Logistic function from Lakshay Garg

Percentage Accurate: 52.7% → 99.9%

Time: 10.0s

Alternatives: 13

Speedup: 123.0×

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: 52.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + e^{-2 \cdot x}\\ t_1 := \frac{2}{t\_0}\\ \mathbf{if}\;-2 \cdot x \leq -5:\\ \;\;\;\;t\_1 + -1\\ \mathbf{elif}\;-2 \cdot x \leq 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.05396825396825397, 0.13333333333333333\right), -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{t\_1}, \sqrt{2} \cdot \sqrt{\frac{1}{t\_0}}, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (exp (* -2.0 x)))) (t_1 (/ 2.0 t_0)))
   (if (<= (* -2.0 x) -5.0)
     (+ t_1 -1.0)
     (if (<= (* -2.0 x) 1e-6)
       (fma
        (fma
         (* x x)
         (fma (* x x) -0.05396825396825397 0.13333333333333333)
         -0.3333333333333333)
        (* x (* x x))
        x)
       (fma (sqrt t_1) (* (sqrt 2.0) (sqrt (/ 1.0 t_0))) -1.0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 + exp((-2.0 * x));
	double t_1 = 2.0 / t_0;
	double tmp;
	if ((-2.0 * x) <= -5.0) {
		tmp = t_1 + -1.0;
	} else if ((-2.0 * x) <= 1e-6) {
		tmp = fma(fma((x * x), fma((x * x), -0.05396825396825397, 0.13333333333333333), -0.3333333333333333), (x * (x * x)), x);
	} else {
		tmp = fma(sqrt(t_1), (sqrt(2.0) * sqrt((1.0 / t_0))), -1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 + exp(Float64(-2.0 * x)))
	t_1 = Float64(2.0 / t_0)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -5.0)
		tmp = Float64(t_1 + -1.0);
	elseif (Float64(-2.0 * x) <= 1e-6)
		tmp = fma(fma(Float64(x * x), fma(Float64(x * x), -0.05396825396825397, 0.13333333333333333), -0.3333333333333333), Float64(x * Float64(x * x)), x);
	else
		tmp = fma(sqrt(t_1), Float64(sqrt(2.0) * sqrt(Float64(1.0 / t_0))), -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / t$95$0), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -5.0], N[(t$95$1 + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 1e-6], N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.05396825396825397 + 0.13333333333333333), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[Sqrt[t$95$1], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   if -5 < (*.f64 #s(literal -2 binary64) x) < 9.99999999999999955e-7

   1. Initial program 8.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lft-identity N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.05396825396825397, 0.13333333333333333\right), -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)} \]

   if 9.99999999999999955e-7 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 100.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. inv-pow N/A

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

      6. metadata-eval N/A

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

      7. sqr-pow N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\left(1 + e^{-2 \cdot x}\right) \cdot 0.5\right)}^{-0.5}, {\left(\left(1 + e^{-2 \cdot x}\right) \cdot 0.5\right)}^{-0.5}, -1\right)} \]

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 100.0

         \[\leadsto \mathsf{fma}\left({\left(\left(1 + e^{-2 \cdot x}\right) \cdot 0.5\right)}^{-0.5}, \sqrt{2} \cdot \sqrt{\frac{1}{1 + e^{\color{blue}{x \cdot -2}}}}, -1\right) \]

   7. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left({\left(\left(1 + e^{-2 \cdot x}\right) \cdot 0.5\right)}^{-0.5}, \color{blue}{\sqrt{2} \cdot \sqrt{\frac{1}{1 + e^{x \cdot -2}}}}, -1\right) \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. unpow1 N/A

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

      7. metadata-eval N/A

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

      8. sqrt-pow1 N/A

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

      9. pow2 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-exp.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift-*.f64 N/A

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

   9. Applied rewrites 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -5:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.05396825396825397, 0.13333333333333333\right), -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{2}{1 + e^{-2 \cdot x}}}, \sqrt{2} \cdot \sqrt{\frac{1}{1 + e^{-2 \cdot x}}}, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 + e^{-2 \cdot x} \leq 2.5:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -1.3333333333333333, 2\right), -2\right), 2\right)} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ 1.0 (exp (* -2.0 x))) 2.5)
   x
   (+ (/ 2.0 (fma x (fma x (fma x -1.3333333333333333 2.0) -2.0) 2.0)) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((1.0 + exp((-2.0 * x))) <= 2.5) {
		tmp = x;
	} else {
		tmp = (2.0 / fma(x, fma(x, fma(x, -1.3333333333333333, 2.0), -2.0), 2.0)) + -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(1.0 + exp(Float64(-2.0 * x))) <= 2.5)
		tmp = x;
	else
		tmp = Float64(Float64(2.0 / fma(x, fma(x, fma(x, -1.3333333333333333, 2.0), -2.0), 2.0)) + -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.5], x, N[(N[(2.0 / N[(x * N[(x * N[(x * -1.3333333333333333 + 2.0), $MachinePrecision] + -2.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 #s(literal 1 binary64) (exp.f64 (*.f64 #s(literal -2 binary64) x))) < 2.5

   1. Initial program 44.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. inv-pow N/A

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

      6. metadata-eval N/A

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

      7. sqr-pow N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 43.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\left(1 + e^{-2 \cdot x}\right) \cdot 0.5\right)}^{-0.5}, {\left(\left(1 + e^{-2 \cdot x}\right) \cdot 0.5\right)}^{-0.5}, -1\right)} \]

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. rem-square-sqrt N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. *-lft-identity N/A

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

      15. associate-+r+ N/A

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

      16. metadata-eval N/A

          \[\leadsto \color{blue}{0} + x \]

      17. +-lft-identity 62.5

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

   7. Applied rewrites 62.5%

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

   if 2.5 < (+.f64 #s(literal 1 binary64) (exp.f64 (*.f64 #s(literal -2 binary64) x)))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{x \cdot \left(x \cdot \left(2 + \frac{-4}{3} \cdot x\right) - 2\right) + 2}} - 1 \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, x \cdot \left(2 + \frac{-4}{3} \cdot x\right) - 2, 2\right)}} - 1 \]

      3. sub-neg N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(x, \color{blue}{x \cdot \left(2 + \frac{-4}{3} \cdot x\right) + \left(\mathsf{neg}\left(2\right)\right)}, 2\right)} - 1 \]

      4. metadata-eval N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(x, x \cdot \left(2 + \frac{-4}{3} \cdot x\right) + \color{blue}{-2}, 2\right)} - 1 \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 2 + \frac{-4}{3} \cdot x, -2\right)}, 2\right)} - 1 \]

      6. +-commutative N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{-4}{3} \cdot x + 2}, -2\right), 2\right)} - 1 \]

      7. *-commutative N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-4}{3}} + 2, -2\right), 2\right)} - 1 \]

      8. lower-fma.f64 99.0

         \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, -1.3333333333333333, 2\right)}, -2\right), 2\right)} - 1 \]

   5. Applied rewrites 99.0%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -1.3333333333333333, 2\right), -2\right), 2\right)}} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + e^{-2 \cdot x} \leq 2.5:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -1.3333333333333333, 2\right), -2\right), 2\right)} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{if}\;-2 \cdot x \leq -5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;-2 \cdot x \leq 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.05396825396825397, 0.13333333333333333\right), -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) -1.0)))
   (if (<= (* -2.0 x) -5.0)
     t_0
     (if (<= (* -2.0 x) 1e-6)
       (fma
        (fma
         (* x x)
         (fma (* x x) -0.05396825396825397 0.13333333333333333)
         -0.3333333333333333)
        (* x (* x x))
        x)
       t_0))))

C

double code(double x, double y) {
	double t_0 = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	double tmp;
	if ((-2.0 * x) <= -5.0) {
		tmp = t_0;
	} else if ((-2.0 * x) <= 1e-6) {
		tmp = fma(fma((x * x), fma((x * x), -0.05396825396825397, 0.13333333333333333), -0.3333333333333333), (x * (x * x)), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) + -1.0)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -5.0)
		tmp = t_0;
	elseif (Float64(-2.0 * x) <= 1e-6)
		tmp = fma(fma(Float64(x * x), fma(Float64(x * x), -0.05396825396825397, 0.13333333333333333), -0.3333333333333333), Float64(x * Float64(x * x)), x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -5.0], t$95$0, If[LessEqual[N[(-2.0 * x), $MachinePrecision], 1e-6], N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.05396825396825397 + 0.13333333333333333), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -5 or 9.99999999999999955e-7 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 100.0%

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

   if -5 < (*.f64 #s(literal -2 binary64) x) < 9.99999999999999955e-7

   1. Initial program 8.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lft-identity N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -5:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.05396825396825397, 0.13333333333333333\right), -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 + e^{-2 \cdot x} \leq 5:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(x \cdot \left(x + x\right), x, 2\right)} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ 1.0 (exp (* -2.0 x))) 5.0)
   x
   (+ (/ 2.0 (fma (* x (+ x x)) x 2.0)) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((1.0 + exp((-2.0 * x))) <= 5.0) {
		tmp = x;
	} else {
		tmp = (2.0 / fma((x * (x + x)), x, 2.0)) + -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(1.0 + exp(Float64(-2.0 * x))) <= 5.0)
		tmp = x;
	else
		tmp = Float64(Float64(2.0 / fma(Float64(x * Float64(x + x)), x, 2.0)) + -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5.0], x, N[(N[(2.0 / N[(N[(x * N[(x + x), $MachinePrecision]), $MachinePrecision] * x + 2.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 #s(literal 1 binary64) (exp.f64 (*.f64 #s(literal -2 binary64) x))) < 5

   1. Initial program 44.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. inv-pow N/A

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

      6. metadata-eval N/A

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

      7. sqr-pow N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 43.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\left(1 + e^{-2 \cdot x}\right) \cdot 0.5\right)}^{-0.5}, {\left(\left(1 + e^{-2 \cdot x}\right) \cdot 0.5\right)}^{-0.5}, -1\right)} \]

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. rem-square-sqrt N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. *-lft-identity N/A

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

      15. associate-+r+ N/A

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

      16. metadata-eval N/A

          \[\leadsto \color{blue}{0} + x \]

      17. +-lft-identity 62.3

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

   7. Applied rewrites 62.3%

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

   if 5 < (+.f64 #s(literal 1 binary64) (exp.f64 (*.f64 #s(literal -2 binary64) x)))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

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

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

      3. lower--.f64 N/A

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

      4. count-2 N/A

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

      5. lower-+.f64 99.2

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

   5. Applied rewrites 99.2%

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

   7. Applied rewrites 99.7%

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

   9. Applied rewrites 100.0%

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

4. Final simplification 72.0%

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

Alternative 5: 76.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 + e^{-2 \cdot x} \leq 5:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 - \left(x \cdot x\right) \cdot 4} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ 1.0 (exp (* -2.0 x))) 5.0)
   x
   (+ (/ 2.0 (- 2.0 (* (* x x) 4.0))) -1.0)))

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((1.0d0 + exp(((-2.0d0) * x))) <= 5.0d0) then
        tmp = x
    else
        tmp = (2.0d0 / (2.0d0 - ((x * x) * 4.0d0))) + (-1.0d0)
    end if
    code = tmp
end function

Java

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

Python

def code(x, y):
	tmp = 0
	if (1.0 + math.exp((-2.0 * x))) <= 5.0:
		tmp = x
	else:
		tmp = (2.0 / (2.0 - ((x * x) * 4.0))) + -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(1.0 + exp(Float64(-2.0 * x))) <= 5.0)
		tmp = x;
	else
		tmp = Float64(Float64(2.0 / Float64(2.0 - Float64(Float64(x * x) * 4.0))) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((1.0 + exp((-2.0 * x))) <= 5.0)
		tmp = x;
	else
		tmp = (2.0 / (2.0 - ((x * x) * 4.0))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5.0], x, N[(N[(2.0 / N[(2.0 - N[(N[(x * x), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 #s(literal 1 binary64) (exp.f64 (*.f64 #s(literal -2 binary64) x))) < 5

   1. Initial program 44.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. inv-pow N/A

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

      6. metadata-eval N/A

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

      7. sqr-pow N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 43.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\left(1 + e^{-2 \cdot x}\right) \cdot 0.5\right)}^{-0.5}, {\left(\left(1 + e^{-2 \cdot x}\right) \cdot 0.5\right)}^{-0.5}, -1\right)} \]

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. rem-square-sqrt N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. *-lft-identity N/A

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

      15. associate-+r+ N/A

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

      16. metadata-eval N/A

          \[\leadsto \color{blue}{0} + x \]

      17. +-lft-identity 62.3

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

   7. Applied rewrites 62.3%

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

   if 5 < (+.f64 #s(literal 1 binary64) (exp.f64 (*.f64 #s(literal -2 binary64) x)))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

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

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

      3. lower--.f64 N/A

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

      4. count-2 N/A

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

      5. lower-+.f64 99.2

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

   5. Applied rewrites 99.2%

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

   7. Applied rewrites 99.8%

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

4. Final simplification 72.0%

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

Alternative 6: 76.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 + e^{-2 \cdot x} \leq 5:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot \left(x \cdot -4\right)} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ 1.0 (exp (* -2.0 x))) 5.0) x (+ (/ 2.0 (* x (* x -4.0))) -1.0)))

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((1.0d0 + exp(((-2.0d0) * x))) <= 5.0d0) then
        tmp = x
    else
        tmp = (2.0d0 / (x * (x * (-4.0d0)))) + (-1.0d0)
    end if
    code = tmp
end function

Java

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

Python

def code(x, y):
	tmp = 0
	if (1.0 + math.exp((-2.0 * x))) <= 5.0:
		tmp = x
	else:
		tmp = (2.0 / (x * (x * -4.0))) + -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(1.0 + exp(Float64(-2.0 * x))) <= 5.0)
		tmp = x;
	else
		tmp = Float64(Float64(2.0 / Float64(x * Float64(x * -4.0))) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((1.0 + exp((-2.0 * x))) <= 5.0)
		tmp = x;
	else
		tmp = (2.0 / (x * (x * -4.0))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5.0], x, N[(N[(2.0 / N[(x * N[(x * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 #s(literal 1 binary64) (exp.f64 (*.f64 #s(literal -2 binary64) x))) < 5

   1. Initial program 44.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. inv-pow N/A

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

      6. metadata-eval N/A

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

      7. sqr-pow N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 43.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\left(1 + e^{-2 \cdot x}\right) \cdot 0.5\right)}^{-0.5}, {\left(\left(1 + e^{-2 \cdot x}\right) \cdot 0.5\right)}^{-0.5}, -1\right)} \]

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. rem-square-sqrt N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. *-lft-identity N/A

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

      15. associate-+r+ N/A

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

      16. metadata-eval N/A

          \[\leadsto \color{blue}{0} + x \]

      17. +-lft-identity 62.3

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

   7. Applied rewrites 62.3%

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

   if 5 < (+.f64 #s(literal 1 binary64) (exp.f64 (*.f64 #s(literal -2 binary64) x)))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

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

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

      3. lower--.f64 N/A

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

      4. count-2 N/A

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

      5. lower-+.f64 99.2

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

   5. Applied rewrites 99.2%

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 99.8%

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

4. Final simplification 72.0%

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

Alternative 7: 76.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 + e^{-2 \cdot x} \leq 5:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(x + x, x, 2\right)} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ 1.0 (exp (* -2.0 x))) 5.0) x (+ (/ 2.0 (fma (+ x x) x 2.0)) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((1.0 + exp((-2.0 * x))) <= 5.0) {
		tmp = x;
	} else {
		tmp = (2.0 / fma((x + x), x, 2.0)) + -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(1.0 + exp(Float64(-2.0 * x))) <= 5.0)
		tmp = x;
	else
		tmp = Float64(Float64(2.0 / fma(Float64(x + x), x, 2.0)) + -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5.0], x, N[(N[(2.0 / N[(N[(x + x), $MachinePrecision] * x + 2.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 #s(literal 1 binary64) (exp.f64 (*.f64 #s(literal -2 binary64) x))) < 5

   1. Initial program 44.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. inv-pow N/A

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

      6. metadata-eval N/A

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

      7. sqr-pow N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 43.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\left(1 + e^{-2 \cdot x}\right) \cdot 0.5\right)}^{-0.5}, {\left(\left(1 + e^{-2 \cdot x}\right) \cdot 0.5\right)}^{-0.5}, -1\right)} \]

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. rem-square-sqrt N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. *-lft-identity N/A

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

      15. associate-+r+ N/A

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

      16. metadata-eval N/A

          \[\leadsto \color{blue}{0} + x \]

      17. +-lft-identity 62.3

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

   7. Applied rewrites 62.3%

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

   if 5 < (+.f64 #s(literal 1 binary64) (exp.f64 (*.f64 #s(literal -2 binary64) x)))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

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

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

      3. lower--.f64 N/A

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

      4. count-2 N/A

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

      5. lower-+.f64 99.2

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

   5. Applied rewrites 99.2%

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

   7. Applied rewrites 99.7%

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

4. Final simplification 71.9%

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

Alternative 8: 75.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 + e^{-2 \cdot x} \leq 5:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(4, x, 2\right)} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ 1.0 (exp (* -2.0 x))) 5.0) x (+ (/ 2.0 (fma 4.0 x 2.0)) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((1.0 + exp((-2.0 * x))) <= 5.0) {
		tmp = x;
	} else {
		tmp = (2.0 / fma(4.0, x, 2.0)) + -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(1.0 + exp(Float64(-2.0 * x))) <= 5.0)
		tmp = x;
	else
		tmp = Float64(Float64(2.0 / fma(4.0, x, 2.0)) + -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5.0], x, N[(N[(2.0 / N[(4.0 * x + 2.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 #s(literal 1 binary64) (exp.f64 (*.f64 #s(literal -2 binary64) x))) < 5

   1. Initial program 44.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. inv-pow N/A

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

      6. metadata-eval N/A

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

      7. sqr-pow N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 43.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\left(1 + e^{-2 \cdot x}\right) \cdot 0.5\right)}^{-0.5}, {\left(\left(1 + e^{-2 \cdot x}\right) \cdot 0.5\right)}^{-0.5}, -1\right)} \]

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. rem-square-sqrt N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. *-lft-identity N/A

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

      15. associate-+r+ N/A

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

      16. metadata-eval N/A

          \[\leadsto \color{blue}{0} + x \]

      17. +-lft-identity 62.3

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

   7. Applied rewrites 62.3%

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

   if 5 < (+.f64 #s(literal 1 binary64) (exp.f64 (*.f64 #s(literal -2 binary64) x)))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

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

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

      3. lower--.f64 N/A

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

      4. count-2 N/A

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

      5. lower-+.f64 99.2

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

   5. Applied rewrites 99.2%

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

   7. Applied rewrites 99.3%

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

4. Final simplification 71.8%

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

Alternative 9: 75.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 + e^{-2 \cdot x} \leq 5:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x + x} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ 1.0 (exp (* -2.0 x))) 5.0) x (+ (/ 2.0 (+ x x)) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((1.0 + exp((-2.0 * x))) <= 5.0) {
		tmp = x;
	} else {
		tmp = (2.0 / (x + x)) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((1.0d0 + exp(((-2.0d0) * x))) <= 5.0d0) then
        tmp = x
    else
        tmp = (2.0d0 / (x + x)) + (-1.0d0)
    end if
    code = tmp
end function

Java

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

Python

def code(x, y):
	tmp = 0
	if (1.0 + math.exp((-2.0 * x))) <= 5.0:
		tmp = x
	else:
		tmp = (2.0 / (x + x)) + -1.0
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := If[LessEqual[N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5.0], x, N[(N[(2.0 / N[(x + x), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 #s(literal 1 binary64) (exp.f64 (*.f64 #s(literal -2 binary64) x))) < 5

   1. Initial program 44.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. inv-pow N/A

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

      6. metadata-eval N/A

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

      7. sqr-pow N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 43.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\left(1 + e^{-2 \cdot x}\right) \cdot 0.5\right)}^{-0.5}, {\left(\left(1 + e^{-2 \cdot x}\right) \cdot 0.5\right)}^{-0.5}, -1\right)} \]

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. rem-square-sqrt N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. *-lft-identity N/A

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

      15. associate-+r+ N/A

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

      16. metadata-eval N/A

          \[\leadsto \color{blue}{0} + x \]

      17. +-lft-identity 62.3

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

   7. Applied rewrites 62.3%

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

   if 5 < (+.f64 #s(literal 1 binary64) (exp.f64 (*.f64 #s(literal -2 binary64) x)))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

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

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

      3. lower--.f64 N/A

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

      4. count-2 N/A

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

      5. lower-+.f64 99.2

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 99.2%

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

   10. Applied rewrites 99.3%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + e^{-2 \cdot x} \leq 5:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x + x} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -5:\\ \;\;\;\;\frac{2}{2 + \frac{x + x}{\left(\left(x + x\right) - \left(x + x\right)\right) - \left(x + x\right)}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.05396825396825397, 0.13333333333333333\right), -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(x \cdot \left(x + x\right), x, 2\right)} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -5.0)
   (+ (/ 2.0 (+ 2.0 (/ (+ x x) (- (- (+ x x) (+ x x)) (+ x x))))) -1.0)
   (if (<= (* -2.0 x) 1.0)
     (fma
      (fma
       (* x x)
       (fma (* x x) -0.05396825396825397 0.13333333333333333)
       -0.3333333333333333)
      (* x (* x x))
      x)
     (+ (/ 2.0 (fma (* x (+ x x)) x 2.0)) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -5.0) {
		tmp = (2.0 / (2.0 + ((x + x) / (((x + x) - (x + x)) - (x + x))))) + -1.0;
	} else if ((-2.0 * x) <= 1.0) {
		tmp = fma(fma((x * x), fma((x * x), -0.05396825396825397, 0.13333333333333333), -0.3333333333333333), (x * (x * x)), x);
	} else {
		tmp = (2.0 / fma((x * (x + x)), x, 2.0)) + -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -5.0)
		tmp = Float64(Float64(2.0 / Float64(2.0 + Float64(Float64(x + x) / Float64(Float64(Float64(x + x) - Float64(x + x)) - Float64(x + x))))) + -1.0);
	elseif (Float64(-2.0 * x) <= 1.0)
		tmp = fma(fma(Float64(x * x), fma(Float64(x * x), -0.05396825396825397, 0.13333333333333333), -0.3333333333333333), Float64(x * Float64(x * x)), x);
	else
		tmp = Float64(Float64(2.0 / fma(Float64(x * Float64(x + x)), x, 2.0)) + -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -5.0], N[(N[(2.0 / N[(2.0 + N[(N[(x + x), $MachinePrecision] / N[(N[(N[(x + x), $MachinePrecision] - N[(x + x), $MachinePrecision]), $MachinePrecision] - N[(x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 1.0], N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.05396825396825397 + 0.13333333333333333), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(2.0 / N[(N[(x * N[(x + x), $MachinePrecision]), $MachinePrecision] * x + 2.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

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

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

      3. lower--.f64 N/A

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

      4. count-2 N/A

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

      5. lower-+.f64 1.6

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

   5. Applied rewrites 1.6%

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

   7. Applied rewrites 99.1%

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

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

   1. Initial program 9.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lft-identity N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.05396825396825397, 0.13333333333333333\right), -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)} \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

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

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

      3. lower--.f64 N/A

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

      4. count-2 N/A

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

      5. lower-+.f64 99.2

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

   5. Applied rewrites 99.2%

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

   7. Applied rewrites 99.7%

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

   9. Applied rewrites 100.0%

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

4. Final simplification 99.5%

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

Alternative 11: 99.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -5:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(2, x, -4\right)}{-2 + \left(x + x\right)}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.05396825396825397, 0.13333333333333333\right), -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(x \cdot \left(x + x\right), x, 2\right)} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -5.0)
   (+ (/ 2.0 (/ (fma 2.0 x -4.0) (+ -2.0 (+ x x)))) -1.0)
   (if (<= (* -2.0 x) 1.0)
     (fma
      (fma
       (* x x)
       (fma (* x x) -0.05396825396825397 0.13333333333333333)
       -0.3333333333333333)
      (* x (* x x))
      x)
     (+ (/ 2.0 (fma (* x (+ x x)) x 2.0)) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -5.0) {
		tmp = (2.0 / (fma(2.0, x, -4.0) / (-2.0 + (x + x)))) + -1.0;
	} else if ((-2.0 * x) <= 1.0) {
		tmp = fma(fma((x * x), fma((x * x), -0.05396825396825397, 0.13333333333333333), -0.3333333333333333), (x * (x * x)), x);
	} else {
		tmp = (2.0 / fma((x * (x + x)), x, 2.0)) + -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -5.0)
		tmp = Float64(Float64(2.0 / Float64(fma(2.0, x, -4.0) / Float64(-2.0 + Float64(x + x)))) + -1.0);
	elseif (Float64(-2.0 * x) <= 1.0)
		tmp = fma(fma(Float64(x * x), fma(Float64(x * x), -0.05396825396825397, 0.13333333333333333), -0.3333333333333333), Float64(x * Float64(x * x)), x);
	else
		tmp = Float64(Float64(2.0 / fma(Float64(x * Float64(x + x)), x, 2.0)) + -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -5.0], N[(N[(2.0 / N[(N[(2.0 * x + -4.0), $MachinePrecision] / N[(-2.0 + N[(x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 1.0], N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.05396825396825397 + 0.13333333333333333), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(2.0 / N[(N[(x * N[(x + x), $MachinePrecision]), $MachinePrecision] * x + 2.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

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

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

      3. lower--.f64 N/A

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

      4. count-2 N/A

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

      5. lower-+.f64 1.6

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

   5. Applied rewrites 1.6%

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

   7. Applied rewrites 1.6%

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

   9. Applied rewrites 1.6%

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

   11. Applied rewrites 96.7%

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

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

   1. Initial program 9.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lft-identity N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.05396825396825397, 0.13333333333333333\right), -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)} \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

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

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

      3. lower--.f64 N/A

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

      4. count-2 N/A

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

      5. lower-+.f64 99.2

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

   5. Applied rewrites 99.2%

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

   7. Applied rewrites 99.7%

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

   9. Applied rewrites 100.0%

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

4. Final simplification 98.8%

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

Alternative 12: 99.2% accurate, 2.4× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -5.0)
   (+ (/ 2.0 (/ (fma 2.0 x -4.0) (+ -2.0 (+ x x)))) -1.0)
   (if (<= (* -2.0 x) 1.0)
     (fma
      (fma (* x x) 0.13333333333333333 -0.3333333333333333)
      (* x (* x x))
      x)
     (+ (/ 2.0 (fma (* x (+ x x)) x 2.0)) -1.0))))

C

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

Julia

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

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -5.0], N[(N[(2.0 / N[(N[(2.0 * x + -4.0), $MachinePrecision] / N[(-2.0 + N[(x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 1.0], N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333 + -0.3333333333333333), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(2.0 / N[(N[(x * N[(x + x), $MachinePrecision]), $MachinePrecision] * x + 2.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

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

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

      3. lower--.f64 N/A

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

      4. count-2 N/A

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

      5. lower-+.f64 1.6

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

   5. Applied rewrites 1.6%

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

   7. Applied rewrites 1.6%

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

   9. Applied rewrites 1.6%

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

   11. Applied rewrites 96.7%

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

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

   1. Initial program 9.1%

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

   3. Taylor expanded in x around 0

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 99.4

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right), x \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \]

   5. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)} \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

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

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

      3. lower--.f64 N/A

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

      4. count-2 N/A

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

      5. lower-+.f64 99.2

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

   5. Applied rewrites 99.2%

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

   7. Applied rewrites 99.7%

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

   9. Applied rewrites 100.0%

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

4. Final simplification 98.8%

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

Alternative 13: 53.6% accurate, 123.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 58.8%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-/.f64 N/A

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

   4. clear-num N/A

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

   5. inv-pow N/A

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

   6. metadata-eval N/A

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

   7. sqr-pow N/A

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

   8. lower-fma.f64 N/A

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

4. Applied rewrites 58.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\left(1 + e^{-2 \cdot x}\right) \cdot 0.5\right)}^{-0.5}, {\left(\left(1 + e^{-2 \cdot x}\right) \cdot 0.5\right)}^{-0.5}, -1\right)} \]

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. unpow2 N/A

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

   4. rem-square-sqrt N/A

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

   5. metadata-eval N/A

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

   6. *-commutative N/A

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

   7. unpow2 N/A

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

   8. rem-square-sqrt N/A

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

   9. associate-*r* N/A

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

   10. metadata-eval N/A

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

   11. *-lft-identity N/A

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

   12. sub-neg N/A

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

   13. metadata-eval N/A

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

   14. +-commutative N/A

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

   15. associate-+r+ N/A

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

   16. metadata-eval N/A

       \[\leadsto \color{blue}{0} + x \]

   17. +-lft-identity 47.5

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

7. Applied rewrites 47.5%

   \[\leadsto \color{blue}{x} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024227 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))