Logistic function from Lakshay Garg

Percentage Accurate: 54.6% → 99.0%

Time: 7.2s

Alternatives: 4

Speedup: 18.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.6% 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: 99.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -10000000000:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.05:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.3333333333333333 + \left(x \cdot x\right) \cdot \left(0.13333333333333333 + \left(x \cdot x\right) \cdot -0.05396825396825397\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -10000000000.0)
   (+ (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) -1.0)
   (if (<= (* -2.0 x) 0.05)
     (*
      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) <= -10000000000.0) {
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	} else if ((-2.0 * x) <= 0.05) {
		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) <= (-10000000000.0d0)) then
        tmp = (2.0d0 / (1.0d0 + exp(((-2.0d0) * x)))) + (-1.0d0)
    else if (((-2.0d0) * x) <= 0.05d0) 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) <= -10000000000.0) {
		tmp = (2.0 / (1.0 + Math.exp((-2.0 * x)))) + -1.0;
	} else if ((-2.0 * x) <= 0.05) {
		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) <= -10000000000.0:
		tmp = (2.0 / (1.0 + math.exp((-2.0 * x)))) + -1.0
	elif (-2.0 * x) <= 0.05:
		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) <= -10000000000.0)
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) + -1.0);
	elseif (Float64(-2.0 * x) <= 0.05)
		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) <= -10000000000.0)
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	elseif ((-2.0 * x) <= 0.05)
		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], -10000000000.0], 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], 0.05], 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) < -1e10

   1. Initial program 100.0%

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

   if -1e10 < (*.f64 #s(literal -2 binary64) x) < 0.050000000000000003

   1. Initial program 7.7%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

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

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

      10. *-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 0.050000000000000003 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 97.4%

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

   5. Simplified 97.4%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -10000000000:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.05:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.3333333333333333 + \left(x \cdot x\right) \cdot \left(0.13333333333333333 + \left(x \cdot x\right) \cdot -0.05396825396825397\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.1% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x \cdot \left(x \cdot 0.3333333333333333\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.7) -1.0 (/ x (+ 1.0 (* x (* x 0.3333333333333333))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.7) {
		tmp = -1.0;
	} else {
		tmp = x / (1.0 + (x * (x * 0.3333333333333333)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.7d0)) then
        tmp = -1.0d0
    else
        tmp = x / (1.0d0 + (x * (x * 0.3333333333333333d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.7) {
		tmp = -1.0;
	} else {
		tmp = x / (1.0 + (x * (x * 0.3333333333333333)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.7:
		tmp = -1.0
	else:
		tmp = x / (1.0 + (x * (x * 0.3333333333333333)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.7)
		tmp = -1.0;
	else
		tmp = Float64(x / Float64(1.0 + Float64(x * Float64(x * 0.3333333333333333))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.7)
		tmp = -1.0;
	else
		tmp = x / (1.0 + (x * (x * 0.3333333333333333)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.7], -1.0, N[(x / N[(1.0 + N[(x * N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.69999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 97.4%

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

   5. Simplified 97.4%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 100.0%

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

   if -1.69999999999999996 < x

   1. Initial program 36.7%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

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

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 69.6%

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

   5. Simplified 69.6%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

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

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

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

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

   7. Applied egg-rr 69.6%

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

   8. Taylor expanded in x around 0

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

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

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-lowering-*.f64 69.9%

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

   10. Simplified 69.9%

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

Alternative 3: 75.8% accurate, 18.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 97.4%

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

   5. Simplified 97.4%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 100.0%

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

   if -1 < x

   1. Initial program 36.7%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 69.6%

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

Alternative 4: 27.7% 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 53.5%

   \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
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 29.7%

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

5. Simplified 29.7%

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

6. Taylor expanded in x around inf

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

8. Simplified 29.1%

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

Reproduce

herbie shell --seed 2024139 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))