Quotient of sum of exps

Percentage Accurate: 99.0% → 99.6%

Time: 6.4s

Alternatives: 12

Speedup: 1.0×

• 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: 99.0% 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: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{a}}{e^{b} + e^{a}}\\ \mathbf{if}\;t_0 \leq 2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ (exp a) (+ (exp b) (exp a)))))
   (if (<= t_0 2.0) t_0 (/ 1.0 (+ 1.0 (exp (- a)))))))

C

double code(double a, double b) {
	double t_0 = exp(a) / (exp(b) + exp(a));
	double tmp;
	if (t_0 <= 2.0) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (1.0 + exp(-a));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(a) / (exp(b) + exp(a))
    if (t_0 <= 2.0d0) then
        tmp = t_0
    else
        tmp = 1.0d0 / (1.0d0 + exp(-a))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = Math.exp(a) / (Math.exp(b) + Math.exp(a));
	double tmp;
	if (t_0 <= 2.0) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (1.0 + Math.exp(-a));
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = math.exp(a) / (math.exp(b) + math.exp(a))
	tmp = 0
	if t_0 <= 2.0:
		tmp = t_0
	else:
		tmp = 1.0 / (1.0 + math.exp(-a))
	return tmp

Julia

function code(a, b)
	t_0 = Float64(exp(a) / Float64(exp(b) + exp(a)))
	tmp = 0.0
	if (t_0 <= 2.0)
		tmp = t_0;
	else
		tmp = Float64(1.0 / Float64(1.0 + exp(Float64(-a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = exp(a) / (exp(b) + exp(a));
	tmp = 0.0;
	if (t_0 <= 2.0)
		tmp = t_0;
	else
		tmp = 1.0 / (1.0 + exp(-a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[b], $MachinePrecision] + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2.0], t$95$0, N[(1.0 / N[(1.0 + N[Exp[(-a)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 2

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]

   if 2 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 0.0%

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

      1. clear-num 0.0%

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

      2. inv-pow 0.0%

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

   3. Applied egg-rr 0.0%

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

      1. expm1-log1p-u 0.0%

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

      2. expm1-udef 0.0%

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

   5. Applied egg-rr 0.0%

      \[\leadsto {\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{e^{a} + e^{b}}{e^{a}}\right)} - 1\right)}}^{-1} \]
   6. Step-by-step derivation

      1. expm1-def 0.0%

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

      2. expm1-log1p 0.0%

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

      3. *-lft-identity 0.0%

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

      4. associate-*l/ 0.0%

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

      5. rec-exp 0.0%

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

      6. distribute-rgt-in 0.0%

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

      7. rec-exp 0.0%

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

      8. rgt-mult-inverse 80.0%

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

      9. rec-exp 80.0%

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

      10. associate-*r/ 80.0%

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

      11. *-rgt-identity 80.0%

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

   7. Simplified 80.0%

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

   8. Taylor expanded in b around 0 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + \frac{1}{e^{a}}}} \]
   9. Step-by-step derivation

      1. rec-exp 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{b} + e^{a}} \leq 2:\\ \;\;\;\;\frac{e^{a}}{e^{b} + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \end{array} \]

Alternative 2: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double b) {
	return pow((1.0 + (exp(b) / exp(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) / exp(a))) ** (-1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.0%

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

   1. clear-num 98.0%

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

   2. inv-pow 98.0%

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

3. Applied egg-rr 98.0%

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

   1. expm1-log1p-u 97.5%

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

   2. expm1-udef 97.5%

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

5. Applied egg-rr 97.5%

   \[\leadsto {\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{e^{a} + e^{b}}{e^{a}}\right)} - 1\right)}}^{-1} \]
6. Step-by-step derivation

   1. expm1-def 97.5%

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

   2. expm1-log1p 98.0%

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

   3. *-lft-identity 98.0%

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

   4. associate-*l/ 98.0%

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

   5. rec-exp 98.0%

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

   6. distribute-rgt-in 70.7%

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

   7. rec-exp 70.7%

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

   8. rgt-mult-inverse 99.6%

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

   9. rec-exp 99.6%

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

   10. associate-*r/ 99.6%

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

   11. *-rgt-identity 99.6%

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

7. Simplified 99.6%

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

8. Final simplification 99.6%

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

Alternative 3: 98.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.999999:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.999999)
   (/ 1.0 (+ 1.0 (exp (- a))))
   (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.999999) {
		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 (exp(a) <= 0.999999d0) 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 (Math.exp(a) <= 0.999999) {
		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 math.exp(a) <= 0.999999:
		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 (exp(a) <= 0.999999)
		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 (exp(a) <= 0.999999)
		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[N[Exp[a], $MachinePrecision], 0.999999], 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 (exp.f64 a) < 0.999998999999999971

   1. Initial program 98.6%

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

      1. clear-num 98.6%

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

      2. inv-pow 98.6%

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

   3. Applied egg-rr 98.6%

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

      1. expm1-log1p-u 98.6%

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

      2. expm1-udef 98.6%

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

   5. Applied egg-rr 98.6%

      \[\leadsto {\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{e^{a} + e^{b}}{e^{a}}\right)} - 1\right)}}^{-1} \]
   6. Step-by-step derivation

      1. expm1-def 98.6%

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

      2. expm1-log1p 98.6%

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

      3. *-lft-identity 98.6%

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

      4. associate-*l/ 98.6%

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

      5. rec-exp 98.6%

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

      6. distribute-rgt-in 4.1%

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

      7. rec-exp 4.1%

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

      8. rgt-mult-inverse 98.6%

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

      9. rec-exp 98.6%

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

      10. associate-*r/ 98.6%

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

      11. *-rgt-identity 98.6%

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

   7. Simplified 98.6%

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

   8. Taylor expanded in b around 0 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + \frac{1}{e^{a}}}} \]
   9. Step-by-step derivation

      1. rec-exp 100.0%

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

   10. Simplified 100.0%

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

   if 0.999998999999999971 < (exp.f64 a)

   1. Initial program 97.8%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]

   2. Taylor expanded in a around 0 98.4%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.999999:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]

Alternative 4: 98.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -395000:\\ \;\;\;\;\frac{e^{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -395000.0) (/ (exp a) a) (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -395000.0) {
		tmp = exp(a) / 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 <= (-395000.0d0)) then
        tmp = exp(a) / 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 <= -395000.0) {
		tmp = Math.exp(a) / a;
	} else {
		tmp = 1.0 / (1.0 + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -395000.0:
		tmp = math.exp(a) / a
	else:
		tmp = 1.0 / (1.0 + math.exp(b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -395000.0)
		tmp = Float64(exp(a) / 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 <= -395000.0)
		tmp = exp(a) / a;
	else
		tmp = 1.0 / (1.0 + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -395000.0], N[(N[Exp[a], $MachinePrecision] / a), $MachinePrecision], N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -395000

   1. Initial program 98.6%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]

   2. Taylor expanded in b around 0 100.0%

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

   3. Taylor expanded in a around 0 100.0%

      \[\leadsto \frac{e^{a}}{\color{blue}{2 + a}} \]
   4. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \frac{e^{a}}{\color{blue}{a + 2}} \]

   5. Simplified 100.0%

      \[\leadsto \frac{e^{a}}{\color{blue}{a + 2}} \]

   6. Taylor expanded in a around inf 100.0%

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

   if -395000 < a

   1. Initial program 97.8%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]

   2. Taylor expanded in a around 0 97.2%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -395000:\\ \;\;\;\;\frac{e^{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]

Alternative 5: 98.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.55 \cdot 10^{-6}:\\ \;\;\;\;\frac{e^{a}}{a + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.55e-6) (/ (exp a) (+ a 2.0)) (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.55e-6) {
		tmp = exp(a) / (a + 2.0);
	} 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 <= (-1.55d-6)) then
        tmp = exp(a) / (a + 2.0d0)
    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 <= -1.55e-6) {
		tmp = Math.exp(a) / (a + 2.0);
	} else {
		tmp = 1.0 / (1.0 + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.55e-6:
		tmp = math.exp(a) / (a + 2.0)
	else:
		tmp = 1.0 / (1.0 + math.exp(b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.55e-6)
		tmp = Float64(exp(a) / Float64(a + 2.0));
	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 <= -1.55e-6)
		tmp = exp(a) / (a + 2.0);
	else
		tmp = 1.0 / (1.0 + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.55e-6], N[(N[Exp[a], $MachinePrecision] / N[(a + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.55e-6

   1. Initial program 98.6%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]

   2. Taylor expanded in b around 0 100.0%

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

   3. Taylor expanded in a around 0 97.6%

      \[\leadsto \frac{e^{a}}{\color{blue}{2 + a}} \]
   4. Step-by-step derivation

      1. +-commutative 97.6%

         \[\leadsto \frac{e^{a}}{\color{blue}{a + 2}} \]

   5. Simplified 97.6%

      \[\leadsto \frac{e^{a}}{\color{blue}{a + 2}} \]

   if -1.55e-6 < a

   1. Initial program 97.8%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]

   2. Taylor expanded in a around 0 98.4%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.55 \cdot 10^{-6}:\\ \;\;\;\;\frac{e^{a}}{a + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]

Alternative 6: 71.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -390000:\\ \;\;\;\;\frac{e^{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(b + 0.5 \cdot \left(b \cdot b\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -390000.0) (/ (exp a) a) (/ 1.0 (+ 2.0 (+ b (* 0.5 (* b b)))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -390000.0) {
		tmp = exp(a) / a;
	} else {
		tmp = 1.0 / (2.0 + (b + (0.5 * (b * 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 <= (-390000.0d0)) then
        tmp = exp(a) / a
    else
        tmp = 1.0d0 / (2.0d0 + (b + (0.5d0 * (b * b))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -390000.0) {
		tmp = Math.exp(a) / a;
	} else {
		tmp = 1.0 / (2.0 + (b + (0.5 * (b * b))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -390000.0:
		tmp = math.exp(a) / a
	else:
		tmp = 1.0 / (2.0 + (b + (0.5 * (b * b))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -390000.0)
		tmp = Float64(exp(a) / a);
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b + Float64(0.5 * Float64(b * b)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -390000.0)
		tmp = exp(a) / a;
	else
		tmp = 1.0 / (2.0 + (b + (0.5 * (b * b))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -390000.0], N[(N[Exp[a], $MachinePrecision] / a), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b + N[(0.5 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.9e5

   1. Initial program 98.6%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]

   2. Taylor expanded in b around 0 100.0%

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

   3. Taylor expanded in a around 0 100.0%

      \[\leadsto \frac{e^{a}}{\color{blue}{2 + a}} \]
   4. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \frac{e^{a}}{\color{blue}{a + 2}} \]

   5. Simplified 100.0%

      \[\leadsto \frac{e^{a}}{\color{blue}{a + 2}} \]

   6. Taylor expanded in a around inf 100.0%

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

   if -3.9e5 < a

   1. Initial program 97.8%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]

   2. Taylor expanded in a around 0 97.2%

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

   3. Taylor expanded in b around 0 61.5%

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

      1. unpow2 61.5%

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

   5. Simplified 61.5%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -390000:\\ \;\;\;\;\frac{e^{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(b + 0.5 \cdot \left(b \cdot b\right)\right)}\\ \end{array} \]

Alternative 7: 57.6% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(-b\right)\\ \mathbf{if}\;b \leq 5.5 \cdot 10^{+63}:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{t_0 - b \cdot -2}{b \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* b (- b))))
   (if (<= b 5.5e+63)
     (/ 1.0 (- 2.0 a))
     (if (<= b 1.35e+154) (/ (- t_0 (* b -2.0)) (* b t_0)) (/ -2.0 (* b b))))))

C

double code(double a, double b) {
	double t_0 = b * -b;
	double tmp;
	if (b <= 5.5e+63) {
		tmp = 1.0 / (2.0 - a);
	} else if (b <= 1.35e+154) {
		tmp = (t_0 - (b * -2.0)) / (b * t_0);
	} else {
		tmp = -2.0 / (b * b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = b * -b
    if (b <= 5.5d+63) then
        tmp = 1.0d0 / (2.0d0 - a)
    else if (b <= 1.35d+154) then
        tmp = (t_0 - (b * (-2.0d0))) / (b * t_0)
    else
        tmp = (-2.0d0) / (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = b * -b;
	double tmp;
	if (b <= 5.5e+63) {
		tmp = 1.0 / (2.0 - a);
	} else if (b <= 1.35e+154) {
		tmp = (t_0 - (b * -2.0)) / (b * t_0);
	} else {
		tmp = -2.0 / (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = b * -b
	tmp = 0
	if b <= 5.5e+63:
		tmp = 1.0 / (2.0 - a)
	elif b <= 1.35e+154:
		tmp = (t_0 - (b * -2.0)) / (b * t_0)
	else:
		tmp = -2.0 / (b * b)
	return tmp

Julia

function code(a, b)
	t_0 = Float64(b * Float64(-b))
	tmp = 0.0
	if (b <= 5.5e+63)
		tmp = Float64(1.0 / Float64(2.0 - a));
	elseif (b <= 1.35e+154)
		tmp = Float64(Float64(t_0 - Float64(b * -2.0)) / Float64(b * t_0));
	else
		tmp = Float64(-2.0 / Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = b * -b;
	tmp = 0.0;
	if (b <= 5.5e+63)
		tmp = 1.0 / (2.0 - a);
	elseif (b <= 1.35e+154)
		tmp = (t_0 - (b * -2.0)) / (b * t_0);
	else
		tmp = -2.0 / (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(b * (-b)), $MachinePrecision]}, If[LessEqual[b, 5.5e+63], N[(1.0 / N[(2.0 - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.35e+154], N[(N[(t$95$0 - N[(b * -2.0), $MachinePrecision]), $MachinePrecision] / N[(b * t$95$0), $MachinePrecision]), $MachinePrecision], N[(-2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < 5.50000000000000004e63

   1. Initial program 97.3%

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

      1. clear-num 97.3%

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

      2. inv-pow 97.3%

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

   3. Applied egg-rr 97.3%

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

      1. expm1-log1p-u 96.6%

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

      2. expm1-udef 96.7%

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

   5. Applied egg-rr 96.7%

      \[\leadsto {\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{e^{a} + e^{b}}{e^{a}}\right)} - 1\right)}}^{-1} \]
   6. Step-by-step derivation

      1. expm1-def 96.6%

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

      2. expm1-log1p 97.3%

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

      3. *-lft-identity 97.3%

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

      4. associate-*l/ 97.3%

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

      5. rec-exp 97.3%

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

      6. distribute-rgt-in 72.1%

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

      7. rec-exp 72.1%

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

      8. rgt-mult-inverse 99.5%

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

      9. rec-exp 99.5%

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

      10. associate-*r/ 99.5%

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

      11. *-rgt-identity 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in b around 0 77.5%

      \[\leadsto \color{blue}{\frac{1}{1 + \frac{1}{e^{a}}}} \]
   9. Step-by-step derivation

      1. rec-exp 77.5%

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

   10. Simplified 77.5%

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

   11. Taylor expanded in a around 0 49.7%

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

       1. mul-1-neg 49.7%

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

       2. unsub-neg 49.7%

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

   13. Simplified 49.7%

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

   if 5.50000000000000004e63 < b < 1.35000000000000003e154

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]

   2. Taylor expanded in a around 0 100.0%

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

   3. Taylor expanded in b around 0 4.1%

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

      1. +-commutative 4.1%

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

   5. Simplified 4.1%

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

   6. Taylor expanded in b around inf 4.1%

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

      1. associate-*r/ 4.1%

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

      2. metadata-eval 4.1%

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

      3. unpow2 4.1%

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

   8. Simplified 4.1%

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

      1. frac-2neg 4.1%

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

      2. frac-sub 48.5%

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

      3. *-un-lft-identity 48.5%

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

      4. distribute-rgt-neg-in 48.5%

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

      5. metadata-eval 48.5%

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

      6. distribute-rgt-neg-in 48.5%

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

   10. Applied egg-rr 48.5%

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

   if 1.35000000000000003e154 < b

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]

   2. Taylor expanded in a around 0 100.0%

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

   3. Taylor expanded in b around 0 6.8%

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

      1. +-commutative 6.8%

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

   5. Simplified 6.8%

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

   6. Taylor expanded in b around inf 6.8%

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

      1. associate-*r/ 6.8%

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

      2. metadata-eval 6.8%

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

      3. unpow2 6.8%

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

   8. Simplified 6.8%

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

   9. Taylor expanded in b around 0 100.0%

      \[\leadsto \color{blue}{\frac{-2}{{b}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 100.0%

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

   11. Simplified 100.0%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.5 \cdot 10^{+63}:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{b \cdot \left(-b\right) - b \cdot -2}{b \cdot \left(b \cdot \left(-b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{b \cdot b}\\ \end{array} \]

Alternative 8: 53.9% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 10^{-14}:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(b + 0.5 \cdot \left(b \cdot b\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1e-14) (/ 1.0 (- 2.0 a)) (/ 1.0 (+ 2.0 (+ b (* 0.5 (* b b)))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1e-14) {
		tmp = 1.0 / (2.0 - a);
	} else {
		tmp = 1.0 / (2.0 + (b + (0.5 * (b * b))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 1d-14) then
        tmp = 1.0d0 / (2.0d0 - a)
    else
        tmp = 1.0d0 / (2.0d0 + (b + (0.5d0 * (b * b))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1e-14) {
		tmp = 1.0 / (2.0 - a);
	} else {
		tmp = 1.0 / (2.0 + (b + (0.5 * (b * b))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1e-14:
		tmp = 1.0 / (2.0 - a)
	else:
		tmp = 1.0 / (2.0 + (b + (0.5 * (b * b))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1e-14)
		tmp = Float64(1.0 / Float64(2.0 - a));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b + Float64(0.5 * Float64(b * b)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1e-14)
		tmp = 1.0 / (2.0 - a);
	else
		tmp = 1.0 / (2.0 + (b + (0.5 * (b * b))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1e-14], N[(1.0 / N[(2.0 - a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b + N[(0.5 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9.99999999999999999e-15

   1. Initial program 97.0%

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

      1. clear-num 97.1%

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

      2. inv-pow 97.1%

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

   3. Applied egg-rr 97.1%

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

      1. expm1-log1p-u 96.3%

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

      2. expm1-udef 96.3%

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

   5. Applied egg-rr 96.3%

      \[\leadsto {\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{e^{a} + e^{b}}{e^{a}}\right)} - 1\right)}}^{-1} \]
   6. Step-by-step derivation

      1. expm1-def 96.3%

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

      2. expm1-log1p 97.1%

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

      3. *-lft-identity 97.1%

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

      4. associate-*l/ 97.1%

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

      5. rec-exp 97.1%

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

      6. distribute-rgt-in 71.9%

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

      7. rec-exp 71.9%

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

      8. rgt-mult-inverse 99.4%

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

      9. rec-exp 99.4%

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

      10. associate-*r/ 99.4%

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

      11. *-rgt-identity 99.4%

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

   7. Simplified 99.4%

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

   8. Taylor expanded in b around 0 81.9%

      \[\leadsto \color{blue}{\frac{1}{1 + \frac{1}{e^{a}}}} \]
   9. Step-by-step derivation

      1. rec-exp 81.9%

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

   10. Simplified 81.9%

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

   11. Taylor expanded in a around 0 53.7%

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

       1. mul-1-neg 53.7%

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

       2. unsub-neg 53.7%

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

   13. Simplified 53.7%

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

   if 9.99999999999999999e-15 < b

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]

   2. Taylor expanded in a around 0 97.7%

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

   3. Taylor expanded in b around 0 51.2%

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

      1. unpow2 51.2%

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

   5. Simplified 51.2%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 10^{-14}:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(b + 0.5 \cdot \left(b \cdot b\right)\right)}\\ \end{array} \]

Alternative 9: 53.3% accurate, 43.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 620:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 620.0) (+ 0.5 (* a 0.25)) (/ -2.0 (* b b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 620.0) {
		tmp = 0.5 + (a * 0.25);
	} else {
		tmp = -2.0 / (b * b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 620.0d0) then
        tmp = 0.5d0 + (a * 0.25d0)
    else
        tmp = (-2.0d0) / (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 620.0) {
		tmp = 0.5 + (a * 0.25);
	} else {
		tmp = -2.0 / (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 620.0:
		tmp = 0.5 + (a * 0.25)
	else:
		tmp = -2.0 / (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 620.0)
		tmp = Float64(0.5 + Float64(a * 0.25));
	else
		tmp = Float64(-2.0 / Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 620.0)
		tmp = 0.5 + (a * 0.25);
	else
		tmp = -2.0 / (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 620.0], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision], N[(-2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 620

   1. Initial program 97.1%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]

   2. Taylor expanded in b around 0 79.0%

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

   3. Taylor expanded in a around 0 52.7%

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

      1. *-commutative 52.7%

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

   5. Simplified 52.7%

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

   if 620 < b

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]

   2. Taylor expanded in a around 0 100.0%

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

   3. Taylor expanded in b around 0 5.3%

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

      1. +-commutative 5.3%

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

   5. Simplified 5.3%

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

   6. Taylor expanded in b around inf 5.3%

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

      1. associate-*r/ 5.3%

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

      2. metadata-eval 5.3%

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

      3. unpow2 5.3%

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

   8. Simplified 5.3%

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

   9. Taylor expanded in b around 0 50.9%

      \[\leadsto \color{blue}{\frac{-2}{{b}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 50.9%

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

   11. Simplified 50.9%

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

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 620:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{b \cdot b}\\ \end{array} \]

Alternative 10: 53.7% accurate, 43.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.8 \cdot 10^{+62}:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 4.8e+62) (/ 1.0 (- 2.0 a)) (/ -2.0 (* b b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 4.8e+62) {
		tmp = 1.0 / (2.0 - a);
	} else {
		tmp = -2.0 / (b * b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 4.8d+62) then
        tmp = 1.0d0 / (2.0d0 - a)
    else
        tmp = (-2.0d0) / (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 4.8e+62) {
		tmp = 1.0 / (2.0 - a);
	} else {
		tmp = -2.0 / (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 4.8e+62:
		tmp = 1.0 / (2.0 - a)
	else:
		tmp = -2.0 / (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 4.8e+62)
		tmp = Float64(1.0 / Float64(2.0 - a));
	else
		tmp = Float64(-2.0 / Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 4.8e+62)
		tmp = 1.0 / (2.0 - a);
	else
		tmp = -2.0 / (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 4.8e+62], N[(1.0 / N[(2.0 - a), $MachinePrecision]), $MachinePrecision], N[(-2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.8e62

   1. Initial program 97.3%

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

      1. clear-num 97.3%

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

      2. inv-pow 97.3%

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

   3. Applied egg-rr 97.3%

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

      1. expm1-log1p-u 96.6%

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

      2. expm1-udef 96.7%

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

   5. Applied egg-rr 96.7%

      \[\leadsto {\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{e^{a} + e^{b}}{e^{a}}\right)} - 1\right)}}^{-1} \]
   6. Step-by-step derivation

      1. expm1-def 96.6%

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

      2. expm1-log1p 97.3%

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

      3. *-lft-identity 97.3%

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

      4. associate-*l/ 97.3%

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

      5. rec-exp 97.3%

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

      6. distribute-rgt-in 72.1%

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

      7. rec-exp 72.1%

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

      8. rgt-mult-inverse 99.5%

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

      9. rec-exp 99.5%

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

      10. associate-*r/ 99.5%

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

      11. *-rgt-identity 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in b around 0 77.5%

      \[\leadsto \color{blue}{\frac{1}{1 + \frac{1}{e^{a}}}} \]
   9. Step-by-step derivation

      1. rec-exp 77.5%

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

   10. Simplified 77.5%

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

   11. Taylor expanded in a around 0 49.7%

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

       1. mul-1-neg 49.7%

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

       2. unsub-neg 49.7%

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

   13. Simplified 49.7%

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

   if 4.8e62 < b

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]

   2. Taylor expanded in a around 0 100.0%

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

   3. Taylor expanded in b around 0 5.7%

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

      1. +-commutative 5.7%

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

   5. Simplified 5.7%

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

   6. Taylor expanded in b around inf 5.7%

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

      1. associate-*r/ 5.7%

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

      2. metadata-eval 5.7%

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

      3. unpow2 5.7%

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

   8. Simplified 5.7%

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

   9. Taylor expanded in b around 0 60.2%

      \[\leadsto \color{blue}{\frac{-2}{{b}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 60.2%

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

   11. Simplified 60.2%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.8 \cdot 10^{+62}:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{b \cdot b}\\ \end{array} \]

Alternative 11: 39.8% 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 98.0%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]

2. Taylor expanded in b around 0 65.1%

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

3. Taylor expanded in a around 0 37.3%

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

   1. *-commutative 37.3%

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

5. Simplified 37.3%

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

6. Final simplification 37.3%

   \[\leadsto 0.5 + a \cdot 0.25 \]

Alternative 12: 39.6% 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 98.0%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]

2. Taylor expanded in a around 0 80.6%

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

3. Taylor expanded in b around 0 37.1%

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

4. Final simplification 37.1%

   \[\leadsto 0.5 \]

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 2023187 
(FPCore (a b)
  :name "Quotient of sum of exps"
  :precision binary64

  :herbie-target
  (/ 1.0 (+ 1.0 (exp (- b a))))

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