Logistic function from Lakshay Garg

Percentage Accurate: 53.8% → 99.7%

Time: 10.6s

Alternatives: 7

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: 53.8% 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, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -50:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.004:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.3333333333333333 + \left(x \cdot x\right) \cdot \left(0.13333333333333333 + \left(x \cdot x\right) \cdot -0.05396825396825397\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -50.0)
   (+ (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) -1.0)
   (if (<= (* -2.0 x) 0.004)
     (*
      x
      (+
       1.0
       (*
        (* x x)
        (+
         -0.3333333333333333
         (*
          (* x x)
          (+ 0.13333333333333333 (* (* x x) -0.05396825396825397)))))))
     -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -50.0) {
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	} else if ((-2.0 * x) <= 0.004) {
		tmp = x * (1.0 + ((x * x) * (-0.3333333333333333 + ((x * x) * (0.13333333333333333 + ((x * x) * -0.05396825396825397))))));
	} 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) <= (-50.0d0)) then
        tmp = (2.0d0 / (1.0d0 + exp(((-2.0d0) * x)))) + (-1.0d0)
    else if (((-2.0d0) * x) <= 0.004d0) then
        tmp = x * (1.0d0 + ((x * x) * ((-0.3333333333333333d0) + ((x * x) * (0.13333333333333333d0 + ((x * x) * (-0.05396825396825397d0)))))))
    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) <= -50.0) {
		tmp = (2.0 / (1.0 + Math.exp((-2.0 * x)))) + -1.0;
	} else if ((-2.0 * x) <= 0.004) {
		tmp = x * (1.0 + ((x * x) * (-0.3333333333333333 + ((x * x) * (0.13333333333333333 + ((x * x) * -0.05396825396825397))))));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (-2.0 * x) <= -50.0:
		tmp = (2.0 / (1.0 + math.exp((-2.0 * x)))) + -1.0
	elif (-2.0 * x) <= 0.004:
		tmp = x * (1.0 + ((x * x) * (-0.3333333333333333 + ((x * x) * (0.13333333333333333 + ((x * x) * -0.05396825396825397))))))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -50.0)
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) + -1.0);
	elseif (Float64(-2.0 * x) <= 0.004)
		tmp = Float64(x * Float64(1.0 + Float64(Float64(x * x) * Float64(-0.3333333333333333 + Float64(Float64(x * x) * Float64(0.13333333333333333 + Float64(Float64(x * x) * -0.05396825396825397)))))));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((-2.0 * x) <= -50.0)
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	elseif ((-2.0 * x) <= 0.004)
		tmp = x * (1.0 + ((x * x) * (-0.3333333333333333 + ((x * x) * (0.13333333333333333 + ((x * x) * -0.05396825396825397))))));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -50.0], N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.004], N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(-0.3333333333333333 + N[(N[(x * x), $MachinePrecision] * N[(0.13333333333333333 + N[(N[(x * x), $MachinePrecision] * -0.05396825396825397), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -50

   1. Initial program 100.0%

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

   if -50 < (*.f64 #s(literal -2 binary64) x) < 0.0040000000000000001

   1. Initial program 10.0%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

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

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

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

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

      11. unpow2 N/A

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

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

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

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

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

      14. *-commutative N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   if 0.0040000000000000001 < (*.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. metadata-eval N/A

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

      2. cancel-sign-sub-inv N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 95.4%

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

   5. Simplified 95.4%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -50:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.004:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.3333333333333333 + \left(x \cdot x\right) \cdot \left(0.13333333333333333 + \left(x \cdot x\right) \cdot -0.05396825396825397\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.8% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 1.85:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;2 + \frac{-4}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.15)
   -1.0
   (if (<= x 1.85)
     (* x (+ 1.0 (* (* x x) -0.3333333333333333)))
     (+ 2.0 (/ -4.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.15) {
		tmp = -1.0;
	} else if (x <= 1.85) {
		tmp = x * (1.0 + ((x * x) * -0.3333333333333333));
	} 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 <= (-1.15d0)) then
        tmp = -1.0d0
    else if (x <= 1.85d0) then
        tmp = x * (1.0d0 + ((x * x) * (-0.3333333333333333d0)))
    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 <= -1.15) {
		tmp = -1.0;
	} else if (x <= 1.85) {
		tmp = x * (1.0 + ((x * x) * -0.3333333333333333));
	} else {
		tmp = 2.0 + (-4.0 / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.15:
		tmp = -1.0
	elif x <= 1.85:
		tmp = x * (1.0 + ((x * x) * -0.3333333333333333))
	else:
		tmp = 2.0 + (-4.0 / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.15)
		tmp = -1.0;
	elseif (x <= 1.85)
		tmp = Float64(x * Float64(1.0 + Float64(Float64(x * x) * -0.3333333333333333)));
	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 <= -1.15)
		tmp = -1.0;
	elseif (x <= 1.85)
		tmp = x * (1.0 + ((x * x) * -0.3333333333333333));
	else
		tmp = 2.0 + (-4.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.15], -1.0, If[LessEqual[x, 1.85], N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 + N[(-4.0 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.1499999999999999

   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. metadata-eval N/A

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

      2. cancel-sign-sub-inv N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 95.4%

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

   5. Simplified 95.4%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 100.0%

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

   if -1.1499999999999999 < x < 1.8500000000000001

   1. Initial program 10.0%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 99.6%

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

   5. Simplified 99.6%

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

   if 1.8500000000000001 < 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 5.5%

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

   5. Simplified 5.5%

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

      1. flip-- N/A

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

      2. div-inv N/A

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

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

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

      4. metadata-eval N/A

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

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

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

      6. associate--l+ N/A

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

      7. metadata-eval N/A

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

      8. +-rgt-identity N/A

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

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

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

      10. associate-+l+ N/A

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

      11. metadata-eval N/A

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

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

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

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

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

      14. associate-+l+ N/A

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

      15. metadata-eval N/A

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

      16. +-lowering-+.f64 5.1%

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

   7. Applied egg-rr 5.1%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 18.8%

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

   10. Simplified 18.8%

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

   11. Taylor expanded in x around inf

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

       1. sub-neg N/A

          \[\leadsto 2 + \color{blue}{\left(\mathsf{neg}\left(4 \cdot \frac{1}{x}\right)\right)} \]

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

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

       3. associate-*r/ N/A

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

       4. metadata-eval N/A

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

       5. distribute-neg-frac N/A

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

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

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

       7. metadata-eval 18.8%

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

   13. Simplified 18.8%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 1.85:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;2 + \frac{-4}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.5% 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. metadata-eval N/A

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

      2. cancel-sign-sub-inv N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 95.4%

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

   5. Simplified 95.4%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 100.0%

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

   if -1 < x < 2.60000000000000009

   1. Initial program 10.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 98.6%

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

   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 5.5%

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

   5. Simplified 5.5%

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

      1. flip-- N/A

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

      2. div-inv N/A

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

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

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

      4. metadata-eval N/A

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

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

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

      6. associate--l+ N/A

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

      7. metadata-eval N/A

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

      8. +-rgt-identity N/A

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

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

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

      10. associate-+l+ N/A

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

      11. metadata-eval N/A

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

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

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

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

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

      14. associate-+l+ N/A

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

      15. metadata-eval N/A

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

      16. +-lowering-+.f64 5.1%

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

   7. Applied egg-rr 5.1%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 18.8%

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

   10. Simplified 18.8%

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

   11. Taylor expanded in x around inf

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

       1. sub-neg N/A

          \[\leadsto 2 + \color{blue}{\left(\mathsf{neg}\left(4 \cdot \frac{1}{x}\right)\right)} \]

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

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

       3. associate-*r/ N/A

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

       4. metadata-eval N/A

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

       5. distribute-neg-frac N/A

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

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

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

       7. metadata-eval 18.8%

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

   13. Simplified 18.8%

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

Alternative 4: 79.0% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.65:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2\right) \cdot \frac{1}{x + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.65) -1.0 (* (* x 2.0) (/ 1.0 (+ x 2.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.65) {
		tmp = -1.0;
	} else {
		tmp = (x * 2.0) * (1.0 / (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 <= (-0.65d0)) then
        tmp = -1.0d0
    else
        tmp = (x * 2.0d0) * (1.0d0 / (x + 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.65) {
		tmp = -1.0;
	} else {
		tmp = (x * 2.0) * (1.0 / (x + 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.65:
		tmp = -1.0
	else:
		tmp = (x * 2.0) * (1.0 / (x + 2.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.65)
		tmp = -1.0;
	else
		tmp = Float64(Float64(x * 2.0) * Float64(1.0 / Float64(x + 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.65)
		tmp = -1.0;
	else
		tmp = (x * 2.0) * (1.0 / (x + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.65], -1.0, N[(N[(x * 2.0), $MachinePrecision] * N[(1.0 / N[(x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.650000000000000022

   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. metadata-eval N/A

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

      2. cancel-sign-sub-inv N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 95.4%

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

   5. Simplified 95.4%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 100.0%

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

   if -0.650000000000000022 < x

   1. Initial program 38.0%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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 8.0%

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

   5. Simplified 8.0%

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

      1. flip-- N/A

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

      2. div-inv N/A

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

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

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

      4. metadata-eval N/A

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

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

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

      6. associate--l+ N/A

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

      7. metadata-eval N/A

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

      8. +-rgt-identity N/A

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

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

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

      10. associate-+l+ N/A

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

      11. metadata-eval N/A

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

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

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

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

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

      14. associate-+l+ N/A

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

      15. metadata-eval N/A

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

      16. +-lowering-+.f64 69.6%

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

   7. Applied egg-rr 69.6%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 72.7%

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

   10. Simplified 72.7%

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

Alternative 5: 79.5% 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. metadata-eval N/A

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

      2. cancel-sign-sub-inv N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 95.4%

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

   5. Simplified 95.4%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 100.0%

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

   if -1 < x < 2

   1. Initial program 10.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 98.6%

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

   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 5.5%

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

   5. Simplified 5.5%

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

      1. flip-- N/A

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

      2. div-inv N/A

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

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

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

      4. metadata-eval N/A

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

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

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

      6. associate--l+ N/A

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

      7. metadata-eval N/A

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

      8. +-rgt-identity N/A

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

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

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

      10. associate-+l+ N/A

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

      11. metadata-eval N/A

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

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

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

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

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

      14. associate-+l+ N/A

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

      15. metadata-eval N/A

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

      16. +-lowering-+.f64 5.1%

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

   7. Applied egg-rr 5.1%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 18.8%

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

   10. Simplified 18.8%

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

   11. Taylor expanded in x around inf

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

   13. Simplified 18.8%

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

Alternative 6: 32.3% accurate, 18.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001e-308

   1. Initial program 54.0%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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. metadata-eval N/A

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

      2. cancel-sign-sub-inv N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 51.3%

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

   5. Simplified 51.3%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 52.2%

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

   if 1.1000000000000001e-308 < x

   1. Initial program 54.0%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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 7.9%

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

   5. Simplified 7.9%

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

      1. flip-- N/A

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

      2. div-inv N/A

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

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

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

      4. metadata-eval N/A

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

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

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

      6. associate--l+ N/A

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

      7. metadata-eval N/A

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

      8. +-rgt-identity N/A

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

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

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

      10. associate-+l+ N/A

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

      11. metadata-eval N/A

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

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

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

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

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

      14. associate-+l+ N/A

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

      15. metadata-eval N/A

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

      16. +-lowering-+.f64 53.4%

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

   7. Applied egg-rr 53.4%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 58.7%

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

   10. Simplified 58.7%

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

   11. Taylor expanded in x around inf

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

   13. Simplified 12.1%

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

Alternative 7: 27.3% 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 54.0%

   \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
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. metadata-eval N/A

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

   2. cancel-sign-sub-inv N/A

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

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

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

   4. *-commutative N/A

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

   5. *-lowering-*.f64 29.3%

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

5. Simplified 29.3%

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

6. Taylor expanded in x around inf

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

8. Simplified 28.0%

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

Reproduce

herbie shell --seed 2024192 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))