Logistic function from Lakshay Garg

Percentage Accurate: 54.4% → 99.9%

Time: 11.1s

Alternatives: 13

Speedup: 21.3×

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, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + \mathsf{expm1}\left(-2 \cdot x\right)\\ \mathbf{if}\;-2 \cdot x \leq -0.05:\\ \;\;\;\;\frac{1 - \frac{4}{{t_0}^{2}}}{-1 - \frac{2}{1 + {\left(e^{x}\right)}^{-2}}}\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-7}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{2}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 2.0 (expm1 (* -2.0 x)))))
   (if (<= (* -2.0 x) -0.05)
     (/
      (- 1.0 (/ 4.0 (pow t_0 2.0)))
      (- -1.0 (/ 2.0 (+ 1.0 (pow (exp x) -2.0)))))
     (if (<= (* -2.0 x) 5e-7)
       (+ x (* -0.3333333333333333 (pow x 3.0)))
       (+ -1.0 (/ 2.0 t_0))))))

C

double code(double x, double y) {
	double t_0 = 2.0 + expm1((-2.0 * x));
	double tmp;
	if ((-2.0 * x) <= -0.05) {
		tmp = (1.0 - (4.0 / pow(t_0, 2.0))) / (-1.0 - (2.0 / (1.0 + pow(exp(x), -2.0))));
	} else if ((-2.0 * x) <= 5e-7) {
		tmp = x + (-0.3333333333333333 * pow(x, 3.0));
	} else {
		tmp = -1.0 + (2.0 / t_0);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = 2.0 + Math.expm1((-2.0 * x));
	double tmp;
	if ((-2.0 * x) <= -0.05) {
		tmp = (1.0 - (4.0 / Math.pow(t_0, 2.0))) / (-1.0 - (2.0 / (1.0 + Math.pow(Math.exp(x), -2.0))));
	} else if ((-2.0 * x) <= 5e-7) {
		tmp = x + (-0.3333333333333333 * Math.pow(x, 3.0));
	} else {
		tmp = -1.0 + (2.0 / t_0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 2.0 + math.expm1((-2.0 * x))
	tmp = 0
	if (-2.0 * x) <= -0.05:
		tmp = (1.0 - (4.0 / math.pow(t_0, 2.0))) / (-1.0 - (2.0 / (1.0 + math.pow(math.exp(x), -2.0))))
	elif (-2.0 * x) <= 5e-7:
		tmp = x + (-0.3333333333333333 * math.pow(x, 3.0))
	else:
		tmp = -1.0 + (2.0 / t_0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(2.0 + expm1(Float64(-2.0 * x)))
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.05)
		tmp = Float64(Float64(1.0 - Float64(4.0 / (t_0 ^ 2.0))) / Float64(-1.0 - Float64(2.0 / Float64(1.0 + (exp(x) ^ -2.0)))));
	elseif (Float64(-2.0 * x) <= 5e-7)
		tmp = Float64(x + Float64(-0.3333333333333333 * (x ^ 3.0)));
	else
		tmp = Float64(-1.0 + Float64(2.0 / t_0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(2.0 + N[(Exp[N[(-2.0 * x), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.05], N[(N[(1.0 - N[(4.0 / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - N[(2.0 / N[(1.0 + N[Power[N[Exp[x], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 5e-7], N[(x + N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(2.0 / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 -2 x) < -0.050000000000000003

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. exp-prod 99.8%

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

      3. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. add-cbrt-cube 99.9%

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

      2. pow3 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. pow-exp 99.9%

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

      2. *-commutative 99.9%

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

      3. pow-exp 99.9%

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

      4. associate-*l* 99.9%

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

      5. metadata-eval 99.9%

         \[\leadsto \frac{2}{1 + \sqrt[3]{e^{x \cdot \color{blue}{-6}}}} + -1 \]

   7. Applied egg-rr 99.9%

      \[\leadsto \frac{2}{1 + \sqrt[3]{\color{blue}{e^{x \cdot -6}}}} + -1 \]
   8. Step-by-step derivation

      1. pow1/3 99.8%

         \[\leadsto \frac{2}{1 + \color{blue}{{\left(e^{x \cdot -6}\right)}^{0.3333333333333333}}} + -1 \]

      2. exp-prod 99.8%

         \[\leadsto \frac{2}{1 + {\color{blue}{\left({\left(e^{x}\right)}^{-6}\right)}}^{0.3333333333333333}} + -1 \]

      3. pow-pow 99.8%

         \[\leadsto \frac{2}{1 + \color{blue}{{\left(e^{x}\right)}^{\left(-6 \cdot 0.3333333333333333\right)}}} + -1 \]

      4. metadata-eval 99.8%

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

      5. exp-prod 99.8%

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

      6. log1p-expm1-u 99.8%

         \[\leadsto \frac{2}{1 + e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot -2\right)\right)}}} + -1 \]

      7. log1p-udef 99.8%

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

      8. add-exp-log 99.8%

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

      9. +-commutative 99.8%

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

      10. flip-+ 99.8%

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

   9. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{1 - \frac{4}{{\left(2 + \mathsf{expm1}\left(x \cdot -2\right)\right)}^{2}}}{-1 - \frac{2}{2 + \mathsf{expm1}\left(x \cdot -2\right)}}} \]

   10. Taylor expanded in x around inf 99.9%

       \[\leadsto \frac{1 - \frac{4}{{\left(2 + \mathsf{expm1}\left(x \cdot -2\right)\right)}^{2}}}{-1 - \color{blue}{\frac{2}{1 + e^{-2 \cdot x}}}} \]
   11. Step-by-step derivation

       1. *-commutative 99.9%

          \[\leadsto \frac{1 - \frac{4}{{\left(2 + \mathsf{expm1}\left(x \cdot -2\right)\right)}^{2}}}{-1 - \frac{2}{1 + e^{\color{blue}{x \cdot -2}}}} \]

       2. exp-prod 100.0%

          \[\leadsto \frac{1 - \frac{4}{{\left(2 + \mathsf{expm1}\left(x \cdot -2\right)\right)}^{2}}}{-1 - \frac{2}{1 + \color{blue}{{\left(e^{x}\right)}^{-2}}}} \]

   12. Simplified 100.0%

       \[\leadsto \frac{1 - \frac{4}{{\left(2 + \mathsf{expm1}\left(x \cdot -2\right)\right)}^{2}}}{-1 - \color{blue}{\frac{2}{1 + {\left(e^{x}\right)}^{-2}}}} \]

   if -0.050000000000000003 < (*.f64 -2 x) < 4.99999999999999977e-7

   1. Initial program 7.1%

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

      1. sub-neg 7.1%

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

      2. exp-prod 7.1%

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

      3. metadata-eval 7.1%

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

   3. Simplified 7.1%

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

   4. Taylor expanded in x around 0 100.0%

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

   if 4.99999999999999977e-7 < (*.f64 -2 x)

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. exp-prod 100.0%

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

      3. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. add-cbrt-cube 100.0%

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

      2. pow3 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. rem-cbrt-cube 100.0%

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

      2. expm1-log1p-u 100.0%

         \[\leadsto \frac{2}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)}} + -1 \]

      3. expm1-udef 100.0%

         \[\leadsto \frac{2}{1 + \color{blue}{\left(e^{\mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)} - 1\right)}} + -1 \]

      4. log1p-udef 100.0%

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

      5. add-exp-log 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. associate--l+ 100.0%

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

      2. exp-prod 100.0%

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

      3. expm1-def 100.0%

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

      4. *-commutative 100.0%

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

   9. Simplified 100.0%

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

       1. add-exp-log 100.0%

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

       2. log1p-udef 100.0%

          \[\leadsto \frac{2}{1 + e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot -2\right)\right)}}} + -1 \]

       3. log1p-expm1-u 100.0%

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

       4. exp-prod 100.0%

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

       5. metadata-eval 100.0%

          \[\leadsto \frac{2}{1 + {\left(e^{x}\right)}^{\color{blue}{\left(-6 \cdot 0.3333333333333333\right)}}} + -1 \]

       6. pow-pow 99.9%

          \[\leadsto \frac{2}{1 + \color{blue}{{\left({\left(e^{x}\right)}^{-6}\right)}^{0.3333333333333333}}} + -1 \]

       7. exp-prod 100.0%

          \[\leadsto \frac{2}{1 + {\color{blue}{\left(e^{x \cdot -6}\right)}}^{0.3333333333333333}} + -1 \]

       8. pow1/3 100.0%

          \[\leadsto \frac{2}{1 + \color{blue}{\sqrt[3]{e^{x \cdot -6}}}} + -1 \]

       9. *-un-lft-identity 100.0%

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

       10. *-commutative 100.0%

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

       11. pow1/3 100.0%

           \[\leadsto \frac{2}{\left(1 + \color{blue}{{\left(e^{x \cdot -6}\right)}^{0.3333333333333333}}\right) \cdot 1} + -1 \]

       12. exp-prod 99.9%

           \[\leadsto \frac{2}{\left(1 + {\color{blue}{\left({\left(e^{x}\right)}^{-6}\right)}}^{0.3333333333333333}\right) \cdot 1} + -1 \]

       13. pow-pow 100.0%

           \[\leadsto \frac{2}{\left(1 + \color{blue}{{\left(e^{x}\right)}^{\left(-6 \cdot 0.3333333333333333\right)}}\right) \cdot 1} + -1 \]

       14. metadata-eval 100.0%

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

       15. exp-prod 100.0%

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

       16. log1p-expm1-u 100.0%

           \[\leadsto \frac{2}{\left(1 + e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot -2\right)\right)}}\right) \cdot 1} + -1 \]

       17. log1p-udef 100.0%

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

       18. add-exp-log 100.0%

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

       19. associate-+r+ 100.0%

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

       20. metadata-eval 100.0%

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

   11. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.05:\\ \;\;\;\;\frac{1 - \frac{4}{{\left(2 + \mathsf{expm1}\left(-2 \cdot x\right)\right)}^{2}}}{-1 - \frac{2}{1 + {\left(e^{x}\right)}^{-2}}}\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-7}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{2}{2 + \mathsf{expm1}\left(-2 \cdot x\right)}\\ \end{array} \]

Alternative 2: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + \mathsf{expm1}\left(-2 \cdot x\right)\\ t_1 := \frac{2}{t_0}\\ \mathbf{if}\;-2 \cdot x \leq -0.05:\\ \;\;\;\;\frac{1 - \frac{4}{{t_0}^{2}}}{-1 - t_1}\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-7}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1 + t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 2.0 (expm1 (* -2.0 x)))) (t_1 (/ 2.0 t_0)))
   (if (<= (* -2.0 x) -0.05)
     (/ (- 1.0 (/ 4.0 (pow t_0 2.0))) (- -1.0 t_1))
     (if (<= (* -2.0 x) 5e-7)
       (+ x (* -0.3333333333333333 (pow x 3.0)))
       (+ -1.0 t_1)))))

C

double code(double x, double y) {
	double t_0 = 2.0 + expm1((-2.0 * x));
	double t_1 = 2.0 / t_0;
	double tmp;
	if ((-2.0 * x) <= -0.05) {
		tmp = (1.0 - (4.0 / pow(t_0, 2.0))) / (-1.0 - t_1);
	} else if ((-2.0 * x) <= 5e-7) {
		tmp = x + (-0.3333333333333333 * pow(x, 3.0));
	} else {
		tmp = -1.0 + t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = 2.0 + Math.expm1((-2.0 * x));
	double t_1 = 2.0 / t_0;
	double tmp;
	if ((-2.0 * x) <= -0.05) {
		tmp = (1.0 - (4.0 / Math.pow(t_0, 2.0))) / (-1.0 - t_1);
	} else if ((-2.0 * x) <= 5e-7) {
		tmp = x + (-0.3333333333333333 * Math.pow(x, 3.0));
	} else {
		tmp = -1.0 + t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 2.0 + math.expm1((-2.0 * x))
	t_1 = 2.0 / t_0
	tmp = 0
	if (-2.0 * x) <= -0.05:
		tmp = (1.0 - (4.0 / math.pow(t_0, 2.0))) / (-1.0 - t_1)
	elif (-2.0 * x) <= 5e-7:
		tmp = x + (-0.3333333333333333 * math.pow(x, 3.0))
	else:
		tmp = -1.0 + t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(2.0 + expm1(Float64(-2.0 * x)))
	t_1 = Float64(2.0 / t_0)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.05)
		tmp = Float64(Float64(1.0 - Float64(4.0 / (t_0 ^ 2.0))) / Float64(-1.0 - t_1));
	elseif (Float64(-2.0 * x) <= 5e-7)
		tmp = Float64(x + Float64(-0.3333333333333333 * (x ^ 3.0)));
	else
		tmp = Float64(-1.0 + t_1);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(2.0 + N[(Exp[N[(-2.0 * x), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / t$95$0), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.05], N[(N[(1.0 - N[(4.0 / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 5e-7], N[(x + N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 -2 x) < -0.050000000000000003

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. exp-prod 99.8%

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

      3. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. add-cbrt-cube 99.9%

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

      2. pow3 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. pow-exp 99.9%

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

      2. *-commutative 99.9%

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

      3. pow-exp 99.9%

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

      4. associate-*l* 99.9%

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

      5. metadata-eval 99.9%

         \[\leadsto \frac{2}{1 + \sqrt[3]{e^{x \cdot \color{blue}{-6}}}} + -1 \]

   7. Applied egg-rr 99.9%

      \[\leadsto \frac{2}{1 + \sqrt[3]{\color{blue}{e^{x \cdot -6}}}} + -1 \]
   8. Step-by-step derivation

      1. pow1/3 99.8%

         \[\leadsto \frac{2}{1 + \color{blue}{{\left(e^{x \cdot -6}\right)}^{0.3333333333333333}}} + -1 \]

      2. exp-prod 99.8%

         \[\leadsto \frac{2}{1 + {\color{blue}{\left({\left(e^{x}\right)}^{-6}\right)}}^{0.3333333333333333}} + -1 \]

      3. pow-pow 99.8%

         \[\leadsto \frac{2}{1 + \color{blue}{{\left(e^{x}\right)}^{\left(-6 \cdot 0.3333333333333333\right)}}} + -1 \]

      4. metadata-eval 99.8%

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

      5. exp-prod 99.8%

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

      6. log1p-expm1-u 99.8%

         \[\leadsto \frac{2}{1 + e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot -2\right)\right)}}} + -1 \]

      7. log1p-udef 99.8%

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

      8. add-exp-log 99.8%

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

      9. +-commutative 99.8%

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

      10. flip-+ 99.8%

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

   9. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{1 - \frac{4}{{\left(2 + \mathsf{expm1}\left(x \cdot -2\right)\right)}^{2}}}{-1 - \frac{2}{2 + \mathsf{expm1}\left(x \cdot -2\right)}}} \]

   if -0.050000000000000003 < (*.f64 -2 x) < 4.99999999999999977e-7

   1. Initial program 7.1%

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

      1. sub-neg 7.1%

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

      2. exp-prod 7.1%

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

      3. metadata-eval 7.1%

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

   3. Simplified 7.1%

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

   4. Taylor expanded in x around 0 100.0%

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

   if 4.99999999999999977e-7 < (*.f64 -2 x)

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. exp-prod 100.0%

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

      3. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. add-cbrt-cube 100.0%

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

      2. pow3 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. rem-cbrt-cube 100.0%

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

      2. expm1-log1p-u 100.0%

         \[\leadsto \frac{2}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)}} + -1 \]

      3. expm1-udef 100.0%

         \[\leadsto \frac{2}{1 + \color{blue}{\left(e^{\mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)} - 1\right)}} + -1 \]

      4. log1p-udef 100.0%

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

      5. add-exp-log 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. associate--l+ 100.0%

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

      2. exp-prod 100.0%

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

      3. expm1-def 100.0%

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

      4. *-commutative 100.0%

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

   9. Simplified 100.0%

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

       1. add-exp-log 100.0%

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

       2. log1p-udef 100.0%

          \[\leadsto \frac{2}{1 + e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot -2\right)\right)}}} + -1 \]

       3. log1p-expm1-u 100.0%

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

       4. exp-prod 100.0%

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

       5. metadata-eval 100.0%

          \[\leadsto \frac{2}{1 + {\left(e^{x}\right)}^{\color{blue}{\left(-6 \cdot 0.3333333333333333\right)}}} + -1 \]

       6. pow-pow 99.9%

          \[\leadsto \frac{2}{1 + \color{blue}{{\left({\left(e^{x}\right)}^{-6}\right)}^{0.3333333333333333}}} + -1 \]

       7. exp-prod 100.0%

          \[\leadsto \frac{2}{1 + {\color{blue}{\left(e^{x \cdot -6}\right)}}^{0.3333333333333333}} + -1 \]

       8. pow1/3 100.0%

          \[\leadsto \frac{2}{1 + \color{blue}{\sqrt[3]{e^{x \cdot -6}}}} + -1 \]

       9. *-un-lft-identity 100.0%

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

       10. *-commutative 100.0%

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

       11. pow1/3 100.0%

           \[\leadsto \frac{2}{\left(1 + \color{blue}{{\left(e^{x \cdot -6}\right)}^{0.3333333333333333}}\right) \cdot 1} + -1 \]

       12. exp-prod 99.9%

           \[\leadsto \frac{2}{\left(1 + {\color{blue}{\left({\left(e^{x}\right)}^{-6}\right)}}^{0.3333333333333333}\right) \cdot 1} + -1 \]

       13. pow-pow 100.0%

           \[\leadsto \frac{2}{\left(1 + \color{blue}{{\left(e^{x}\right)}^{\left(-6 \cdot 0.3333333333333333\right)}}\right) \cdot 1} + -1 \]

       14. metadata-eval 100.0%

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

       15. exp-prod 100.0%

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

       16. log1p-expm1-u 100.0%

           \[\leadsto \frac{2}{\left(1 + e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot -2\right)\right)}}\right) \cdot 1} + -1 \]

       17. log1p-udef 100.0%

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

       18. add-exp-log 100.0%

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

       19. associate-+r+ 100.0%

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

       20. metadata-eval 100.0%

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

   11. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.05:\\ \;\;\;\;\frac{1 - \frac{4}{{\left(2 + \mathsf{expm1}\left(-2 \cdot x\right)\right)}^{2}}}{-1 - \frac{2}{2 + \mathsf{expm1}\left(-2 \cdot x\right)}}\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-7}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{2}{2 + \mathsf{expm1}\left(-2 \cdot x\right)}\\ \end{array} \]

Alternative 3: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 + \frac{2}{2 + \mathsf{expm1}\left(-2 \cdot x\right)}\\ \mathbf{if}\;-2 \cdot x \leq -0.05:\\ \;\;\;\;{\left(\sqrt[3]{t_0}\right)}^{3}\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-7}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ -1.0 (/ 2.0 (+ 2.0 (expm1 (* -2.0 x)))))))
   (if (<= (* -2.0 x) -0.05)
     (pow (cbrt t_0) 3.0)
     (if (<= (* -2.0 x) 5e-7) (+ x (* -0.3333333333333333 (pow x 3.0))) t_0))))

C

double code(double x, double y) {
	double t_0 = -1.0 + (2.0 / (2.0 + expm1((-2.0 * x))));
	double tmp;
	if ((-2.0 * x) <= -0.05) {
		tmp = pow(cbrt(t_0), 3.0);
	} else if ((-2.0 * x) <= 5e-7) {
		tmp = x + (-0.3333333333333333 * pow(x, 3.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = -1.0 + (2.0 / (2.0 + Math.expm1((-2.0 * x))));
	double tmp;
	if ((-2.0 * x) <= -0.05) {
		tmp = Math.pow(Math.cbrt(t_0), 3.0);
	} else if ((-2.0 * x) <= 5e-7) {
		tmp = x + (-0.3333333333333333 * Math.pow(x, 3.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(-1.0 + Float64(2.0 / Float64(2.0 + expm1(Float64(-2.0 * x)))))
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.05)
		tmp = cbrt(t_0) ^ 3.0;
	elseif (Float64(-2.0 * x) <= 5e-7)
		tmp = Float64(x + Float64(-0.3333333333333333 * (x ^ 3.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(-1.0 + N[(2.0 / N[(2.0 + N[(Exp[N[(-2.0 * x), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.05], N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 3.0], $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 5e-7], N[(x + N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 -2 x) < -0.050000000000000003

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. exp-prod 99.8%

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

      3. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. add-cbrt-cube 99.9%

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

      2. pow3 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. pow-exp 99.9%

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

      2. *-commutative 99.9%

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

      3. pow-exp 99.9%

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

      4. associate-*l* 99.9%

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

      5. metadata-eval 99.9%

         \[\leadsto \frac{2}{1 + \sqrt[3]{e^{x \cdot \color{blue}{-6}}}} + -1 \]

   7. Applied egg-rr 99.9%

      \[\leadsto \frac{2}{1 + \sqrt[3]{\color{blue}{e^{x \cdot -6}}}} + -1 \]
   8. Step-by-step derivation

      1. add-cube-cbrt 99.9%

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

      2. pow3 99.9%

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

   9. Applied egg-rr 99.9%

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

   if -0.050000000000000003 < (*.f64 -2 x) < 4.99999999999999977e-7

   1. Initial program 7.1%

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

      1. sub-neg 7.1%

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

      2. exp-prod 7.1%

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

      3. metadata-eval 7.1%

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

   3. Simplified 7.1%

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

   4. Taylor expanded in x around 0 100.0%

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

   if 4.99999999999999977e-7 < (*.f64 -2 x)

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. exp-prod 100.0%

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

      3. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. add-cbrt-cube 100.0%

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

      2. pow3 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. rem-cbrt-cube 100.0%

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

      2. expm1-log1p-u 100.0%

         \[\leadsto \frac{2}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)}} + -1 \]

      3. expm1-udef 100.0%

         \[\leadsto \frac{2}{1 + \color{blue}{\left(e^{\mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)} - 1\right)}} + -1 \]

      4. log1p-udef 100.0%

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

      5. add-exp-log 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. associate--l+ 100.0%

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

      2. exp-prod 100.0%

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

      3. expm1-def 100.0%

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

      4. *-commutative 100.0%

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

   9. Simplified 100.0%

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

       1. add-exp-log 100.0%

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

       2. log1p-udef 100.0%

          \[\leadsto \frac{2}{1 + e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot -2\right)\right)}}} + -1 \]

       3. log1p-expm1-u 100.0%

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

       4. exp-prod 100.0%

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

       5. metadata-eval 100.0%

          \[\leadsto \frac{2}{1 + {\left(e^{x}\right)}^{\color{blue}{\left(-6 \cdot 0.3333333333333333\right)}}} + -1 \]

       6. pow-pow 99.9%

          \[\leadsto \frac{2}{1 + \color{blue}{{\left({\left(e^{x}\right)}^{-6}\right)}^{0.3333333333333333}}} + -1 \]

       7. exp-prod 100.0%

          \[\leadsto \frac{2}{1 + {\color{blue}{\left(e^{x \cdot -6}\right)}}^{0.3333333333333333}} + -1 \]

       8. pow1/3 100.0%

          \[\leadsto \frac{2}{1 + \color{blue}{\sqrt[3]{e^{x \cdot -6}}}} + -1 \]

       9. *-un-lft-identity 100.0%

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

       10. *-commutative 100.0%

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

       11. pow1/3 100.0%

           \[\leadsto \frac{2}{\left(1 + \color{blue}{{\left(e^{x \cdot -6}\right)}^{0.3333333333333333}}\right) \cdot 1} + -1 \]

       12. exp-prod 99.9%

           \[\leadsto \frac{2}{\left(1 + {\color{blue}{\left({\left(e^{x}\right)}^{-6}\right)}}^{0.3333333333333333}\right) \cdot 1} + -1 \]

       13. pow-pow 100.0%

           \[\leadsto \frac{2}{\left(1 + \color{blue}{{\left(e^{x}\right)}^{\left(-6 \cdot 0.3333333333333333\right)}}\right) \cdot 1} + -1 \]

       14. metadata-eval 100.0%

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

       15. exp-prod 100.0%

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

       16. log1p-expm1-u 100.0%

           \[\leadsto \frac{2}{\left(1 + e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot -2\right)\right)}}\right) \cdot 1} + -1 \]

       17. log1p-udef 100.0%

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

       18. add-exp-log 100.0%

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

       19. associate-+r+ 100.0%

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

       20. metadata-eval 100.0%

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

   11. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.05:\\ \;\;\;\;{\left(\sqrt[3]{-1 + \frac{2}{2 + \mathsf{expm1}\left(-2 \cdot x\right)}}\right)}^{3}\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-7}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{2}{2 + \mathsf{expm1}\left(-2 \cdot x\right)}\\ \end{array} \]

Alternative 4: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 + \frac{2}{2 + \mathsf{expm1}\left(-2 \cdot x\right)}\\ \mathbf{if}\;-2 \cdot x \leq -0.05:\\ \;\;\;\;e^{\log t_0}\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-7}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ -1.0 (/ 2.0 (+ 2.0 (expm1 (* -2.0 x)))))))
   (if (<= (* -2.0 x) -0.05)
     (exp (log t_0))
     (if (<= (* -2.0 x) 5e-7) (+ x (* -0.3333333333333333 (pow x 3.0))) t_0))))

C

double code(double x, double y) {
	double t_0 = -1.0 + (2.0 / (2.0 + expm1((-2.0 * x))));
	double tmp;
	if ((-2.0 * x) <= -0.05) {
		tmp = exp(log(t_0));
	} else if ((-2.0 * x) <= 5e-7) {
		tmp = x + (-0.3333333333333333 * pow(x, 3.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = -1.0 + (2.0 / (2.0 + Math.expm1((-2.0 * x))));
	double tmp;
	if ((-2.0 * x) <= -0.05) {
		tmp = Math.exp(Math.log(t_0));
	} else if ((-2.0 * x) <= 5e-7) {
		tmp = x + (-0.3333333333333333 * Math.pow(x, 3.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = -1.0 + (2.0 / (2.0 + math.expm1((-2.0 * x))))
	tmp = 0
	if (-2.0 * x) <= -0.05:
		tmp = math.exp(math.log(t_0))
	elif (-2.0 * x) <= 5e-7:
		tmp = x + (-0.3333333333333333 * math.pow(x, 3.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(-1.0 + Float64(2.0 / Float64(2.0 + expm1(Float64(-2.0 * x)))))
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.05)
		tmp = exp(log(t_0));
	elseif (Float64(-2.0 * x) <= 5e-7)
		tmp = Float64(x + Float64(-0.3333333333333333 * (x ^ 3.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(-1.0 + N[(2.0 / N[(2.0 + N[(Exp[N[(-2.0 * x), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.05], N[Exp[N[Log[t$95$0], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 5e-7], N[(x + N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 -2 x) < -0.050000000000000003

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. exp-prod 99.8%

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

      3. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. add-cbrt-cube 99.9%

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

      2. pow3 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. pow-exp 99.9%

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

      2. *-commutative 99.9%

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

      3. pow-exp 99.9%

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

      4. associate-*l* 99.9%

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

      5. metadata-eval 99.9%

         \[\leadsto \frac{2}{1 + \sqrt[3]{e^{x \cdot \color{blue}{-6}}}} + -1 \]

   7. Applied egg-rr 99.9%

      \[\leadsto \frac{2}{1 + \sqrt[3]{\color{blue}{e^{x \cdot -6}}}} + -1 \]
   8. Step-by-step derivation

      1. add-exp-log 99.9%

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

      2. pow1/3 99.8%

         \[\leadsto e^{\log \left(\frac{2}{1 + \color{blue}{{\left(e^{x \cdot -6}\right)}^{0.3333333333333333}}} + -1\right)} \]

      3. exp-prod 99.8%

         \[\leadsto e^{\log \left(\frac{2}{1 + {\color{blue}{\left({\left(e^{x}\right)}^{-6}\right)}}^{0.3333333333333333}} + -1\right)} \]

      4. pow-pow 99.8%

         \[\leadsto e^{\log \left(\frac{2}{1 + \color{blue}{{\left(e^{x}\right)}^{\left(-6 \cdot 0.3333333333333333\right)}}} + -1\right)} \]

      5. metadata-eval 99.8%

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

      6. exp-prod 99.8%

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

      7. log1p-expm1-u 99.8%

         \[\leadsto e^{\log \left(\frac{2}{1 + e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot -2\right)\right)}}} + -1\right)} \]

      8. log1p-udef 99.8%

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

      9. add-exp-log 99.8%

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

      10. associate-+r+ 99.9%

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

      11. metadata-eval 99.9%

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

   9. Applied egg-rr 99.9%

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

   if -0.050000000000000003 < (*.f64 -2 x) < 4.99999999999999977e-7

   1. Initial program 7.1%

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

      1. sub-neg 7.1%

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

      2. exp-prod 7.1%

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

      3. metadata-eval 7.1%

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

   3. Simplified 7.1%

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

   4. Taylor expanded in x around 0 100.0%

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

   if 4.99999999999999977e-7 < (*.f64 -2 x)

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. exp-prod 100.0%

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

      3. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. add-cbrt-cube 100.0%

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

      2. pow3 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. rem-cbrt-cube 100.0%

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

      2. expm1-log1p-u 100.0%

         \[\leadsto \frac{2}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)}} + -1 \]

      3. expm1-udef 100.0%

         \[\leadsto \frac{2}{1 + \color{blue}{\left(e^{\mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)} - 1\right)}} + -1 \]

      4. log1p-udef 100.0%

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

      5. add-exp-log 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. associate--l+ 100.0%

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

      2. exp-prod 100.0%

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

      3. expm1-def 100.0%

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

      4. *-commutative 100.0%

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

   9. Simplified 100.0%

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

       1. add-exp-log 100.0%

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

       2. log1p-udef 100.0%

          \[\leadsto \frac{2}{1 + e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot -2\right)\right)}}} + -1 \]

       3. log1p-expm1-u 100.0%

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

       4. exp-prod 100.0%

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

       5. metadata-eval 100.0%

          \[\leadsto \frac{2}{1 + {\left(e^{x}\right)}^{\color{blue}{\left(-6 \cdot 0.3333333333333333\right)}}} + -1 \]

       6. pow-pow 99.9%

          \[\leadsto \frac{2}{1 + \color{blue}{{\left({\left(e^{x}\right)}^{-6}\right)}^{0.3333333333333333}}} + -1 \]

       7. exp-prod 100.0%

          \[\leadsto \frac{2}{1 + {\color{blue}{\left(e^{x \cdot -6}\right)}}^{0.3333333333333333}} + -1 \]

       8. pow1/3 100.0%

          \[\leadsto \frac{2}{1 + \color{blue}{\sqrt[3]{e^{x \cdot -6}}}} + -1 \]

       9. *-un-lft-identity 100.0%

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

       10. *-commutative 100.0%

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

       11. pow1/3 100.0%

           \[\leadsto \frac{2}{\left(1 + \color{blue}{{\left(e^{x \cdot -6}\right)}^{0.3333333333333333}}\right) \cdot 1} + -1 \]

       12. exp-prod 99.9%

           \[\leadsto \frac{2}{\left(1 + {\color{blue}{\left({\left(e^{x}\right)}^{-6}\right)}}^{0.3333333333333333}\right) \cdot 1} + -1 \]

       13. pow-pow 100.0%

           \[\leadsto \frac{2}{\left(1 + \color{blue}{{\left(e^{x}\right)}^{\left(-6 \cdot 0.3333333333333333\right)}}\right) \cdot 1} + -1 \]

       14. metadata-eval 100.0%

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

       15. exp-prod 100.0%

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

       16. log1p-expm1-u 100.0%

           \[\leadsto \frac{2}{\left(1 + e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot -2\right)\right)}}\right) \cdot 1} + -1 \]

       17. log1p-udef 100.0%

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

       18. add-exp-log 100.0%

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

       19. associate-+r+ 100.0%

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

       20. metadata-eval 100.0%

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

   11. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.05:\\ \;\;\;\;e^{\log \left(-1 + \frac{2}{2 + \mathsf{expm1}\left(-2 \cdot x\right)}\right)}\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-7}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{2}{2 + \mathsf{expm1}\left(-2 \cdot x\right)}\\ \end{array} \]

Alternative 5: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.05:\\ \;\;\;\;\left|-1 + \frac{2}{1 + {\left(e^{2}\right)}^{x}}\right|\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-7}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{2}{2 + \mathsf{expm1}\left(-2 \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -0.05)
   (fabs (+ -1.0 (/ 2.0 (+ 1.0 (pow (exp 2.0) x)))))
   (if (<= (* -2.0 x) 5e-7)
     (+ x (* -0.3333333333333333 (pow x 3.0)))
     (+ -1.0 (/ 2.0 (+ 2.0 (expm1 (* -2.0 x))))))))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.05) {
		tmp = fabs((-1.0 + (2.0 / (1.0 + pow(exp(2.0), x)))));
	} else if ((-2.0 * x) <= 5e-7) {
		tmp = x + (-0.3333333333333333 * pow(x, 3.0));
	} else {
		tmp = -1.0 + (2.0 / (2.0 + expm1((-2.0 * x))));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.05) {
		tmp = Math.abs((-1.0 + (2.0 / (1.0 + Math.pow(Math.exp(2.0), x)))));
	} else if ((-2.0 * x) <= 5e-7) {
		tmp = x + (-0.3333333333333333 * Math.pow(x, 3.0));
	} else {
		tmp = -1.0 + (2.0 / (2.0 + Math.expm1((-2.0 * x))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (-2.0 * x) <= -0.05:
		tmp = math.fabs((-1.0 + (2.0 / (1.0 + math.pow(math.exp(2.0), x)))))
	elif (-2.0 * x) <= 5e-7:
		tmp = x + (-0.3333333333333333 * math.pow(x, 3.0))
	else:
		tmp = -1.0 + (2.0 / (2.0 + math.expm1((-2.0 * x))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.05)
		tmp = abs(Float64(-1.0 + Float64(2.0 / Float64(1.0 + (exp(2.0) ^ x)))));
	elseif (Float64(-2.0 * x) <= 5e-7)
		tmp = Float64(x + Float64(-0.3333333333333333 * (x ^ 3.0)));
	else
		tmp = Float64(-1.0 + Float64(2.0 / Float64(2.0 + expm1(Float64(-2.0 * x)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.05], N[Abs[N[(-1.0 + N[(2.0 / N[(1.0 + N[Power[N[Exp[2.0], $MachinePrecision], x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 5e-7], N[(x + N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(2.0 / N[(2.0 + N[(Exp[N[(-2.0 * x), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 -2 x) < -0.050000000000000003

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. exp-prod 99.8%

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

      3. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. add-cbrt-cube 99.9%

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

      2. pow3 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. rem-cbrt-cube 99.8%

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

      2. expm1-log1p-u 99.8%

         \[\leadsto \frac{2}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)}} + -1 \]

      3. expm1-udef 99.8%

         \[\leadsto \frac{2}{1 + \color{blue}{\left(e^{\mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)} - 1\right)}} + -1 \]

      4. log1p-udef 99.8%

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

      5. add-exp-log 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. associate--l+ 99.8%

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

      2. exp-prod 99.8%

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

      3. expm1-def 99.8%

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

      4. *-commutative 99.8%

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

   9. Simplified 99.8%

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

       1. add-exp-log 99.8%

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

       2. log1p-udef 99.8%

          \[\leadsto \frac{2}{1 + e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot -2\right)\right)}}} + -1 \]

       3. log1p-expm1-u 99.8%

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

       4. exp-prod 99.8%

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

       5. metadata-eval 99.8%

          \[\leadsto \frac{2}{1 + {\left(e^{x}\right)}^{\color{blue}{\left(-6 \cdot 0.3333333333333333\right)}}} + -1 \]

       6. pow-pow 99.8%

          \[\leadsto \frac{2}{1 + \color{blue}{{\left({\left(e^{x}\right)}^{-6}\right)}^{0.3333333333333333}}} + -1 \]

       7. exp-prod 99.8%

          \[\leadsto \frac{2}{1 + {\color{blue}{\left(e^{x \cdot -6}\right)}}^{0.3333333333333333}} + -1 \]

       8. pow1/3 99.9%

          \[\leadsto \frac{2}{1 + \color{blue}{\sqrt[3]{e^{x \cdot -6}}}} + -1 \]

       9. *-un-lft-identity 99.9%

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

       10. *-commutative 99.9%

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

       11. pow1/3 99.8%

           \[\leadsto \frac{2}{\left(1 + \color{blue}{{\left(e^{x \cdot -6}\right)}^{0.3333333333333333}}\right) \cdot 1} + -1 \]

       12. exp-prod 99.8%

           \[\leadsto \frac{2}{\left(1 + {\color{blue}{\left({\left(e^{x}\right)}^{-6}\right)}}^{0.3333333333333333}\right) \cdot 1} + -1 \]

       13. pow-pow 99.8%

           \[\leadsto \frac{2}{\left(1 + \color{blue}{{\left(e^{x}\right)}^{\left(-6 \cdot 0.3333333333333333\right)}}\right) \cdot 1} + -1 \]

       14. metadata-eval 99.8%

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

       15. exp-prod 99.8%

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

       16. log1p-expm1-u 99.8%

           \[\leadsto \frac{2}{\left(1 + e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot -2\right)\right)}}\right) \cdot 1} + -1 \]

       17. log1p-udef 99.8%

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

       18. add-exp-log 99.8%

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

       19. associate-+r+ 99.9%

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

       20. metadata-eval 99.9%

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

   11. Applied egg-rr 99.9%

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

       1. add-sqr-sqrt 99.9%

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

       2. sqrt-unprod 99.9%

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

       3. pow2 99.9%

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

   13. Applied egg-rr 99.9%

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

       1. unpow2 99.9%

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

       2. rem-sqrt-square 99.9%

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

       3. sub-neg 99.9%

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

       4. metadata-eval 99.9%

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

       5. +-commutative 99.9%

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

   15. Simplified 99.9%

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

   if -0.050000000000000003 < (*.f64 -2 x) < 4.99999999999999977e-7

   1. Initial program 7.1%

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

      1. sub-neg 7.1%

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

      2. exp-prod 7.1%

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

      3. metadata-eval 7.1%

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

   3. Simplified 7.1%

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

   4. Taylor expanded in x around 0 100.0%

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

   if 4.99999999999999977e-7 < (*.f64 -2 x)

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. exp-prod 100.0%

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

      3. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. add-cbrt-cube 100.0%

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

      2. pow3 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. rem-cbrt-cube 100.0%

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

      2. expm1-log1p-u 100.0%

         \[\leadsto \frac{2}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)}} + -1 \]

      3. expm1-udef 100.0%

         \[\leadsto \frac{2}{1 + \color{blue}{\left(e^{\mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)} - 1\right)}} + -1 \]

      4. log1p-udef 100.0%

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

      5. add-exp-log 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. associate--l+ 100.0%

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

      2. exp-prod 100.0%

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

      3. expm1-def 100.0%

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

      4. *-commutative 100.0%

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

   9. Simplified 100.0%

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

       1. add-exp-log 100.0%

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

       2. log1p-udef 100.0%

          \[\leadsto \frac{2}{1 + e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot -2\right)\right)}}} + -1 \]

       3. log1p-expm1-u 100.0%

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

       4. exp-prod 100.0%

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

       5. metadata-eval 100.0%

          \[\leadsto \frac{2}{1 + {\left(e^{x}\right)}^{\color{blue}{\left(-6 \cdot 0.3333333333333333\right)}}} + -1 \]

       6. pow-pow 99.9%

          \[\leadsto \frac{2}{1 + \color{blue}{{\left({\left(e^{x}\right)}^{-6}\right)}^{0.3333333333333333}}} + -1 \]

       7. exp-prod 100.0%

          \[\leadsto \frac{2}{1 + {\color{blue}{\left(e^{x \cdot -6}\right)}}^{0.3333333333333333}} + -1 \]

       8. pow1/3 100.0%

          \[\leadsto \frac{2}{1 + \color{blue}{\sqrt[3]{e^{x \cdot -6}}}} + -1 \]

       9. *-un-lft-identity 100.0%

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

       10. *-commutative 100.0%

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

       11. pow1/3 100.0%

           \[\leadsto \frac{2}{\left(1 + \color{blue}{{\left(e^{x \cdot -6}\right)}^{0.3333333333333333}}\right) \cdot 1} + -1 \]

       12. exp-prod 99.9%

           \[\leadsto \frac{2}{\left(1 + {\color{blue}{\left({\left(e^{x}\right)}^{-6}\right)}}^{0.3333333333333333}\right) \cdot 1} + -1 \]

       13. pow-pow 100.0%

           \[\leadsto \frac{2}{\left(1 + \color{blue}{{\left(e^{x}\right)}^{\left(-6 \cdot 0.3333333333333333\right)}}\right) \cdot 1} + -1 \]

       14. metadata-eval 100.0%

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

       15. exp-prod 100.0%

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

       16. log1p-expm1-u 100.0%

           \[\leadsto \frac{2}{\left(1 + e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot -2\right)\right)}}\right) \cdot 1} + -1 \]

       17. log1p-udef 100.0%

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

       18. add-exp-log 100.0%

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

       19. associate-+r+ 100.0%

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

       20. metadata-eval 100.0%

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

   11. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.05:\\ \;\;\;\;\left|-1 + \frac{2}{1 + {\left(e^{2}\right)}^{x}}\right|\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-7}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{2}{2 + \mathsf{expm1}\left(-2 \cdot x\right)}\\ \end{array} \]

Alternative 6: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.05 \lor \neg \left(-2 \cdot x \leq 5 \cdot 10^{-7}\right):\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* -2.0 x) -0.05) (not (<= (* -2.0 x) 5e-7)))
   (+ -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) <= -0.05) || !((-2.0 * x) <= 5e-7)) {
		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) <= (-0.05d0)) .or. (.not. (((-2.0d0) * x) <= 5d-7))) 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) <= -0.05) || !((-2.0 * x) <= 5e-7)) {
		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) <= -0.05) or not ((-2.0 * x) <= 5e-7):
		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) <= -0.05) || !(Float64(-2.0 * x) <= 5e-7))
		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) <= -0.05) || ~(((-2.0 * x) <= 5e-7)))
		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], -0.05], N[Not[LessEqual[N[(-2.0 * x), $MachinePrecision], 5e-7]], $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 -2 x) < -0.050000000000000003 or 4.99999999999999977e-7 < (*.f64 -2 x)

   1. Initial program 99.9%

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

   if -0.050000000000000003 < (*.f64 -2 x) < 4.99999999999999977e-7

   1. Initial program 7.1%

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

      1. sub-neg 7.1%

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

      2. exp-prod 7.1%

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

      3. metadata-eval 7.1%

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

   3. Simplified 7.1%

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

   4. Taylor expanded in x around 0 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.05 \lor \neg \left(-2 \cdot x \leq 5 \cdot 10^{-7}\right):\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \end{array} \]

Alternative 7: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.05 \lor \neg \left(-2 \cdot x \leq 5 \cdot 10^{-7}\right):\\ \;\;\;\;-1 + \frac{2}{2 + \mathsf{expm1}\left(-2 \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* -2.0 x) -0.05) (not (<= (* -2.0 x) 5e-7)))
   (+ -1.0 (/ 2.0 (+ 2.0 (expm1 (* -2.0 x)))))
   (+ x (* -0.3333333333333333 (pow x 3.0)))))

C

double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -0.05) || !((-2.0 * x) <= 5e-7)) {
		tmp = -1.0 + (2.0 / (2.0 + expm1((-2.0 * x))));
	} else {
		tmp = x + (-0.3333333333333333 * pow(x, 3.0));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -0.05) || !((-2.0 * x) <= 5e-7)) {
		tmp = -1.0 + (2.0 / (2.0 + Math.expm1((-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) <= -0.05) or not ((-2.0 * x) <= 5e-7):
		tmp = -1.0 + (2.0 / (2.0 + math.expm1((-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) <= -0.05) || !(Float64(-2.0 * x) <= 5e-7))
		tmp = Float64(-1.0 + Float64(2.0 / Float64(2.0 + expm1(Float64(-2.0 * x)))));
	else
		tmp = Float64(x + Float64(-0.3333333333333333 * (x ^ 3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.05], N[Not[LessEqual[N[(-2.0 * x), $MachinePrecision], 5e-7]], $MachinePrecision]], N[(-1.0 + N[(2.0 / N[(2.0 + N[(Exp[N[(-2.0 * x), $MachinePrecision]] - 1), $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 -2 x) < -0.050000000000000003 or 4.99999999999999977e-7 < (*.f64 -2 x)

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. exp-prod 99.9%

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

      3. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. add-cbrt-cube 99.9%

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

      2. pow3 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. rem-cbrt-cube 99.9%

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

      2. expm1-log1p-u 99.9%

         \[\leadsto \frac{2}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)}} + -1 \]

      3. expm1-udef 99.9%

         \[\leadsto \frac{2}{1 + \color{blue}{\left(e^{\mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)} - 1\right)}} + -1 \]

      4. log1p-udef 99.9%

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

      5. add-exp-log 99.9%

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

   7. Applied egg-rr 99.9%

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

      1. associate--l+ 99.9%

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

      2. exp-prod 99.9%

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

      3. expm1-def 99.9%

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

      4. *-commutative 99.9%

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

   9. Simplified 99.9%

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

       1. add-exp-log 99.9%

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

       2. log1p-udef 99.9%

          \[\leadsto \frac{2}{1 + e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot -2\right)\right)}}} + -1 \]

       3. log1p-expm1-u 99.9%

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

       4. exp-prod 99.9%

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

       5. metadata-eval 99.9%

          \[\leadsto \frac{2}{1 + {\left(e^{x}\right)}^{\color{blue}{\left(-6 \cdot 0.3333333333333333\right)}}} + -1 \]

       6. pow-pow 99.9%

          \[\leadsto \frac{2}{1 + \color{blue}{{\left({\left(e^{x}\right)}^{-6}\right)}^{0.3333333333333333}}} + -1 \]

       7. exp-prod 99.9%

          \[\leadsto \frac{2}{1 + {\color{blue}{\left(e^{x \cdot -6}\right)}}^{0.3333333333333333}} + -1 \]

       8. pow1/3 99.9%

          \[\leadsto \frac{2}{1 + \color{blue}{\sqrt[3]{e^{x \cdot -6}}}} + -1 \]

       9. *-un-lft-identity 99.9%

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

       10. *-commutative 99.9%

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

       11. pow1/3 99.9%

           \[\leadsto \frac{2}{\left(1 + \color{blue}{{\left(e^{x \cdot -6}\right)}^{0.3333333333333333}}\right) \cdot 1} + -1 \]

       12. exp-prod 99.9%

           \[\leadsto \frac{2}{\left(1 + {\color{blue}{\left({\left(e^{x}\right)}^{-6}\right)}}^{0.3333333333333333}\right) \cdot 1} + -1 \]

       13. pow-pow 99.9%

           \[\leadsto \frac{2}{\left(1 + \color{blue}{{\left(e^{x}\right)}^{\left(-6 \cdot 0.3333333333333333\right)}}\right) \cdot 1} + -1 \]

       14. metadata-eval 99.9%

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

       15. exp-prod 99.9%

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

       16. log1p-expm1-u 99.9%

           \[\leadsto \frac{2}{\left(1 + e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot -2\right)\right)}}\right) \cdot 1} + -1 \]

       17. log1p-udef 99.9%

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

       18. add-exp-log 99.9%

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

       19. associate-+r+ 99.9%

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

       20. metadata-eval 99.9%

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

   11. Applied egg-rr 99.9%

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

   if -0.050000000000000003 < (*.f64 -2 x) < 4.99999999999999977e-7

   1. Initial program 7.1%

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

      1. sub-neg 7.1%

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

      2. exp-prod 7.1%

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

      3. metadata-eval 7.1%

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

   3. Simplified 7.1%

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

   4. Taylor expanded in x around 0 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.05 \lor \neg \left(-2 \cdot x \leq 5 \cdot 10^{-7}\right):\\ \;\;\;\;-1 + \frac{2}{2 + \mathsf{expm1}\left(-2 \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \end{array} \]

Alternative 8: 77.9% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.58:\\ \;\;\;\;-1 + \frac{2}{2 + -2 \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{x + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.58) (+ -1.0 (/ 2.0 (+ 2.0 (* -2.0 x)))) (/ (* x 2.0) (+ x 2.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.58) {
		tmp = -1.0 + (2.0 / (2.0 + (-2.0 * x)));
	} 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 <= (-0.58d0)) then
        tmp = (-1.0d0) + (2.0d0 / (2.0d0 + ((-2.0d0) * x)))
    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 <= -0.58) {
		tmp = -1.0 + (2.0 / (2.0 + (-2.0 * x)));
	} else {
		tmp = (x * 2.0) / (x + 2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.58:
		tmp = -1.0 + (2.0 / (2.0 + (-2.0 * x)))
	else:
		tmp = (x * 2.0) / (x + 2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.58)
		tmp = Float64(-1.0 + Float64(2.0 / Float64(2.0 + Float64(-2.0 * x))));
	else
		tmp = Float64(Float64(x * 2.0) / Float64(x + 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.58)
		tmp = -1.0 + (2.0 / (2.0 + (-2.0 * x)));
	else
		tmp = (x * 2.0) / (x + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.58], N[(-1.0 + N[(2.0 / N[(2.0 + N[(-2.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 2.0), $MachinePrecision] / N[(x + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.57999999999999996

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. exp-prod 100.0%

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

      3. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 98.7%

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

      1. *-commutative 98.7%

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

   6. Simplified 98.7%

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

   if -0.57999999999999996 < x

   1. Initial program 39.6%

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

      1. sub-neg 39.6%

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

      2. exp-prod 39.6%

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

      3. metadata-eval 39.6%

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

   3. Simplified 39.6%

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

   4. Taylor expanded in x around 0 6.8%

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

      1. +-commutative 6.8%

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

   6. Simplified 6.8%

      \[\leadsto \color{blue}{\left(x + 1\right)} + -1 \]
   7. 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 6.7%

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

      4. metadata-eval 6.7%

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

      5. sub-neg 6.7%

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

      6. associate--l+ 6.7%

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

      7. metadata-eval 6.7%

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

      8. add-exp-log 6.7%

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

      9. +-commutative 6.7%

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

      10. log1p-udef 6.7%

          \[\leadsto \frac{\left(x + 2\right) \cdot \left(e^{\color{blue}{\mathsf{log1p}\left(x\right)}} - 1\right)}{\left(x + 1\right) - -1} \]

      11. expm1-udef 66.8%

          \[\leadsto \frac{\left(x + 2\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)}}{\left(x + 1\right) - -1} \]

      12. expm1-log1p-u 66.8%

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

      13. associate--l+ 66.8%

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

      14. metadata-eval 66.8%

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

   8. Applied egg-rr 66.8%

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

   9. Taylor expanded in x around 0 70.9%

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

       1. *-commutative 70.9%

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

   11. Simplified 70.9%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.58:\\ \;\;\;\;-1 + \frac{2}{2 + -2 \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{x + 2}\\ \end{array} \]

Alternative 9: 78.2% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.66:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{1 + \frac{2}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.66) -1.0 (/ 2.0 (+ 1.0 (/ 2.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.66) {
		tmp = -1.0;
	} else {
		tmp = 2.0 / (1.0 + (2.0 / 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 <= (-0.66d0)) then
        tmp = -1.0d0
    else
        tmp = 2.0d0 / (1.0d0 + (2.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.66) {
		tmp = -1.0;
	} else {
		tmp = 2.0 / (1.0 + (2.0 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.66:
		tmp = -1.0
	else:
		tmp = 2.0 / (1.0 + (2.0 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.66)
		tmp = -1.0;
	else
		tmp = Float64(2.0 / Float64(1.0 + Float64(2.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.66)
		tmp = -1.0;
	else
		tmp = 2.0 / (1.0 + (2.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.66], -1.0, N[(2.0 / N[(1.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.660000000000000031

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. exp-prod 100.0%

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

      3. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 98.7%

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

      1. *-commutative 98.7%

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

   6. Simplified 98.7%

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

   7. Taylor expanded in x around inf 98.7%

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

   if -0.660000000000000031 < x

   1. Initial program 39.6%

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

      1. sub-neg 39.6%

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

      2. exp-prod 39.6%

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

      3. metadata-eval 39.6%

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

   3. Simplified 39.6%

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

   4. Taylor expanded in x around 0 6.8%

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

      1. +-commutative 6.8%

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

   6. Simplified 6.8%

      \[\leadsto \color{blue}{\left(x + 1\right)} + -1 \]
   7. 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. div-inv 6.7%

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

      3. metadata-eval 6.7%

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

      4. difference-of-sqr-1 6.7%

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

      5. metadata-eval 6.7%

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

      6. sub-neg 6.7%

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

      7. associate--l+ 6.7%

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

      8. metadata-eval 6.7%

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

      9. add-exp-log 6.7%

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

      10. +-commutative 6.7%

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

      11. log1p-udef 6.7%

          \[\leadsto \left(\left(x + 2\right) \cdot \left(e^{\color{blue}{\mathsf{log1p}\left(x\right)}} - 1\right)\right) \cdot \frac{1}{\left(x + 1\right) - -1} \]

      12. expm1-udef 66.8%

          \[\leadsto \left(\left(x + 2\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)}\right) \cdot \frac{1}{\left(x + 1\right) - -1} \]

      13. expm1-log1p-u 66.8%

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

      14. associate--l+ 66.8%

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

      15. metadata-eval 66.8%

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

   8. Applied egg-rr 66.8%

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

   9. Taylor expanded in x around 0 70.9%

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

       1. *-commutative 70.9%

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

   11. Simplified 70.9%

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

       1. expm1-log1p-u 70.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x \cdot 2\right) \cdot \frac{1}{x + 2}\right)\right)} \]

       2. expm1-udef 11.1%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(x \cdot 2\right) \cdot \frac{1}{x + 2}\right)} - 1} \]

       3. *-commutative 11.1%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(2 \cdot x\right)} \cdot \frac{1}{x + 2}\right)} - 1 \]

       4. associate-*l* 11.1%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{2 \cdot \left(x \cdot \frac{1}{x + 2}\right)}\right)} - 1 \]

       5. +-commutative 11.1%

          \[\leadsto e^{\mathsf{log1p}\left(2 \cdot \left(x \cdot \frac{1}{\color{blue}{2 + x}}\right)\right)} - 1 \]

       6. div-inv 11.1%

          \[\leadsto e^{\mathsf{log1p}\left(2 \cdot \color{blue}{\frac{x}{2 + x}}\right)} - 1 \]

   13. Applied egg-rr 11.1%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(2 \cdot \frac{x}{2 + x}\right)} - 1} \]
   14. Step-by-step derivation

       1. expm1-def 70.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \frac{x}{2 + x}\right)\right)} \]

       2. expm1-log1p 70.9%

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

       3. associate-*r/ 70.9%

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

       4. /-rgt-identity 70.9%

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

       5. associate-/l* 70.7%

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

       6. associate-/r* 70.7%

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

       7. +-commutative 70.7%

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

       8. distribute-rgt-in 70.7%

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

       9. rgt-mult-inverse 70.7%

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

       10. associate-*r/ 70.7%

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

       11. metadata-eval 70.7%

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

   15. Simplified 70.7%

       \[\leadsto \color{blue}{\frac{2}{1 + \frac{2}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.66:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{1 + \frac{2}{x}}\\ \end{array} \]

Alternative 10: 78.3% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.66:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{x + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.66) -1.0 (/ (* x 2.0) (+ x 2.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.66) {
		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 <= (-0.66d0)) 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 <= -0.66) {
		tmp = -1.0;
	} else {
		tmp = (x * 2.0) / (x + 2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.66:
		tmp = -1.0
	else:
		tmp = (x * 2.0) / (x + 2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.66)
		tmp = -1.0;
	else
		tmp = Float64(Float64(x * 2.0) / Float64(x + 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.66)
		tmp = -1.0;
	else
		tmp = (x * 2.0) / (x + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.66], -1.0, N[(N[(x * 2.0), $MachinePrecision] / N[(x + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.660000000000000031

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. exp-prod 100.0%

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

      3. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 98.7%

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

      1. *-commutative 98.7%

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

   6. Simplified 98.7%

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

   7. Taylor expanded in x around inf 98.7%

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

   if -0.660000000000000031 < x

   1. Initial program 39.6%

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

      1. sub-neg 39.6%

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

      2. exp-prod 39.6%

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

      3. metadata-eval 39.6%

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

   3. Simplified 39.6%

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

   4. Taylor expanded in x around 0 6.8%

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

      1. +-commutative 6.8%

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

   6. Simplified 6.8%

      \[\leadsto \color{blue}{\left(x + 1\right)} + -1 \]
   7. 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 6.7%

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

      4. metadata-eval 6.7%

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

      5. sub-neg 6.7%

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

      6. associate--l+ 6.7%

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

      7. metadata-eval 6.7%

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

      8. add-exp-log 6.7%

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

      9. +-commutative 6.7%

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

      10. log1p-udef 6.7%

          \[\leadsto \frac{\left(x + 2\right) \cdot \left(e^{\color{blue}{\mathsf{log1p}\left(x\right)}} - 1\right)}{\left(x + 1\right) - -1} \]

      11. expm1-udef 66.8%

          \[\leadsto \frac{\left(x + 2\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)}}{\left(x + 1\right) - -1} \]

      12. expm1-log1p-u 66.8%

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

      13. associate--l+ 66.8%

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

      14. metadata-eval 66.8%

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

   8. Applied egg-rr 66.8%

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

   9. Taylor expanded in x around 0 70.9%

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

       1. *-commutative 70.9%

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

   11. Simplified 70.9%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.66:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{x + 2}\\ \end{array} \]

Alternative 11: 78.8% accurate, 21.3× 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. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. exp-prod 100.0%

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

      3. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 100.0%

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

      1. *-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around inf 100.0%

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

   if -1 < x < 2

   1. Initial program 9.2%

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

      1. sub-neg 9.2%

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

      2. exp-prod 9.2%

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

      3. metadata-eval 9.2%

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

   3. Simplified 9.2%

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

   4. Taylor expanded in x around 0 98.0%

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

   if 2 < x

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. exp-prod 100.0%

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

      3. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 5.5%

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

      1. +-commutative 5.5%

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

   6. Simplified 5.5%

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

      1. flip-+ 5.2%

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

      2. div-inv 5.2%

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

      3. metadata-eval 5.2%

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

      4. difference-of-sqr-1 5.2%

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

      5. metadata-eval 5.2%

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

      6. sub-neg 5.2%

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

      7. associate--l+ 5.2%

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

      8. metadata-eval 5.2%

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

      9. add-exp-log 5.2%

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

      10. +-commutative 5.2%

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

      11. log1p-udef 5.2%

          \[\leadsto \left(\left(x + 2\right) \cdot \left(e^{\color{blue}{\mathsf{log1p}\left(x\right)}} - 1\right)\right) \cdot \frac{1}{\left(x + 1\right) - -1} \]

      12. expm1-udef 5.2%

          \[\leadsto \left(\left(x + 2\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)}\right) \cdot \frac{1}{\left(x + 1\right) - -1} \]

      13. expm1-log1p-u 5.2%

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

      14. associate--l+ 5.2%

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

      15. metadata-eval 5.2%

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

   8. Applied egg-rr 5.2%

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

   9. Taylor expanded in x around 0 18.8%

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

       1. *-commutative 18.8%

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

   11. Simplified 18.8%

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

   12. Taylor expanded in x around inf 18.7%

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

4. Final simplification 78.0%

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

Alternative 12: 32.3% accurate, 35.6× 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 48.5%

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

      1. sub-neg 48.5%

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

      2. exp-prod 48.5%

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

      3. metadata-eval 48.5%

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

   3. Simplified 48.5%

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

   4. Taylor expanded in x around 0 47.3%

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

      1. *-commutative 47.3%

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

   6. Simplified 47.3%

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

   7. Taylor expanded in x around inf 46.2%

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

   if 1.1000000000000001e-308 < x

   1. Initial program 61.2%

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

      1. sub-neg 61.2%

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

      2. exp-prod 61.2%

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

      3. metadata-eval 61.2%

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

   3. Simplified 61.2%

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

   4. Taylor expanded in x around 0 6.5%

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

      1. +-commutative 6.5%

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

   6. Simplified 6.5%

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

      1. flip-+ 6.3%

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

      2. div-inv 6.3%

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

      3. metadata-eval 6.3%

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

      4. difference-of-sqr-1 6.3%

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

      5. metadata-eval 6.3%

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

      6. sub-neg 6.3%

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

      7. associate--l+ 6.3%

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

      8. metadata-eval 6.3%

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

      9. add-exp-log 6.3%

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

      10. +-commutative 6.3%

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

      11. log1p-udef 6.3%

          \[\leadsto \left(\left(x + 2\right) \cdot \left(e^{\color{blue}{\mathsf{log1p}\left(x\right)}} - 1\right)\right) \cdot \frac{1}{\left(x + 1\right) - -1} \]

      12. expm1-udef 44.8%

          \[\leadsto \left(\left(x + 2\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)}\right) \cdot \frac{1}{\left(x + 1\right) - -1} \]

      13. expm1-log1p-u 44.8%

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

      14. associate--l+ 44.8%

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

      15. metadata-eval 44.8%

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

   8. Applied egg-rr 44.8%

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

   9. Taylor expanded in x around 0 52.1%

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

       1. *-commutative 52.1%

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

   11. Simplified 52.1%

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

   12. Taylor expanded in x around inf 13.2%

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

4. Final simplification 31.4%

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

Alternative 13: 27.2% accurate, 109.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 -1.0)

C

double code(double x, double y) {
	return -1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = -1.0d0
end function

Java

public static double code(double x, double y) {
	return -1.0;
}

Python

def code(x, y):
	return -1.0

Julia

function code(x, y)
	return -1.0
end

MATLAB

function tmp = code(x, y)
	tmp = -1.0;
end

Wolfram

code[x_, y_] := -1.0

Derivation

1. Initial program 54.2%

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

   1. sub-neg 54.2%

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

   2. exp-prod 54.2%

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

   3. metadata-eval 54.2%

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

3. Simplified 54.2%

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

4. Taylor expanded in x around 0 27.8%

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

   1. *-commutative 27.8%

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

6. Simplified 27.8%

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

7. Taylor expanded in x around inf 26.3%

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

8. Final simplification 26.3%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023320 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))