}^{-2}, 4, 1\right) + {\left(0.5 \cdot \left({\left(e^{x}\right)}^{-2} + 1\right)\right)}^{-4}\right) \cdot \left(\frac{2}{{\left(e^{x}\right)}^{-2} + 1} - -1\right)}} \]

   6. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\left(1 + {\left(e^{x}\right)}^{-2}\right)}^{-6}, 64, -1\right)}{\mathsf{fma}\left({\left(1 + {\left(e^{x}\right)}^{-2}\right)}^{-4}, 16, \mathsf{fma}\left(4, {\left(1 + {\left(e^{x}\right)}^{-2}\right)}^{-2}, 1\right)\right) \cdot \left(\frac{2}{1 + {\left(e^{x}\right)}^{-2}} - -1\right)}} \]

   if -0.10000000000000001 < (*.f64 #s(literal -2 binary64) x) < 4.00000000000000029e-17

   1. Initial program 6.4%

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

   4. Applied rewrites 6.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(64, {\left({\left(e^{x}\right)}^{-2} + 1\right)}^{-6}, -1\right)}{\left(\frac{2}{{\left(e^{x}\right)}^{-2} + 1} - -1\right) \cdot \left(\mathsf{fma}\left({\left({\left(e^{x}\right)}^{-2} + 1\right)}^{-2}, 4, 1\right) + {\left(0.5 \cdot \left({\left(e^{x}\right)}^{-2} + 1\right)\right)}^{-4}\right)}} \]

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. unpow3 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-pow.f64 100.0

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

   7. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, -0.3333333333333333, x\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 100.0%

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

   if 4.00000000000000029e-17 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 100.0%

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

Alternative 2: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.1:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{\mathsf{expm1}\left(-4 \cdot x\right)}, \mathsf{expm1}\left(x \cdot -2\right), -1\right)\\ \mathbf{elif}\;-2 \cdot x \leq 4 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, -0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -0.1)
   (fma (/ 2.0 (expm1 (* -4.0 x))) (expm1 (* x -2.0)) -1.0)
   (if (<= (* -2.0 x) 4e-17)
     (fma (* (* x x) x) -0.3333333333333333 x)
     (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.1) {
		tmp = fma((2.0 / expm1((-4.0 * x))), expm1((x * -2.0)), -1.0);
	} else if ((-2.0 * x) <= 4e-17) {
		tmp = fma(((x * x) * x), -0.3333333333333333, x);
	} else {
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) - 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.1)
		tmp = fma(Float64(2.0 / expm1(Float64(-4.0 * x))), expm1(Float64(x * -2.0)), -1.0);
	elseif (Float64(-2.0 * x) <= 4e-17)
		tmp = fma(Float64(Float64(x * x) * x), -0.3333333333333333, x);
	else
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) - 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.1], N[(N[(2.0 / N[(Exp[N[(-4.0 * x), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] * N[(Exp[N[(x * -2.0), $MachinePrecision]] - 1), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 4e-17], N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * -0.3333333333333333 + x), $MachinePrecision], N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -0.10000000000000001

   1. Initial program 99.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-+ N/A

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

      7. associate-/r/ N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   if -0.10000000000000001 < (*.f64 #s(literal -2 binary64) x) < 4.00000000000000029e-17

   1. Initial program 6.4%

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

   4. Applied rewrites 6.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(64, {\left({\left(e^{x}\right)}^{-2} + 1\right)}^{-6}, -1\right)}{\left(\frac{2}{{\left(e^{x}\right)}^{-2} + 1} - -1\right) \cdot \left(\mathsf{fma}\left({\left({\left(e^{x}\right)}^{-2} + 1\right)}^{-2}, 4, 1\right) + {\left(0.5 \cdot \left({\left(e^{x}\right)}^{-2} + 1\right)\right)}^{-4}\right)}} \]

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. unpow3 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-pow.f64 100.0

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

   7. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, -0.3333333333333333, x\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 100.0%

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

   if 4.00000000000000029e-17 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 100.0%

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

Alternative 3: 99.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.1 \lor \neg \left(-2 \cdot x \leq 4 \cdot 10^{-17}\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, -0.3333333333333333, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* -2.0 x) -0.1) (not (<= (* -2.0 x) 4e-17)))
   (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0)
   (fma (* (* x x) x) -0.3333333333333333 x)))

C

double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -0.1) || !((-2.0 * x) <= 4e-17)) {
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) - 1.0;
	} else {
		tmp = fma(((x * x) * x), -0.3333333333333333, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((Float64(-2.0 * x) <= -0.1) || !(Float64(-2.0 * x) <= 4e-17))
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) - 1.0);
	else
		tmp = fma(Float64(Float64(x * x) * x), -0.3333333333333333, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.1], N[Not[LessEqual[N[(-2.0 * x), $MachinePrecision], 4e-17]], $MachinePrecision]], N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * -0.3333333333333333 + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -0.10000000000000001 or 4.00000000000000029e-17 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 99.9%

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

   if -0.10000000000000001 < (*.f64 #s(literal -2 binary64) x) < 4.00000000000000029e-17

   1. Initial program 6.4%

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

   4. Applied rewrites 6.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(64, {\left({\left(e^{x}\right)}^{-2} + 1\right)}^{-6}, -1\right)}{\left(\frac{2}{{\left(e^{x}\right)}^{-2} + 1} - -1\right) \cdot \left(\mathsf{fma}\left({\left({\left(e^{x}\right)}^{-2} + 1\right)}^{-2}, 4, 1\right) + {\left(0.5 \cdot \left({\left(e^{x}\right)}^{-2} + 1\right)\right)}^{-4}\right)}} \]

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. unpow3 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-pow.f64 100.0

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

   7. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, -0.3333333333333333, x\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.1 \lor \neg \left(-2 \cdot x \leq 4 \cdot 10^{-17}\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, -0.3333333333333333, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq 4 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.13333333333333333, x \cdot x, -0.3333333333333333\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(x, x, 1\right) \cdot \left(1 - x\right)\right)}^{-1} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) 4e-17)
   (fma (pow x 3.0) (fma 0.13333333333333333 (* x x) -0.3333333333333333) x)
   (- (pow (* (fma x x 1.0) (- 1.0 x)) -1.0) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= 4e-17) {
		tmp = fma(pow(x, 3.0), fma(0.13333333333333333, (x * x), -0.3333333333333333), x);
	} else {
		tmp = pow((fma(x, x, 1.0) * (1.0 - x)), -1.0) - 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= 4e-17)
		tmp = fma((x ^ 3.0), fma(0.13333333333333333, Float64(x * x), -0.3333333333333333), x);
	else
		tmp = Float64((Float64(fma(x, x, 1.0) * Float64(1.0 - x)) ^ -1.0) - 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], 4e-17], N[(N[Power[x, 3.0], $MachinePrecision] * N[(0.13333333333333333 * N[(x * x), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] + x), $MachinePrecision], N[(N[Power[N[(N[(x * x + 1.0), $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal -2 binary64) x) < 4.00000000000000029e-17

   1. Initial program 37.4%

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

   4. Applied rewrites 37.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(64, {\left({\left(e^{x}\right)}^{-2} + 1\right)}^{-6}, -1\right)}{\left(\frac{2}{{\left(e^{x}\right)}^{-2} + 1} - -1\right) \cdot \left(\mathsf{fma}\left({\left({\left(e^{x}\right)}^{-2} + 1\right)}^{-2}, 4, 1\right) + {\left(0.5 \cdot \left({\left(e^{x}\right)}^{-2} + 1\right)\right)}^{-4}\right)}} \]

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. cube-mult N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 68.6

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

   7. Applied rewrites 68.6%

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

   if 4.00000000000000029e-17 < (*.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

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

      1. lower-+.f64 5.3

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

   5. Applied rewrites 5.3%

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

   7. Applied rewrites 4.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 99.0%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq 4 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.13333333333333333, x \cdot x, -0.3333333333333333\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(x, x, 1\right) \cdot \left(1 - x\right)\right)}^{-1} - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.52:\\ \;\;\;\;{\left(\mathsf{fma}\left(x, x, 1\right) \cdot \left(1 - x\right)\right)}^{-1} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, -0.3333333333333333, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.52)
   (- (pow (* (fma x x 1.0) (- 1.0 x)) -1.0) 1.0)
   (fma (* (* x x) x) -0.3333333333333333 x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.52) {
		tmp = pow((fma(x, x, 1.0) * (1.0 - x)), -1.0) - 1.0;
	} else {
		tmp = fma(((x * x) * x), -0.3333333333333333, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.52)
		tmp = Float64((Float64(fma(x, x, 1.0) * Float64(1.0 - x)) ^ -1.0) - 1.0);
	else
		tmp = fma(Float64(Float64(x * x) * x), -0.3333333333333333, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.52], N[(N[Power[N[(N[(x * x + 1.0), $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * -0.3333333333333333 + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.52000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-+.f64 5.3

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

   5. Applied rewrites 5.3%

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

   7. Applied rewrites 4.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 99.0%

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

   if -0.52000000000000002 < x

   1. Initial program 37.4%

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

   4. Applied rewrites 37.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(64, {\left({\left(e^{x}\right)}^{-2} + 1\right)}^{-6}, -1\right)}{\left(\frac{2}{{\left(e^{x}\right)}^{-2} + 1} - -1\right) \cdot \left(\mathsf{fma}\left({\left({\left(e^{x}\right)}^{-2} + 1\right)}^{-2}, 4, 1\right) + {\left(0.5 \cdot \left({\left(e^{x}\right)}^{-2} + 1\right)\right)}^{-4}\right)}} \]

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. unpow3 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-pow.f64 67.5

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

   7. Applied rewrites 67.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, -0.3333333333333333, x\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 67.5%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.52:\\ \;\;\;\;{\left(\mathsf{fma}\left(x, x, 1\right) \cdot \left(1 - x\right)\right)}^{-1} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, -0.3333333333333333, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-+.f64 5.3

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

   5. Applied rewrites 5.3%

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

   7. Applied rewrites 4.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 98.9%

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

   if -1 < x

   1. Initial program 37.4%

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

   4. Applied rewrites 37.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(64, {\left({\left(e^{x}\right)}^{-2} + 1\right)}^{-6}, -1\right)}{\left(\frac{2}{{\left(e^{x}\right)}^{-2} + 1} - -1\right) \cdot \left(\mathsf{fma}\left({\left({\left(e^{x}\right)}^{-2} + 1\right)}^{-2}, 4, 1\right) + {\left(0.5 \cdot \left({\left(e^{x}\right)}^{-2} + 1\right)\right)}^{-4}\right)}} \]

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. unpow3 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-pow.f64 67.5

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

   7. Applied rewrites 67.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, -0.3333333333333333, x\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 67.5%

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

4. Final simplification 76.0%

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

Alternative 7: 73.5% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;\frac{-1}{x - 1} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, -0.3333333333333333, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.3)
   (- (/ -1.0 (- x 1.0)) 1.0)
   (fma (* (* x x) x) -0.3333333333333333 x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.3) {
		tmp = (-1.0 / (x - 1.0)) - 1.0;
	} else {
		tmp = fma(((x * x) * x), -0.3333333333333333, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.3)
		tmp = Float64(Float64(-1.0 / Float64(x - 1.0)) - 1.0);
	else
		tmp = fma(Float64(Float64(x * x) * x), -0.3333333333333333, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.3], N[(N[(-1.0 / N[(x - 1.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * -0.3333333333333333 + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.30000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-+.f64 5.3

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

   5. Applied rewrites 5.3%

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

   7. Applied rewrites 4.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 98.4%

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

   if -1.30000000000000004 < x

   1. Initial program 37.4%

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

   4. Applied rewrites 37.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(64, {\left({\left(e^{x}\right)}^{-2} + 1\right)}^{-6}, -1\right)}{\left(\frac{2}{{\left(e^{x}\right)}^{-2} + 1} - -1\right) \cdot \left(\mathsf{fma}\left({\left({\left(e^{x}\right)}^{-2} + 1\right)}^{-2}, 4, 1\right) + {\left(0.5 \cdot \left({\left(e^{x}\right)}^{-2} + 1\right)\right)}^{-4}\right)}} \]

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. unpow3 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-pow.f64 67.5

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

   7. Applied rewrites 67.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, -0.3333333333333333, x\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 67.5%

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

Alternative 8: 50.2% accurate, 7.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.3%

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

4. Applied rewrites 54.3%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(64, {\left({\left(e^{x}\right)}^{-2} + 1\right)}^{-6}, -1\right)}{\left(\frac{2}{{\left(e^{x}\right)}^{-2} + 1} - -1\right) \cdot \left(\mathsf{fma}\left({\left({\left(e^{x}\right)}^{-2} + 1\right)}^{-2}, 4, 1\right) + {\left(0.5 \cdot \left({\left(e^{x}\right)}^{-2} + 1\right)\right)}^{-4}\right)}} \]

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. distribute-lft1-in N/A

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

   4. associate-*r* N/A

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

   5. unpow2 N/A

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

   6. unpow3 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-pow.f64 49.5

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

7. Applied rewrites 49.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, -0.3333333333333333, x\right)} \]
8. Step-by-step derivation

9. Applied rewrites 49.5%

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

Alternative 9: 6.5% accurate, 17.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.3%

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

3. Taylor expanded in x around 0

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

   1. lower-+.f64 6.1

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

5. Applied rewrites 6.1%

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

Alternative 10: 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 54.3%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 4.4%

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

Reproduce

herbie shell --seed 2024309 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))