Logistic function from Lakshay Garg

Percentage Accurate: 54.4% → 99.9%

Time: 10.2s

Alternatives: 12

Speedup: 9.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.4% 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: 99.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(e^{-2}\right)}^{x}\\ t_1 := t\_0 + 1\\ \mathbf{if}\;-2 \cdot x \leq -5:\\ \;\;\;\;\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(t\_0\right)\right)\\ \mathbf{elif}\;-2 \cdot x \leq 0.002:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left(0.13333333333333333 \cdot \left(x \cdot x\right) - 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1 + \frac{2}{t\_1}}{\frac{4}{{t\_1}^{2}} + -1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (pow (exp -2.0) x)) (t_1 (+ t_0 1.0)))
   (if (<= (* -2.0 x) -5.0)
     (expm1 (- (log 2.0) (log1p t_0)))
     (if (<= (* -2.0 x) 0.002)
       (*
        x
        (+
         1.0
         (*
          (pow x 2.0)
          (- (* 0.13333333333333333 (* x x)) 0.3333333333333333))))
       (/ 1.0 (/ (+ 1.0 (/ 2.0 t_1)) (+ (/ 4.0 (pow t_1 2.0)) -1.0)))))))

C

double code(double x, double y) {
	double t_0 = pow(exp(-2.0), x);
	double t_1 = t_0 + 1.0;
	double tmp;
	if ((-2.0 * x) <= -5.0) {
		tmp = expm1((log(2.0) - log1p(t_0)));
	} else if ((-2.0 * x) <= 0.002) {
		tmp = x * (1.0 + (pow(x, 2.0) * ((0.13333333333333333 * (x * x)) - 0.3333333333333333)));
	} else {
		tmp = 1.0 / ((1.0 + (2.0 / t_1)) / ((4.0 / pow(t_1, 2.0)) + -1.0));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = Math.pow(Math.exp(-2.0), x);
	double t_1 = t_0 + 1.0;
	double tmp;
	if ((-2.0 * x) <= -5.0) {
		tmp = Math.expm1((Math.log(2.0) - Math.log1p(t_0)));
	} else if ((-2.0 * x) <= 0.002) {
		tmp = x * (1.0 + (Math.pow(x, 2.0) * ((0.13333333333333333 * (x * x)) - 0.3333333333333333)));
	} else {
		tmp = 1.0 / ((1.0 + (2.0 / t_1)) / ((4.0 / Math.pow(t_1, 2.0)) + -1.0));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.pow(math.exp(-2.0), x)
	t_1 = t_0 + 1.0
	tmp = 0
	if (-2.0 * x) <= -5.0:
		tmp = math.expm1((math.log(2.0) - math.log1p(t_0)))
	elif (-2.0 * x) <= 0.002:
		tmp = x * (1.0 + (math.pow(x, 2.0) * ((0.13333333333333333 * (x * x)) - 0.3333333333333333)))
	else:
		tmp = 1.0 / ((1.0 + (2.0 / t_1)) / ((4.0 / math.pow(t_1, 2.0)) + -1.0))
	return tmp

Julia

function code(x, y)
	t_0 = exp(-2.0) ^ x
	t_1 = Float64(t_0 + 1.0)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -5.0)
		tmp = expm1(Float64(log(2.0) - log1p(t_0)));
	elseif (Float64(-2.0 * x) <= 0.002)
		tmp = Float64(x * Float64(1.0 + Float64((x ^ 2.0) * Float64(Float64(0.13333333333333333 * Float64(x * x)) - 0.3333333333333333))));
	else
		tmp = Float64(1.0 / Float64(Float64(1.0 + Float64(2.0 / t_1)) / Float64(Float64(4.0 / (t_1 ^ 2.0)) + -1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Power[N[Exp[-2.0], $MachinePrecision], x], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + 1.0), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -5.0], N[(Exp[N[(N[Log[2.0], $MachinePrecision] - N[Log[1 + t$95$0], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.002], N[(x * N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(0.13333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(1.0 + N[(2.0 / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(N[(4.0 / N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

      1. add-exp-log 100.0%

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

      2. expm1-define 100.0%

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

      3. log-div 100.0%

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

      4. log1p-define 100.0%

         \[\leadsto \mathsf{expm1}\left(\log 2 - \color{blue}{\mathsf{log1p}\left(e^{-2 \cdot x}\right)}\right) \]

      5. exp-prod 100.0%

         \[\leadsto \mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(\color{blue}{{\left(e^{-2}\right)}^{x}}\right)\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)} \]

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

   1. Initial program 7.5%

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

   3. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.13333333333333333 \cdot {x}^{2} - 0.3333333333333333\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.13333333333333333 \cdot \color{blue}{\left(x \cdot x\right)} - 0.3333333333333333\right)\right) \]

   5. Applied egg-rr 100.0%

      \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.13333333333333333 \cdot \color{blue}{\left(x \cdot x\right)} - 0.3333333333333333\right)\right) \]

   if 2e-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-- 100.0%

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

      2. clear-num 100.0%

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

      3. +-commutative 100.0%

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

      4. exp-prod 100.0%

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

      5. metadata-eval 100.0%

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

      6. sub-neg 100.0%

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

      7. frac-times 100.0%

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

      8. metadata-eval 100.0%

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

      9. pow2 100.0%

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

      10. exp-prod 100.0%

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

      11. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -5:\\ \;\;\;\;\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)\\ \mathbf{elif}\;-2 \cdot x \leq 0.002:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left(0.13333333333333333 \cdot \left(x \cdot x\right) - 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1 + \frac{2}{{\left(e^{-2}\right)}^{x} + 1}}{\frac{4}{{\left({\left(e^{-2}\right)}^{x} + 1\right)}^{2}} + -1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -5:\\ \;\;\;\;\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)\\ \mathbf{elif}\;-2 \cdot x \leq 0.002:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left(0.13333333333333333 \cdot \left(x \cdot x\right) - 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + \frac{4}{{\left(\mathsf{hypot}\left(1, e^{-x}\right)\right)}^{4}}}{1 + \frac{-2}{-1 - e^{-2 \cdot x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -5.0)
   (expm1 (- (log 2.0) (log1p (pow (exp -2.0) x))))
   (if (<= (* -2.0 x) 0.002)
     (*
      x
      (+
       1.0
       (* (pow x 2.0) (- (* 0.13333333333333333 (* x x)) 0.3333333333333333))))
     (/
      (+ -1.0 (/ 4.0 (pow (hypot 1.0 (exp (- x))) 4.0)))
      (+ 1.0 (/ -2.0 (- -1.0 (exp (* -2.0 x)))))))))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -5.0) {
		tmp = expm1((log(2.0) - log1p(pow(exp(-2.0), x))));
	} else if ((-2.0 * x) <= 0.002) {
		tmp = x * (1.0 + (pow(x, 2.0) * ((0.13333333333333333 * (x * x)) - 0.3333333333333333)));
	} else {
		tmp = (-1.0 + (4.0 / pow(hypot(1.0, exp(-x)), 4.0))) / (1.0 + (-2.0 / (-1.0 - exp((-2.0 * x)))));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -5.0) {
		tmp = Math.expm1((Math.log(2.0) - Math.log1p(Math.pow(Math.exp(-2.0), x))));
	} else if ((-2.0 * x) <= 0.002) {
		tmp = x * (1.0 + (Math.pow(x, 2.0) * ((0.13333333333333333 * (x * x)) - 0.3333333333333333)));
	} else {
		tmp = (-1.0 + (4.0 / Math.pow(Math.hypot(1.0, Math.exp(-x)), 4.0))) / (1.0 + (-2.0 / (-1.0 - Math.exp((-2.0 * x)))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (-2.0 * x) <= -5.0:
		tmp = math.expm1((math.log(2.0) - math.log1p(math.pow(math.exp(-2.0), x))))
	elif (-2.0 * x) <= 0.002:
		tmp = x * (1.0 + (math.pow(x, 2.0) * ((0.13333333333333333 * (x * x)) - 0.3333333333333333)))
	else:
		tmp = (-1.0 + (4.0 / math.pow(math.hypot(1.0, math.exp(-x)), 4.0))) / (1.0 + (-2.0 / (-1.0 - math.exp((-2.0 * x)))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -5.0)
		tmp = expm1(Float64(log(2.0) - log1p((exp(-2.0) ^ x))));
	elseif (Float64(-2.0 * x) <= 0.002)
		tmp = Float64(x * Float64(1.0 + Float64((x ^ 2.0) * Float64(Float64(0.13333333333333333 * Float64(x * x)) - 0.3333333333333333))));
	else
		tmp = Float64(Float64(-1.0 + Float64(4.0 / (hypot(1.0, exp(Float64(-x))) ^ 4.0))) / Float64(1.0 + Float64(-2.0 / Float64(-1.0 - exp(Float64(-2.0 * x))))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -5.0], N[(Exp[N[(N[Log[2.0], $MachinePrecision] - N[Log[1 + N[Power[N[Exp[-2.0], $MachinePrecision], x], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.002], N[(x * N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(0.13333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 + N[(4.0 / N[Power[N[Sqrt[1.0 ^ 2 + N[Exp[(-x)], $MachinePrecision] ^ 2], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(-2.0 / N[(-1.0 - N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

      1. add-exp-log 100.0%

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

      2. expm1-define 100.0%

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

      3. log-div 100.0%

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

      4. log1p-define 100.0%

         \[\leadsto \mathsf{expm1}\left(\log 2 - \color{blue}{\mathsf{log1p}\left(e^{-2 \cdot x}\right)}\right) \]

      5. exp-prod 100.0%

         \[\leadsto \mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(\color{blue}{{\left(e^{-2}\right)}^{x}}\right)\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)} \]

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

   1. Initial program 7.5%

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

   3. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.13333333333333333 \cdot {x}^{2} - 0.3333333333333333\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.13333333333333333 \cdot \color{blue}{\left(x \cdot x\right)} - 0.3333333333333333\right)\right) \]

   5. Applied egg-rr 100.0%

      \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.13333333333333333 \cdot \color{blue}{\left(x \cdot x\right)} - 0.3333333333333333\right)\right) \]

   if 2e-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-- 100.0%

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

      2. div-inv 100.0%

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

      3. metadata-eval 100.0%

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

      4. sub-neg 100.0%

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

      5. frac-times 100.0%

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

      6. metadata-eval 100.0%

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

      7. pow2 100.0%

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

      8. exp-prod 100.0%

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

      9. metadata-eval 100.0%

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

      10. +-commutative 100.0%

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

      11. exp-prod 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. associate-*r/ 100.0%

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

   6. Simplified 100.0%

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

      1. add-exp-log 100.0%

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

      2. log-pow 100.0%

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

      3. rem-log-exp 100.0%

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

   8. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -5:\\ \;\;\;\;\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)\\ \mathbf{elif}\;-2 \cdot x \leq 0.002:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left(0.13333333333333333 \cdot \left(x \cdot x\right) - 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + \frac{4}{{\left(\mathsf{hypot}\left(1, e^{-x}\right)\right)}^{4}}}{1 + \frac{-2}{-1 - e^{-2 \cdot x}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-2 \cdot x}\\ \mathbf{if}\;-2 \cdot x \leq -5:\\ \;\;\;\;-1 + \frac{2}{1 + t\_0}\\ \mathbf{elif}\;-2 \cdot x \leq 0.002:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left(0.13333333333333333 \cdot \left(x \cdot x\right) - 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + \frac{4}{{\left(\mathsf{hypot}\left(1, e^{-x}\right)\right)}^{4}}}{1 + \frac{-2}{-1 - t\_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (exp (* -2.0 x))))
   (if (<= (* -2.0 x) -5.0)
     (+ -1.0 (/ 2.0 (+ 1.0 t_0)))
     (if (<= (* -2.0 x) 0.002)
       (*
        x
        (+
         1.0
         (*
          (pow x 2.0)
          (- (* 0.13333333333333333 (* x x)) 0.3333333333333333))))
       (/
        (+ -1.0 (/ 4.0 (pow (hypot 1.0 (exp (- x))) 4.0)))
        (+ 1.0 (/ -2.0 (- -1.0 t_0))))))))

C

double code(double x, double y) {
	double t_0 = exp((-2.0 * x));
	double tmp;
	if ((-2.0 * x) <= -5.0) {
		tmp = -1.0 + (2.0 / (1.0 + t_0));
	} else if ((-2.0 * x) <= 0.002) {
		tmp = x * (1.0 + (pow(x, 2.0) * ((0.13333333333333333 * (x * x)) - 0.3333333333333333)));
	} else {
		tmp = (-1.0 + (4.0 / pow(hypot(1.0, exp(-x)), 4.0))) / (1.0 + (-2.0 / (-1.0 - t_0)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = Math.exp((-2.0 * x));
	double tmp;
	if ((-2.0 * x) <= -5.0) {
		tmp = -1.0 + (2.0 / (1.0 + t_0));
	} else if ((-2.0 * x) <= 0.002) {
		tmp = x * (1.0 + (Math.pow(x, 2.0) * ((0.13333333333333333 * (x * x)) - 0.3333333333333333)));
	} else {
		tmp = (-1.0 + (4.0 / Math.pow(Math.hypot(1.0, Math.exp(-x)), 4.0))) / (1.0 + (-2.0 / (-1.0 - t_0)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.exp((-2.0 * x))
	tmp = 0
	if (-2.0 * x) <= -5.0:
		tmp = -1.0 + (2.0 / (1.0 + t_0))
	elif (-2.0 * x) <= 0.002:
		tmp = x * (1.0 + (math.pow(x, 2.0) * ((0.13333333333333333 * (x * x)) - 0.3333333333333333)))
	else:
		tmp = (-1.0 + (4.0 / math.pow(math.hypot(1.0, math.exp(-x)), 4.0))) / (1.0 + (-2.0 / (-1.0 - t_0)))
	return tmp

Julia

function code(x, y)
	t_0 = exp(Float64(-2.0 * x))
	tmp = 0.0
	if (Float64(-2.0 * x) <= -5.0)
		tmp = Float64(-1.0 + Float64(2.0 / Float64(1.0 + t_0)));
	elseif (Float64(-2.0 * x) <= 0.002)
		tmp = Float64(x * Float64(1.0 + Float64((x ^ 2.0) * Float64(Float64(0.13333333333333333 * Float64(x * x)) - 0.3333333333333333))));
	else
		tmp = Float64(Float64(-1.0 + Float64(4.0 / (hypot(1.0, exp(Float64(-x))) ^ 4.0))) / Float64(1.0 + Float64(-2.0 / Float64(-1.0 - t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = exp((-2.0 * x));
	tmp = 0.0;
	if ((-2.0 * x) <= -5.0)
		tmp = -1.0 + (2.0 / (1.0 + t_0));
	elseif ((-2.0 * x) <= 0.002)
		tmp = x * (1.0 + ((x ^ 2.0) * ((0.13333333333333333 * (x * x)) - 0.3333333333333333)));
	else
		tmp = (-1.0 + (4.0 / (hypot(1.0, exp(-x)) ^ 4.0))) / (1.0 + (-2.0 / (-1.0 - t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -5.0], N[(-1.0 + N[(2.0 / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.002], N[(x * N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(0.13333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 + N[(4.0 / N[Power[N[Sqrt[1.0 ^ 2 + N[Exp[(-x)], $MachinePrecision] ^ 2], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(-2.0 / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

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

   1. Initial program 7.5%

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

   3. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.13333333333333333 \cdot {x}^{2} - 0.3333333333333333\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.13333333333333333 \cdot \color{blue}{\left(x \cdot x\right)} - 0.3333333333333333\right)\right) \]

   5. Applied egg-rr 100.0%

      \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.13333333333333333 \cdot \color{blue}{\left(x \cdot x\right)} - 0.3333333333333333\right)\right) \]

   if 2e-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-- 100.0%

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

      2. div-inv 100.0%

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

      3. metadata-eval 100.0%

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

      4. sub-neg 100.0%

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

      5. frac-times 100.0%

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

      6. metadata-eval 100.0%

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

      7. pow2 100.0%

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

      8. exp-prod 100.0%

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

      9. metadata-eval 100.0%

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

      10. +-commutative 100.0%

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

      11. exp-prod 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. associate-*r/ 100.0%

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

   6. Simplified 100.0%

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

      1. add-exp-log 100.0%

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

      2. log-pow 100.0%

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

      3. rem-log-exp 100.0%

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

   8. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

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

Alternative 4: 99.9% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* -2.0 x) -5.0) (not (<= (* -2.0 x) 0.002)))
   (+ -1.0 (/ 2.0 (+ 1.0 (exp (* -2.0 x)))))
   (*
    x
    (+
     1.0
     (* (pow x 2.0) (- (* 0.13333333333333333 (* x x)) 0.3333333333333333))))))

C

double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -5.0) || !((-2.0 * x) <= 0.002)) {
		tmp = -1.0 + (2.0 / (1.0 + exp((-2.0 * x))));
	} else {
		tmp = x * (1.0 + (pow(x, 2.0) * ((0.13333333333333333 * (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 ((((-2.0d0) * x) <= (-5.0d0)) .or. (.not. (((-2.0d0) * x) <= 0.002d0))) then
        tmp = (-1.0d0) + (2.0d0 / (1.0d0 + exp(((-2.0d0) * x))))
    else
        tmp = x * (1.0d0 + ((x ** 2.0d0) * ((0.13333333333333333d0 * (x * x)) - 0.3333333333333333d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -5.0) || !((-2.0 * x) <= 0.002)) {
		tmp = -1.0 + (2.0 / (1.0 + Math.exp((-2.0 * x))));
	} else {
		tmp = x * (1.0 + (Math.pow(x, 2.0) * ((0.13333333333333333 * (x * x)) - 0.3333333333333333)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((-2.0 * x) <= -5.0) or not ((-2.0 * x) <= 0.002):
		tmp = -1.0 + (2.0 / (1.0 + math.exp((-2.0 * x))))
	else:
		tmp = x * (1.0 + (math.pow(x, 2.0) * ((0.13333333333333333 * (x * x)) - 0.3333333333333333)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((Float64(-2.0 * x) <= -5.0) || !(Float64(-2.0 * x) <= 0.002))
		tmp = Float64(-1.0 + Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))));
	else
		tmp = Float64(x * Float64(1.0 + Float64((x ^ 2.0) * Float64(Float64(0.13333333333333333 * Float64(x * x)) - 0.3333333333333333))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((-2.0 * x) <= -5.0) || ~(((-2.0 * x) <= 0.002)))
		tmp = -1.0 + (2.0 / (1.0 + exp((-2.0 * x))));
	else
		tmp = x * (1.0 + ((x ^ 2.0) * ((0.13333333333333333 * (x * x)) - 0.3333333333333333)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[(-2.0 * x), $MachinePrecision], -5.0], N[Not[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.002]], $MachinePrecision]], N[(-1.0 + N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(0.13333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

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

   1. Initial program 7.5%

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

   3. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.13333333333333333 \cdot {x}^{2} - 0.3333333333333333\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.13333333333333333 \cdot \color{blue}{\left(x \cdot x\right)} - 0.3333333333333333\right)\right) \]

   5. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

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

Alternative 5: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -5 \lor \neg \left(-2 \cdot x \leq 5 \cdot 10^{-6}\right):\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* -2.0 x) -5.0) (not (<= (* -2.0 x) 5e-6)))
   (+ -1.0 (/ 2.0 (+ 1.0 (exp (* -2.0 x)))))
   (+ x (* -0.3333333333333333 (pow x 3.0)))))

C

double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -5.0) || !((-2.0 * x) <= 5e-6)) {
		tmp = -1.0 + (2.0 / (1.0 + exp((-2.0 * x))));
	} else {
		tmp = x + (-0.3333333333333333 * pow(x, 3.0));
	}
	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) <= (-5.0d0)) .or. (.not. (((-2.0d0) * x) <= 5d-6))) then
        tmp = (-1.0d0) + (2.0d0 / (1.0d0 + exp(((-2.0d0) * x))))
    else
        tmp = x + ((-0.3333333333333333d0) * (x ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -5.0) || !((-2.0 * x) <= 5e-6)) {
		tmp = -1.0 + (2.0 / (1.0 + Math.exp((-2.0 * x))));
	} else {
		tmp = x + (-0.3333333333333333 * Math.pow(x, 3.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((-2.0 * x) <= -5.0) or not ((-2.0 * x) <= 5e-6):
		tmp = -1.0 + (2.0 / (1.0 + math.exp((-2.0 * x))))
	else:
		tmp = x + (-0.3333333333333333 * math.pow(x, 3.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((Float64(-2.0 * x) <= -5.0) || !(Float64(-2.0 * x) <= 5e-6))
		tmp = Float64(-1.0 + Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))));
	else
		tmp = Float64(x + Float64(-0.3333333333333333 * (x ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((-2.0 * x) <= -5.0) || ~(((-2.0 * x) <= 5e-6)))
		tmp = -1.0 + (2.0 / (1.0 + exp((-2.0 * x))));
	else
		tmp = x + (-0.3333333333333333 * (x ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[(-2.0 * x), $MachinePrecision], -5.0], N[Not[LessEqual[N[(-2.0 * x), $MachinePrecision], 5e-6]], $MachinePrecision]], N[(-1.0 + N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -5 or 5.00000000000000041e-6 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 99.9%

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

   if -5 < (*.f64 #s(literal -2 binary64) x) < 5.00000000000000041e-6

   1. Initial program 6.9%

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

   3. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{x \cdot \left(1 + -0.3333333333333333 \cdot {x}^{2}\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \color{blue}{1 \cdot x + \left(-0.3333333333333333 \cdot {x}^{2}\right) \cdot x} \]

      2. *-lft-identity 100.0%

         \[\leadsto \color{blue}{x} + \left(-0.3333333333333333 \cdot {x}^{2}\right) \cdot x \]

      3. associate-*l* 100.0%

         \[\leadsto x + \color{blue}{-0.3333333333333333 \cdot \left({x}^{2} \cdot x\right)} \]

      4. unpow2 100.0%

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

      5. unpow3 100.0%

         \[\leadsto x + -0.3333333333333333 \cdot \color{blue}{{x}^{3}} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{x + -0.3333333333333333 \cdot {x}^{3}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -5 \lor \neg \left(-2 \cdot x \leq 5 \cdot 10^{-6}\right):\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.3% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-8}:\\ \;\;\;\;-1 + \frac{2}{2 + x \cdot \left(x \cdot \left(2 + x \cdot -1.3333333333333333\right) - 2\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{x + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.35e-8)
   (+
    -1.0
    (/ 2.0 (+ 2.0 (* x (- (* x (+ 2.0 (* x -1.3333333333333333))) 2.0)))))
   (* x (/ 2.0 (+ x 2.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.35e-8) {
		tmp = -1.0 + (2.0 / (2.0 + (x * ((x * (2.0 + (x * -1.3333333333333333))) - 2.0))));
	} else {
		tmp = x * (2.0 / (x + 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.35d-8)) then
        tmp = (-1.0d0) + (2.0d0 / (2.0d0 + (x * ((x * (2.0d0 + (x * (-1.3333333333333333d0)))) - 2.0d0))))
    else
        tmp = x * (2.0d0 / (x + 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.35e-8) {
		tmp = -1.0 + (2.0 / (2.0 + (x * ((x * (2.0 + (x * -1.3333333333333333))) - 2.0))));
	} else {
		tmp = x * (2.0 / (x + 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.35e-8:
		tmp = -1.0 + (2.0 / (2.0 + (x * ((x * (2.0 + (x * -1.3333333333333333))) - 2.0))))
	else:
		tmp = x * (2.0 / (x + 2.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.35e-8)
		tmp = Float64(-1.0 + Float64(2.0 / Float64(2.0 + Float64(x * Float64(Float64(x * Float64(2.0 + Float64(x * -1.3333333333333333))) - 2.0)))));
	else
		tmp = Float64(x * Float64(2.0 / Float64(x + 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.35e-8)
		tmp = -1.0 + (2.0 / (2.0 + (x * ((x * (2.0 + (x * -1.3333333333333333))) - 2.0))));
	else
		tmp = x * (2.0 / (x + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.35e-8], N[(-1.0 + N[(2.0 / N[(2.0 + N[(x * N[(N[(x * N[(2.0 + N[(x * -1.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(2.0 / N[(x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.35000000000000001e-8

   1. Initial program 99.2%

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

   3. Taylor expanded in x around 0 98.0%

      \[\leadsto \frac{2}{\color{blue}{2 + x \cdot \left(x \cdot \left(2 + -1.3333333333333333 \cdot x\right) - 2\right)}} - 1 \]

   if -1.35000000000000001e-8 < x

   1. Initial program 37.0%

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

   3. Taylor expanded in x around 0 5.7%

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

      1. +-commutative 5.7%

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

   5. Simplified 5.7%

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

      1. flip-- 5.6%

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

      2. metadata-eval 5.6%

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

      3. difference-of-sqr-1 5.6%

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

      4. associate-+l+ 5.6%

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

      5. metadata-eval 5.6%

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

      6. associate--l+ 68.6%

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

      7. metadata-eval 68.6%

         \[\leadsto \frac{\left(x + 2\right) \cdot \left(x + \color{blue}{0}\right)}{\left(x + 1\right) + 1} \]

      8. +-rgt-identity 68.6%

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

      9. associate-+l+ 68.6%

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

      10. metadata-eval 68.6%

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

   7. Applied egg-rr 68.6%

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

   8. Taylor expanded in x around 0 72.9%

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

      1. *-commutative 72.9%

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

   10. Simplified 72.9%

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

       1. associate-/l* 72.9%

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

   12. Applied egg-rr 72.9%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-8}:\\ \;\;\;\;-1 + \frac{2}{2 + x \cdot \left(x \cdot \left(2 + x \cdot -1.3333333333333333\right) - 2\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{x + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.1% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-8}:\\ \;\;\;\;-1 + \frac{2}{2 + x \cdot \left(x \cdot 2 - 2\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{x + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.35e-8)
   (+ -1.0 (/ 2.0 (+ 2.0 (* x (- (* x 2.0) 2.0)))))
   (* x (/ 2.0 (+ x 2.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.35e-8) {
		tmp = -1.0 + (2.0 / (2.0 + (x * ((x * 2.0) - 2.0))));
	} else {
		tmp = x * (2.0 / (x + 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.35d-8)) then
        tmp = (-1.0d0) + (2.0d0 / (2.0d0 + (x * ((x * 2.0d0) - 2.0d0))))
    else
        tmp = x * (2.0d0 / (x + 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.35e-8) {
		tmp = -1.0 + (2.0 / (2.0 + (x * ((x * 2.0) - 2.0))));
	} else {
		tmp = x * (2.0 / (x + 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.35e-8:
		tmp = -1.0 + (2.0 / (2.0 + (x * ((x * 2.0) - 2.0))))
	else:
		tmp = x * (2.0 / (x + 2.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.35e-8)
		tmp = Float64(-1.0 + Float64(2.0 / Float64(2.0 + Float64(x * Float64(Float64(x * 2.0) - 2.0)))));
	else
		tmp = Float64(x * Float64(2.0 / Float64(x + 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.35e-8)
		tmp = -1.0 + (2.0 / (2.0 + (x * ((x * 2.0) - 2.0))));
	else
		tmp = x * (2.0 / (x + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.35e-8], N[(-1.0 + N[(2.0 / N[(2.0 + N[(x * N[(N[(x * 2.0), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(2.0 / N[(x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.35000000000000001e-8

   1. Initial program 99.2%

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

   3. Taylor expanded in x around 0 97.6%

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

   if -1.35000000000000001e-8 < x

   1. Initial program 37.0%

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

   3. Taylor expanded in x around 0 5.7%

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

      1. +-commutative 5.7%

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

   5. Simplified 5.7%

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

      1. flip-- 5.6%

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

      2. metadata-eval 5.6%

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

      3. difference-of-sqr-1 5.6%

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

      4. associate-+l+ 5.6%

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

      5. metadata-eval 5.6%

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

      6. associate--l+ 68.6%

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

      7. metadata-eval 68.6%

         \[\leadsto \frac{\left(x + 2\right) \cdot \left(x + \color{blue}{0}\right)}{\left(x + 1\right) + 1} \]

      8. +-rgt-identity 68.6%

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

      9. associate-+l+ 68.6%

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

      10. metadata-eval 68.6%

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

   7. Applied egg-rr 68.6%

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

   8. Taylor expanded in x around 0 72.9%

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

      1. *-commutative 72.9%

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

   10. Simplified 72.9%

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

       1. associate-/l* 72.9%

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

   12. Applied egg-rr 72.9%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-8}:\\ \;\;\;\;-1 + \frac{2}{2 + x \cdot \left(x \cdot 2 - 2\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{x + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.8% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 2.5:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;2 - \frac{4}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0) -1.0 (if (<= x 2.5) x (- 2.0 (/ 4.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = -1.0;
	} else if (x <= 2.5) {
		tmp = x;
	} else {
		tmp = 2.0 - (4.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = -1.0d0
    else if (x <= 2.5d0) then
        tmp = x
    else
        tmp = 2.0d0 - (4.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = -1.0;
	} else if (x <= 2.5) {
		tmp = x;
	} else {
		tmp = 2.0 - (4.0 / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = -1.0
	elif x <= 2.5:
		tmp = x
	else:
		tmp = 2.0 - (4.0 / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = -1.0;
	elseif (x <= 2.5)
		tmp = x;
	else
		tmp = Float64(2.0 - Float64(4.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = -1.0;
	elseif (x <= 2.5)
		tmp = x;
	else
		tmp = 2.0 - (4.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], -1.0, If[LessEqual[x, 2.5], x, N[(2.0 - N[(4.0 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.9%

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

   4. Taylor expanded in x around inf 100.0%

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

   if -1 < x < 2.5

   1. Initial program 8.2%

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

   3. Taylor expanded in x around 0 98.9%

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

   if 2.5 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 5.7%

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

      1. +-commutative 5.7%

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

   5. Simplified 5.7%

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

      1. flip-- 5.4%

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

      2. metadata-eval 5.4%

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

      3. difference-of-sqr-1 5.4%

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

      4. associate-+l+ 5.4%

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

      5. metadata-eval 5.4%

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

      6. associate--l+ 5.4%

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

      7. metadata-eval 5.4%

         \[\leadsto \frac{\left(x + 2\right) \cdot \left(x + \color{blue}{0}\right)}{\left(x + 1\right) + 1} \]

      8. +-rgt-identity 5.4%

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

      9. associate-+l+ 5.4%

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

      10. metadata-eval 5.4%

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

   7. Applied egg-rr 5.4%

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

   8. Taylor expanded in x around 0 18.8%

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

      1. *-commutative 18.8%

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

   10. Simplified 18.8%

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

   11. Taylor expanded in x around inf 18.9%

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

       1. associate-*r/ 18.9%

          \[\leadsto 2 - \color{blue}{\frac{4 \cdot 1}{x}} \]

       2. metadata-eval 18.9%

          \[\leadsto 2 - \frac{\color{blue}{4}}{x} \]

   13. Simplified 18.9%

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

Alternative 9: 78.3% accurate, 9.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-0.68d0)) then
        tmp = -1.0d0
    else
        tmp = x * (2.0d0 / (x + 2.0d0))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.68)
		tmp = -1.0;
	else
		tmp = Float64(x * Float64(2.0 / Float64(x + 2.0)));
	end
	return tmp
end

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -0.680000000000000049

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.9%

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

   4. Taylor expanded in x around inf 100.0%

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

   if -0.680000000000000049 < x

   1. Initial program 38.0%

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

   3. Taylor expanded in x around 0 6.9%

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

      1. +-commutative 6.9%

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

   5. Simplified 6.9%

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

      1. flip-- 6.7%

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

      2. metadata-eval 6.7%

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

      3. difference-of-sqr-1 6.8%

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

      4. associate-+l+ 6.8%

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

      5. metadata-eval 6.8%

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

      6. associate--l+ 68.5%

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

      7. metadata-eval 68.5%

         \[\leadsto \frac{\left(x + 2\right) \cdot \left(x + \color{blue}{0}\right)}{\left(x + 1\right) + 1} \]

      8. +-rgt-identity 68.5%

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

      9. associate-+l+ 68.5%

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

      10. metadata-eval 68.5%

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

   7. Applied egg-rr 68.5%

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

   8. Taylor expanded in x around 0 72.3%

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

      1. *-commutative 72.3%

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

   10. Simplified 72.3%

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

       1. associate-/l* 72.3%

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

   12. Applied egg-rr 72.3%

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

Alternative 10: 78.8% accurate, 9.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.9%

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

   4. Taylor expanded in x around inf 100.0%

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

   if -1 < x < 2

   1. Initial program 8.2%

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

   3. Taylor expanded in x around 0 98.9%

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

   if 2 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 5.7%

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

      1. +-commutative 5.7%

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

   5. Simplified 5.7%

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

      1. flip-- 5.4%

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

      2. metadata-eval 5.4%

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

      3. difference-of-sqr-1 5.4%

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

      4. associate-+l+ 5.4%

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

      5. metadata-eval 5.4%

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

      6. associate--l+ 5.4%

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

      7. metadata-eval 5.4%

         \[\leadsto \frac{\left(x + 2\right) \cdot \left(x + \color{blue}{0}\right)}{\left(x + 1\right) + 1} \]

      8. +-rgt-identity 5.4%

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

      9. associate-+l+ 5.4%

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

      10. metadata-eval 5.4%

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

   7. Applied egg-rr 5.4%

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

   8. Taylor expanded in x around 0 18.8%

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

      1. *-commutative 18.8%

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

   10. Simplified 18.8%

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

   11. Taylor expanded in x around inf 18.7%

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

Alternative 11: 32.2% accurate, 18.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001e-308

   1. Initial program 53.1%

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

   3. Taylor expanded in x around 0 52.3%

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

   4. Taylor expanded in x around inf 50.6%

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

   if 1.1000000000000001e-308 < x

   1. Initial program 52.9%

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

   3. Taylor expanded in x around 0 5.8%

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

      1. +-commutative 5.8%

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

   5. Simplified 5.8%

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

      1. flip-- 5.6%

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

      2. metadata-eval 5.6%

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

      3. difference-of-sqr-1 5.6%

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

      4. associate-+l+ 5.6%

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

      5. metadata-eval 5.6%

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

      6. associate--l+ 52.7%

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

      7. metadata-eval 52.7%

         \[\leadsto \frac{\left(x + 2\right) \cdot \left(x + \color{blue}{0}\right)}{\left(x + 1\right) + 1} \]

      8. +-rgt-identity 52.7%

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

      9. associate-+l+ 52.7%

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

      10. metadata-eval 52.7%

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

   7. Applied egg-rr 52.7%

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

   8. Taylor expanded in x around 0 59.2%

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

      1. *-commutative 59.2%

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

   10. Simplified 59.2%

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

   11. Taylor expanded in x around inf 12.1%

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

Alternative 12: 27.1% 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 53.0%

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

3. Taylor expanded in x around 0 28.4%

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

4. Taylor expanded in x around inf 26.6%

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

Reproduce

herbie shell --seed 2024172 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))