Quotient of sum of exps

Percentage Accurate: 99.1% → 100.0%

Time: 12.9s

Alternatives: 15

Speedup: 2.9×

• could not determine a ground truth

  1. a = -2.00000000000000007e154
  2. b = -2.00000000000000007e154

Specification

Math

\[\begin{array}{l} \\ \frac{e^{a}}{e^{a} + e^{b}} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ (exp a) (+ (exp a) (exp b))))

C

double code(double a, double b) {
	return exp(a) / (exp(a) + exp(b));
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = exp(a) / (exp(a) + exp(b))
end function

Java

public static double code(double a, double b) {
	return Math.exp(a) / (Math.exp(a) + Math.exp(b));
}

Python

def code(a, b):
	return math.exp(a) / (math.exp(a) + math.exp(b))

Julia

function code(a, b)
	return Float64(exp(a) / Float64(exp(a) + exp(b)))
end

MATLAB

function tmp = code(a, b)
	tmp = exp(a) / (exp(a) + exp(b));
end

Wolfram

code[a_, b_] := N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{a}}{e^{a} + e^{b}} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ (exp a) (+ (exp a) (exp b))))

C

double code(double a, double b) {
	return exp(a) / (exp(a) + exp(b));
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = exp(a) / (exp(a) + exp(b))
end function

Java

public static double code(double a, double b) {
	return Math.exp(a) / (Math.exp(a) + Math.exp(b));
}

Python

def code(a, b):
	return math.exp(a) / (math.exp(a) + math.exp(b))

Julia

function code(a, b)
	return Float64(exp(a) / Float64(exp(a) + exp(b)))
end

MATLAB

function tmp = code(a, b)
	tmp = exp(a) / (exp(a) + exp(b));
end

Wolfram

code[a_, b_] := N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{-\mathsf{log1p}\left(e^{b - a}\right)} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (exp (- (log1p (exp (- b a))))))

C

double code(double a, double b) {
	return exp(-log1p(exp((b - a))));
}

Java

public static double code(double a, double b) {
	return Math.exp(-Math.log1p(Math.exp((b - a))));
}

Python

def code(a, b):
	return math.exp(-math.log1p(math.exp((b - a))))

Julia

function code(a, b)
	return exp(Float64(-log1p(exp(Float64(b - a)))))
end

Wolfram

code[a_, b_] := N[Exp[(-N[Log[1 + N[Exp[N[(b - a), $MachinePrecision]], $MachinePrecision]], $MachinePrecision])], $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
2. Step-by-step derivation

   1. *-lft-identity 99.2%

      \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

   2. associate-*l/ 99.2%

      \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

   3. associate-/r/ 99.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

   4. remove-double-neg 99.2%

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

   5. unsub-neg 99.2%

      \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

   6. div-sub 77.3%

      \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

   7. *-lft-identity 77.3%

      \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

   8. associate-*l/ 77.3%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

   9. lft-mult-inverse 99.2%

      \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

   10. sub-neg 99.2%

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

   11. distribute-frac-neg 99.2%

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

   12. remove-double-neg 99.2%

       \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

   13. div-exp 100.0%

       \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-exp-log 100.0%

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

   2. log-rec 100.0%

      \[\leadsto e^{\color{blue}{-\log \left(1 + e^{b - a}\right)}} \]

   3. log1p-define 100.0%

      \[\leadsto e^{-\color{blue}{\mathsf{log1p}\left(e^{b - a}\right)}} \]

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{e^{-\mathsf{log1p}\left(e^{b - a}\right)}} \]

7. Final simplification 100.0%

   \[\leadsto e^{-\mathsf{log1p}\left(e^{b - a}\right)} \]
8. Add Preprocessing

Alternative 2: 98.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -54000000:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -54000000.0) (/ 1.0 (+ 1.0 (exp (- a)))) (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -54000000.0) {
		tmp = 1.0 / (1.0 + exp(-a));
	} else {
		tmp = 1.0 / (1.0 + exp(b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-54000000.0d0)) then
        tmp = 1.0d0 / (1.0d0 + exp(-a))
    else
        tmp = 1.0d0 / (1.0d0 + exp(b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -54000000.0) {
		tmp = 1.0 / (1.0 + Math.exp(-a));
	} else {
		tmp = 1.0 / (1.0 + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -54000000.0:
		tmp = 1.0 / (1.0 + math.exp(-a))
	else:
		tmp = 1.0 / (1.0 + math.exp(b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -54000000.0)
		tmp = Float64(1.0 / Float64(1.0 + exp(Float64(-a))));
	else
		tmp = Float64(1.0 / Float64(1.0 + exp(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -54000000.0)
		tmp = 1.0 / (1.0 + exp(-a));
	else
		tmp = 1.0 / (1.0 + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -54000000.0], N[(1.0 / N[(1.0 + N[Exp[(-a)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.4e7

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 100.0%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 100.0%

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

      5. unsub-neg 100.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 0.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 0.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 0.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 100.0%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. remove-double-neg 100.0%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0 100.0%

      \[\leadsto \frac{1}{\color{blue}{1 + e^{-a}}} \]

   if -5.4e7 < a

   1. Initial program 99.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 99.0%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 99.0%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 99.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 99.0%

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

      5. unsub-neg 99.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 98.5%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 98.5%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 98.5%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 99.0%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 99.0%

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

      11. distribute-frac-neg 99.0%

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

      12. remove-double-neg 99.0%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 98.9%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -54000000:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.5 \cdot 10^{+77}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot \left(0.5 + a \cdot \left(a \cdot 0.041666666666666664 - 0.16666666666666666\right)\right) + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.5e+77)
   (/
    1.0
    (+
     2.0
     (*
      a
      (+
       (* a (+ 0.5 (* a (- (* a 0.041666666666666664) 0.16666666666666666))))
       -1.0))))
   (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.5e+77) {
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * ((a * 0.041666666666666664) - 0.16666666666666666)))) + -1.0)));
	} else {
		tmp = 1.0 / (1.0 + exp(b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-2.5d+77)) then
        tmp = 1.0d0 / (2.0d0 + (a * ((a * (0.5d0 + (a * ((a * 0.041666666666666664d0) - 0.16666666666666666d0)))) + (-1.0d0))))
    else
        tmp = 1.0d0 / (1.0d0 + exp(b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -2.5e+77) {
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * ((a * 0.041666666666666664) - 0.16666666666666666)))) + -1.0)));
	} else {
		tmp = 1.0 / (1.0 + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2.5e+77:
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * ((a * 0.041666666666666664) - 0.16666666666666666)))) + -1.0)))
	else:
		tmp = 1.0 / (1.0 + math.exp(b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.5e+77)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(Float64(a * Float64(0.5 + Float64(a * Float64(Float64(a * 0.041666666666666664) - 0.16666666666666666)))) + -1.0))));
	else
		tmp = Float64(1.0 / Float64(1.0 + exp(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.5e+77)
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * ((a * 0.041666666666666664) - 0.16666666666666666)))) + -1.0)));
	else
		tmp = 1.0 / (1.0 + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.5e+77], N[(1.0 / N[(2.0 + N[(a * N[(N[(a * N[(0.5 + N[(a * N[(N[(a * 0.041666666666666664), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.50000000000000002e77

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 100.0%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 100.0%

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

      5. unsub-neg 100.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 0.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 0.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 0.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 100.0%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. remove-double-neg 100.0%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0 100.0%

      \[\leadsto \frac{1}{\color{blue}{1 + e^{-a}}} \]

   6. Taylor expanded in a around 0 100.0%

      \[\leadsto \frac{1}{\color{blue}{2 + a \cdot \left(a \cdot \left(0.5 + a \cdot \left(0.041666666666666664 \cdot a - 0.16666666666666666\right)\right) - 1\right)}} \]

   if -2.50000000000000002e77 < a

   1. Initial program 99.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 99.0%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 99.0%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 99.1%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 99.1%

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

      5. unsub-neg 99.1%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 92.1%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 92.1%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 92.1%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 99.1%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 99.1%

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

      11. distribute-frac-neg 99.1%

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

      12. remove-double-neg 99.1%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 96.7%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.5 \cdot 10^{+77}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot \left(0.5 + a \cdot \left(a \cdot 0.041666666666666664 - 0.16666666666666666\right)\right) + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 100.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{e^{b - a} + 1} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ 1.0 (+ (exp (- b a)) 1.0)))

C

double code(double a, double b) {
	return 1.0 / (exp((b - a)) + 1.0);
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 1.0d0 / (exp((b - a)) + 1.0d0)
end function

Java

public static double code(double a, double b) {
	return 1.0 / (Math.exp((b - a)) + 1.0);
}

Python

def code(a, b):
	return 1.0 / (math.exp((b - a)) + 1.0)

Julia

function code(a, b)
	return Float64(1.0 / Float64(exp(Float64(b - a)) + 1.0))
end

MATLAB

function tmp = code(a, b)
	tmp = 1.0 / (exp((b - a)) + 1.0);
end

Wolfram

code[a_, b_] := N[(1.0 / N[(N[Exp[N[(b - a), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
2. Step-by-step derivation

   1. *-lft-identity 99.2%

      \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

   2. associate-*l/ 99.2%

      \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

   3. associate-/r/ 99.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

   4. remove-double-neg 99.2%

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

   5. unsub-neg 99.2%

      \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

   6. div-sub 77.3%

      \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

   7. *-lft-identity 77.3%

      \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

   8. associate-*l/ 77.3%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

   9. lft-mult-inverse 99.2%

      \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

   10. sub-neg 99.2%

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

   11. distribute-frac-neg 99.2%

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

   12. remove-double-neg 99.2%

       \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

   13. div-exp 100.0%

       \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
4. Add Preprocessing

5. Final simplification 100.0%

   \[\leadsto \frac{1}{e^{b - a} + 1} \]
6. Add Preprocessing

Alternative 5: 75.3% accurate, 10.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4 \cdot 10^{+67}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot \left(0.5 + a \cdot \left(a \cdot \left(0.041666666666666664 + a \cdot -0.008333333333333333\right) - 0.16666666666666666\right)\right) + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot \left(0.16666666666666666 + b \cdot 0.041666666666666664\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 4e+67)
   (/
    1.0
    (+
     2.0
     (*
      a
      (+
       (*
        a
        (+
         0.5
         (*
          a
          (-
           (* a (+ 0.041666666666666664 (* a -0.008333333333333333)))
           0.16666666666666666))))
       -1.0))))
   (/
    1.0
    (+
     2.0
     (*
      b
      (+
       1.0
       (*
        b
        (+ 0.5 (* b (+ 0.16666666666666666 (* b 0.041666666666666664)))))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 4e+67) {
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * ((a * (0.041666666666666664 + (a * -0.008333333333333333))) - 0.16666666666666666)))) + -1.0)));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * (0.16666666666666666 + (b * 0.041666666666666664))))))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 4d+67) then
        tmp = 1.0d0 / (2.0d0 + (a * ((a * (0.5d0 + (a * ((a * (0.041666666666666664d0 + (a * (-0.008333333333333333d0)))) - 0.16666666666666666d0)))) + (-1.0d0))))
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (b * (0.5d0 + (b * (0.16666666666666666d0 + (b * 0.041666666666666664d0))))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 4e+67) {
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * ((a * (0.041666666666666664 + (a * -0.008333333333333333))) - 0.16666666666666666)))) + -1.0)));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * (0.16666666666666666 + (b * 0.041666666666666664))))))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 4e+67:
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * ((a * (0.041666666666666664 + (a * -0.008333333333333333))) - 0.16666666666666666)))) + -1.0)))
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * (0.16666666666666666 + (b * 0.041666666666666664))))))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 4e+67)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(Float64(a * Float64(0.5 + Float64(a * Float64(Float64(a * Float64(0.041666666666666664 + Float64(a * -0.008333333333333333))) - 0.16666666666666666)))) + -1.0))));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(b * Float64(0.5 + Float64(b * Float64(0.16666666666666666 + Float64(b * 0.041666666666666664)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 4e+67)
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * ((a * (0.041666666666666664 + (a * -0.008333333333333333))) - 0.16666666666666666)))) + -1.0)));
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * (0.16666666666666666 + (b * 0.041666666666666664))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 4e+67], N[(1.0 / N[(2.0 + N[(a * N[(N[(a * N[(0.5 + N[(a * N[(N[(a * N[(0.041666666666666664 + N[(a * -0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(b * N[(0.5 + N[(b * N[(0.16666666666666666 + N[(b * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.99999999999999993e67

   1. Initial program 98.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 98.9%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 98.9%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 98.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 98.9%

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

      5. unsub-neg 98.9%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 78.6%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 78.6%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 78.6%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 98.9%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 98.9%

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

      11. distribute-frac-neg 98.9%

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

      12. remove-double-neg 98.9%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0 74.0%

      \[\leadsto \frac{1}{\color{blue}{1 + e^{-a}}} \]

   6. Taylor expanded in a around 0 70.0%

      \[\leadsto \frac{1}{\color{blue}{2 + a \cdot \left(a \cdot \left(0.5 + a \cdot \left(a \cdot \left(0.041666666666666664 + -0.008333333333333333 \cdot a\right) - 0.16666666666666666\right)\right) - 1\right)}} \]

   if 3.99999999999999993e67 < b

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 100.0%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 100.0%

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

      5. unsub-neg 100.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 73.9%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 73.9%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 73.9%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 100.0%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. remove-double-neg 100.0%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 100.0%

      \[\leadsto \frac{1}{\color{blue}{1 + e^{b}}} \]

   6. Taylor expanded in b around 0 94.8%

      \[\leadsto \frac{1}{\color{blue}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot \left(0.16666666666666666 + 0.041666666666666664 \cdot b\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. *-commutative 94.8%

         \[\leadsto \frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot \left(0.16666666666666666 + \color{blue}{b \cdot 0.041666666666666664}\right)\right)\right)} \]

   8. Simplified 94.8%

      \[\leadsto \frac{1}{\color{blue}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot \left(0.16666666666666666 + b \cdot 0.041666666666666664\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4 \cdot 10^{+67}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot \left(0.5 + a \cdot \left(a \cdot \left(0.041666666666666664 + a \cdot -0.008333333333333333\right) - 0.16666666666666666\right)\right) + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot \left(0.16666666666666666 + b \cdot 0.041666666666666664\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.3% accurate, 10.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.35 \cdot 10^{+48}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot \left(0.5 + a \cdot \left(a \cdot \left(0.041666666666666664 + a \cdot -0.008333333333333333\right) - 0.16666666666666666\right)\right) + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot \left(0.16666666666666666 + b \cdot \left(0.041666666666666664 + b \cdot 0.008333333333333333\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.35e+48)
   (/
    1.0
    (+
     2.0
     (*
      a
      (+
       (*
        a
        (+
         0.5
         (*
          a
          (-
           (* a (+ 0.041666666666666664 (* a -0.008333333333333333)))
           0.16666666666666666))))
       -1.0))))
   (/
    1.0
    (+
     2.0
     (*
      b
      (+
       1.0
       (*
        b
        (+
         0.5
         (*
          b
          (+
           0.16666666666666666
           (* b (+ 0.041666666666666664 (* b 0.008333333333333333)))))))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.35e+48) {
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * ((a * (0.041666666666666664 + (a * -0.008333333333333333))) - 0.16666666666666666)))) + -1.0)));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * (0.16666666666666666 + (b * (0.041666666666666664 + (b * 0.008333333333333333))))))))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 1.35d+48) then
        tmp = 1.0d0 / (2.0d0 + (a * ((a * (0.5d0 + (a * ((a * (0.041666666666666664d0 + (a * (-0.008333333333333333d0)))) - 0.16666666666666666d0)))) + (-1.0d0))))
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (b * (0.5d0 + (b * (0.16666666666666666d0 + (b * (0.041666666666666664d0 + (b * 0.008333333333333333d0))))))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.35e+48) {
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * ((a * (0.041666666666666664 + (a * -0.008333333333333333))) - 0.16666666666666666)))) + -1.0)));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * (0.16666666666666666 + (b * (0.041666666666666664 + (b * 0.008333333333333333))))))))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.35e+48:
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * ((a * (0.041666666666666664 + (a * -0.008333333333333333))) - 0.16666666666666666)))) + -1.0)))
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * (0.16666666666666666 + (b * (0.041666666666666664 + (b * 0.008333333333333333))))))))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.35e+48)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(Float64(a * Float64(0.5 + Float64(a * Float64(Float64(a * Float64(0.041666666666666664 + Float64(a * -0.008333333333333333))) - 0.16666666666666666)))) + -1.0))));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(b * Float64(0.5 + Float64(b * Float64(0.16666666666666666 + Float64(b * Float64(0.041666666666666664 + Float64(b * 0.008333333333333333)))))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.35e+48)
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * ((a * (0.041666666666666664 + (a * -0.008333333333333333))) - 0.16666666666666666)))) + -1.0)));
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * (0.16666666666666666 + (b * (0.041666666666666664 + (b * 0.008333333333333333))))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.35e+48], N[(1.0 / N[(2.0 + N[(a * N[(N[(a * N[(0.5 + N[(a * N[(N[(a * N[(0.041666666666666664 + N[(a * -0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(b * N[(0.5 + N[(b * N[(0.16666666666666666 + N[(b * N[(0.041666666666666664 + N[(b * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.35000000000000002e48

   1. Initial program 98.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 98.9%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 98.9%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 98.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 98.9%

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

      5. unsub-neg 98.9%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 79.1%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 79.1%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 79.1%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 98.9%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 98.9%

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

      11. distribute-frac-neg 98.9%

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

      12. remove-double-neg 98.9%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0 74.9%

      \[\leadsto \frac{1}{\color{blue}{1 + e^{-a}}} \]

   6. Taylor expanded in a around 0 71.3%

      \[\leadsto \frac{1}{\color{blue}{2 + a \cdot \left(a \cdot \left(0.5 + a \cdot \left(a \cdot \left(0.041666666666666664 + -0.008333333333333333 \cdot a\right) - 0.16666666666666666\right)\right) - 1\right)}} \]

   if 1.35000000000000002e48 < b

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 100.0%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 100.0%

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

      5. unsub-neg 100.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 73.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 73.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 73.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 100.0%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. remove-double-neg 100.0%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 100.0%

      \[\leadsto \frac{1}{\color{blue}{1 + e^{b}}} \]

   6. Taylor expanded in b around 0 96.3%

      \[\leadsto \frac{1}{\color{blue}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot \left(0.16666666666666666 + b \cdot \left(0.041666666666666664 + 0.008333333333333333 \cdot b\right)\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. *-commutative 96.3%

         \[\leadsto \frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot \left(0.16666666666666666 + b \cdot \left(0.041666666666666664 + \color{blue}{b \cdot 0.008333333333333333}\right)\right)\right)\right)} \]

   8. Simplified 96.3%

      \[\leadsto \frac{1}{\color{blue}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot \left(0.16666666666666666 + b \cdot \left(0.041666666666666664 + b \cdot 0.008333333333333333\right)\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.35 \cdot 10^{+48}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot \left(0.5 + a \cdot \left(a \cdot \left(0.041666666666666664 + a \cdot -0.008333333333333333\right) - 0.16666666666666666\right)\right) + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot \left(0.16666666666666666 + b \cdot \left(0.041666666666666664 + b \cdot 0.008333333333333333\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.7% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.35 \cdot 10^{+48}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot \left(0.5 + a \cdot \left(a \cdot 0.041666666666666664 - 0.16666666666666666\right)\right) + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.35e+48)
   (/
    1.0
    (+
     2.0
     (*
      a
      (+
       (* a (+ 0.5 (* a (- (* a 0.041666666666666664) 0.16666666666666666))))
       -1.0))))
   (/ 1.0 (+ 2.0 (* b (+ 1.0 (* b (+ 0.5 (* b 0.16666666666666666)))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.35e+48) {
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * ((a * 0.041666666666666664) - 0.16666666666666666)))) + -1.0)));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 1.35d+48) then
        tmp = 1.0d0 / (2.0d0 + (a * ((a * (0.5d0 + (a * ((a * 0.041666666666666664d0) - 0.16666666666666666d0)))) + (-1.0d0))))
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (b * (0.5d0 + (b * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.35e+48) {
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * ((a * 0.041666666666666664) - 0.16666666666666666)))) + -1.0)));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.35e+48:
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * ((a * 0.041666666666666664) - 0.16666666666666666)))) + -1.0)))
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.35e+48)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(Float64(a * Float64(0.5 + Float64(a * Float64(Float64(a * 0.041666666666666664) - 0.16666666666666666)))) + -1.0))));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(b * Float64(0.5 + Float64(b * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.35e+48)
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * ((a * 0.041666666666666664) - 0.16666666666666666)))) + -1.0)));
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.35e+48], N[(1.0 / N[(2.0 + N[(a * N[(N[(a * N[(0.5 + N[(a * N[(N[(a * 0.041666666666666664), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(b * N[(0.5 + N[(b * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.35000000000000002e48

   1. Initial program 98.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 98.9%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 98.9%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 98.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 98.9%

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

      5. unsub-neg 98.9%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 79.1%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 79.1%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 79.1%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 98.9%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 98.9%

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

      11. distribute-frac-neg 98.9%

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

      12. remove-double-neg 98.9%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0 74.9%

      \[\leadsto \frac{1}{\color{blue}{1 + e^{-a}}} \]

   6. Taylor expanded in a around 0 71.2%

      \[\leadsto \frac{1}{\color{blue}{2 + a \cdot \left(a \cdot \left(0.5 + a \cdot \left(0.041666666666666664 \cdot a - 0.16666666666666666\right)\right) - 1\right)}} \]

   if 1.35000000000000002e48 < b

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 100.0%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 100.0%

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

      5. unsub-neg 100.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 73.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 73.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 73.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 100.0%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. remove-double-neg 100.0%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 100.0%

      \[\leadsto \frac{1}{\color{blue}{1 + e^{b}}} \]

   6. Taylor expanded in b around 0 86.1%

      \[\leadsto \frac{1}{\color{blue}{2 + b \cdot \left(1 + b \cdot \left(0.5 + 0.16666666666666666 \cdot b\right)\right)}} \]
   7. Step-by-step derivation

      1. *-commutative 86.1%

         \[\leadsto \frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + \color{blue}{b \cdot 0.16666666666666666}\right)\right)} \]

   8. Simplified 86.1%

      \[\leadsto \frac{1}{\color{blue}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.35 \cdot 10^{+48}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot \left(0.5 + a \cdot \left(a \cdot 0.041666666666666664 - 0.16666666666666666\right)\right) + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.1% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.35 \cdot 10^{+48}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot \left(0.5 + a \cdot \left(a \cdot 0.041666666666666664 - 0.16666666666666666\right)\right) + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot \left(0.16666666666666666 + b \cdot 0.041666666666666664\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.35e+48)
   (/
    1.0
    (+
     2.0
     (*
      a
      (+
       (* a (+ 0.5 (* a (- (* a 0.041666666666666664) 0.16666666666666666))))
       -1.0))))
   (/
    1.0
    (+
     2.0
     (*
      b
      (+
       1.0
       (*
        b
        (+ 0.5 (* b (+ 0.16666666666666666 (* b 0.041666666666666664)))))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.35e+48) {
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * ((a * 0.041666666666666664) - 0.16666666666666666)))) + -1.0)));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * (0.16666666666666666 + (b * 0.041666666666666664))))))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 1.35d+48) then
        tmp = 1.0d0 / (2.0d0 + (a * ((a * (0.5d0 + (a * ((a * 0.041666666666666664d0) - 0.16666666666666666d0)))) + (-1.0d0))))
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (b * (0.5d0 + (b * (0.16666666666666666d0 + (b * 0.041666666666666664d0))))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.35e+48) {
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * ((a * 0.041666666666666664) - 0.16666666666666666)))) + -1.0)));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * (0.16666666666666666 + (b * 0.041666666666666664))))))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.35e+48:
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * ((a * 0.041666666666666664) - 0.16666666666666666)))) + -1.0)))
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * (0.16666666666666666 + (b * 0.041666666666666664))))))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.35e+48)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(Float64(a * Float64(0.5 + Float64(a * Float64(Float64(a * 0.041666666666666664) - 0.16666666666666666)))) + -1.0))));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(b * Float64(0.5 + Float64(b * Float64(0.16666666666666666 + Float64(b * 0.041666666666666664)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.35e+48)
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * ((a * 0.041666666666666664) - 0.16666666666666666)))) + -1.0)));
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * (0.16666666666666666 + (b * 0.041666666666666664))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.35e+48], N[(1.0 / N[(2.0 + N[(a * N[(N[(a * N[(0.5 + N[(a * N[(N[(a * 0.041666666666666664), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(b * N[(0.5 + N[(b * N[(0.16666666666666666 + N[(b * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.35000000000000002e48

   1. Initial program 98.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 98.9%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 98.9%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 98.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 98.9%

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

      5. unsub-neg 98.9%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 79.1%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 79.1%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 79.1%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 98.9%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 98.9%

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

      11. distribute-frac-neg 98.9%

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

      12. remove-double-neg 98.9%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0 74.9%

      \[\leadsto \frac{1}{\color{blue}{1 + e^{-a}}} \]

   6. Taylor expanded in a around 0 71.2%

      \[\leadsto \frac{1}{\color{blue}{2 + a \cdot \left(a \cdot \left(0.5 + a \cdot \left(0.041666666666666664 \cdot a - 0.16666666666666666\right)\right) - 1\right)}} \]

   if 1.35000000000000002e48 < b

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 100.0%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 100.0%

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

      5. unsub-neg 100.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 73.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 73.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 73.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 100.0%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. remove-double-neg 100.0%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 100.0%

      \[\leadsto \frac{1}{\color{blue}{1 + e^{b}}} \]

   6. Taylor expanded in b around 0 88.9%

      \[\leadsto \frac{1}{\color{blue}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot \left(0.16666666666666666 + 0.041666666666666664 \cdot b\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. *-commutative 88.9%

         \[\leadsto \frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot \left(0.16666666666666666 + \color{blue}{b \cdot 0.041666666666666664}\right)\right)\right)} \]

   8. Simplified 88.9%

      \[\leadsto \frac{1}{\color{blue}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot \left(0.16666666666666666 + b \cdot 0.041666666666666664\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.35 \cdot 10^{+48}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot \left(0.5 + a \cdot \left(a \cdot 0.041666666666666664 - 0.16666666666666666\right)\right) + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot \left(0.16666666666666666 + b \cdot 0.041666666666666664\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.5% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.85 \cdot 10^{+127}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot \left(0.5 + a \cdot -0.16666666666666666\right) + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.85e+127)
   (/ 1.0 (+ 2.0 (* a (+ (* a (+ 0.5 (* a -0.16666666666666666))) -1.0))))
   (/ 1.0 (+ 2.0 (* b (+ 1.0 (* b 0.5)))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.85e+127) {
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * -0.16666666666666666))) + -1.0)));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 1.85d+127) then
        tmp = 1.0d0 / (2.0d0 + (a * ((a * (0.5d0 + (a * (-0.16666666666666666d0)))) + (-1.0d0))))
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (b * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.85e+127) {
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * -0.16666666666666666))) + -1.0)));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.85e+127:
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * -0.16666666666666666))) + -1.0)))
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.85e+127)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(Float64(a * Float64(0.5 + Float64(a * -0.16666666666666666))) + -1.0))));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(b * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.85e+127)
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * -0.16666666666666666))) + -1.0)));
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.85e+127], N[(1.0 / N[(2.0 + N[(a * N[(N[(a * N[(0.5 + N[(a * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(b * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.8499999999999999e127

   1. Initial program 99.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 99.0%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 99.0%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 99.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 99.0%

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

      5. unsub-neg 99.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 79.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 79.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 79.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 99.0%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 99.0%

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

      11. distribute-frac-neg 99.0%

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

      12. remove-double-neg 99.0%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0 70.4%

      \[\leadsto \frac{1}{\color{blue}{1 + e^{-a}}} \]

   6. Taylor expanded in a around 0 63.2%

      \[\leadsto \frac{1}{\color{blue}{2 + a \cdot \left(a \cdot \left(0.5 + -0.16666666666666666 \cdot a\right) - 1\right)}} \]

   if 1.8499999999999999e127 < b

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 100.0%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 100.0%

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

      5. unsub-neg 100.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 71.4%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 71.4%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 71.4%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 100.0%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. remove-double-neg 100.0%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 100.0%

      \[\leadsto \frac{1}{\color{blue}{1 + e^{b}}} \]

   6. Taylor expanded in b around 0 95.1%

      \[\leadsto \frac{1}{\color{blue}{2 + b \cdot \left(1 + 0.5 \cdot b\right)}} \]
   7. Step-by-step derivation

      1. *-commutative 95.1%

         \[\leadsto \frac{1}{2 + b \cdot \left(1 + \color{blue}{b \cdot 0.5}\right)} \]

   8. Simplified 95.1%

      \[\leadsto \frac{1}{\color{blue}{2 + b \cdot \left(1 + b \cdot 0.5\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.85 \cdot 10^{+127}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot \left(0.5 + a \cdot -0.16666666666666666\right) + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.1% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.1 \cdot 10^{+48}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot \left(0.5 + a \cdot -0.16666666666666666\right) + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.1e+48)
   (/ 1.0 (+ 2.0 (* a (+ (* a (+ 0.5 (* a -0.16666666666666666))) -1.0))))
   (/ 1.0 (+ 2.0 (* b (+ 1.0 (* b (+ 0.5 (* b 0.16666666666666666)))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.1e+48) {
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * -0.16666666666666666))) + -1.0)));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 1.1d+48) then
        tmp = 1.0d0 / (2.0d0 + (a * ((a * (0.5d0 + (a * (-0.16666666666666666d0)))) + (-1.0d0))))
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (b * (0.5d0 + (b * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.1e+48) {
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * -0.16666666666666666))) + -1.0)));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.1e+48:
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * -0.16666666666666666))) + -1.0)))
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.1e+48)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(Float64(a * Float64(0.5 + Float64(a * -0.16666666666666666))) + -1.0))));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(b * Float64(0.5 + Float64(b * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.1e+48)
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * -0.16666666666666666))) + -1.0)));
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.1e+48], N[(1.0 / N[(2.0 + N[(a * N[(N[(a * N[(0.5 + N[(a * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(b * N[(0.5 + N[(b * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.1e48

   1. Initial program 98.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 98.9%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 98.9%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 98.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 98.9%

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

      5. unsub-neg 98.9%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 79.1%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 79.1%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 79.1%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 98.9%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 98.9%

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

      11. distribute-frac-neg 98.9%

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

      12. remove-double-neg 98.9%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0 74.9%

      \[\leadsto \frac{1}{\color{blue}{1 + e^{-a}}} \]

   6. Taylor expanded in a around 0 68.6%

      \[\leadsto \frac{1}{\color{blue}{2 + a \cdot \left(a \cdot \left(0.5 + -0.16666666666666666 \cdot a\right) - 1\right)}} \]

   if 1.1e48 < b

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 100.0%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 100.0%

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

      5. unsub-neg 100.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 73.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 73.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 73.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 100.0%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. remove-double-neg 100.0%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 100.0%

      \[\leadsto \frac{1}{\color{blue}{1 + e^{b}}} \]

   6. Taylor expanded in b around 0 86.1%

      \[\leadsto \frac{1}{\color{blue}{2 + b \cdot \left(1 + b \cdot \left(0.5 + 0.16666666666666666 \cdot b\right)\right)}} \]
   7. Step-by-step derivation

      1. *-commutative 86.1%

         \[\leadsto \frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + \color{blue}{b \cdot 0.16666666666666666}\right)\right)} \]

   8. Simplified 86.1%

      \[\leadsto \frac{1}{\color{blue}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.1 \cdot 10^{+48}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot \left(0.5 + a \cdot -0.16666666666666666\right) + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 63.9% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.85 \cdot 10^{+127}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot 0.5 + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.85e+127)
   (/ 1.0 (+ 2.0 (* a (+ (* a 0.5) -1.0))))
   (/ 1.0 (+ 2.0 (* b (+ 1.0 (* b 0.5)))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.85e+127) {
		tmp = 1.0 / (2.0 + (a * ((a * 0.5) + -1.0)));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 1.85d+127) then
        tmp = 1.0d0 / (2.0d0 + (a * ((a * 0.5d0) + (-1.0d0))))
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (b * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.85e+127) {
		tmp = 1.0 / (2.0 + (a * ((a * 0.5) + -1.0)));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.85e+127:
		tmp = 1.0 / (2.0 + (a * ((a * 0.5) + -1.0)))
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.85e+127)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(Float64(a * 0.5) + -1.0))));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(b * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.85e+127)
		tmp = 1.0 / (2.0 + (a * ((a * 0.5) + -1.0)));
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.85e+127], N[(1.0 / N[(2.0 + N[(a * N[(N[(a * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(b * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.8499999999999999e127

   1. Initial program 99.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 99.0%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 99.0%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 99.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 99.0%

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

      5. unsub-neg 99.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 79.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 79.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 79.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 99.0%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 99.0%

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

      11. distribute-frac-neg 99.0%

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

      12. remove-double-neg 99.0%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0 70.4%

      \[\leadsto \frac{1}{\color{blue}{1 + e^{-a}}} \]

   6. Taylor expanded in a around 0 59.3%

      \[\leadsto \frac{1}{\color{blue}{2 + a \cdot \left(0.5 \cdot a - 1\right)}} \]

   if 1.8499999999999999e127 < b

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 100.0%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 100.0%

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

      5. unsub-neg 100.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 71.4%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 71.4%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 71.4%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 100.0%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. remove-double-neg 100.0%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 100.0%

      \[\leadsto \frac{1}{\color{blue}{1 + e^{b}}} \]

   6. Taylor expanded in b around 0 95.1%

      \[\leadsto \frac{1}{\color{blue}{2 + b \cdot \left(1 + 0.5 \cdot b\right)}} \]
   7. Step-by-step derivation

      1. *-commutative 95.1%

         \[\leadsto \frac{1}{2 + b \cdot \left(1 + \color{blue}{b \cdot 0.5}\right)} \]

   8. Simplified 95.1%

      \[\leadsto \frac{1}{\color{blue}{2 + b \cdot \left(1 + b \cdot 0.5\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.85 \cdot 10^{+127}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot 0.5 + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 53.3% accurate, 27.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{2 + a \cdot \left(a \cdot 0.5 + -1\right)} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ 1.0 (+ 2.0 (* a (+ (* a 0.5) -1.0)))))

C

double code(double a, double b) {
	return 1.0 / (2.0 + (a * ((a * 0.5) + -1.0)));
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 1.0d0 / (2.0d0 + (a * ((a * 0.5d0) + (-1.0d0))))
end function

Java

public static double code(double a, double b) {
	return 1.0 / (2.0 + (a * ((a * 0.5) + -1.0)));
}

Python

def code(a, b):
	return 1.0 / (2.0 + (a * ((a * 0.5) + -1.0)))

Julia

function code(a, b)
	return Float64(1.0 / Float64(2.0 + Float64(a * Float64(Float64(a * 0.5) + -1.0))))
end

MATLAB

function tmp = code(a, b)
	tmp = 1.0 / (2.0 + (a * ((a * 0.5) + -1.0)));
end

Wolfram

code[a_, b_] := N[(1.0 / N[(2.0 + N[(a * N[(N[(a * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
2. Step-by-step derivation

   1. *-lft-identity 99.2%

      \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

   2. associate-*l/ 99.2%

      \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

   3. associate-/r/ 99.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

   4. remove-double-neg 99.2%

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

   5. unsub-neg 99.2%

      \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

   6. div-sub 77.3%

      \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

   7. *-lft-identity 77.3%

      \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

   8. associate-*l/ 77.3%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

   9. lft-mult-inverse 99.2%

      \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

   10. sub-neg 99.2%

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

   11. distribute-frac-neg 99.2%

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

   12. remove-double-neg 99.2%

       \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

   13. div-exp 100.0%

       \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
4. Add Preprocessing

5. Taylor expanded in b around 0 61.7%

   \[\leadsto \frac{1}{\color{blue}{1 + e^{-a}}} \]

6. Taylor expanded in a around 0 50.0%

   \[\leadsto \frac{1}{\color{blue}{2 + a \cdot \left(0.5 \cdot a - 1\right)}} \]

7. Final simplification 50.0%

   \[\leadsto \frac{1}{2 + a \cdot \left(a \cdot 0.5 + -1\right)} \]
8. Add Preprocessing

Alternative 13: 39.5% accurate, 61.0× speedup

Math

\[\begin{array}{l} \\ 0.5 + a \cdot 0.25 \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (+ 0.5 (* a 0.25)))

C

double code(double a, double b) {
	return 0.5 + (a * 0.25);
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 0.5d0 + (a * 0.25d0)
end function

Java

public static double code(double a, double b) {
	return 0.5 + (a * 0.25);
}

Python

def code(a, b):
	return 0.5 + (a * 0.25)

Julia

function code(a, b)
	return Float64(0.5 + Float64(a * 0.25))
end

MATLAB

function tmp = code(a, b)
	tmp = 0.5 + (a * 0.25);
end

Wolfram

code[a_, b_] := N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
2. Step-by-step derivation

   1. *-lft-identity 99.2%

      \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

   2. associate-*l/ 99.2%

      \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

   3. associate-/r/ 99.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

   4. remove-double-neg 99.2%

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

   5. unsub-neg 99.2%

      \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

   6. div-sub 77.3%

      \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

   7. *-lft-identity 77.3%

      \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

   8. associate-*l/ 77.3%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

   9. lft-mult-inverse 99.2%

      \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

   10. sub-neg 99.2%

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

   11. distribute-frac-neg 99.2%

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

   12. remove-double-neg 99.2%

       \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

   13. div-exp 100.0%

       \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
4. Add Preprocessing

5. Taylor expanded in b around 0 61.7%

   \[\leadsto \frac{1}{\color{blue}{1 + e^{-a}}} \]

6. Taylor expanded in a around 0 40.1%

   \[\leadsto \color{blue}{0.5 + 0.25 \cdot a} \]
7. Step-by-step derivation

   1. *-commutative 40.1%

      \[\leadsto 0.5 + \color{blue}{a \cdot 0.25} \]

8. Simplified 40.1%

   \[\leadsto \color{blue}{0.5 + a \cdot 0.25} \]

9. Final simplification 40.1%

   \[\leadsto 0.5 + a \cdot 0.25 \]
10. Add Preprocessing

Alternative 14: 40.1% accurate, 61.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{2 - a} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ 1.0 (- 2.0 a)))

C

double code(double a, double b) {
	return 1.0 / (2.0 - a);
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 1.0d0 / (2.0d0 - a)
end function

Java

public static double code(double a, double b) {
	return 1.0 / (2.0 - a);
}

Python

def code(a, b):
	return 1.0 / (2.0 - a)

Julia

function code(a, b)
	return Float64(1.0 / Float64(2.0 - a))
end

MATLAB

function tmp = code(a, b)
	tmp = 1.0 / (2.0 - a);
end

Wolfram

code[a_, b_] := N[(1.0 / N[(2.0 - a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
2. Step-by-step derivation

   1. *-lft-identity 99.2%

      \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

   2. associate-*l/ 99.2%

      \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

   3. associate-/r/ 99.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

   4. remove-double-neg 99.2%

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

   5. unsub-neg 99.2%

      \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

   6. div-sub 77.3%

      \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

   7. *-lft-identity 77.3%

      \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

   8. associate-*l/ 77.3%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

   9. lft-mult-inverse 99.2%

      \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

   10. sub-neg 99.2%

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

   11. distribute-frac-neg 99.2%

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

   12. remove-double-neg 99.2%

       \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

   13. div-exp 100.0%

       \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
4. Add Preprocessing

5. Taylor expanded in b around 0 61.7%

   \[\leadsto \frac{1}{\color{blue}{1 + e^{-a}}} \]

6. Taylor expanded in a around 0 40.6%

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

   1. neg-mul-1 40.6%

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

   2. unsub-neg 40.6%

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

8. Simplified 40.6%

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

9. Final simplification 40.6%

   \[\leadsto \frac{1}{2 - a} \]
10. Add Preprocessing

Alternative 15: 39.3% accurate, 305.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (a b) :precision binary64 0.5)

C

double code(double a, double b) {
	return 0.5;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 0.5d0
end function

Java

public static double code(double a, double b) {
	return 0.5;
}

Python

def code(a, b):
	return 0.5

Julia

function code(a, b)
	return 0.5
end

MATLAB

function tmp = code(a, b)
	tmp = 0.5;
end

Wolfram

code[a_, b_] := 0.5

Derivation

1. Initial program 99.2%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
2. Step-by-step derivation

   1. *-lft-identity 99.2%

      \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

   2. associate-*l/ 99.2%

      \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

   3. associate-/r/ 99.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

   4. remove-double-neg 99.2%

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

   5. unsub-neg 99.2%

      \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

   6. div-sub 77.3%

      \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

   7. *-lft-identity 77.3%

      \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

   8. associate-*l/ 77.3%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

   9. lft-mult-inverse 99.2%

      \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

   10. sub-neg 99.2%

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

   11. distribute-frac-neg 99.2%

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

   12. remove-double-neg 99.2%

       \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

   13. div-exp 100.0%

       \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
4. Add Preprocessing

5. Taylor expanded in a around 0 87.0%

   \[\leadsto \frac{1}{\color{blue}{1 + e^{b}}} \]

6. Taylor expanded in b around 0 39.8%

   \[\leadsto \color{blue}{0.5} \]

7. Final simplification 39.8%

   \[\leadsto 0.5 \]
8. Add Preprocessing

Developer target: 100.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + e^{b - a}} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ 1.0 (+ 1.0 (exp (- b a)))))

C

double code(double a, double b) {
	return 1.0 / (1.0 + exp((b - a)));
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 1.0d0 / (1.0d0 + exp((b - a)))
end function

Java

public static double code(double a, double b) {
	return 1.0 / (1.0 + Math.exp((b - a)));
}

Python

def code(a, b):
	return 1.0 / (1.0 + math.exp((b - a)))

Julia

function code(a, b)
	return Float64(1.0 / Float64(1.0 + exp(Float64(b - a))))
end

MATLAB

function tmp = code(a, b)
	tmp = 1.0 / (1.0 + exp((b - a)));
end

Wolfram

code[a_, b_] := N[(1.0 / N[(1.0 + N[Exp[N[(b - a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024053 
(FPCore (a b)
  :name "Quotient of sum of exps"
  :precision binary64

  :alt
  (/ 1.0 (+ 1.0 (exp (- b a))))

  (/ (exp a) (+ (exp a) (exp b))))