Logistic function from Lakshay Garg

Percentage Accurate: 54.9% → 99.8%

Time: 10.9s

Alternatives: 9

Speedup: 18.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.9% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + e^{-2 \cdot x}\\ \mathbf{if}\;-2 \cdot x \leq -0.04:\\ \;\;\;\;\frac{1 - \frac{4}{{t\_0}^{2}}}{-1 - \frac{2}{{\left({t\_0}^{0.5}\right)}^{2}}}\\ \mathbf{elif}\;-2 \cdot x \leq 0.002:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(x \cdot \left(-0.3333333333333333 + \left(x \cdot x\right) \cdot 0.13333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (exp (* -2.0 x)))))
   (if (<= (* -2.0 x) -0.04)
     (/ (- 1.0 (/ 4.0 (pow t_0 2.0))) (- -1.0 (/ 2.0 (pow (pow t_0 0.5) 2.0))))
     (if (<= (* -2.0 x) 0.002)
       (*
        x
        (+
         1.0
         (* x (* x (+ -0.3333333333333333 (* (* x x) 0.13333333333333333))))))
       -1.0))))

C

double code(double x, double y) {
	double t_0 = 1.0 + exp((-2.0 * x));
	double tmp;
	if ((-2.0 * x) <= -0.04) {
		tmp = (1.0 - (4.0 / pow(t_0, 2.0))) / (-1.0 - (2.0 / pow(pow(t_0, 0.5), 2.0)));
	} else if ((-2.0 * x) <= 0.002) {
		tmp = x * (1.0 + (x * (x * (-0.3333333333333333 + ((x * x) * 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) :: tmp
    t_0 = 1.0d0 + exp(((-2.0d0) * x))
    if (((-2.0d0) * x) <= (-0.04d0)) then
        tmp = (1.0d0 - (4.0d0 / (t_0 ** 2.0d0))) / ((-1.0d0) - (2.0d0 / ((t_0 ** 0.5d0) ** 2.0d0)))
    else if (((-2.0d0) * x) <= 0.002d0) then
        tmp = x * (1.0d0 + (x * (x * ((-0.3333333333333333d0) + ((x * x) * 0.13333333333333333d0)))))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + Math.exp((-2.0 * x));
	double tmp;
	if ((-2.0 * x) <= -0.04) {
		tmp = (1.0 - (4.0 / Math.pow(t_0, 2.0))) / (-1.0 - (2.0 / Math.pow(Math.pow(t_0, 0.5), 2.0)));
	} else if ((-2.0 * x) <= 0.002) {
		tmp = x * (1.0 + (x * (x * (-0.3333333333333333 + ((x * x) * 0.13333333333333333)))));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + math.exp((-2.0 * x))
	tmp = 0
	if (-2.0 * x) <= -0.04:
		tmp = (1.0 - (4.0 / math.pow(t_0, 2.0))) / (-1.0 - (2.0 / math.pow(math.pow(t_0, 0.5), 2.0)))
	elif (-2.0 * x) <= 0.002:
		tmp = x * (1.0 + (x * (x * (-0.3333333333333333 + ((x * x) * 0.13333333333333333)))))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + exp(Float64(-2.0 * x)))
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.04)
		tmp = Float64(Float64(1.0 - Float64(4.0 / (t_0 ^ 2.0))) / Float64(-1.0 - Float64(2.0 / ((t_0 ^ 0.5) ^ 2.0))));
	elseif (Float64(-2.0 * x) <= 0.002)
		tmp = Float64(x * Float64(1.0 + Float64(x * Float64(x * Float64(-0.3333333333333333 + Float64(Float64(x * x) * 0.13333333333333333))))));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + exp((-2.0 * x));
	tmp = 0.0;
	if ((-2.0 * x) <= -0.04)
		tmp = (1.0 - (4.0 / (t_0 ^ 2.0))) / (-1.0 - (2.0 / ((t_0 ^ 0.5) ^ 2.0)));
	elseif ((-2.0 * x) <= 0.002)
		tmp = x * (1.0 + (x * (x * (-0.3333333333333333 + ((x * x) * 0.13333333333333333)))));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.04], N[(N[(1.0 - N[(4.0 / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - N[(2.0 / N[Power[N[Power[t$95$0, 0.5], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.002], N[(x * N[(1.0 + N[(x * N[(x * N[(-0.3333333333333333 + N[(N[(x * x), $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) < -0.0400000000000000008

   1. Initial program 99.8%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(1 \cdot 1 - \frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right), \color{blue}{\left(\left(\mathsf{neg}\left(1\right)\right) - \frac{2}{1 + e^{-2 \cdot x}}\right)}\right) \]

   4. Applied egg-rr 99.9%

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

      1. unpow1 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(4, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-2, x\right)\right)\right), 2\right)\right)\right), \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(2, \left({\left(1 + e^{-2 \cdot x}\right)}^{\color{blue}{1}}\right)\right)\right)\right) \]

      2. sqr-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(4, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-2, x\right)\right)\right), 2\right)\right)\right), \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(2, \left({\left(1 + e^{-2 \cdot x}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(1 + e^{-2 \cdot x}\right)}^{\left(\frac{1}{2}\right)}}\right)\right)\right)\right) \]

      3. pow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(4, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-2, x\right)\right)\right), 2\right)\right)\right), \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(2, \left({\left({\left(1 + e^{-2 \cdot x}\right)}^{\left(\frac{1}{2}\right)}\right)}^{\color{blue}{2}}\right)\right)\right)\right) \]

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(4, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-2, x\right)\right)\right), 2\right)\right)\right), \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(2, \mathsf{pow.f64}\left(\left({\left(1 + e^{-2 \cdot x}\right)}^{\left(\frac{1}{2}\right)}\right), \color{blue}{2}\right)\right)\right)\right) \]

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

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

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

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

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

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

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

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

      9. metadata-eval 99.9%

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

   6. Applied egg-rr 99.9%

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

   if -0.0400000000000000008 < (*.f64 #s(literal -2 binary64) x) < 2e-3

   1. Initial program 6.5%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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. *-lowering-*.f64 N/A

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

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

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

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

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   if 2e-3 < (*.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 97.8%

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

   5. Simplified 97.8%

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

   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. Add Preprocessing

Alternative 2: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-2 \cdot x}\\ \mathbf{if}\;-2 \cdot x \leq -0.04:\\ \;\;\;\;\frac{1 - \frac{4}{{\left(1 + t\_0\right)}^{2}}}{-1 + \frac{2}{-1 - t\_0}}\\ \mathbf{elif}\;-2 \cdot x \leq 0.002:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(x \cdot \left(-0.3333333333333333 + \left(x \cdot x\right) \cdot 0.13333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (exp (* -2.0 x))))
   (if (<= (* -2.0 x) -0.04)
     (/ (- 1.0 (/ 4.0 (pow (+ 1.0 t_0) 2.0))) (+ -1.0 (/ 2.0 (- -1.0 t_0))))
     (if (<= (* -2.0 x) 0.002)
       (*
        x
        (+
         1.0
         (* x (* x (+ -0.3333333333333333 (* (* x x) 0.13333333333333333))))))
       -1.0))))

C

double code(double x, double y) {
	double t_0 = exp((-2.0 * x));
	double tmp;
	if ((-2.0 * x) <= -0.04) {
		tmp = (1.0 - (4.0 / pow((1.0 + t_0), 2.0))) / (-1.0 + (2.0 / (-1.0 - t_0)));
	} else if ((-2.0 * x) <= 0.002) {
		tmp = x * (1.0 + (x * (x * (-0.3333333333333333 + ((x * x) * 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) :: tmp
    t_0 = exp(((-2.0d0) * x))
    if (((-2.0d0) * x) <= (-0.04d0)) then
        tmp = (1.0d0 - (4.0d0 / ((1.0d0 + t_0) ** 2.0d0))) / ((-1.0d0) + (2.0d0 / ((-1.0d0) - t_0)))
    else if (((-2.0d0) * x) <= 0.002d0) then
        tmp = x * (1.0d0 + (x * (x * ((-0.3333333333333333d0) + ((x * x) * 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 tmp;
	if ((-2.0 * x) <= -0.04) {
		tmp = (1.0 - (4.0 / Math.pow((1.0 + t_0), 2.0))) / (-1.0 + (2.0 / (-1.0 - t_0)));
	} else if ((-2.0 * x) <= 0.002) {
		tmp = x * (1.0 + (x * (x * (-0.3333333333333333 + ((x * x) * 0.13333333333333333)))));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.exp((-2.0 * x))
	tmp = 0
	if (-2.0 * x) <= -0.04:
		tmp = (1.0 - (4.0 / math.pow((1.0 + t_0), 2.0))) / (-1.0 + (2.0 / (-1.0 - t_0)))
	elif (-2.0 * x) <= 0.002:
		tmp = x * (1.0 + (x * (x * (-0.3333333333333333 + ((x * x) * 0.13333333333333333)))))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	t_0 = exp(Float64(-2.0 * x))
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.04)
		tmp = Float64(Float64(1.0 - Float64(4.0 / (Float64(1.0 + t_0) ^ 2.0))) / Float64(-1.0 + Float64(2.0 / Float64(-1.0 - t_0))));
	elseif (Float64(-2.0 * x) <= 0.002)
		tmp = Float64(x * Float64(1.0 + Float64(x * Float64(x * Float64(-0.3333333333333333 + Float64(Float64(x * x) * 0.13333333333333333))))));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = exp((-2.0 * x));
	tmp = 0.0;
	if ((-2.0 * x) <= -0.04)
		tmp = (1.0 - (4.0 / ((1.0 + t_0) ^ 2.0))) / (-1.0 + (2.0 / (-1.0 - t_0)));
	elseif ((-2.0 * x) <= 0.002)
		tmp = x * (1.0 + (x * (x * (-0.3333333333333333 + ((x * x) * 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]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.04], N[(N[(1.0 - N[(4.0 / N[Power[N[(1.0 + t$95$0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + N[(2.0 / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.002], N[(x * N[(1.0 + N[(x * N[(x * N[(-0.3333333333333333 + N[(N[(x * x), $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) < -0.0400000000000000008

   1. Initial program 99.8%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(1 \cdot 1 - \frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right), \color{blue}{\left(\left(\mathsf{neg}\left(1\right)\right) - \frac{2}{1 + e^{-2 \cdot x}}\right)}\right) \]

   4. Applied egg-rr 99.9%

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

   if -0.0400000000000000008 < (*.f64 #s(literal -2 binary64) x) < 2e-3

   1. Initial program 6.5%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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. *-lowering-*.f64 N/A

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

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

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

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

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   if 2e-3 < (*.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 97.8%

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

   5. Simplified 97.8%

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

   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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.04:\\ \;\;\;\;\frac{1 - \frac{4}{{\left(1 + e^{-2 \cdot x}\right)}^{2}}}{-1 + \frac{2}{-1 - e^{-2 \cdot x}}}\\ \mathbf{elif}\;-2 \cdot x \leq 0.002:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(x \cdot \left(-0.3333333333333333 + \left(x \cdot x\right) \cdot 0.13333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.1:\\ \;\;\;\;\mathsf{expm1}\left(0 - \log \left(\frac{1 + e^{-2 \cdot x}}{2}\right)\right)\\ \mathbf{elif}\;-2 \cdot x \leq 0.002:\\ \;\;\;\;x \cdot \left(-0.3333333333333333 \cdot \left(x \cdot x\right) + \left(1 + \left(0.13333333333333333 + \left(x \cdot x\right) \cdot -0.05396825396825397\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -0.1)
   (expm1 (- 0.0 (log (/ (+ 1.0 (exp (* -2.0 x))) 2.0))))
   (if (<= (* -2.0 x) 0.002)
     (*
      x
      (+
       (* -0.3333333333333333 (* x x))
       (+
        1.0
        (*
         (+ 0.13333333333333333 (* (* x x) -0.05396825396825397))
         (* (* x x) (* x x))))))
     -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.1) {
		tmp = expm1((0.0 - log(((1.0 + exp((-2.0 * x))) / 2.0))));
	} else if ((-2.0 * x) <= 0.002) {
		tmp = x * ((-0.3333333333333333 * (x * x)) + (1.0 + ((0.13333333333333333 + ((x * x) * -0.05396825396825397)) * ((x * x) * (x * x)))));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.1) {
		tmp = Math.expm1((0.0 - Math.log(((1.0 + Math.exp((-2.0 * x))) / 2.0))));
	} else if ((-2.0 * x) <= 0.002) {
		tmp = x * ((-0.3333333333333333 * (x * x)) + (1.0 + ((0.13333333333333333 + ((x * x) * -0.05396825396825397)) * ((x * x) * (x * x)))));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (-2.0 * x) <= -0.1:
		tmp = math.expm1((0.0 - math.log(((1.0 + math.exp((-2.0 * x))) / 2.0))))
	elif (-2.0 * x) <= 0.002:
		tmp = x * ((-0.3333333333333333 * (x * x)) + (1.0 + ((0.13333333333333333 + ((x * x) * -0.05396825396825397)) * ((x * x) * (x * x)))))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.1)
		tmp = expm1(Float64(0.0 - log(Float64(Float64(1.0 + exp(Float64(-2.0 * x))) / 2.0))));
	elseif (Float64(-2.0 * x) <= 0.002)
		tmp = Float64(x * Float64(Float64(-0.3333333333333333 * Float64(x * x)) + Float64(1.0 + Float64(Float64(0.13333333333333333 + Float64(Float64(x * x) * -0.05396825396825397)) * Float64(Float64(x * x) * Float64(x * x))))));
	else
		tmp = -1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.1], N[(Exp[N[(0.0 - N[Log[N[(N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.002], N[(x * N[(N[(-0.3333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(N[(0.13333333333333333 + N[(N[(x * x), $MachinePrecision] * -0.05396825396825397), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.9%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. metadata-eval N/A

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

      4. pow-to-exp N/A

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

      5. expm1-define N/A

         \[\leadsto \mathsf{expm1}\left(\log \left(\frac{1 + e^{-2 \cdot x}}{2}\right) \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \]

      6. expm1-lowering-expm1.f64 N/A

         \[\leadsto \mathsf{expm1.f64}\left(\left(\log \left(\frac{1 + e^{-2 \cdot x}}{2}\right) \cdot \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\log \left(\frac{1 + e^{-2 \cdot x}}{2}\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\frac{1 + e^{-2 \cdot x}}{2}\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(1 + e^{-2 \cdot x}\right), 2\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(e^{-2 \cdot x}\right)\right), 2\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      11. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(-2 \cdot x\right)\right)\right), 2\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-2, x\right)\right)\right), 2\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

      13. metadata-eval 100.0%

          \[\leadsto \mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-2, x\right)\right)\right), 2\right)\right), -1\right)\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\log \left(\frac{1 + e^{-2 \cdot x}}{2}\right) \cdot -1\right)} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{expm1.f64}\left(\left(-1 \cdot \log \left(\frac{1 + e^{-2 \cdot x}}{2}\right)\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{expm1.f64}\left(\left(\mathsf{neg}\left(\log \left(\frac{1 + e^{-2 \cdot x}}{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{expm1.f64}\left(\mathsf{neg.f64}\left(\log \left(\frac{1 + e^{-2 \cdot x}}{2}\right)\right)\right) \]

      4. rem-exp-log N/A

         \[\leadsto \mathsf{expm1.f64}\left(\mathsf{neg.f64}\left(\log \left(e^{\log \left(\frac{1 + e^{-2 \cdot x}}{2}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{expm1.f64}\left(\mathsf{neg.f64}\left(\mathsf{log.f64}\left(\left(e^{\log \left(\frac{1 + e^{-2 \cdot x}}{2}\right)}\right)\right)\right)\right) \]

      6. rem-exp-log N/A

         \[\leadsto \mathsf{expm1.f64}\left(\mathsf{neg.f64}\left(\mathsf{log.f64}\left(\left(\frac{1 + e^{-2 \cdot x}}{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{expm1.f64}\left(\mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(1 + e^{-2 \cdot x}\right), 2\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{expm1.f64}\left(\mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(e^{-2 \cdot x}\right)\right), 2\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{expm1.f64}\left(\mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(e^{x \cdot -2}\right)\right), 2\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{expm1.f64}\left(\mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(x \cdot -2\right)\right)\right), 2\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{expm1.f64}\left(\mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(-2 \cdot x\right)\right)\right), 2\right)\right)\right)\right) \]

      12. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{expm1.f64}\left(\mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-2, x\right)\right)\right), 2\right)\right)\right)\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{-\log \left(\frac{1 + e^{-2 \cdot x}}{2}\right)}\right) \]

   if -0.10000000000000001 < (*.f64 #s(literal -2 binary64) x) < 2e-3

   1. Initial program 7.3%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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. *-lowering-*.f64 N/A

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

      11. unpow2 N/A

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

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

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

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

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

      14. *-commutative N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 99.9%

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

   5. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.3333333333333333 + \left(x \cdot x\right) \cdot \left(0.13333333333333333 + \left(x \cdot x\right) \cdot -0.05396825396825397\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-+l+ N/A

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

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

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

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

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

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

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

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

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

   7. Applied egg-rr 99.9%

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

   if 2e-3 < (*.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 97.8%

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

   5. Simplified 97.8%

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

   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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.1:\\ \;\;\;\;\mathsf{expm1}\left(0 - \log \left(\frac{1 + e^{-2 \cdot x}}{2}\right)\right)\\ \mathbf{elif}\;-2 \cdot x \leq 0.002:\\ \;\;\;\;x \cdot \left(-0.3333333333333333 \cdot \left(x \cdot x\right) + \left(1 + \left(0.13333333333333333 + \left(x \cdot x\right) \cdot -0.05396825396825397\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.04:\\ \;\;\;\;\frac{1}{\frac{1}{-1 + \frac{2}{1 + e^{-2 \cdot x}}}}\\ \mathbf{elif}\;-2 \cdot x \leq 0.002:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(x \cdot \left(-0.3333333333333333 + \left(x \cdot x\right) \cdot 0.13333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -0.04)
   (/ 1.0 (/ 1.0 (+ -1.0 (/ 2.0 (+ 1.0 (exp (* -2.0 x)))))))
   (if (<= (* -2.0 x) 0.002)
     (*
      x
      (+
       1.0
       (* x (* x (+ -0.3333333333333333 (* (* x x) 0.13333333333333333))))))
     -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.04) {
		tmp = 1.0 / (1.0 / (-1.0 + (2.0 / (1.0 + exp((-2.0 * x))))));
	} else if ((-2.0 * x) <= 0.002) {
		tmp = x * (1.0 + (x * (x * (-0.3333333333333333 + ((x * x) * 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) <= (-0.04d0)) then
        tmp = 1.0d0 / (1.0d0 / ((-1.0d0) + (2.0d0 / (1.0d0 + exp(((-2.0d0) * x))))))
    else if (((-2.0d0) * x) <= 0.002d0) then
        tmp = x * (1.0d0 + (x * (x * ((-0.3333333333333333d0) + ((x * x) * 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) <= -0.04) {
		tmp = 1.0 / (1.0 / (-1.0 + (2.0 / (1.0 + Math.exp((-2.0 * x))))));
	} else if ((-2.0 * x) <= 0.002) {
		tmp = x * (1.0 + (x * (x * (-0.3333333333333333 + ((x * x) * 0.13333333333333333)))));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (-2.0 * x) <= -0.04:
		tmp = 1.0 / (1.0 / (-1.0 + (2.0 / (1.0 + math.exp((-2.0 * x))))))
	elif (-2.0 * x) <= 0.002:
		tmp = x * (1.0 + (x * (x * (-0.3333333333333333 + ((x * x) * 0.13333333333333333)))))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.04)
		tmp = Float64(1.0 / Float64(1.0 / Float64(-1.0 + Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))))));
	elseif (Float64(-2.0 * x) <= 0.002)
		tmp = Float64(x * Float64(1.0 + Float64(x * Float64(x * Float64(-0.3333333333333333 + Float64(Float64(x * x) * 0.13333333333333333))))));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((-2.0 * x) <= -0.04)
		tmp = 1.0 / (1.0 / (-1.0 + (2.0 / (1.0 + exp((-2.0 * x))))));
	elseif ((-2.0 * x) <= 0.002)
		tmp = x * (1.0 + (x * (x * (-0.3333333333333333 + ((x * x) * 0.13333333333333333)))));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.04], N[(1.0 / N[(1.0 / N[(-1.0 + N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.002], N[(x * N[(1.0 + N[(x * N[(x * N[(-0.3333333333333333 + N[(N[(x * x), $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) < -0.0400000000000000008

   1. Initial program 99.8%

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

      1. flip3-- N/A

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

      2. clear-num N/A

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

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

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

      4. clear-num N/A

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

      5. flip3-- N/A

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

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

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

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \left(\frac{2}{1 + e^{-2 \cdot x}} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(\frac{2}{1 + e^{-2 \cdot x}}\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

   4. Applied egg-rr 99.8%

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

   if -0.0400000000000000008 < (*.f64 #s(literal -2 binary64) x) < 2e-3

   1. Initial program 6.5%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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. *-lowering-*.f64 N/A

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

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

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

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

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   if 2e-3 < (*.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 97.8%

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

   5. Simplified 97.8%

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

   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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.04:\\ \;\;\;\;\frac{1}{\frac{1}{-1 + \frac{2}{1 + e^{-2 \cdot x}}}}\\ \mathbf{elif}\;-2 \cdot x \leq 0.002:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(x \cdot \left(-0.3333333333333333 + \left(x \cdot x\right) \cdot 0.13333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.04:\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{elif}\;-2 \cdot x \leq 0.002:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(x \cdot \left(-0.3333333333333333 + \left(x \cdot x\right) \cdot 0.13333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -0.04)
   (+ -1.0 (/ 2.0 (+ 1.0 (exp (* -2.0 x)))))
   (if (<= (* -2.0 x) 0.002)
     (*
      x
      (+
       1.0
       (* x (* x (+ -0.3333333333333333 (* (* x x) 0.13333333333333333))))))
     -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.04) {
		tmp = -1.0 + (2.0 / (1.0 + exp((-2.0 * x))));
	} else if ((-2.0 * x) <= 0.002) {
		tmp = x * (1.0 + (x * (x * (-0.3333333333333333 + ((x * x) * 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) <= (-0.04d0)) then
        tmp = (-1.0d0) + (2.0d0 / (1.0d0 + exp(((-2.0d0) * x))))
    else if (((-2.0d0) * x) <= 0.002d0) then
        tmp = x * (1.0d0 + (x * (x * ((-0.3333333333333333d0) + ((x * x) * 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) <= -0.04) {
		tmp = -1.0 + (2.0 / (1.0 + Math.exp((-2.0 * x))));
	} else if ((-2.0 * x) <= 0.002) {
		tmp = x * (1.0 + (x * (x * (-0.3333333333333333 + ((x * x) * 0.13333333333333333)))));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (-2.0 * x) <= -0.04:
		tmp = -1.0 + (2.0 / (1.0 + math.exp((-2.0 * x))))
	elif (-2.0 * x) <= 0.002:
		tmp = x * (1.0 + (x * (x * (-0.3333333333333333 + ((x * x) * 0.13333333333333333)))))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.04)
		tmp = Float64(-1.0 + Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))));
	elseif (Float64(-2.0 * x) <= 0.002)
		tmp = Float64(x * Float64(1.0 + Float64(x * Float64(x * Float64(-0.3333333333333333 + Float64(Float64(x * x) * 0.13333333333333333))))));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((-2.0 * x) <= -0.04)
		tmp = -1.0 + (2.0 / (1.0 + exp((-2.0 * x))));
	elseif ((-2.0 * x) <= 0.002)
		tmp = x * (1.0 + (x * (x * (-0.3333333333333333 + ((x * x) * 0.13333333333333333)))));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.04], 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], 0.002], N[(x * N[(1.0 + N[(x * N[(x * N[(-0.3333333333333333 + N[(N[(x * x), $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) < -0.0400000000000000008

   1. Initial program 99.8%

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

   if -0.0400000000000000008 < (*.f64 #s(literal -2 binary64) x) < 2e-3

   1. Initial program 6.5%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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. *-lowering-*.f64 N/A

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

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

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

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

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   if 2e-3 < (*.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 97.8%

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

   5. Simplified 97.8%

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

   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 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.04:\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{elif}\;-2 \cdot x \leq 0.002:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(x \cdot \left(-0.3333333333333333 + \left(x \cdot x\right) \cdot 0.13333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.6% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.1:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1 + \left(x \cdot x\right) \cdot \left(0.3333333333333333 + \left(x \cdot x\right) \cdot \left(-0.022222222222222223 + \left(x \cdot x\right) \cdot 0.0021164021164021165\right)\right)}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.1)
   -1.0
   (/
    1.0
    (/
     (+
      1.0
      (*
       (* x x)
       (+
        0.3333333333333333
        (*
         (* x x)
         (+ -0.022222222222222223 (* (* x x) 0.0021164021164021165))))))
     x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.1) {
		tmp = -1.0;
	} else {
		tmp = 1.0 / ((1.0 + ((x * x) * (0.3333333333333333 + ((x * x) * (-0.022222222222222223 + ((x * x) * 0.0021164021164021165)))))) / 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 <= (-2.1d0)) then
        tmp = -1.0d0
    else
        tmp = 1.0d0 / ((1.0d0 + ((x * x) * (0.3333333333333333d0 + ((x * x) * ((-0.022222222222222223d0) + ((x * x) * 0.0021164021164021165d0)))))) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.1) {
		tmp = -1.0;
	} else {
		tmp = 1.0 / ((1.0 + ((x * x) * (0.3333333333333333 + ((x * x) * (-0.022222222222222223 + ((x * x) * 0.0021164021164021165)))))) / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.1:
		tmp = -1.0
	else:
		tmp = 1.0 / ((1.0 + ((x * x) * (0.3333333333333333 + ((x * x) * (-0.022222222222222223 + ((x * x) * 0.0021164021164021165)))))) / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.1)
		tmp = -1.0;
	else
		tmp = Float64(1.0 / Float64(Float64(1.0 + Float64(Float64(x * x) * Float64(0.3333333333333333 + Float64(Float64(x * x) * Float64(-0.022222222222222223 + Float64(Float64(x * x) * 0.0021164021164021165)))))) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.1)
		tmp = -1.0;
	else
		tmp = 1.0 / ((1.0 + ((x * x) * (0.3333333333333333 + ((x * x) * (-0.022222222222222223 + ((x * x) * 0.0021164021164021165)))))) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.1], -1.0, N[(1.0 / N[(N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.3333333333333333 + N[(N[(x * x), $MachinePrecision] * N[(-0.022222222222222223 + N[(N[(x * x), $MachinePrecision] * 0.0021164021164021165), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.10000000000000009

   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 97.8%

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

   5. Simplified 97.8%

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

   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 -2.10000000000000009 < x

   1. Initial program 41.4%

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

      1. flip3-- N/A

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

      2. clear-num N/A

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

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

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

      4. clear-num N/A

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

      5. flip3-- N/A

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

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

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

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \left(\frac{2}{1 + e^{-2 \cdot x}} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(\frac{2}{1 + e^{-2 \cdot x}}\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

   4. Applied egg-rr 41.4%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1 + {x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{2}{945} \cdot {x}^{2} - \frac{1}{45}\right)\right)}{x}\right)}\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(1 + {x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{2}{945} \cdot {x}^{2} - \frac{1}{45}\right)\right)\right), \color{blue}{x}\right)\right) \]

   7. Simplified 65.1%

      \[\leadsto \frac{1}{\color{blue}{\frac{1 + \left(x \cdot x\right) \cdot \left(0.3333333333333333 + \left(x \cdot x\right) \cdot \left(-0.022222222222222223 + \left(x \cdot x\right) \cdot 0.0021164021164021165\right)\right)}{x}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 74.8% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.65:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1 + \left(x \cdot x\right) \cdot 0.3333333333333333}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.65) -1.0 (/ 1.0 (/ (+ 1.0 (* (* x x) 0.3333333333333333)) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.65) {
		tmp = -1.0;
	} else {
		tmp = 1.0 / ((1.0 + ((x * x) * 0.3333333333333333)) / 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 <= (-1.65d0)) then
        tmp = -1.0d0
    else
        tmp = 1.0d0 / ((1.0d0 + ((x * x) * 0.3333333333333333d0)) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.65) {
		tmp = -1.0;
	} else {
		tmp = 1.0 / ((1.0 + ((x * x) * 0.3333333333333333)) / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.65:
		tmp = -1.0
	else:
		tmp = 1.0 / ((1.0 + ((x * x) * 0.3333333333333333)) / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.65)
		tmp = -1.0;
	else
		tmp = Float64(1.0 / Float64(Float64(1.0 + Float64(Float64(x * x) * 0.3333333333333333)) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.65)
		tmp = -1.0;
	else
		tmp = 1.0 / ((1.0 + ((x * x) * 0.3333333333333333)) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.65], -1.0, N[(1.0 / N[(N[(1.0 + N[(N[(x * x), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.6499999999999999

   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 97.8%

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

   5. Simplified 97.8%

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

   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 -1.6499999999999999 < x

   1. Initial program 41.4%

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

      1. flip3-- N/A

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

      2. clear-num N/A

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

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

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

      4. clear-num N/A

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

      5. flip3-- N/A

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

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

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

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \left(\frac{2}{1 + e^{-2 \cdot x}} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(\frac{2}{1 + e^{-2 \cdot x}}\right), \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

   4. Applied egg-rr 41.4%

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

   5. Taylor expanded in x around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 65.1%

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

   7. Simplified 65.1%

      \[\leadsto \frac{1}{\color{blue}{\frac{1 + \left(x \cdot x\right) \cdot 0.3333333333333333}{x}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 74.8% accurate, 18.1× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \color{blue}{\left(2 + -2 \cdot x\right)}\right), 1\right) \]
   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 97.8%

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

   5. Simplified 97.8%

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

   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

   1. Initial program 41.4%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 64.8%

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

Alternative 9: 26.9% 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 59.1%

   \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
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 32.7%

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

5. Simplified 32.7%

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

6. Taylor expanded in x around inf

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

8. Simplified 32.1%

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

Reproduce

herbie shell --seed 2024138 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))