Logistic function from Lakshay Garg

Percentage Accurate: 53.6% → 99.7%

Time: 10.5s

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: 53.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 - e^{-2 \cdot x}\\ \mathbf{if}\;-2 \cdot x \leq -0.1:\\ \;\;\;\;\frac{1 - \frac{4}{{t\_0}^{2}}}{-1 + \frac{2}{t\_0}}\\ \mathbf{elif}\;-2 \cdot x \leq 0.005:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.3333333333333333 + \left(x \cdot x\right) \cdot 0.13333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- -1.0 (exp (* -2.0 x)))))
   (if (<= (* -2.0 x) -0.1)
     (/ (- 1.0 (/ 4.0 (pow t_0 2.0))) (+ -1.0 (/ 2.0 t_0)))
     (if (<= (* -2.0 x) 0.005)
       (*
        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) <= -0.1) {
		tmp = (1.0 - (4.0 / pow(t_0, 2.0))) / (-1.0 + (2.0 / t_0));
	} else if ((-2.0 * x) <= 0.005) {
		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) <= (-0.1d0)) then
        tmp = (1.0d0 - (4.0d0 / (t_0 ** 2.0d0))) / ((-1.0d0) + (2.0d0 / t_0))
    else if (((-2.0d0) * x) <= 0.005d0) 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) <= -0.1) {
		tmp = (1.0 - (4.0 / Math.pow(t_0, 2.0))) / (-1.0 + (2.0 / t_0));
	} else if ((-2.0 * x) <= 0.005) {
		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) <= -0.1:
		tmp = (1.0 - (4.0 / math.pow(t_0, 2.0))) / (-1.0 + (2.0 / t_0))
	elif (-2.0 * x) <= 0.005:
		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) <= -0.1)
		tmp = Float64(Float64(1.0 - Float64(4.0 / (t_0 ^ 2.0))) / Float64(-1.0 + Float64(2.0 / t_0)));
	elseif (Float64(-2.0 * x) <= 0.005)
		tmp = Float64(x * Float64(1.0 + Float64(Float64(x * 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) <= -0.1)
		tmp = (1.0 - (4.0 / (t_0 ^ 2.0))) / (-1.0 + (2.0 / t_0));
	elseif ((-2.0 * x) <= 0.005)
		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], -0.1], N[(N[(1.0 - N[(4.0 / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + N[(2.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.005], N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(-0.3333333333333333 + N[(N[(x * x), $MachinePrecision] * 0.13333333333333333), $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 100.0%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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 100.0%

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

   if -0.10000000000000001 < (*.f64 #s(literal -2 binary64) x) < 0.0050000000000000001

   1. Initial program 7.8%

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

   3. Taylor expanded in x around 0

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

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

      4. unpow2 N/A

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

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

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

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

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

      10. *-commutative N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   if 0.0050000000000000001 < (*.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 98.8%

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

   5. Simplified 98.8%

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

   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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.1:\\ \;\;\;\;\frac{1 - \frac{4}{{\left(-1 - e^{-2 \cdot x}\right)}^{2}}}{-1 + \frac{2}{-1 - e^{-2 \cdot x}}}\\ \mathbf{elif}\;-2 \cdot x \leq 0.005:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.3333333333333333 + \left(x \cdot x\right) \cdot 0.13333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.1:\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{elif}\;-2 \cdot x \leq 0.005:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.3333333333333333 + \left(x \cdot x\right) \cdot 0.13333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -0.1)
   (+ -1.0 (/ 2.0 (+ 1.0 (exp (* -2.0 x)))))
   (if (<= (* -2.0 x) 0.005)
     (*
      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) <= -0.1) {
		tmp = -1.0 + (2.0 / (1.0 + exp((-2.0 * x))));
	} else if ((-2.0 * x) <= 0.005) {
		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) <= (-0.1d0)) then
        tmp = (-1.0d0) + (2.0d0 / (1.0d0 + exp(((-2.0d0) * x))))
    else if (((-2.0d0) * x) <= 0.005d0) 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) <= -0.1) {
		tmp = -1.0 + (2.0 / (1.0 + Math.exp((-2.0 * x))));
	} else if ((-2.0 * x) <= 0.005) {
		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) <= -0.1:
		tmp = -1.0 + (2.0 / (1.0 + math.exp((-2.0 * x))))
	elif (-2.0 * x) <= 0.005:
		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) <= -0.1)
		tmp = Float64(-1.0 + Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))));
	elseif (Float64(-2.0 * x) <= 0.005)
		tmp = Float64(x * Float64(1.0 + Float64(Float64(x * 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) <= -0.1)
		tmp = -1.0 + (2.0 / (1.0 + exp((-2.0 * x))));
	elseif ((-2.0 * x) <= 0.005)
		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], -0.1], 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], 0.005], N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(-0.3333333333333333 + N[(N[(x * x), $MachinePrecision] * 0.13333333333333333), $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 100.0%

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

   if -0.10000000000000001 < (*.f64 #s(literal -2 binary64) x) < 0.0050000000000000001

   1. Initial program 7.8%

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

   3. Taylor expanded in x around 0

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

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

      4. unpow2 N/A

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

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

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

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

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

      10. *-commutative N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   if 0.0050000000000000001 < (*.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 98.8%

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

   5. Simplified 98.8%

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

   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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.1:\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{elif}\;-2 \cdot x \leq 0.005:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.3333333333333333 + \left(x \cdot x\right) \cdot 0.13333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.3333333333333333 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot -0.05396825396825397\right) - -0.13333333333333333\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;1 + \frac{-1 + x \cdot x}{1 + x \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.3)
   -1.0
   (if (<= x 1.4)
     (*
      x
      (+
       1.0
       (*
        (* x x)
        (+
         -0.3333333333333333
         (*
          (* x x)
          (- (* x (* x -0.05396825396825397)) -0.13333333333333333))))))
     (if (<= x 1.35e+154)
       (+ 1.0 (/ (+ -1.0 (* x x)) (+ 1.0 (* x (* x x)))))
       2.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.3) {
		tmp = -1.0;
	} else if (x <= 1.4) {
		tmp = x * (1.0 + ((x * x) * (-0.3333333333333333 + ((x * x) * ((x * (x * -0.05396825396825397)) - -0.13333333333333333)))));
	} else if (x <= 1.35e+154) {
		tmp = 1.0 + ((-1.0 + (x * x)) / (1.0 + (x * (x * 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 <= (-1.3d0)) then
        tmp = -1.0d0
    else if (x <= 1.4d0) then
        tmp = x * (1.0d0 + ((x * x) * ((-0.3333333333333333d0) + ((x * x) * ((x * (x * (-0.05396825396825397d0))) - (-0.13333333333333333d0))))))
    else if (x <= 1.35d+154) then
        tmp = 1.0d0 + (((-1.0d0) + (x * x)) / (1.0d0 + (x * (x * x))))
    else
        tmp = 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.3) {
		tmp = -1.0;
	} else if (x <= 1.4) {
		tmp = x * (1.0 + ((x * x) * (-0.3333333333333333 + ((x * x) * ((x * (x * -0.05396825396825397)) - -0.13333333333333333)))));
	} else if (x <= 1.35e+154) {
		tmp = 1.0 + ((-1.0 + (x * x)) / (1.0 + (x * (x * x))));
	} else {
		tmp = 2.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.3:
		tmp = -1.0
	elif x <= 1.4:
		tmp = x * (1.0 + ((x * x) * (-0.3333333333333333 + ((x * x) * ((x * (x * -0.05396825396825397)) - -0.13333333333333333)))))
	elif x <= 1.35e+154:
		tmp = 1.0 + ((-1.0 + (x * x)) / (1.0 + (x * (x * x))))
	else:
		tmp = 2.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.3)
		tmp = -1.0;
	elseif (x <= 1.4)
		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))))));
	elseif (x <= 1.35e+154)
		tmp = Float64(1.0 + Float64(Float64(-1.0 + Float64(x * x)) / Float64(1.0 + Float64(x * Float64(x * x)))));
	else
		tmp = 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.3)
		tmp = -1.0;
	elseif (x <= 1.4)
		tmp = x * (1.0 + ((x * x) * (-0.3333333333333333 + ((x * x) * ((x * (x * -0.05396825396825397)) - -0.13333333333333333)))));
	elseif (x <= 1.35e+154)
		tmp = 1.0 + ((-1.0 + (x * x)) / (1.0 + (x * (x * x))));
	else
		tmp = 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.3], -1.0, If[LessEqual[x, 1.4], 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], If[LessEqual[x, 1.35e+154], N[(1.0 + N[(N[(-1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.30000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \color{blue}{\left(2 + -2 \cdot x\right)}\right), 1\right) \]
   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 98.8%

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

   5. Simplified 98.8%

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

   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 -1.30000000000000004 < x < 1.3999999999999999

   1. Initial program 8.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)\right)} \]
   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 99.8%

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

   if 1.3999999999999999 < x < 1.35000000000000003e154

   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 6.8%

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

   5. Simplified 6.8%

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

      1. +-commutative N/A

         \[\leadsto \left(1 + x\right) - 1 \]

      2. associate--l+ N/A

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

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

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

      4. sub-neg N/A

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

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

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

      6. metadata-eval 6.8%

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

   7. Applied egg-rr 6.8%

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

      1. *-lft-identity N/A

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

      2. flip-+ N/A

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

      3. associate-*r/ N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. flip3-+ N/A

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

      8. metadata-eval N/A

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

      9. cube-unmult N/A

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

      10. metadata-eval N/A

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

      11. *-lft-identity N/A

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

      12. associate--l+ N/A

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

      13. +-commutative N/A

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

      14. un-div-inv N/A

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

      15. *-commutative N/A

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

      16. times-frac N/A

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

      17. clear-num N/A

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

      18. /-rgt-identity N/A

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

   9. Applied egg-rr 40.3%

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

   10. Taylor expanded in x around 0

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

   12. Simplified 96.7%

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

   if 1.35000000000000003e154 < 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 3.6%

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

   5. Simplified 3.6%

      \[\leadsto \color{blue}{\left(x + 1\right)} - 1 \]
   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. clear-num N/A

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

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

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

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

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

      5. associate-+l+ N/A

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

      6. metadata-eval N/A

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

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

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

      8. metadata-eval N/A

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

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

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

      10. associate--l+ N/A

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

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, 2\right), \left(\left(\left(x + 1\right) + 1\right) \cdot \left(x + 0\right)\right)\right)\right) \]

      12. +-rgt-identity N/A

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

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

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

      14. associate-+l+ N/A

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

      15. metadata-eval N/A

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

      16. +-lowering-+.f64 3.1%

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

   7. Applied egg-rr 3.1%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 18.8%

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

   10. Simplified 18.8%

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

   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 4 regimes into one program.

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.3333333333333333 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot -0.05396825396825397\right) - -0.13333333333333333\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;1 + \frac{-1 + x \cdot x}{1 + x \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.2% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 1.45:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.3333333333333333 + \left(x \cdot x\right) \cdot 0.13333333333333333\right)\right)\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;1 + \frac{-1 + x \cdot x}{1 + x \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.25)
   -1.0
   (if (<= x 1.45)
     (*
      x
      (+
       1.0
       (* (* x x) (+ -0.3333333333333333 (* (* x x) 0.13333333333333333)))))
     (if (<= x 1.35e+154)
       (+ 1.0 (/ (+ -1.0 (* x x)) (+ 1.0 (* x (* x x)))))
       2.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.25) {
		tmp = -1.0;
	} else if (x <= 1.45) {
		tmp = x * (1.0 + ((x * x) * (-0.3333333333333333 + ((x * x) * 0.13333333333333333))));
	} else if (x <= 1.35e+154) {
		tmp = 1.0 + ((-1.0 + (x * x)) / (1.0 + (x * (x * 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 <= (-1.25d0)) then
        tmp = -1.0d0
    else if (x <= 1.45d0) then
        tmp = x * (1.0d0 + ((x * x) * ((-0.3333333333333333d0) + ((x * x) * 0.13333333333333333d0))))
    else if (x <= 1.35d+154) then
        tmp = 1.0d0 + (((-1.0d0) + (x * x)) / (1.0d0 + (x * (x * x))))
    else
        tmp = 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.25) {
		tmp = -1.0;
	} else if (x <= 1.45) {
		tmp = x * (1.0 + ((x * x) * (-0.3333333333333333 + ((x * x) * 0.13333333333333333))));
	} else if (x <= 1.35e+154) {
		tmp = 1.0 + ((-1.0 + (x * x)) / (1.0 + (x * (x * x))));
	} else {
		tmp = 2.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.25:
		tmp = -1.0
	elif x <= 1.45:
		tmp = x * (1.0 + ((x * x) * (-0.3333333333333333 + ((x * x) * 0.13333333333333333))))
	elif x <= 1.35e+154:
		tmp = 1.0 + ((-1.0 + (x * x)) / (1.0 + (x * (x * x))))
	else:
		tmp = 2.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.25)
		tmp = -1.0;
	elseif (x <= 1.45)
		tmp = Float64(x * Float64(1.0 + Float64(Float64(x * x) * Float64(-0.3333333333333333 + Float64(Float64(x * x) * 0.13333333333333333)))));
	elseif (x <= 1.35e+154)
		tmp = Float64(1.0 + Float64(Float64(-1.0 + Float64(x * x)) / Float64(1.0 + Float64(x * Float64(x * x)))));
	else
		tmp = 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.25)
		tmp = -1.0;
	elseif (x <= 1.45)
		tmp = x * (1.0 + ((x * x) * (-0.3333333333333333 + ((x * x) * 0.13333333333333333))));
	elseif (x <= 1.35e+154)
		tmp = 1.0 + ((-1.0 + (x * x)) / (1.0 + (x * (x * x))));
	else
		tmp = 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.25], -1.0, If[LessEqual[x, 1.45], N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(-0.3333333333333333 + N[(N[(x * x), $MachinePrecision] * 0.13333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.35e+154], N[(1.0 + N[(N[(-1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.25

   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 98.8%

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

   5. Simplified 98.8%

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

   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 -1.25 < x < 1.44999999999999996

   1. Initial program 8.5%

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

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

      4. unpow2 N/A

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

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

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

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

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

      10. *-commutative N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 99.7%

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

   5. Simplified 99.7%

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

   if 1.44999999999999996 < x < 1.35000000000000003e154

   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 6.8%

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

   5. Simplified 6.8%

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

      1. +-commutative N/A

         \[\leadsto \left(1 + x\right) - 1 \]

      2. associate--l+ N/A

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

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

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

      4. sub-neg N/A

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

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

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

      6. metadata-eval 6.8%

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

   7. Applied egg-rr 6.8%

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

      1. *-lft-identity N/A

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

      2. flip-+ N/A

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

      3. associate-*r/ N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. flip3-+ N/A

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

      8. metadata-eval N/A

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

      9. cube-unmult N/A

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

      10. metadata-eval N/A

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

      11. *-lft-identity N/A

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

      12. associate--l+ N/A

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

      13. +-commutative N/A

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

      14. un-div-inv N/A

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

      15. *-commutative N/A

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

      16. times-frac N/A

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

      17. clear-num N/A

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

      18. /-rgt-identity N/A

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

   9. Applied egg-rr 40.3%

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

   10. Taylor expanded in x around 0

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

   12. Simplified 96.7%

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

   if 1.35000000000000003e154 < 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 3.6%

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

   5. Simplified 3.6%

      \[\leadsto \color{blue}{\left(x + 1\right)} - 1 \]
   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. clear-num N/A

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

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

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

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

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

      5. associate-+l+ N/A

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

      6. metadata-eval N/A

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

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

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

      8. metadata-eval N/A

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

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

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

      10. associate--l+ N/A

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

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, 2\right), \left(\left(\left(x + 1\right) + 1\right) \cdot \left(x + 0\right)\right)\right)\right) \]

      12. +-rgt-identity N/A

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

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

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

      14. associate-+l+ N/A

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

      15. metadata-eval N/A

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

      16. +-lowering-+.f64 3.1%

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

   7. Applied egg-rr 3.1%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 18.8%

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

   10. Simplified 18.8%

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

   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 4 regimes into one program.

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 1.45:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.3333333333333333 + \left(x \cdot x\right) \cdot 0.13333333333333333\right)\right)\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;1 + \frac{-1 + x \cdot x}{1 + x \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.4% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 2.6:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;2 - \frac{4}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0) -1.0 (if (<= x 2.6) 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 <= 2.6) {
		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 <= 2.6d0) 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 <= 2.6) {
		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 <= 2.6:
		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 <= 2.6)
		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 <= 2.6)
		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, 2.6], 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 98.8%

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

   5. Simplified 98.8%

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

   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 < 2.60000000000000009

   1. Initial program 8.5%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 98.8%

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

   if 2.60000000000000009 < 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 5.3%

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

   5. Simplified 5.3%

      \[\leadsto \color{blue}{\left(x + 1\right)} - 1 \]
   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. clear-num N/A

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

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

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

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

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

      5. associate-+l+ N/A

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

      6. metadata-eval N/A

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

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

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

      8. metadata-eval N/A

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

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

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

      10. associate--l+ N/A

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

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, 2\right), \left(\left(\left(x + 1\right) + 1\right) \cdot \left(x + 0\right)\right)\right)\right) \]

      12. +-rgt-identity N/A

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

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

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

      14. associate-+l+ N/A

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

      15. metadata-eval N/A

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

      16. +-lowering-+.f64 5.0%

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

   7. Applied egg-rr 5.0%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 18.8%

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

   10. Simplified 18.8%

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

   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: 79.9% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.68:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{x + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.68) -1.0 (* x (/ 2.0 (+ x 2.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.68) {
		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 <= (-0.68d0)) 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 <= -0.68) {
		tmp = -1.0;
	} else {
		tmp = x * (2.0 / (x + 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.68:
		tmp = -1.0
	else:
		tmp = x * (2.0 / (x + 2.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.68)
		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 <= -0.68)
		tmp = -1.0;
	else
		tmp = x * (2.0 / (x + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.68], -1.0, N[(x * N[(2.0 / N[(x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.680000000000000049

   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 98.8%

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

   5. Simplified 98.8%

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

   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 -0.680000000000000049 < x

   1. Initial program 38.7%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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 6.8%

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

   5. Simplified 6.8%

      \[\leadsto \color{blue}{\left(x + 1\right)} - 1 \]
   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. clear-num N/A

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

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

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

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

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

      5. associate-+l+ N/A

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

      6. metadata-eval N/A

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

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

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

      8. metadata-eval N/A

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

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

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

      10. associate--l+ N/A

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

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, 2\right), \left(\left(\left(x + 1\right) + 1\right) \cdot \left(x + 0\right)\right)\right)\right) \]

      12. +-rgt-identity N/A

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

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

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

      14. associate-+l+ N/A

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

      15. metadata-eval N/A

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

      16. +-lowering-+.f64 67.7%

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

   7. Applied egg-rr 67.7%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 71.6%

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

   10. Simplified 71.6%

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

       1. clear-num N/A

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

       2. associate-/l* N/A

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

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

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

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

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

       5. +-lowering-+.f64 71.7%

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

   12. Applied egg-rr 71.7%

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

Alternative 7: 80.4% 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 98.8%

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

   5. Simplified 98.8%

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

   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 < 2

   1. Initial program 8.5%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 98.8%

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

   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 5.3%

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

   5. Simplified 5.3%

      \[\leadsto \color{blue}{\left(x + 1\right)} - 1 \]
   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. clear-num N/A

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

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

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

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

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

      5. associate-+l+ N/A

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

      6. metadata-eval N/A

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

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

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

      8. metadata-eval N/A

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

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

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

      10. associate--l+ N/A

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

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, 2\right), \left(\left(\left(x + 1\right) + 1\right) \cdot \left(x + 0\right)\right)\right)\right) \]

      12. +-rgt-identity N/A

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

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

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

      14. associate-+l+ N/A

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

      15. metadata-eval N/A

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

      16. +-lowering-+.f64 5.0%

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

   7. Applied egg-rr 5.0%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 18.8%

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

   10. Simplified 18.8%

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

   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: 33.0% accurate, 18.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.2 \cdot 10^{-308}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 1.2e-308) -1.0 2.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.2e-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 <= 1.2d-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 <= 1.2e-308) {
		tmp = -1.0;
	} else {
		tmp = 2.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.2e-308:
		tmp = -1.0
	else:
		tmp = 2.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.2e-308)
		tmp = -1.0;
	else
		tmp = 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.2e-308)
		tmp = -1.0;
	else
		tmp = 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.2e-308], -1.0, 2.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.1999999999999998e-308

   1. Initial program 48.4%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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 47.1%

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

   5. Simplified 47.1%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 46.4%

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

   if 1.1999999999999998e-308 < x

   1. Initial program 55.8%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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 6.1%

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

   5. Simplified 6.1%

      \[\leadsto \color{blue}{\left(x + 1\right)} - 1 \]
   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. clear-num N/A

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

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

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

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

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

      5. associate-+l+ N/A

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

      6. metadata-eval N/A

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

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

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

      8. metadata-eval N/A

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

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

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

      10. associate--l+ N/A

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

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, 2\right), \left(\left(\left(x + 1\right) + 1\right) \cdot \left(x + 0\right)\right)\right)\right) \]

      12. +-rgt-identity N/A

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

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

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

      14. associate-+l+ N/A

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

      15. metadata-eval N/A

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

      16. +-lowering-+.f64 50.1%

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

   7. Applied egg-rr 50.1%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 56.8%

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

   10. Simplified 56.8%

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

   11. Taylor expanded in x around inf

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

   13. Simplified 12.2%

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

Alternative 9: 28.2% 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 52.1%

   \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
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 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 24.3%

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

Reproduce

herbie shell --seed 2024158 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))