Logistic function from Lakshay Garg

Percentage Accurate: 54.5% → 99.2%

Time: 8.7s

Alternatives: 8

Speedup: 18.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.5% 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: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + e^{-2 \cdot x}\\ \mathbf{if}\;-2 \cdot x \leq -0.5:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{\mathsf{expm1}\left(x \cdot -4\right)}, \mathsf{expm1}\left(-2 \cdot x\right), -1\right)\\ \mathbf{elif}\;-2 \cdot x \leq 10^{-18}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\left(-1 + \frac{4}{{t\_0}^{2}}\right) \cdot \frac{1}{1 + \frac{2}{t\_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (exp (* -2.0 x)))))
   (if (<= (* -2.0 x) -0.5)
     (fma (/ 2.0 (expm1 (* x -4.0))) (expm1 (* -2.0 x)) -1.0)
     (if (<= (* -2.0 x) 1e-18)
       x
       (* (+ -1.0 (/ 4.0 (pow t_0 2.0))) (/ 1.0 (+ 1.0 (/ 2.0 t_0))))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + exp((-2.0 * x));
	double tmp;
	if ((-2.0 * x) <= -0.5) {
		tmp = fma((2.0 / expm1((x * -4.0))), expm1((-2.0 * x)), -1.0);
	} else if ((-2.0 * x) <= 1e-18) {
		tmp = x;
	} else {
		tmp = (-1.0 + (4.0 / pow(t_0, 2.0))) * (1.0 / (1.0 + (2.0 / t_0)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 + exp(Float64(-2.0 * x)))
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.5)
		tmp = fma(Float64(2.0 / expm1(Float64(x * -4.0))), expm1(Float64(-2.0 * x)), -1.0);
	elseif (Float64(-2.0 * x) <= 1e-18)
		tmp = x;
	else
		tmp = Float64(Float64(-1.0 + Float64(4.0 / (t_0 ^ 2.0))) * Float64(1.0 / Float64(1.0 + Float64(2.0 / t_0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.5], N[(N[(2.0 / N[(Exp[N[(x * -4.0), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] * N[(Exp[N[(-2.0 * x), $MachinePrecision]] - 1), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 1e-18], x, N[(N[(-1.0 + N[(4.0 / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 + N[(2.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -0.5

   1. Initial program 100.0%

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

      1. flip-+ 100.0%

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

      2. associate-/r/ 100.0%

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

      3. fma-neg 100.0%

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

      4. metadata-eval 100.0%

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

      5. pow2 100.0%

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

      6. *-commutative 100.0%

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

      7. exp-prod 100.0%

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

      8. pow-pow 100.0%

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

      9. metadata-eval 100.0%

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

      10. exp-prod 100.0%

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

      11. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{-\mathsf{expm1}\left(x \cdot -4\right)}, -\mathsf{expm1}\left(x \cdot -2\right), -1\right)} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 100.0%

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

      2. add-sqr-sqrt 100.0%

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

      3. sqrt-unprod 100.0%

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

      4. sqr-neg 100.0%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 1.6%

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

      7. add-sqr-sqrt 1.6%

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

      8. sqrt-unprod 1.6%

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

      9. sqr-neg 1.6%

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

      10. sqrt-unprod 0.0%

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

      11. add-sqr-sqrt 100.0%

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

   8. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{1 \cdot \mathsf{fma}\left(\frac{2}{\mathsf{expm1}\left(x \cdot -4\right)}, \mathsf{expm1}\left(x \cdot -2\right), -1\right)} \]
   9. Step-by-step derivation

      1. *-lft-identity 100.0%

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

      2. *-commutative 100.0%

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

   10. Simplified 100.0%

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

   if -0.5 < (*.f64 #s(literal -2 binary64) x) < 1.0000000000000001e-18

   1. Initial program 5.1%

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

   3. Taylor expanded in x around 0 100.0%

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

   if 1.0000000000000001e-18 < (*.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. flip-- 100.0%

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

      2. div-inv 100.0%

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

      3. metadata-eval 100.0%

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

      4. sub-neg 100.0%

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

      5. frac-times 100.0%

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

      6. metadata-eval 100.0%

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

      7. pow2 100.0%

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

      8. exp-prod 100.0%

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

      9. metadata-eval 100.0%

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

      10. +-commutative 100.0%

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

      11. exp-prod 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

   6. Taylor expanded in x around inf 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.5:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{\mathsf{expm1}\left(x \cdot -4\right)}, \mathsf{expm1}\left(-2 \cdot x\right), -1\right)\\ \mathbf{elif}\;-2 \cdot x \leq 10^{-18}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\left(-1 + \frac{4}{{\left(1 + e^{-2 \cdot x}\right)}^{2}}\right) \cdot \frac{1}{1 + \frac{2}{1 + e^{-2 \cdot x}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.5:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{\mathsf{expm1}\left(x \cdot -4\right)}, \mathsf{expm1}\left(-2 \cdot x\right), -1\right)\\ \mathbf{elif}\;-2 \cdot x \leq 10^{-18}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -0.5)
   (fma (/ 2.0 (expm1 (* x -4.0))) (expm1 (* -2.0 x)) -1.0)
   (if (<= (* -2.0 x) 1e-18) x (+ -1.0 (/ 2.0 (+ 1.0 (exp (* -2.0 x))))))))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.5) {
		tmp = fma((2.0 / expm1((x * -4.0))), expm1((-2.0 * x)), -1.0);
	} else if ((-2.0 * x) <= 1e-18) {
		tmp = x;
	} else {
		tmp = -1.0 + (2.0 / (1.0 + exp((-2.0 * x))));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.5)
		tmp = fma(Float64(2.0 / expm1(Float64(x * -4.0))), expm1(Float64(-2.0 * x)), -1.0);
	elseif (Float64(-2.0 * x) <= 1e-18)
		tmp = x;
	else
		tmp = Float64(-1.0 + Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.5], N[(N[(2.0 / N[(Exp[N[(x * -4.0), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] * N[(Exp[N[(-2.0 * x), $MachinePrecision]] - 1), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 1e-18], x, N[(-1.0 + N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -0.5

   1. Initial program 100.0%

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

      1. flip-+ 100.0%

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

      2. associate-/r/ 100.0%

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

      3. fma-neg 100.0%

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

      4. metadata-eval 100.0%

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

      5. pow2 100.0%

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

      6. *-commutative 100.0%

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

      7. exp-prod 100.0%

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

      8. pow-pow 100.0%

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

      9. metadata-eval 100.0%

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

      10. exp-prod 100.0%

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

      11. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{-\mathsf{expm1}\left(x \cdot -4\right)}, -\mathsf{expm1}\left(x \cdot -2\right), -1\right)} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 100.0%

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

      2. add-sqr-sqrt 100.0%

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

      3. sqrt-unprod 100.0%

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

      4. sqr-neg 100.0%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 1.6%

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

      7. add-sqr-sqrt 1.6%

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

      8. sqrt-unprod 1.6%

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

      9. sqr-neg 1.6%

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

      10. sqrt-unprod 0.0%

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

      11. add-sqr-sqrt 100.0%

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

   8. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{1 \cdot \mathsf{fma}\left(\frac{2}{\mathsf{expm1}\left(x \cdot -4\right)}, \mathsf{expm1}\left(x \cdot -2\right), -1\right)} \]
   9. Step-by-step derivation

      1. *-lft-identity 100.0%

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

      2. *-commutative 100.0%

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

   10. Simplified 100.0%

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

   if -0.5 < (*.f64 #s(literal -2 binary64) x) < 1.0000000000000001e-18

   1. Initial program 5.1%

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

   3. Taylor expanded in x around 0 100.0%

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

   if 1.0000000000000001e-18 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 100.0%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Add Preprocessing
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.5:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{\mathsf{expm1}\left(x \cdot -4\right)}, \mathsf{expm1}\left(-2 \cdot x\right), -1\right)\\ \mathbf{elif}\;-2 \cdot x \leq 10^{-18}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.5 \lor \neg \left(-2 \cdot x \leq 10^{-18}\right):\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* -2.0 x) -0.5) (not (<= (* -2.0 x) 1e-18)))
   (+ -1.0 (/ 2.0 (+ 1.0 (exp (* -2.0 x)))))
   x))

C

double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -0.5) || !((-2.0 * x) <= 1e-18)) {
		tmp = -1.0 + (2.0 / (1.0 + exp((-2.0 * x))));
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((((-2.0d0) * x) <= (-0.5d0)) .or. (.not. (((-2.0d0) * x) <= 1d-18))) then
        tmp = (-1.0d0) + (2.0d0 / (1.0d0 + exp(((-2.0d0) * x))))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -0.5) || !((-2.0 * x) <= 1e-18)) {
		tmp = -1.0 + (2.0 / (1.0 + Math.exp((-2.0 * x))));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((-2.0 * x) <= -0.5) or not ((-2.0 * x) <= 1e-18):
		tmp = -1.0 + (2.0 / (1.0 + math.exp((-2.0 * x))))
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((Float64(-2.0 * x) <= -0.5) || !(Float64(-2.0 * x) <= 1e-18))
		tmp = Float64(-1.0 + Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((-2.0 * x) <= -0.5) || ~(((-2.0 * x) <= 1e-18)))
		tmp = -1.0 + (2.0 / (1.0 + exp((-2.0 * x))));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.5], N[Not[LessEqual[N[(-2.0 * x), $MachinePrecision], 1e-18]], $MachinePrecision]], N[(-1.0 + N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -0.5 or 1.0000000000000001e-18 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 100.0%

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

   if -0.5 < (*.f64 #s(literal -2 binary64) x) < 1.0000000000000001e-18

   1. Initial program 5.1%

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

   3. Taylor expanded in x around 0 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.5 \lor \neg \left(-2 \cdot x \leq 10^{-18}\right):\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.0% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1.35000000000000001e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.1%

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

   if -1.35000000000000001e-8 < x

   1. Initial program 41.4%

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

   3. Taylor expanded in x around 0 5.3%

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

      1. +-commutative 5.3%

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

   5. Simplified 5.3%

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

      1. flip-- 5.2%

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

      2. div-inv 5.2%

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

      3. metadata-eval 5.2%

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

      4. difference-of-sqr-1 5.2%

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

      5. associate-+l+ 5.2%

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

      6. metadata-eval 5.2%

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

      7. associate--l+ 63.8%

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

      8. metadata-eval 63.8%

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

      9. +-rgt-identity 63.8%

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

      10. associate-+l+ 63.8%

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

      11. metadata-eval 63.8%

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

   7. Applied egg-rr 63.8%

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

   8. Taylor expanded in x around 0 69.0%

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

4. Final simplification 78.2%

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

Alternative 5: 77.9% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1.35000000000000001e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.1%

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

   if -1.35000000000000001e-8 < x

   1. Initial program 41.4%

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

   3. Taylor expanded in x around 0 5.3%

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

      1. +-commutative 5.3%

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

   5. Simplified 5.3%

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

      1. flip-- 5.2%

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

      2. div-inv 5.2%

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

      3. metadata-eval 5.2%

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

      4. difference-of-sqr-1 5.2%

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

      5. associate-+l+ 5.2%

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

      6. metadata-eval 5.2%

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

      7. associate--l+ 63.8%

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

      8. metadata-eval 63.8%

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

      9. +-rgt-identity 63.8%

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

      10. associate-+l+ 63.8%

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

      11. metadata-eval 63.8%

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

   7. Applied egg-rr 63.8%

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

   8. Taylor expanded in x around 0 69.0%

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

4. Final simplification 78.1%

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

Alternative 6: 78.0% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.67:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2\right) \cdot \frac{1}{x + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.67) -1.0 (* (* x 2.0) (/ 1.0 (+ x 2.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.67) {
		tmp = -1.0;
	} else {
		tmp = (x * 2.0) * (1.0 / (x + 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-0.67d0)) then
        tmp = -1.0d0
    else
        tmp = (x * 2.0d0) * (1.0d0 / (x + 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.67) {
		tmp = -1.0;
	} else {
		tmp = (x * 2.0) * (1.0 / (x + 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.67:
		tmp = -1.0
	else:
		tmp = (x * 2.0) * (1.0 / (x + 2.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.67)
		tmp = -1.0;
	else
		tmp = Float64(Float64(x * 2.0) * Float64(1.0 / Float64(x + 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.67)
		tmp = -1.0;
	else
		tmp = (x * 2.0) * (1.0 / (x + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.67], -1.0, N[(N[(x * 2.0), $MachinePrecision] * N[(1.0 / N[(x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.67000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.2%

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

      1. *-commutative 99.2%

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

   5. Simplified 99.2%

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

   6. Taylor expanded in x around inf 100.0%

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

   if -0.67000000000000004 < x

   1. Initial program 41.7%

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

   3. Taylor expanded in x around 0 5.4%

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

      1. +-commutative 5.4%

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

   5. Simplified 5.4%

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

      1. flip-- 5.3%

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

      2. div-inv 5.3%

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

      3. metadata-eval 5.3%

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

      4. difference-of-sqr-1 5.3%

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

      5. associate-+l+ 5.3%

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

      6. metadata-eval 5.3%

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

      7. associate--l+ 63.6%

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

      8. metadata-eval 63.6%

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

      9. +-rgt-identity 63.6%

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

      10. associate-+l+ 63.6%

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

      11. metadata-eval 63.6%

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

   7. Applied egg-rr 63.6%

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

   8. Taylor expanded in x around 0 68.7%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.67:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2\right) \cdot \frac{1}{x + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.0% accurate, 18.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.2%

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

      1. *-commutative 99.2%

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

   5. Simplified 99.2%

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

   6. Taylor expanded in x around inf 100.0%

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

   if -1 < x

   1. Initial program 41.7%

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

   3. Taylor expanded in x around 0 63.7%

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

Alternative 8: 26.8% accurate, 109.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return -1.0

Julia

function code(x, y)
	return -1.0
end

MATLAB

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

Wolfram

code[x_, y_] := -1.0

Derivation

1. Initial program 59.2%

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

3. Taylor expanded in x around 0 32.6%

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

   1. *-commutative 32.6%

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

5. Simplified 32.6%

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

6. Taylor expanded in x around inf 32.2%

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

Reproduce

herbie shell --seed 2024137 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))