Logistic function from Lakshay Garg

Percentage Accurate: 54.5% → 99.7%

Time: 10.5s

Alternatives: 5

Speedup: 10.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.5% 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: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + e^{\mathsf{fma}\left(-2, x, 0\right)}\\ \mathbf{if}\;-2 \cdot x \leq -0.2:\\ \;\;\;\;\frac{\mathsf{fma}\left(4, {t\_0}^{-2}, -1\right)}{1 + \frac{2}{t\_0}}\\ \mathbf{elif}\;-2 \cdot x \leq 0.01:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.05396825396825397, 0.13333333333333333\right), -0.3333333333333333\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (exp (fma -2.0 x 0.0)))))
   (if (<= (* -2.0 x) -0.2)
     (/ (fma 4.0 (pow t_0 -2.0) -1.0) (+ 1.0 (/ 2.0 t_0)))
     (if (<= (* -2.0 x) 0.01)
       (*
        x
        (fma
         (* x x)
         (fma
          (* x x)
          (fma x (* x -0.05396825396825397) 0.13333333333333333)
          -0.3333333333333333)
         1.0))
       -1.0))))

C

double code(double x, double y) {
	double t_0 = 1.0 + exp(fma(-2.0, x, 0.0));
	double tmp;
	if ((-2.0 * x) <= -0.2) {
		tmp = fma(4.0, pow(t_0, -2.0), -1.0) / (1.0 + (2.0 / t_0));
	} else if ((-2.0 * x) <= 0.01) {
		tmp = x * fma((x * x), fma((x * x), fma(x, (x * -0.05396825396825397), 0.13333333333333333), -0.3333333333333333), 1.0);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 + exp(fma(-2.0, x, 0.0)))
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.2)
		tmp = Float64(fma(4.0, (t_0 ^ -2.0), -1.0) / Float64(1.0 + Float64(2.0 / t_0)));
	elseif (Float64(-2.0 * x) <= 0.01)
		tmp = Float64(x * fma(Float64(x * x), fma(Float64(x * x), fma(x, Float64(x * -0.05396825396825397), 0.13333333333333333), -0.3333333333333333), 1.0));
	else
		tmp = -1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[Exp[N[(-2.0 * x + 0.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.2], N[(N[(4.0 * N[Power[t$95$0, -2.0], $MachinePrecision] + -1.0), $MachinePrecision] / N[(1.0 + N[(2.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.01], N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * -0.05396825396825397), $MachinePrecision] + 0.13333333333333333), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], -1.0]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -0.20000000000000001

   1. Initial program 100.0%

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

      1. flip-- N/A

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

      2. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(4, {\left(1 + e^{\mathsf{fma}\left(-2, x, 0\right)}\right)}^{-2}, -1\right)}{1 + \frac{2}{1 + e^{\mathsf{fma}\left(-2, x, 0\right)}}}} \]

   if -0.20000000000000001 < (*.f64 #s(literal -2 binary64) x) < 0.0100000000000000002

   1. Initial program 8.1%

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

      1. flip-- N/A

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

      2. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 8.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(4, {\left(1 + e^{\mathsf{fma}\left(-2, x, 0\right)}\right)}^{-2}, -1\right)}{1 + \frac{2}{1 + e^{\mathsf{fma}\left(-2, x, 0\right)}}}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)\right)} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right) + 1\right)} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, {x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}, 1\right)} \]

      4. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}, 1\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}, 1\right) \]

      6. sub-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, 1\right) \]

      7. metadata-eval N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, {x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) + \color{blue}{\frac{-1}{3}}, 1\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}, \frac{-1}{3}\right)}, 1\right) \]

      9. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}, \frac{-1}{3}\right), 1\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}, \frac{-1}{3}\right), 1\right) \]

      11. +-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{-17}{315} \cdot {x}^{2} + \frac{2}{15}}, \frac{-1}{3}\right), 1\right) \]

      12. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-17}{315} \cdot \color{blue}{\left(x \cdot x\right)} + \frac{2}{15}, \frac{-1}{3}\right), 1\right) \]

      13. associate-*r* N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\left(\frac{-17}{315} \cdot x\right) \cdot x} + \frac{2}{15}, \frac{-1}{3}\right), 1\right) \]

      14. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(\frac{-17}{315} \cdot x\right)} + \frac{2}{15}, \frac{-1}{3}\right), 1\right) \]

      15. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, \frac{-17}{315} \cdot x, \frac{2}{15}\right)}, \frac{-1}{3}\right), 1\right) \]

      16. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-17}{315}}, \frac{2}{15}\right), \frac{-1}{3}\right), 1\right) \]

      17. *-lowering-*.f64 100.0

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot -0.05396825396825397}, 0.13333333333333333\right), -0.3333333333333333\right), 1\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.05396825396825397, 0.13333333333333333\right), -0.3333333333333333\right), 1\right)} \]

   if 0.0100000000000000002 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 99.7

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

   5. Simplified 99.7%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 100.0%

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

Alternative 2: 75.5% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (exp (* -2.0 x)) 2.0)
   (*
    x
    (fma (* x x) (fma (* x x) 0.13333333333333333 -0.3333333333333333) 1.0))
   -1.0))

C

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

Julia

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

Wolfram

code[x_, y_] := If[LessEqual[N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision], 2.0], N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.13333333333333333 + -0.3333333333333333), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 36.9%

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

   3. Taylor expanded in x around 0

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right) + 0} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1 + {x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right), 0\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{{x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + 1}, 0\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, 1\right)}, 0\right) \]

      5. +-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(\color{blue}{{x}^{2} + 0}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, 1\right), 0\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(\color{blue}{x \cdot x} + 0, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, 1\right), 0\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, x, 0\right)}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, 1\right), 0\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\frac{2}{15} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, 1\right), 0\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{{x}^{2} \cdot \frac{2}{15}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), 1\right), 0\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), {x}^{2} \cdot \frac{2}{15} + \color{blue}{\frac{-1}{3}}, 1\right), 0\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{2}{15}, \frac{-1}{3}\right)}, 1\right), 0\right) \]

      12. +-rgt-identity N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(\color{blue}{{x}^{2} + 0}, \frac{2}{15}, \frac{-1}{3}\right), 1\right), 0\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(\color{blue}{x \cdot x} + 0, \frac{2}{15}, \frac{-1}{3}\right), 1\right), 0\right) \]

      14. accelerator-lowering-fma.f64 70.0

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, x, 0\right)}, 0.13333333333333333, -0.3333333333333333\right), 1\right), 0\right) \]

   5. Simplified 70.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 0.13333333333333333, -0.3333333333333333\right), 1\right), 0\right)} \]

   6. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 70.0

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.13333333333333333, -0.3333333333333333\right), 1\right) \]

   8. Simplified 70.0%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right), 1\right)} \]

   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. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 99.7

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

   5. Simplified 99.7%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 100.0%

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

Alternative 3: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.2:\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{elif}\;-2 \cdot x \leq 0.01:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.05396825396825397, 0.13333333333333333\right), -0.3333333333333333\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -0.2)
   (+ -1.0 (/ 2.0 (+ 1.0 (exp (* -2.0 x)))))
   (if (<= (* -2.0 x) 0.01)
     (*
      x
      (fma
       (* x x)
       (fma
        (* x x)
        (fma x (* x -0.05396825396825397) 0.13333333333333333)
        -0.3333333333333333)
       1.0))
     -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.2) {
		tmp = -1.0 + (2.0 / (1.0 + exp((-2.0 * x))));
	} else if ((-2.0 * x) <= 0.01) {
		tmp = x * fma((x * x), fma((x * x), fma(x, (x * -0.05396825396825397), 0.13333333333333333), -0.3333333333333333), 1.0);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.2)
		tmp = Float64(-1.0 + Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))));
	elseif (Float64(-2.0 * x) <= 0.01)
		tmp = Float64(x * fma(Float64(x * x), fma(Float64(x * x), fma(x, Float64(x * -0.05396825396825397), 0.13333333333333333), -0.3333333333333333), 1.0));
	else
		tmp = -1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.2], N[(-1.0 + N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.01], N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * -0.05396825396825397), $MachinePrecision] + 0.13333333333333333), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -0.20000000000000001

   1. Initial program 100.0%

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

   if -0.20000000000000001 < (*.f64 #s(literal -2 binary64) x) < 0.0100000000000000002

   1. Initial program 8.1%

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

      1. flip-- N/A

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

      2. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 8.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(4, {\left(1 + e^{\mathsf{fma}\left(-2, x, 0\right)}\right)}^{-2}, -1\right)}{1 + \frac{2}{1 + e^{\mathsf{fma}\left(-2, x, 0\right)}}}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)\right)} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right) + 1\right)} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, {x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}, 1\right)} \]

      4. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}, 1\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}, 1\right) \]

      6. sub-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, 1\right) \]

      7. metadata-eval N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, {x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) + \color{blue}{\frac{-1}{3}}, 1\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}, \frac{-1}{3}\right)}, 1\right) \]

      9. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}, \frac{-1}{3}\right), 1\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}, \frac{-1}{3}\right), 1\right) \]

      11. +-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{-17}{315} \cdot {x}^{2} + \frac{2}{15}}, \frac{-1}{3}\right), 1\right) \]

      12. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-17}{315} \cdot \color{blue}{\left(x \cdot x\right)} + \frac{2}{15}, \frac{-1}{3}\right), 1\right) \]

      13. associate-*r* N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\left(\frac{-17}{315} \cdot x\right) \cdot x} + \frac{2}{15}, \frac{-1}{3}\right), 1\right) \]

      14. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(\frac{-17}{315} \cdot x\right)} + \frac{2}{15}, \frac{-1}{3}\right), 1\right) \]

      15. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, \frac{-17}{315} \cdot x, \frac{2}{15}\right)}, \frac{-1}{3}\right), 1\right) \]

      16. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-17}{315}}, \frac{2}{15}\right), \frac{-1}{3}\right), 1\right) \]

      17. *-lowering-*.f64 100.0

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot -0.05396825396825397}, 0.13333333333333333\right), -0.3333333333333333\right), 1\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.05396825396825397, 0.13333333333333333\right), -0.3333333333333333\right), 1\right)} \]

   if 0.0100000000000000002 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 99.7

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

   5. Simplified 99.7%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.2:\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{elif}\;-2 \cdot x \leq 0.01:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.05396825396825397, 0.13333333333333333\right), -0.3333333333333333\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.6% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq 0.01:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* -2.0 x) 0.01) x -1.0))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= 0.01) {
		tmp = x;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((-2.0d0) * x) <= 0.01d0) then
        tmp = x
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= 0.01) {
		tmp = x;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (-2.0 * x) <= 0.01:
		tmp = x
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= 0.01)
		tmp = x;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((-2.0 * x) <= 0.01)
		tmp = x;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.01], x, -1.0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal -2 binary64) x) < 0.0100000000000000002

   1. Initial program 36.9%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 69.8%

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

   if 0.0100000000000000002 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 99.7

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

   5. Simplified 99.7%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 100.0%

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

Alternative 5: 27.4% accurate, 123.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return -1.0

Julia

function code(x, y)
	return -1.0
end

MATLAB

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

Wolfram

code[x_, y_] := -1.0

Derivation

1. Initial program 51.2%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. accelerator-lowering-fma.f64 26.8

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

5. Simplified 26.8%

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

6. Taylor expanded in x around inf

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

8. Simplified 24.9%

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

Reproduce

herbie shell --seed 2024194 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))