Logistic function from Lakshay Garg

Percentage Accurate: 54.7% → 100.0%

Time: 15.3s

Alternatives: 8

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: 54.7% 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: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + e^{-2 \cdot x}\\ t_1 := \frac{2}{t\_0}\\ t_2 := 1 + t\_1 \cdot \left(1 + t\_1\right)\\ \mathbf{if}\;-2 \cdot x \leq -0.02:\\ \;\;\;\;\frac{{t\_1}^{6} + -1}{{t\_2}^{2}} \cdot \frac{1}{\frac{\frac{8}{{t\_0}^{3}} - -1}{t\_2}}\\ \mathbf{elif}\;-2 \cdot x \leq 0.001:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.3333333333333333 + \left(x \cdot x\right) \cdot 0.13333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + \frac{4}{{t\_0}^{2}}}{t\_1 - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (exp (* -2.0 x))))
        (t_1 (/ 2.0 t_0))
        (t_2 (+ 1.0 (* t_1 (+ 1.0 t_1)))))
   (if (<= (* -2.0 x) -0.02)
     (*
      (/ (+ (pow t_1 6.0) -1.0) (pow t_2 2.0))
      (/ 1.0 (/ (- (/ 8.0 (pow t_0 3.0)) -1.0) t_2)))
     (if (<= (* -2.0 x) 0.001)
       (*
        x
        (+
         1.0
         (* (* x x) (+ -0.3333333333333333 (* (* x x) 0.13333333333333333)))))
       (/ (+ -1.0 (/ 4.0 (pow t_0 2.0))) (- t_1 -1.0))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + exp((-2.0 * x));
	double t_1 = 2.0 / t_0;
	double t_2 = 1.0 + (t_1 * (1.0 + t_1));
	double tmp;
	if ((-2.0 * x) <= -0.02) {
		tmp = ((pow(t_1, 6.0) + -1.0) / pow(t_2, 2.0)) * (1.0 / (((8.0 / pow(t_0, 3.0)) - -1.0) / t_2));
	} else if ((-2.0 * x) <= 0.001) {
		tmp = x * (1.0 + ((x * x) * (-0.3333333333333333 + ((x * x) * 0.13333333333333333))));
	} else {
		tmp = (-1.0 + (4.0 / pow(t_0, 2.0))) / (t_1 - -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 + exp(((-2.0d0) * x))
    t_1 = 2.0d0 / t_0
    t_2 = 1.0d0 + (t_1 * (1.0d0 + t_1))
    if (((-2.0d0) * x) <= (-0.02d0)) then
        tmp = (((t_1 ** 6.0d0) + (-1.0d0)) / (t_2 ** 2.0d0)) * (1.0d0 / (((8.0d0 / (t_0 ** 3.0d0)) - (-1.0d0)) / t_2))
    else if (((-2.0d0) * x) <= 0.001d0) then
        tmp = x * (1.0d0 + ((x * x) * ((-0.3333333333333333d0) + ((x * x) * 0.13333333333333333d0))))
    else
        tmp = ((-1.0d0) + (4.0d0 / (t_0 ** 2.0d0))) / (t_1 - (-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 t_1 = 2.0 / t_0;
	double t_2 = 1.0 + (t_1 * (1.0 + t_1));
	double tmp;
	if ((-2.0 * x) <= -0.02) {
		tmp = ((Math.pow(t_1, 6.0) + -1.0) / Math.pow(t_2, 2.0)) * (1.0 / (((8.0 / Math.pow(t_0, 3.0)) - -1.0) / t_2));
	} else if ((-2.0 * x) <= 0.001) {
		tmp = x * (1.0 + ((x * x) * (-0.3333333333333333 + ((x * x) * 0.13333333333333333))));
	} else {
		tmp = (-1.0 + (4.0 / Math.pow(t_0, 2.0))) / (t_1 - -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + math.exp((-2.0 * x))
	t_1 = 2.0 / t_0
	t_2 = 1.0 + (t_1 * (1.0 + t_1))
	tmp = 0
	if (-2.0 * x) <= -0.02:
		tmp = ((math.pow(t_1, 6.0) + -1.0) / math.pow(t_2, 2.0)) * (1.0 / (((8.0 / math.pow(t_0, 3.0)) - -1.0) / t_2))
	elif (-2.0 * x) <= 0.001:
		tmp = x * (1.0 + ((x * x) * (-0.3333333333333333 + ((x * x) * 0.13333333333333333))))
	else:
		tmp = (-1.0 + (4.0 / math.pow(t_0, 2.0))) / (t_1 - -1.0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + exp(Float64(-2.0 * x)))
	t_1 = Float64(2.0 / t_0)
	t_2 = Float64(1.0 + Float64(t_1 * Float64(1.0 + t_1)))
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.02)
		tmp = Float64(Float64(Float64((t_1 ^ 6.0) + -1.0) / (t_2 ^ 2.0)) * Float64(1.0 / Float64(Float64(Float64(8.0 / (t_0 ^ 3.0)) - -1.0) / t_2)));
	elseif (Float64(-2.0 * x) <= 0.001)
		tmp = Float64(x * Float64(1.0 + Float64(Float64(x * x) * Float64(-0.3333333333333333 + Float64(Float64(x * x) * 0.13333333333333333)))));
	else
		tmp = Float64(Float64(-1.0 + Float64(4.0 / (t_0 ^ 2.0))) / Float64(t_1 - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + exp((-2.0 * x));
	t_1 = 2.0 / t_0;
	t_2 = 1.0 + (t_1 * (1.0 + t_1));
	tmp = 0.0;
	if ((-2.0 * x) <= -0.02)
		tmp = (((t_1 ^ 6.0) + -1.0) / (t_2 ^ 2.0)) * (1.0 / (((8.0 / (t_0 ^ 3.0)) - -1.0) / t_2));
	elseif ((-2.0 * x) <= 0.001)
		tmp = x * (1.0 + ((x * x) * (-0.3333333333333333 + ((x * x) * 0.13333333333333333))));
	else
		tmp = (-1.0 + (4.0 / (t_0 ^ 2.0))) / (t_1 - -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]}, Block[{t$95$1 = N[(2.0 / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(t$95$1 * N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.02], N[(N[(N[(N[Power[t$95$1, 6.0], $MachinePrecision] + -1.0), $MachinePrecision] / N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(N[(8.0 / N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.001], 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], N[(N[(-1.0 + N[(4.0 / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 - -1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -0.0200000000000000004

   1. Initial program 100.0%

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

      1. flip3-- N/A

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

      2. metadata-eval N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

   4. Applied egg-rr 100.0%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6} - 1}{{\left(1 + \left(1 + \frac{2}{1 + e^{-2 \cdot x}}\right) \cdot \frac{2}{1 + e^{-2 \cdot x}}\right)}^{2}} \cdot \frac{1}{\frac{\frac{8}{{\left(1 + e^{-2 \cdot x}\right)}^{3}} - -1}{1 + \left(1 + \frac{2}{1 + e^{-2 \cdot x}}\right) \cdot \frac{2}{1 + e^{-2 \cdot x}}}}} \]

   if -0.0200000000000000004 < (*.f64 #s(literal -2 binary64) x) < 1e-3

   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 1e-3 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 99.9%

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

      1. flip-- N/A

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

      2. metadata-eval N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

   4. Applied egg-rr 100.0%

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.02:\\ \;\;\;\;\frac{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6} + -1}{{\left(1 + \frac{2}{1 + e^{-2 \cdot x}} \cdot \left(1 + \frac{2}{1 + e^{-2 \cdot x}}\right)\right)}^{2}} \cdot \frac{1}{\frac{\frac{8}{{\left(1 + e^{-2 \cdot x}\right)}^{3}} - -1}{1 + \frac{2}{1 + e^{-2 \cdot x}} \cdot \left(1 + \frac{2}{1 + e^{-2 \cdot x}}\right)}}\\ \mathbf{elif}\;-2 \cdot x \leq 0.001:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.3333333333333333 + \left(x \cdot x\right) \cdot 0.13333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + \frac{4}{{\left(1 + e^{-2 \cdot x}\right)}^{2}}}{\frac{2}{1 + e^{-2 \cdot x}} - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 100.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-2 \cdot x}\\ t_1 := 1 + t\_0\\ \mathbf{if}\;-2 \cdot x \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{1 + e^{-2 \cdot \left(x \cdot 3\right)}}, 1 + t\_0 \cdot \mathsf{expm1}\left(-2 \cdot x\right), -1\right)\\ \mathbf{elif}\;-2 \cdot x \leq 0.001:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.3333333333333333 + \left(x \cdot x\right) \cdot 0.13333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + \frac{4}{{t\_1}^{2}}}{\frac{2}{t\_1} - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (exp (* -2.0 x))) (t_1 (+ 1.0 t_0)))
   (if (<= (* -2.0 x) -0.02)
     (fma
      (/ 2.0 (+ 1.0 (exp (* -2.0 (* x 3.0)))))
      (+ 1.0 (* t_0 (expm1 (* -2.0 x))))
      -1.0)
     (if (<= (* -2.0 x) 0.001)
       (*
        x
        (+
         1.0
         (* (* x x) (+ -0.3333333333333333 (* (* x x) 0.13333333333333333)))))
       (/ (+ -1.0 (/ 4.0 (pow t_1 2.0))) (- (/ 2.0 t_1) -1.0))))))

C

double code(double x, double y) {
	double t_0 = exp((-2.0 * x));
	double t_1 = 1.0 + t_0;
	double tmp;
	if ((-2.0 * x) <= -0.02) {
		tmp = fma((2.0 / (1.0 + exp((-2.0 * (x * 3.0))))), (1.0 + (t_0 * expm1((-2.0 * x)))), -1.0);
	} else if ((-2.0 * x) <= 0.001) {
		tmp = x * (1.0 + ((x * x) * (-0.3333333333333333 + ((x * x) * 0.13333333333333333))));
	} else {
		tmp = (-1.0 + (4.0 / pow(t_1, 2.0))) / ((2.0 / t_1) - -1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = exp(Float64(-2.0 * x))
	t_1 = Float64(1.0 + t_0)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.02)
		tmp = fma(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * Float64(x * 3.0))))), Float64(1.0 + Float64(t_0 * expm1(Float64(-2.0 * x)))), -1.0);
	elseif (Float64(-2.0 * x) <= 0.001)
		tmp = Float64(x * Float64(1.0 + Float64(Float64(x * x) * Float64(-0.3333333333333333 + Float64(Float64(x * x) * 0.13333333333333333)))));
	else
		tmp = Float64(Float64(-1.0 + Float64(4.0 / (t_1 ^ 2.0))) / Float64(Float64(2.0 / t_1) - -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + t$95$0), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.02], N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * N[(x * 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(t$95$0 * N[(Exp[N[(-2.0 * x), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.001], 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], N[(N[(-1.0 + N[(4.0 / N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 / t$95$1), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -0.0200000000000000004

   1. Initial program 100.0%

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

      1. flip-- N/A

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

      2. metadata-eval N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

   4. Applied egg-rr 100.0%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{1 + e^{-2 \cdot \left(x \cdot 3\right)}}, 1 + e^{-2 \cdot x} \cdot \mathsf{expm1}\left(-2 \cdot x\right), -1\right)} \]

   if -0.0200000000000000004 < (*.f64 #s(literal -2 binary64) x) < 1e-3

   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 1e-3 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 99.9%

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

      1. flip-- N/A

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

      2. metadata-eval N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

   4. Applied egg-rr 100.0%

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{1 + e^{-2 \cdot \left(x \cdot 3\right)}}, 1 + e^{-2 \cdot x} \cdot \mathsf{expm1}\left(-2 \cdot x\right), -1\right)\\ \mathbf{elif}\;-2 \cdot x \leq 0.001:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.3333333333333333 + \left(x \cdot x\right) \cdot 0.13333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + \frac{4}{{\left(1 + e^{-2 \cdot x}\right)}^{2}}}{\frac{2}{1 + e^{-2 \cdot x}} - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + e^{-2 \cdot x}\\ t_1 := \frac{2}{t\_0}\\ \mathbf{if}\;-2 \cdot x \leq -5000000000:\\ \;\;\;\;t\_1 + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.001:\\ \;\;\;\;x \cdot \left(\left(x \cdot x\right) \cdot -0.3333333333333333 + \left(1 + \left(0.13333333333333333 + x \cdot \left(x \cdot -0.05396825396825397\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + \frac{4}{{t\_0}^{2}}}{t\_1 - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (exp (* -2.0 x)))) (t_1 (/ 2.0 t_0)))
   (if (<= (* -2.0 x) -5000000000.0)
     (+ t_1 -1.0)
     (if (<= (* -2.0 x) 0.001)
       (*
        x
        (+
         (* (* x x) -0.3333333333333333)
         (+
          1.0
          (*
           (+ 0.13333333333333333 (* x (* x -0.05396825396825397)))
           (* (* x x) (* x x))))))
       (/ (+ -1.0 (/ 4.0 (pow t_0 2.0))) (- t_1 -1.0))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + exp((-2.0 * x));
	double t_1 = 2.0 / t_0;
	double tmp;
	if ((-2.0 * x) <= -5000000000.0) {
		tmp = t_1 + -1.0;
	} else if ((-2.0 * x) <= 0.001) {
		tmp = x * (((x * x) * -0.3333333333333333) + (1.0 + ((0.13333333333333333 + (x * (x * -0.05396825396825397))) * ((x * x) * (x * x)))));
	} else {
		tmp = (-1.0 + (4.0 / pow(t_0, 2.0))) / (t_1 - -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + exp(((-2.0d0) * x))
    t_1 = 2.0d0 / t_0
    if (((-2.0d0) * x) <= (-5000000000.0d0)) then
        tmp = t_1 + (-1.0d0)
    else if (((-2.0d0) * x) <= 0.001d0) then
        tmp = x * (((x * x) * (-0.3333333333333333d0)) + (1.0d0 + ((0.13333333333333333d0 + (x * (x * (-0.05396825396825397d0)))) * ((x * x) * (x * x)))))
    else
        tmp = ((-1.0d0) + (4.0d0 / (t_0 ** 2.0d0))) / (t_1 - (-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 t_1 = 2.0 / t_0;
	double tmp;
	if ((-2.0 * x) <= -5000000000.0) {
		tmp = t_1 + -1.0;
	} else if ((-2.0 * x) <= 0.001) {
		tmp = x * (((x * x) * -0.3333333333333333) + (1.0 + ((0.13333333333333333 + (x * (x * -0.05396825396825397))) * ((x * x) * (x * x)))));
	} else {
		tmp = (-1.0 + (4.0 / Math.pow(t_0, 2.0))) / (t_1 - -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + math.exp((-2.0 * x))
	t_1 = 2.0 / t_0
	tmp = 0
	if (-2.0 * x) <= -5000000000.0:
		tmp = t_1 + -1.0
	elif (-2.0 * x) <= 0.001:
		tmp = x * (((x * x) * -0.3333333333333333) + (1.0 + ((0.13333333333333333 + (x * (x * -0.05396825396825397))) * ((x * x) * (x * x)))))
	else:
		tmp = (-1.0 + (4.0 / math.pow(t_0, 2.0))) / (t_1 - -1.0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + exp(Float64(-2.0 * x)))
	t_1 = Float64(2.0 / t_0)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -5000000000.0)
		tmp = Float64(t_1 + -1.0);
	elseif (Float64(-2.0 * x) <= 0.001)
		tmp = Float64(x * Float64(Float64(Float64(x * x) * -0.3333333333333333) + Float64(1.0 + Float64(Float64(0.13333333333333333 + Float64(x * Float64(x * -0.05396825396825397))) * Float64(Float64(x * x) * Float64(x * x))))));
	else
		tmp = Float64(Float64(-1.0 + Float64(4.0 / (t_0 ^ 2.0))) / Float64(t_1 - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + exp((-2.0 * x));
	t_1 = 2.0 / t_0;
	tmp = 0.0;
	if ((-2.0 * x) <= -5000000000.0)
		tmp = t_1 + -1.0;
	elseif ((-2.0 * x) <= 0.001)
		tmp = x * (((x * x) * -0.3333333333333333) + (1.0 + ((0.13333333333333333 + (x * (x * -0.05396825396825397))) * ((x * x) * (x * x)))));
	else
		tmp = (-1.0 + (4.0 / (t_0 ^ 2.0))) / (t_1 - -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]}, Block[{t$95$1 = N[(2.0 / t$95$0), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -5000000000.0], N[(t$95$1 + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.001], N[(x * N[(N[(N[(x * x), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] + N[(1.0 + N[(N[(0.13333333333333333 + N[(x * N[(x * -0.05396825396825397), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 + N[(4.0 / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 - -1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -5e9

   1. Initial program 100.0%

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

   if -5e9 < (*.f64 #s(literal -2 binary64) x) < 1e-3

   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. *-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%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-+l+ N/A

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

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

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

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

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

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

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

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

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

   7. Applied egg-rr 100.0%

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

   if 1e-3 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 99.9%

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

      1. flip-- N/A

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

      2. metadata-eval N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

   4. Applied egg-rr 100.0%

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -5000000000:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.001:\\ \;\;\;\;x \cdot \left(\left(x \cdot x\right) \cdot -0.3333333333333333 + \left(1 + \left(0.13333333333333333 + x \cdot \left(x \cdot -0.05396825396825397\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + \frac{4}{{\left(1 + e^{-2 \cdot x}\right)}^{2}}}{\frac{2}{1 + e^{-2 \cdot x}} - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + e^{-2 \cdot x}\\ \mathbf{if}\;-2 \cdot x \leq -5000000000:\\ \;\;\;\;\frac{2}{t\_0} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.001:\\ \;\;\;\;x \cdot \left(\left(x \cdot x\right) \cdot -0.3333333333333333 + \left(1 + \left(0.13333333333333333 + x \cdot \left(x \cdot -0.05396825396825397\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(0 - \log \left(\frac{t\_0}{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (exp (* -2.0 x)))))
   (if (<= (* -2.0 x) -5000000000.0)
     (+ (/ 2.0 t_0) -1.0)
     (if (<= (* -2.0 x) 0.001)
       (*
        x
        (+
         (* (* x x) -0.3333333333333333)
         (+
          1.0
          (*
           (+ 0.13333333333333333 (* x (* x -0.05396825396825397)))
           (* (* x x) (* x x))))))
       (expm1 (- 0.0 (log (/ t_0 2.0))))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + exp((-2.0 * x));
	double tmp;
	if ((-2.0 * x) <= -5000000000.0) {
		tmp = (2.0 / t_0) + -1.0;
	} else if ((-2.0 * x) <= 0.001) {
		tmp = x * (((x * x) * -0.3333333333333333) + (1.0 + ((0.13333333333333333 + (x * (x * -0.05396825396825397))) * ((x * x) * (x * x)))));
	} else {
		tmp = expm1((0.0 - log((t_0 / 2.0))));
	}
	return tmp;
}

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) <= -5000000000.0) {
		tmp = (2.0 / t_0) + -1.0;
	} else if ((-2.0 * x) <= 0.001) {
		tmp = x * (((x * x) * -0.3333333333333333) + (1.0 + ((0.13333333333333333 + (x * (x * -0.05396825396825397))) * ((x * x) * (x * x)))));
	} else {
		tmp = Math.expm1((0.0 - Math.log((t_0 / 2.0))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + math.exp((-2.0 * x))
	tmp = 0
	if (-2.0 * x) <= -5000000000.0:
		tmp = (2.0 / t_0) + -1.0
	elif (-2.0 * x) <= 0.001:
		tmp = x * (((x * x) * -0.3333333333333333) + (1.0 + ((0.13333333333333333 + (x * (x * -0.05396825396825397))) * ((x * x) * (x * x)))))
	else:
		tmp = math.expm1((0.0 - math.log((t_0 / 2.0))))
	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) <= -5000000000.0)
		tmp = Float64(Float64(2.0 / t_0) + -1.0);
	elseif (Float64(-2.0 * x) <= 0.001)
		tmp = Float64(x * Float64(Float64(Float64(x * x) * -0.3333333333333333) + Float64(1.0 + Float64(Float64(0.13333333333333333 + Float64(x * Float64(x * -0.05396825396825397))) * Float64(Float64(x * x) * Float64(x * x))))));
	else
		tmp = expm1(Float64(0.0 - log(Float64(t_0 / 2.0))));
	end
	return 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], -5000000000.0], N[(N[(2.0 / t$95$0), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.001], N[(x * N[(N[(N[(x * x), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] + N[(1.0 + N[(N[(0.13333333333333333 + N[(x * N[(x * -0.05396825396825397), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Exp[N[(0.0 - N[Log[N[(t$95$0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -5e9

   1. Initial program 100.0%

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

   if -5e9 < (*.f64 #s(literal -2 binary64) x) < 1e-3

   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. *-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%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-+l+ N/A

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

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

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

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

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

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

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

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

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

   7. Applied egg-rr 100.0%

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

   if 1e-3 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 99.9%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. metadata-eval N/A

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

      4. pow-to-exp N/A

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

      5. accelerator-lowering-expm1.f64 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. *-commutative N/A

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

      2. mul-1-neg N/A

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

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

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

      4. rem-exp-log N/A

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

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

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

      6. rem-exp-log N/A

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

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

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

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

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

      9. metadata-eval N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. *-commutative N/A

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

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

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

      15. metadata-eval N/A

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

      16. *-lowering-*.f64 100.0%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{-\log \left(\frac{1 + e^{-2 \cdot x}}{2}\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -5000000000:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.001:\\ \;\;\;\;x \cdot \left(\left(x \cdot x\right) \cdot -0.3333333333333333 + \left(1 + \left(0.13333333333333333 + x \cdot \left(x \cdot -0.05396825396825397\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(0 - \log \left(\frac{1 + e^{-2 \cdot x}}{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{if}\;-2 \cdot x \leq -5000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;-2 \cdot x \leq 0.001:\\ \;\;\;\;x \cdot \left(\left(x \cdot x\right) \cdot -0.3333333333333333 + \left(1 + \left(0.13333333333333333 + x \cdot \left(x \cdot -0.05396825396825397\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) -1.0)))
   (if (<= (* -2.0 x) -5000000000.0)
     t_0
     (if (<= (* -2.0 x) 0.001)
       (*
        x
        (+
         (* (* x x) -0.3333333333333333)
         (+
          1.0
          (*
           (+ 0.13333333333333333 (* x (* x -0.05396825396825397)))
           (* (* x x) (* x x))))))
       t_0))))

C

double code(double x, double y) {
	double t_0 = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	double tmp;
	if ((-2.0 * x) <= -5000000000.0) {
		tmp = t_0;
	} else if ((-2.0 * x) <= 0.001) {
		tmp = x * (((x * x) * -0.3333333333333333) + (1.0 + ((0.13333333333333333 + (x * (x * -0.05396825396825397))) * ((x * x) * (x * x)))));
	} else {
		tmp = t_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 = (2.0d0 / (1.0d0 + exp(((-2.0d0) * x)))) + (-1.0d0)
    if (((-2.0d0) * x) <= (-5000000000.0d0)) then
        tmp = t_0
    else if (((-2.0d0) * x) <= 0.001d0) then
        tmp = x * (((x * x) * (-0.3333333333333333d0)) + (1.0d0 + ((0.13333333333333333d0 + (x * (x * (-0.05396825396825397d0)))) * ((x * x) * (x * x)))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (2.0 / (1.0 + Math.exp((-2.0 * x)))) + -1.0;
	double tmp;
	if ((-2.0 * x) <= -5000000000.0) {
		tmp = t_0;
	} else if ((-2.0 * x) <= 0.001) {
		tmp = x * (((x * x) * -0.3333333333333333) + (1.0 + ((0.13333333333333333 + (x * (x * -0.05396825396825397))) * ((x * x) * (x * x)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (2.0 / (1.0 + math.exp((-2.0 * x)))) + -1.0
	tmp = 0
	if (-2.0 * x) <= -5000000000.0:
		tmp = t_0
	elif (-2.0 * x) <= 0.001:
		tmp = x * (((x * x) * -0.3333333333333333) + (1.0 + ((0.13333333333333333 + (x * (x * -0.05396825396825397))) * ((x * x) * (x * x)))))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) + -1.0)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -5000000000.0)
		tmp = t_0;
	elseif (Float64(-2.0 * x) <= 0.001)
		tmp = Float64(x * Float64(Float64(Float64(x * x) * -0.3333333333333333) + Float64(1.0 + Float64(Float64(0.13333333333333333 + Float64(x * Float64(x * -0.05396825396825397))) * Float64(Float64(x * x) * Float64(x * x))))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	tmp = 0.0;
	if ((-2.0 * x) <= -5000000000.0)
		tmp = t_0;
	elseif ((-2.0 * x) <= 0.001)
		tmp = x * (((x * x) * -0.3333333333333333) + (1.0 + ((0.13333333333333333 + (x * (x * -0.05396825396825397))) * ((x * x) * (x * x)))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$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], -5000000000.0], t$95$0, If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.001], N[(x * N[(N[(N[(x * x), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] + N[(1.0 + N[(N[(0.13333333333333333 + N[(x * N[(x * -0.05396825396825397), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -5e9 or 1e-3 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 100.0%

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

   if -5e9 < (*.f64 #s(literal -2 binary64) x) < 1e-3

   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. *-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%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-+l+ N/A

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

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

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

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

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

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

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

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

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

   7. Applied egg-rr 100.0%

      \[\leadsto x \cdot \color{blue}{\left(-0.3333333333333333 \cdot \left(x \cdot x\right) + \left(\left(0.13333333333333333 + x \cdot \left(x \cdot -0.05396825396825397\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) + 1\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -5000000000:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.001:\\ \;\;\;\;x \cdot \left(\left(x \cdot x\right) \cdot -0.3333333333333333 + \left(1 + \left(0.13333333333333333 + x \cdot \left(x \cdot -0.05396825396825397\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.3% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.22:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.3333333333333333 + \left(x \cdot x\right) \cdot 0.13333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.22)
   -1.0
   (*
    x
    (+
     1.0
     (* (* x x) (+ -0.3333333333333333 (* (* x x) 0.13333333333333333)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.22) {
		tmp = -1.0;
	} else {
		tmp = x * (1.0 + ((x * x) * (-0.3333333333333333 + ((x * x) * 0.13333333333333333))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.22d0)) then
        tmp = -1.0d0
    else
        tmp = x * (1.0d0 + ((x * x) * ((-0.3333333333333333d0) + ((x * x) * 0.13333333333333333d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.22) {
		tmp = -1.0;
	} else {
		tmp = x * (1.0 + ((x * x) * (-0.3333333333333333 + ((x * x) * 0.13333333333333333))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.22:
		tmp = -1.0
	else:
		tmp = x * (1.0 + ((x * x) * (-0.3333333333333333 + ((x * x) * 0.13333333333333333))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.22)
		tmp = -1.0;
	else
		tmp = Float64(x * Float64(1.0 + Float64(Float64(x * x) * Float64(-0.3333333333333333 + Float64(Float64(x * x) * 0.13333333333333333)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.22)
		tmp = -1.0;
	else
		tmp = x * (1.0 + ((x * x) * (-0.3333333333333333 + ((x * x) * 0.13333333333333333))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.22], -1.0, 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]]

Derivation

1. Split input into 2 regimes

2. if x < -1.21999999999999997

   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 97.2%

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

   5. Simplified 97.2%

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

   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 -1.21999999999999997 < x

   1. Initial program 40.6%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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 66.3%

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

   5. Simplified 66.3%

      \[\leadsto \color{blue}{x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.3333333333333333 + \left(x \cdot x\right) \cdot 0.13333333333333333\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 75.5% 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. 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 97.2%

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

   5. Simplified 97.2%

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

   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 40.6%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 66.2%

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

Alternative 8: 27.4% 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 55.7%

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

3. Taylor expanded in x around 0

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

   1. 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 28.7%

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

5. Simplified 28.7%

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

6. Taylor expanded in x around inf

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

8. Simplified 27.7%

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

Reproduce

herbie shell --seed 2024184 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))