Logistic function from Lakshay Garg

Percentage Accurate: 54.1% → 99.8%

Time: 9.1s

Alternatives: 9

Speedup: 9.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-2 \cdot x}\\ t_1 := -1 - t\_0\\ \mathbf{if}\;-2 \cdot x \leq -2:\\ \;\;\;\;\frac{-1 + \frac{-8}{{t\_1}^{3}}}{1 + \left(\frac{4}{{t\_1}^{2}} + \frac{2}{t\_0 - -1}\right)}\\ \mathbf{elif}\;-2 \cdot x \leq 0.04:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.3333333333333333 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot -0.05396825396825397\right) - -0.13333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (exp (* -2.0 x))) (t_1 (- -1.0 t_0)))
   (if (<= (* -2.0 x) -2.0)
     (/
      (+ -1.0 (/ -8.0 (pow t_1 3.0)))
      (+ 1.0 (+ (/ 4.0 (pow t_1 2.0)) (/ 2.0 (- t_0 -1.0)))))
     (if (<= (* -2.0 x) 0.04)
       (*
        x
        (+
         1.0
         (*
          (* x x)
          (+
           -0.3333333333333333
           (*
            (* x x)
            (- (* x (* x -0.05396825396825397)) -0.13333333333333333))))))
       -1.0))))

C

double code(double x, double y) {
	double t_0 = exp((-2.0 * x));
	double t_1 = -1.0 - t_0;
	double tmp;
	if ((-2.0 * x) <= -2.0) {
		tmp = (-1.0 + (-8.0 / pow(t_1, 3.0))) / (1.0 + ((4.0 / pow(t_1, 2.0)) + (2.0 / (t_0 - -1.0))));
	} else if ((-2.0 * x) <= 0.04) {
		tmp = x * (1.0 + ((x * x) * (-0.3333333333333333 + ((x * x) * ((x * (x * -0.05396825396825397)) - -0.13333333333333333)))));
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(((-2.0d0) * x))
    t_1 = (-1.0d0) - t_0
    if (((-2.0d0) * x) <= (-2.0d0)) then
        tmp = ((-1.0d0) + ((-8.0d0) / (t_1 ** 3.0d0))) / (1.0d0 + ((4.0d0 / (t_1 ** 2.0d0)) + (2.0d0 / (t_0 - (-1.0d0)))))
    else if (((-2.0d0) * x) <= 0.04d0) then
        tmp = x * (1.0d0 + ((x * x) * ((-0.3333333333333333d0) + ((x * x) * ((x * (x * (-0.05396825396825397d0))) - (-0.13333333333333333d0))))))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.exp((-2.0 * x));
	double t_1 = -1.0 - t_0;
	double tmp;
	if ((-2.0 * x) <= -2.0) {
		tmp = (-1.0 + (-8.0 / Math.pow(t_1, 3.0))) / (1.0 + ((4.0 / Math.pow(t_1, 2.0)) + (2.0 / (t_0 - -1.0))));
	} else if ((-2.0 * x) <= 0.04) {
		tmp = x * (1.0 + ((x * x) * (-0.3333333333333333 + ((x * x) * ((x * (x * -0.05396825396825397)) - -0.13333333333333333)))));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.exp((-2.0 * x))
	t_1 = -1.0 - t_0
	tmp = 0
	if (-2.0 * x) <= -2.0:
		tmp = (-1.0 + (-8.0 / math.pow(t_1, 3.0))) / (1.0 + ((4.0 / math.pow(t_1, 2.0)) + (2.0 / (t_0 - -1.0))))
	elif (-2.0 * x) <= 0.04:
		tmp = x * (1.0 + ((x * x) * (-0.3333333333333333 + ((x * x) * ((x * (x * -0.05396825396825397)) - -0.13333333333333333)))))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	t_0 = exp(Float64(-2.0 * x))
	t_1 = Float64(-1.0 - t_0)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -2.0)
		tmp = Float64(Float64(-1.0 + Float64(-8.0 / (t_1 ^ 3.0))) / Float64(1.0 + Float64(Float64(4.0 / (t_1 ^ 2.0)) + Float64(2.0 / Float64(t_0 - -1.0)))));
	elseif (Float64(-2.0 * x) <= 0.04)
		tmp = Float64(x * Float64(1.0 + Float64(Float64(x * x) * Float64(-0.3333333333333333 + Float64(Float64(x * x) * Float64(Float64(x * Float64(x * -0.05396825396825397)) - -0.13333333333333333))))));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = exp((-2.0 * x));
	t_1 = -1.0 - t_0;
	tmp = 0.0;
	if ((-2.0 * x) <= -2.0)
		tmp = (-1.0 + (-8.0 / (t_1 ^ 3.0))) / (1.0 + ((4.0 / (t_1 ^ 2.0)) + (2.0 / (t_0 - -1.0))));
	elseif ((-2.0 * x) <= 0.04)
		tmp = x * (1.0 + ((x * x) * (-0.3333333333333333 + ((x * x) * ((x * (x * -0.05396825396825397)) - -0.13333333333333333)))));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(-1.0 - t$95$0), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -2.0], N[(N[(-1.0 + N[(-8.0 / N[Power[t$95$1, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(4.0 / N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 / N[(t$95$0 - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.04], N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(-0.3333333333333333 + N[(N[(x * x), $MachinePrecision] * N[(N[(x * N[(x * -0.05396825396825397), $MachinePrecision]), $MachinePrecision] - -0.13333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\frac{-8}{{\left(-1 - e^{-2 \cdot x}\right)}^{3}} + -1}{1 + \left(\frac{4}{{\left(-1 - e^{-2 \cdot x}\right)}^{2}} - \frac{2}{-1 - e^{-2 \cdot x}}\right)}} \]

   if -2 < (*.f64 #s(literal -2 binary64) x) < 0.0400000000000000008

   1. Initial program 7.7%

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

   3. Taylor expanded in x around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right)} - \frac{1}{3}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right)} - \frac{1}{3}\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) + \frac{-1}{3}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-1}{3} + \color{blue}{{x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{3}, \left({x}^{2} \cdot \left(\frac{-17}{315} \cdot {x}^{2} + \color{blue}{\frac{2}{15}}\right)\right)\right)\right)\right)\right) \]

      11. distribute-lft-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{3}, \left({x}^{2} \cdot \left(\frac{-17}{315} \cdot {x}^{2}\right) + \color{blue}{{x}^{2} \cdot \frac{2}{15}}\right)\right)\right)\right)\right) \]

      12. cancel-sign-sub N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{3}, \left({x}^{2} \cdot \left(\frac{-17}{315} \cdot {x}^{2}\right) - \color{blue}{\left(\mathsf{neg}\left({x}^{2}\right)\right) \cdot \frac{2}{15}}\right)\right)\right)\right)\right) \]

      13. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{3}, \left({x}^{2} \cdot \left(\frac{-17}{315} \cdot {x}^{2}\right) - \left(\mathsf{neg}\left({x}^{2} \cdot \frac{2}{15}\right)\right)\right)\right)\right)\right)\right) \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{3}, \left({x}^{2} \cdot \left(\frac{-17}{315} \cdot {x}^{2}\right) - {x}^{2} \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{2}{15}\right)\right)}\right)\right)\right)\right)\right) \]

      15. distribute-lft-out-- N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{3}, \left({x}^{2} \cdot \color{blue}{\left(\frac{-17}{315} \cdot {x}^{2} - \left(\mathsf{neg}\left(\frac{2}{15}\right)\right)\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{-17}{315} \cdot {x}^{2} - \left(\mathsf{neg}\left(\frac{2}{15}\right)\right)\right)}\right)\right)\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{-17}{315} \cdot {x}^{2}} - \left(\mathsf{neg}\left(\frac{2}{15}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{-17}{315} \cdot {x}^{2}} - \left(\mathsf{neg}\left(\frac{2}{15}\right)\right)\right)\right)\right)\right)\right)\right) \]

      19. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\left(\frac{-17}{315} \cdot {x}^{2}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{2}{15}\right)\right)}\right)\right)\right)\right)\right)\right) \]

   5. Simplified 100.0%

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

   if 0.0400000000000000008 < (*.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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \color{blue}{\left(2 + -2 \cdot x\right)}\right), 1\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \left(-2 \cdot x\right)\right)\right), 1\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \left(x \cdot -2\right)\right)\right), 1\right) \]

      3. *-lowering-*.f64 97.4%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), 1\right) \]

   5. Simplified 97.4%

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

   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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -2:\\ \;\;\;\;\frac{-1 + \frac{-8}{{\left(-1 - e^{-2 \cdot x}\right)}^{3}}}{1 + \left(\frac{4}{{\left(-1 - e^{-2 \cdot x}\right)}^{2}} + \frac{2}{e^{-2 \cdot x} - -1}\right)}\\ \mathbf{elif}\;-2 \cdot x \leq 0.04:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.3333333333333333 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot -0.05396825396825397\right) - -0.13333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -2:\\ \;\;\;\;-1 + \frac{2}{e^{-2 \cdot x} + 1}\\ \mathbf{elif}\;-2 \cdot x \leq 0.04:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.3333333333333333 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot -0.05396825396825397\right) - -0.13333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -2.0)
   (+ -1.0 (/ 2.0 (+ (exp (* -2.0 x)) 1.0)))
   (if (<= (* -2.0 x) 0.04)
     (*
      x
      (+
       1.0
       (*
        (* x x)
        (+
         -0.3333333333333333
         (*
          (* x x)
          (- (* x (* x -0.05396825396825397)) -0.13333333333333333))))))
     -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -2.0) {
		tmp = -1.0 + (2.0 / (exp((-2.0 * x)) + 1.0));
	} else if ((-2.0 * x) <= 0.04) {
		tmp = x * (1.0 + ((x * x) * (-0.3333333333333333 + ((x * x) * ((x * (x * -0.05396825396825397)) - -0.13333333333333333)))));
	} 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) <= (-2.0d0)) then
        tmp = (-1.0d0) + (2.0d0 / (exp(((-2.0d0) * x)) + 1.0d0))
    else if (((-2.0d0) * x) <= 0.04d0) then
        tmp = x * (1.0d0 + ((x * x) * ((-0.3333333333333333d0) + ((x * x) * ((x * (x * (-0.05396825396825397d0))) - (-0.13333333333333333d0))))))
    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) <= -2.0) {
		tmp = -1.0 + (2.0 / (Math.exp((-2.0 * x)) + 1.0));
	} else if ((-2.0 * x) <= 0.04) {
		tmp = x * (1.0 + ((x * x) * (-0.3333333333333333 + ((x * x) * ((x * (x * -0.05396825396825397)) - -0.13333333333333333)))));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (-2.0 * x) <= -2.0:
		tmp = -1.0 + (2.0 / (math.exp((-2.0 * x)) + 1.0))
	elif (-2.0 * x) <= 0.04:
		tmp = x * (1.0 + ((x * x) * (-0.3333333333333333 + ((x * x) * ((x * (x * -0.05396825396825397)) - -0.13333333333333333)))))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -2.0)
		tmp = Float64(-1.0 + Float64(2.0 / Float64(exp(Float64(-2.0 * x)) + 1.0)));
	elseif (Float64(-2.0 * x) <= 0.04)
		tmp = Float64(x * Float64(1.0 + Float64(Float64(x * x) * Float64(-0.3333333333333333 + Float64(Float64(x * x) * Float64(Float64(x * Float64(x * -0.05396825396825397)) - -0.13333333333333333))))));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((-2.0 * x) <= -2.0)
		tmp = -1.0 + (2.0 / (exp((-2.0 * x)) + 1.0));
	elseif ((-2.0 * x) <= 0.04)
		tmp = x * (1.0 + ((x * x) * (-0.3333333333333333 + ((x * x) * ((x * (x * -0.05396825396825397)) - -0.13333333333333333)))));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -2.0], N[(-1.0 + N[(2.0 / N[(N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.04], N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(-0.3333333333333333 + N[(N[(x * x), $MachinePrecision] * N[(N[(x * N[(x * -0.05396825396825397), $MachinePrecision]), $MachinePrecision] - -0.13333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   if -2 < (*.f64 #s(literal -2 binary64) x) < 0.0400000000000000008

   1. Initial program 7.7%

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

   3. Taylor expanded in x around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right)} - \frac{1}{3}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right)} - \frac{1}{3}\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) + \frac{-1}{3}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-1}{3} + \color{blue}{{x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{3}, \left({x}^{2} \cdot \left(\frac{-17}{315} \cdot {x}^{2} + \color{blue}{\frac{2}{15}}\right)\right)\right)\right)\right)\right) \]

      11. distribute-lft-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{3}, \left({x}^{2} \cdot \left(\frac{-17}{315} \cdot {x}^{2}\right) + \color{blue}{{x}^{2} \cdot \frac{2}{15}}\right)\right)\right)\right)\right) \]

      12. cancel-sign-sub N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{3}, \left({x}^{2} \cdot \left(\frac{-17}{315} \cdot {x}^{2}\right) - \color{blue}{\left(\mathsf{neg}\left({x}^{2}\right)\right) \cdot \frac{2}{15}}\right)\right)\right)\right)\right) \]

      13. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{3}, \left({x}^{2} \cdot \left(\frac{-17}{315} \cdot {x}^{2}\right) - \left(\mathsf{neg}\left({x}^{2} \cdot \frac{2}{15}\right)\right)\right)\right)\right)\right)\right) \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{3}, \left({x}^{2} \cdot \left(\frac{-17}{315} \cdot {x}^{2}\right) - {x}^{2} \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{2}{15}\right)\right)}\right)\right)\right)\right)\right) \]

      15. distribute-lft-out-- N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{3}, \left({x}^{2} \cdot \color{blue}{\left(\frac{-17}{315} \cdot {x}^{2} - \left(\mathsf{neg}\left(\frac{2}{15}\right)\right)\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{-17}{315} \cdot {x}^{2} - \left(\mathsf{neg}\left(\frac{2}{15}\right)\right)\right)}\right)\right)\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{-17}{315} \cdot {x}^{2}} - \left(\mathsf{neg}\left(\frac{2}{15}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{-17}{315} \cdot {x}^{2}} - \left(\mathsf{neg}\left(\frac{2}{15}\right)\right)\right)\right)\right)\right)\right)\right) \]

      19. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(\left(\frac{-17}{315} \cdot {x}^{2}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{2}{15}\right)\right)}\right)\right)\right)\right)\right)\right) \]

   5. Simplified 100.0%

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

   if 0.0400000000000000008 < (*.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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \color{blue}{\left(2 + -2 \cdot x\right)}\right), 1\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \left(-2 \cdot x\right)\right)\right), 1\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \left(x \cdot -2\right)\right)\right), 1\right) \]

      3. *-lowering-*.f64 97.4%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), 1\right) \]

   5. Simplified 97.4%

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

   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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -2:\\ \;\;\;\;-1 + \frac{2}{e^{-2 \cdot x} + 1}\\ \mathbf{elif}\;-2 \cdot x \leq 0.04:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.3333333333333333 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot -0.05396825396825397\right) - -0.13333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.7% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-8}:\\ \;\;\;\;-1 + \frac{2}{2 + x \cdot \left(-2 + x \cdot \left(2 + x \cdot -1.3333333333333333\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{x + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1e-8)
   (+
    -1.0
    (/ 2.0 (+ 2.0 (* x (+ -2.0 (* x (+ 2.0 (* x -1.3333333333333333))))))))
   (* x (/ 2.0 (+ x 2.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1e-8) {
		tmp = -1.0 + (2.0 / (2.0 + (x * (-2.0 + (x * (2.0 + (x * -1.3333333333333333)))))));
	} else {
		tmp = x * (2.0 / (x + 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1d-8)) then
        tmp = (-1.0d0) + (2.0d0 / (2.0d0 + (x * ((-2.0d0) + (x * (2.0d0 + (x * (-1.3333333333333333d0))))))))
    else
        tmp = x * (2.0d0 / (x + 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1e-8) {
		tmp = -1.0 + (2.0 / (2.0 + (x * (-2.0 + (x * (2.0 + (x * -1.3333333333333333)))))));
	} else {
		tmp = x * (2.0 / (x + 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1e-8:
		tmp = -1.0 + (2.0 / (2.0 + (x * (-2.0 + (x * (2.0 + (x * -1.3333333333333333)))))))
	else:
		tmp = x * (2.0 / (x + 2.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1e-8)
		tmp = Float64(-1.0 + Float64(2.0 / Float64(2.0 + Float64(x * Float64(-2.0 + Float64(x * Float64(2.0 + Float64(x * -1.3333333333333333))))))));
	else
		tmp = Float64(x * Float64(2.0 / Float64(x + 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1e-8)
		tmp = -1.0 + (2.0 / (2.0 + (x * (-2.0 + (x * (2.0 + (x * -1.3333333333333333)))))));
	else
		tmp = x * (2.0 / (x + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1e-8], N[(-1.0 + N[(2.0 / N[(2.0 + N[(x * N[(-2.0 + N[(x * N[(2.0 + N[(x * -1.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(2.0 / N[(x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1e-8

   1. Initial program 98.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \color{blue}{\left(2 + x \cdot \left(x \cdot \left(2 + \frac{-4}{3} \cdot x\right) - 2\right)\right)}\right), 1\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \left(x \cdot \left(x \cdot \left(2 + \frac{-4}{3} \cdot x\right) - 2\right)\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left(x \cdot \left(2 + \frac{-4}{3} \cdot x\right) - 2\right)\right)\right)\right), 1\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left(x \cdot \left(2 + \frac{-4}{3} \cdot x\right) + \left(\mathsf{neg}\left(2\right)\right)\right)\right)\right)\right), 1\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left(x \cdot \left(2 + \frac{-4}{3} \cdot x\right) + -2\right)\right)\right)\right), 1\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left(-2 + x \cdot \left(2 + \frac{-4}{3} \cdot x\right)\right)\right)\right)\right), 1\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-2, \left(x \cdot \left(2 + \frac{-4}{3} \cdot x\right)\right)\right)\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(x, \left(2 + \frac{-4}{3} \cdot x\right)\right)\right)\right)\right)\right), 1\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left(\frac{-4}{3} \cdot x\right)\right)\right)\right)\right)\right)\right), 1\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left(x \cdot \frac{-4}{3}\right)\right)\right)\right)\right)\right)\right), 1\right) \]

      10. *-lowering-*.f64 98.1%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \frac{-4}{3}\right)\right)\right)\right)\right)\right)\right), 1\right) \]

   5. Simplified 98.1%

      \[\leadsto \frac{2}{\color{blue}{2 + x \cdot \left(-2 + x \cdot \left(2 + x \cdot -1.3333333333333333\right)\right)}} - 1 \]

   if -1e-8 < x

   1. Initial program 37.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(1 + x\right)}, 1\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x + 1\right), 1\right) \]

      2. +-lowering-+.f64 5.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(x, 1\right), 1\right) \]

   5. Simplified 5.8%

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

      1. flip-- N/A

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

      2. div-inv N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\left(x + 1\right) \cdot \left(x + 1\right) - 1 \cdot 1\right), \color{blue}{\left(\frac{1}{\left(x + 1\right) + 1}\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(x + 1\right) \cdot \left(x + 1\right) - 1\right), \left(\frac{1}{\left(x + 1\right) + 1}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(x + 1\right) + 1\right) \cdot \left(\left(x + 1\right) - 1\right)\right), \left(\frac{\color{blue}{1}}{\left(x + 1\right) + 1}\right)\right) \]

      6. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(x + 1\right) + 1\right) \cdot \left(x + \left(1 - 1\right)\right)\right), \left(\frac{1}{\left(x + 1\right) + 1}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(x + 1\right) + 1\right) \cdot \left(x + 0\right)\right), \left(\frac{1}{\left(x + 1\right) + 1}\right)\right) \]

      8. +-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(x + 1\right) + 1\right) \cdot x\right), \left(\frac{1}{\left(x + 1\right) + 1}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(x + 1\right) + 1\right), x\right), \left(\frac{\color{blue}{1}}{\left(x + 1\right) + 1}\right)\right) \]

      10. associate-+l+ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x + \left(1 + 1\right)\right), x\right), \left(\frac{1}{\left(x + 1\right) + 1}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x + 2\right), x\right), \left(\frac{1}{\left(x + 1\right) + 1}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 2\right), x\right), \left(\frac{1}{\left(x + 1\right) + 1}\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 2\right), x\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\left(x + 1\right) + 1\right)}\right)\right) \]

      14. associate-+l+ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 2\right), x\right), \mathsf{/.f64}\left(1, \left(x + \color{blue}{\left(1 + 1\right)}\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 2\right), x\right), \mathsf{/.f64}\left(1, \left(x + 2\right)\right)\right) \]

      16. +-lowering-+.f64 68.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 2\right), x\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, \color{blue}{2}\right)\right)\right) \]

   7. Applied egg-rr 68.4%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(2 \cdot x\right)}, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, 2\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x \cdot 2\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(x, 2\right)\right)\right) \]

      2. *-lowering-*.f64 72.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(x, 2\right)\right)\right) \]

   10. Simplified 72.8%

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

       1. associate-*l* N/A

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

       2. *-commutative N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(\left(2 \cdot \frac{1}{x + 2}\right), \color{blue}{x}\right) \]

       4. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{x + 2}\right), x\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(x + 2\right)\right), x\right) \]

       6. +-lowering-+.f64 72.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(x, 2\right)\right), x\right) \]

   12. Applied egg-rr 72.8%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-8}:\\ \;\;\;\;-1 + \frac{2}{2 + x \cdot \left(-2 + x \cdot \left(2 + x \cdot -1.3333333333333333\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{x + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.5% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-8}:\\ \;\;\;\;-1 + \frac{2}{2 + x \cdot \left(-2 + x \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{x + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1e-8)
   (+ -1.0 (/ 2.0 (+ 2.0 (* x (+ -2.0 (* x 2.0))))))
   (* x (/ 2.0 (+ x 2.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1e-8) {
		tmp = -1.0 + (2.0 / (2.0 + (x * (-2.0 + (x * 2.0)))));
	} else {
		tmp = x * (2.0 / (x + 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1d-8)) then
        tmp = (-1.0d0) + (2.0d0 / (2.0d0 + (x * ((-2.0d0) + (x * 2.0d0)))))
    else
        tmp = x * (2.0d0 / (x + 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1e-8) {
		tmp = -1.0 + (2.0 / (2.0 + (x * (-2.0 + (x * 2.0)))));
	} else {
		tmp = x * (2.0 / (x + 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1e-8:
		tmp = -1.0 + (2.0 / (2.0 + (x * (-2.0 + (x * 2.0)))))
	else:
		tmp = x * (2.0 / (x + 2.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1e-8)
		tmp = Float64(-1.0 + Float64(2.0 / Float64(2.0 + Float64(x * Float64(-2.0 + Float64(x * 2.0))))));
	else
		tmp = Float64(x * Float64(2.0 / Float64(x + 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1e-8)
		tmp = -1.0 + (2.0 / (2.0 + (x * (-2.0 + (x * 2.0)))));
	else
		tmp = x * (2.0 / (x + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1e-8], N[(-1.0 + N[(2.0 / N[(2.0 + N[(x * N[(-2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(2.0 / N[(x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1e-8

   1. Initial program 98.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \color{blue}{\left(2 + x \cdot \left(2 \cdot x - 2\right)\right)}\right), 1\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \left(x \cdot \left(2 \cdot x - 2\right)\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left(2 \cdot x - 2\right)\right)\right)\right), 1\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left(2 \cdot x + \left(\mathsf{neg}\left(2\right)\right)\right)\right)\right)\right), 1\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left(2 \cdot x + -2\right)\right)\right)\right), 1\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left(-2 + 2 \cdot x\right)\right)\right)\right), 1\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-2, \left(2 \cdot x\right)\right)\right)\right)\right), 1\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-2, \left(x \cdot 2\right)\right)\right)\right)\right), 1\right) \]

      8. *-lowering-*.f64 97.2%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(x, 2\right)\right)\right)\right)\right), 1\right) \]

   5. Simplified 97.2%

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

   if -1e-8 < x

   1. Initial program 37.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(1 + x\right)}, 1\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x + 1\right), 1\right) \]

      2. +-lowering-+.f64 5.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(x, 1\right), 1\right) \]

   5. Simplified 5.8%

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

      1. flip-- N/A

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

      2. div-inv N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\left(x + 1\right) \cdot \left(x + 1\right) - 1 \cdot 1\right), \color{blue}{\left(\frac{1}{\left(x + 1\right) + 1}\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(x + 1\right) \cdot \left(x + 1\right) - 1\right), \left(\frac{1}{\left(x + 1\right) + 1}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(x + 1\right) + 1\right) \cdot \left(\left(x + 1\right) - 1\right)\right), \left(\frac{\color{blue}{1}}{\left(x + 1\right) + 1}\right)\right) \]

      6. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(x + 1\right) + 1\right) \cdot \left(x + \left(1 - 1\right)\right)\right), \left(\frac{1}{\left(x + 1\right) + 1}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(x + 1\right) + 1\right) \cdot \left(x + 0\right)\right), \left(\frac{1}{\left(x + 1\right) + 1}\right)\right) \]

      8. +-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(x + 1\right) + 1\right) \cdot x\right), \left(\frac{1}{\left(x + 1\right) + 1}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(x + 1\right) + 1\right), x\right), \left(\frac{\color{blue}{1}}{\left(x + 1\right) + 1}\right)\right) \]

      10. associate-+l+ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x + \left(1 + 1\right)\right), x\right), \left(\frac{1}{\left(x + 1\right) + 1}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x + 2\right), x\right), \left(\frac{1}{\left(x + 1\right) + 1}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 2\right), x\right), \left(\frac{1}{\left(x + 1\right) + 1}\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 2\right), x\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\left(x + 1\right) + 1\right)}\right)\right) \]

      14. associate-+l+ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 2\right), x\right), \mathsf{/.f64}\left(1, \left(x + \color{blue}{\left(1 + 1\right)}\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 2\right), x\right), \mathsf{/.f64}\left(1, \left(x + 2\right)\right)\right) \]

      16. +-lowering-+.f64 68.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 2\right), x\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, \color{blue}{2}\right)\right)\right) \]

   7. Applied egg-rr 68.4%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(2 \cdot x\right)}, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, 2\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x \cdot 2\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(x, 2\right)\right)\right) \]

      2. *-lowering-*.f64 72.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(x, 2\right)\right)\right) \]

   10. Simplified 72.8%

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

       1. associate-*l* N/A

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

       2. *-commutative N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(\left(2 \cdot \frac{1}{x + 2}\right), \color{blue}{x}\right) \]

       4. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{x + 2}\right), x\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(x + 2\right)\right), x\right) \]

       6. +-lowering-+.f64 72.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(x, 2\right)\right), x\right) \]

   12. Applied egg-rr 72.8%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-8}:\\ \;\;\;\;-1 + \frac{2}{2 + x \cdot \left(-2 + x \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{x + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.2% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 2.5:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;2 - \frac{4}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0) -1.0 (if (<= x 2.5) x (- 2.0 (/ 4.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = -1.0;
	} else if (x <= 2.5) {
		tmp = x;
	} else {
		tmp = 2.0 - (4.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = -1.0d0
    else if (x <= 2.5d0) then
        tmp = x
    else
        tmp = 2.0d0 - (4.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = -1.0;
	} else if (x <= 2.5) {
		tmp = x;
	} else {
		tmp = 2.0 - (4.0 / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = -1.0
	elif x <= 2.5:
		tmp = x
	else:
		tmp = 2.0 - (4.0 / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = -1.0;
	elseif (x <= 2.5)
		tmp = x;
	else
		tmp = Float64(2.0 - Float64(4.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = -1.0;
	elseif (x <= 2.5)
		tmp = x;
	else
		tmp = 2.0 - (4.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], -1.0, If[LessEqual[x, 2.5], x, N[(2.0 - N[(4.0 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \color{blue}{\left(2 + -2 \cdot x\right)}\right), 1\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \left(-2 \cdot x\right)\right)\right), 1\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \left(x \cdot -2\right)\right)\right), 1\right) \]

      3. *-lowering-*.f64 97.4%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), 1\right) \]

   5. Simplified 97.4%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 100.0%

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

   if -1 < x < 2.5

   1. Initial program 8.4%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 98.7%

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

   if 2.5 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(1 + x\right)}, 1\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x + 1\right), 1\right) \]

      2. +-lowering-+.f64 4.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(x, 1\right), 1\right) \]

   5. Simplified 4.9%

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

      1. flip-- N/A

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

      2. div-inv N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\left(x + 1\right) \cdot \left(x + 1\right) - 1 \cdot 1\right), \color{blue}{\left(\frac{1}{\left(x + 1\right) + 1}\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(x + 1\right) \cdot \left(x + 1\right) - 1\right), \left(\frac{1}{\left(x + 1\right) + 1}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(x + 1\right) + 1\right) \cdot \left(\left(x + 1\right) - 1\right)\right), \left(\frac{\color{blue}{1}}{\left(x + 1\right) + 1}\right)\right) \]

      6. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(x + 1\right) + 1\right) \cdot \left(x + \left(1 - 1\right)\right)\right), \left(\frac{1}{\left(x + 1\right) + 1}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(x + 1\right) + 1\right) \cdot \left(x + 0\right)\right), \left(\frac{1}{\left(x + 1\right) + 1}\right)\right) \]

      8. +-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(x + 1\right) + 1\right) \cdot x\right), \left(\frac{1}{\left(x + 1\right) + 1}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(x + 1\right) + 1\right), x\right), \left(\frac{\color{blue}{1}}{\left(x + 1\right) + 1}\right)\right) \]

      10. associate-+l+ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x + \left(1 + 1\right)\right), x\right), \left(\frac{1}{\left(x + 1\right) + 1}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x + 2\right), x\right), \left(\frac{1}{\left(x + 1\right) + 1}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 2\right), x\right), \left(\frac{1}{\left(x + 1\right) + 1}\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 2\right), x\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\left(x + 1\right) + 1\right)}\right)\right) \]

      14. associate-+l+ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 2\right), x\right), \mathsf{/.f64}\left(1, \left(x + \color{blue}{\left(1 + 1\right)}\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 2\right), x\right), \mathsf{/.f64}\left(1, \left(x + 2\right)\right)\right) \]

      16. +-lowering-+.f64 4.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 2\right), x\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, \color{blue}{2}\right)\right)\right) \]

   7. Applied egg-rr 4.5%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(2 \cdot x\right)}, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, 2\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x \cdot 2\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(x, 2\right)\right)\right) \]

      2. *-lowering-*.f64 18.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(x, 2\right)\right)\right) \]

   10. Simplified 18.8%

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

   11. Taylor expanded in x around inf

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

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

          \[\leadsto \mathsf{\_.f64}\left(2, \color{blue}{\left(4 \cdot \frac{1}{x}\right)}\right) \]

       2. associate-*r/ N/A

          \[\leadsto \mathsf{\_.f64}\left(2, \left(\frac{4 \cdot 1}{\color{blue}{x}}\right)\right) \]

       3. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(2, \left(\frac{4}{x}\right)\right) \]

       4. /-lowering-/.f64 18.8%

          \[\leadsto \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(4, \color{blue}{x}\right)\right) \]

   13. Simplified 18.8%

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

Alternative 6: 78.6% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.66:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{x + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.66) -1.0 (* x (/ 2.0 (+ x 2.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.66) {
		tmp = -1.0;
	} else {
		tmp = x * (2.0 / (x + 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-0.66d0)) then
        tmp = -1.0d0
    else
        tmp = x * (2.0d0 / (x + 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.66) {
		tmp = -1.0;
	} else {
		tmp = x * (2.0 / (x + 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.66:
		tmp = -1.0
	else:
		tmp = x * (2.0 / (x + 2.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.66)
		tmp = -1.0;
	else
		tmp = Float64(x * Float64(2.0 / Float64(x + 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.66)
		tmp = -1.0;
	else
		tmp = x * (2.0 / (x + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.66], -1.0, N[(x * N[(2.0 / N[(x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.660000000000000031

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \color{blue}{\left(2 + -2 \cdot x\right)}\right), 1\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \left(-2 \cdot x\right)\right)\right), 1\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \left(x \cdot -2\right)\right)\right), 1\right) \]

      3. *-lowering-*.f64 97.4%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), 1\right) \]

   5. Simplified 97.4%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 100.0%

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

   if -0.660000000000000031 < x

   1. Initial program 37.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(1 + x\right)}, 1\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x + 1\right), 1\right) \]

      2. +-lowering-+.f64 6.6%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(x, 1\right), 1\right) \]

   5. Simplified 6.6%

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

      1. flip-- N/A

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

      2. div-inv N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\left(x + 1\right) \cdot \left(x + 1\right) - 1 \cdot 1\right), \color{blue}{\left(\frac{1}{\left(x + 1\right) + 1}\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(x + 1\right) \cdot \left(x + 1\right) - 1\right), \left(\frac{1}{\left(x + 1\right) + 1}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(x + 1\right) + 1\right) \cdot \left(\left(x + 1\right) - 1\right)\right), \left(\frac{\color{blue}{1}}{\left(x + 1\right) + 1}\right)\right) \]

      6. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(x + 1\right) + 1\right) \cdot \left(x + \left(1 - 1\right)\right)\right), \left(\frac{1}{\left(x + 1\right) + 1}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(x + 1\right) + 1\right) \cdot \left(x + 0\right)\right), \left(\frac{1}{\left(x + 1\right) + 1}\right)\right) \]

      8. +-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(x + 1\right) + 1\right) \cdot x\right), \left(\frac{1}{\left(x + 1\right) + 1}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(x + 1\right) + 1\right), x\right), \left(\frac{\color{blue}{1}}{\left(x + 1\right) + 1}\right)\right) \]

      10. associate-+l+ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x + \left(1 + 1\right)\right), x\right), \left(\frac{1}{\left(x + 1\right) + 1}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x + 2\right), x\right), \left(\frac{1}{\left(x + 1\right) + 1}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 2\right), x\right), \left(\frac{1}{\left(x + 1\right) + 1}\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 2\right), x\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\left(x + 1\right) + 1\right)}\right)\right) \]

      14. associate-+l+ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 2\right), x\right), \mathsf{/.f64}\left(1, \left(x + \color{blue}{\left(1 + 1\right)}\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 2\right), x\right), \mathsf{/.f64}\left(1, \left(x + 2\right)\right)\right) \]

      16. +-lowering-+.f64 68.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 2\right), x\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, \color{blue}{2}\right)\right)\right) \]

   7. Applied egg-rr 68.4%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(2 \cdot x\right)}, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, 2\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x \cdot 2\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(x, 2\right)\right)\right) \]

      2. *-lowering-*.f64 72.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(x, 2\right)\right)\right) \]

   10. Simplified 72.3%

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

       1. associate-*l* N/A

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

       2. *-commutative N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(\left(2 \cdot \frac{1}{x + 2}\right), \color{blue}{x}\right) \]

       4. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{x + 2}\right), x\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(x + 2\right)\right), x\right) \]

       6. +-lowering-+.f64 72.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(x, 2\right)\right), x\right) \]

   12. Applied egg-rr 72.3%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.66:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{x + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.2% accurate, 9.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], -1.0, If[LessEqual[x, 2.0], x, 2.0]]

Derivation

1. Split input into 3 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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \color{blue}{\left(2 + -2 \cdot x\right)}\right), 1\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \left(-2 \cdot x\right)\right)\right), 1\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \left(x \cdot -2\right)\right)\right), 1\right) \]

      3. *-lowering-*.f64 97.4%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), 1\right) \]

   5. Simplified 97.4%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 100.0%

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

   if -1 < x < 2

   1. Initial program 8.4%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 98.7%

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

   if 2 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(1 + x\right)}, 1\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x + 1\right), 1\right) \]

      2. +-lowering-+.f64 4.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(x, 1\right), 1\right) \]

   5. Simplified 4.9%

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

      1. flip-- N/A

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

      2. div-inv N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\left(x + 1\right) \cdot \left(x + 1\right) - 1 \cdot 1\right), \color{blue}{\left(\frac{1}{\left(x + 1\right) + 1}\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(x + 1\right) \cdot \left(x + 1\right) - 1\right), \left(\frac{1}{\left(x + 1\right) + 1}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(x + 1\right) + 1\right) \cdot \left(\left(x + 1\right) - 1\right)\right), \left(\frac{\color{blue}{1}}{\left(x + 1\right) + 1}\right)\right) \]

      6. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(x + 1\right) + 1\right) \cdot \left(x + \left(1 - 1\right)\right)\right), \left(\frac{1}{\left(x + 1\right) + 1}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(x + 1\right) + 1\right) \cdot \left(x + 0\right)\right), \left(\frac{1}{\left(x + 1\right) + 1}\right)\right) \]

      8. +-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(x + 1\right) + 1\right) \cdot x\right), \left(\frac{1}{\left(x + 1\right) + 1}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(x + 1\right) + 1\right), x\right), \left(\frac{\color{blue}{1}}{\left(x + 1\right) + 1}\right)\right) \]

      10. associate-+l+ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x + \left(1 + 1\right)\right), x\right), \left(\frac{1}{\left(x + 1\right) + 1}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x + 2\right), x\right), \left(\frac{1}{\left(x + 1\right) + 1}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 2\right), x\right), \left(\frac{1}{\left(x + 1\right) + 1}\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 2\right), x\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\left(x + 1\right) + 1\right)}\right)\right) \]

      14. associate-+l+ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 2\right), x\right), \mathsf{/.f64}\left(1, \left(x + \color{blue}{\left(1 + 1\right)}\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 2\right), x\right), \mathsf{/.f64}\left(1, \left(x + 2\right)\right)\right) \]

      16. +-lowering-+.f64 4.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 2\right), x\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, \color{blue}{2}\right)\right)\right) \]

   7. Applied egg-rr 4.5%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(2 \cdot x\right)}, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, 2\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x \cdot 2\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(x, 2\right)\right)\right) \]

      2. *-lowering-*.f64 18.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(x, 2\right)\right)\right) \]

   10. Simplified 18.8%

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

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{2} \]
   12. Step-by-step derivation

   13. Simplified 18.8%

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

Alternative 8: 32.2% accurate, 18.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{-308}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 1.1e-308) -1.0 2.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.1e-308) {
		tmp = -1.0;
	} else {
		tmp = 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 1.1d-308) then
        tmp = -1.0d0
    else
        tmp = 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.1e-308) {
		tmp = -1.0;
	} else {
		tmp = 2.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.1e-308:
		tmp = -1.0
	else:
		tmp = 2.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.1e-308)
		tmp = -1.0;
	else
		tmp = 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.1e-308)
		tmp = -1.0;
	else
		tmp = 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.1e-308], -1.0, 2.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001e-308

   1. Initial program 50.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \color{blue}{\left(2 + -2 \cdot x\right)}\right), 1\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \left(-2 \cdot x\right)\right)\right), 1\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \left(x \cdot -2\right)\right)\right), 1\right) \]

      3. *-lowering-*.f64 48.1%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), 1\right) \]

   5. Simplified 48.1%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 47.8%

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

   if 1.1000000000000001e-308 < x

   1. Initial program 53.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(1 + x\right)}, 1\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x + 1\right), 1\right) \]

      2. +-lowering-+.f64 5.4%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(x, 1\right), 1\right) \]

   5. Simplified 5.4%

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

      1. flip-- N/A

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

      2. div-inv N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\left(x + 1\right) \cdot \left(x + 1\right) - 1 \cdot 1\right), \color{blue}{\left(\frac{1}{\left(x + 1\right) + 1}\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(x + 1\right) \cdot \left(x + 1\right) - 1\right), \left(\frac{1}{\left(x + 1\right) + 1}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(x + 1\right) + 1\right) \cdot \left(\left(x + 1\right) - 1\right)\right), \left(\frac{\color{blue}{1}}{\left(x + 1\right) + 1}\right)\right) \]

      6. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(x + 1\right) + 1\right) \cdot \left(x + \left(1 - 1\right)\right)\right), \left(\frac{1}{\left(x + 1\right) + 1}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(x + 1\right) + 1\right) \cdot \left(x + 0\right)\right), \left(\frac{1}{\left(x + 1\right) + 1}\right)\right) \]

      8. +-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(x + 1\right) + 1\right) \cdot x\right), \left(\frac{1}{\left(x + 1\right) + 1}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(x + 1\right) + 1\right), x\right), \left(\frac{\color{blue}{1}}{\left(x + 1\right) + 1}\right)\right) \]

      10. associate-+l+ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x + \left(1 + 1\right)\right), x\right), \left(\frac{1}{\left(x + 1\right) + 1}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x + 2\right), x\right), \left(\frac{1}{\left(x + 1\right) + 1}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 2\right), x\right), \left(\frac{1}{\left(x + 1\right) + 1}\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 2\right), x\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\left(x + 1\right) + 1\right)}\right)\right) \]

      14. associate-+l+ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 2\right), x\right), \mathsf{/.f64}\left(1, \left(x + \color{blue}{\left(1 + 1\right)}\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 2\right), x\right), \mathsf{/.f64}\left(1, \left(x + 2\right)\right)\right) \]

      16. +-lowering-+.f64 52.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 2\right), x\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, \color{blue}{2}\right)\right)\right) \]

   7. Applied egg-rr 52.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(2 \cdot x\right)}, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, 2\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x \cdot 2\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(x, 2\right)\right)\right) \]

      2. *-lowering-*.f64 59.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(x, 2\right)\right)\right) \]

   10. Simplified 59.0%

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

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{2} \]
   12. Step-by-step derivation

   13. Simplified 12.0%

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

Alternative 9: 27.2% accurate, 109.0× 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 51.7%

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

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \color{blue}{\left(2 + -2 \cdot x\right)}\right), 1\right) \]
4. Step-by-step derivation

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \left(-2 \cdot x\right)\right)\right), 1\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \left(x \cdot -2\right)\right)\right), 1\right) \]

   3. *-lowering-*.f64 25.7%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, -2\right)\right)\right), 1\right) \]

5. Simplified 25.7%

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

6. Taylor expanded in x around inf

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

8. Simplified 24.7%

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

Reproduce

herbie shell --seed 2024155 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))