Quotient of sum of exps

Percentage Accurate: 99.2% → 100.0%

Time: 9.3s

Alternatives: 14

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: 99.2% 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 98.8%

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

   1. *-lft-identity 98.8%

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

   2. associate-/l* 98.8%

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

   3. remove-double-div 98.8%

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

   4. exp-neg 98.8%

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

   5. associate-/r/ 98.8%

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

   6. /-rgt-identity 98.8%

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

   7. *-commutative 98.8%

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

   8. distribute-rgt-in 70.7%

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

   9. exp-neg 70.7%

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

   10. rgt-mult-inverse 99.2%

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

   11. prod-exp 100.0%

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

   12. unsub-neg 100.0%

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

3. Simplified 100.0%

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

   1. inv-pow 100.0%

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

   2. pow-to-exp 100.0%

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

   3. *-commutative 100.0%

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

   4. log1p-def 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 98.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -9e-6) (/ 1.0 (+ 1.0 (exp (- a)))) (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -9e-6) {
		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 <= (-9d-6)) 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 <= -9e-6) {
		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 <= -9e-6:
		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 <= -9e-6)
		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 <= -9e-6)
		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, -9e-6], 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 < -9.00000000000000023e-6

   1. Initial program 98.7%

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

      1. *-lft-identity 98.7%

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

      2. associate-/l* 98.7%

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

      3. remove-double-div 98.6%

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

      4. exp-neg 98.6%

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

      5. associate-/r/ 98.7%

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

      6. /-rgt-identity 98.7%

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

      7. *-commutative 98.7%

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

      8. distribute-rgt-in 6.4%

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

      9. exp-neg 6.4%

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

      10. rgt-mult-inverse 98.7%

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

      11. prod-exp 99.9%

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

      12. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in b around 0 98.1%

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

   if -9.00000000000000023e-6 < a

   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%

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

      3. remove-double-div 98.9%

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

      4. exp-neg 98.9%

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

      5. associate-/r/ 98.9%

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

      6. /-rgt-identity 98.9%

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

      7. *-commutative 98.9%

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

      8. distribute-rgt-in 98.9%

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

      9. exp-neg 98.9%

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

      10. rgt-mult-inverse 99.4%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 99.2%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]

Alternative 3: 98.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2400000:\\ \;\;\;\;\frac{e^{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2400000.0) (/ (exp a) a) (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2400000.0) {
		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 <= (-2400000.0d0)) 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 <= -2400000.0) {
		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 <= -2400000.0:
		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 <= -2400000.0)
		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 <= -2400000.0)
		tmp = exp(a) / a;
	else
		tmp = 1.0 / (1.0 + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2400000.0], 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 < -2.4e6

   1. Initial program 98.6%

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

   2. Taylor expanded in b around 0 98.7%

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

   3. Taylor expanded in a around 0 98.7%

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

      1. +-commutative 98.7%

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

   5. Simplified 98.7%

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

   6. Taylor expanded in a around inf 98.7%

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

   if -2.4e6 < a

   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%

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

      3. remove-double-div 98.9%

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

      4. exp-neg 98.9%

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

      5. associate-/r/ 98.9%

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

      6. /-rgt-identity 98.9%

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

      7. *-commutative 98.9%

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

      8. distribute-rgt-in 98.3%

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

      9. exp-neg 98.3%

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

      10. rgt-mult-inverse 99.4%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 97.4%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2400000:\\ \;\;\;\;\frac{e^{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]

Alternative 4: 98.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.000125:\\ \;\;\;\;\frac{e^{a}}{a + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -0.000125) (/ (exp a) (+ a 2.0)) (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -0.000125) {
		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 <= (-0.000125d0)) 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 <= -0.000125) {
		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 <= -0.000125:
		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 <= -0.000125)
		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 <= -0.000125)
		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, -0.000125], 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 < -1.25e-4

   1. Initial program 98.7%

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

   2. Taylor expanded in b around 0 98.1%

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

   3. Taylor expanded in a around 0 95.1%

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

      1. +-commutative 95.1%

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

   5. Simplified 95.1%

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

   if -1.25e-4 < a

   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%

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

      3. remove-double-div 98.9%

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

      4. exp-neg 98.9%

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

      5. associate-/r/ 98.9%

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

      6. /-rgt-identity 98.9%

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

      7. *-commutative 98.9%

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

      8. distribute-rgt-in 98.9%

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

      9. exp-neg 98.9%

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

      10. rgt-mult-inverse 99.4%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 99.2%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.000125:\\ \;\;\;\;\frac{e^{a}}{a + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]

Alternative 5: 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 98.8%

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

   1. *-lft-identity 98.8%

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

   2. associate-/l* 98.8%

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

   3. remove-double-div 98.8%

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

   4. exp-neg 98.8%

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

   5. associate-/r/ 98.8%

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

   6. /-rgt-identity 98.8%

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

   7. *-commutative 98.8%

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

   8. distribute-rgt-in 70.7%

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

   9. exp-neg 70.7%

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

   10. rgt-mult-inverse 99.2%

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

   11. prod-exp 100.0%

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

   12. unsub-neg 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 6: 71.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -700:\\ \;\;\;\;\frac{e^{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{4 - b \cdot b} \cdot \left(2 - b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -700.0) (/ (exp a) a) (* (/ 1.0 (- 4.0 (* b b))) (- 2.0 b))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -700.0) {
		tmp = exp(a) / a;
	} else {
		tmp = (1.0 / (4.0 - (b * b))) * (2.0 - 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 <= (-700.0d0)) then
        tmp = exp(a) / a
    else
        tmp = (1.0d0 / (4.0d0 - (b * b))) * (2.0d0 - b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -700.0) {
		tmp = Math.exp(a) / a;
	} else {
		tmp = (1.0 / (4.0 - (b * b))) * (2.0 - b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -700.0:
		tmp = math.exp(a) / a
	else:
		tmp = (1.0 / (4.0 - (b * b))) * (2.0 - b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -700.0)
		tmp = Float64(exp(a) / a);
	else
		tmp = Float64(Float64(1.0 / Float64(4.0 - Float64(b * b))) * Float64(2.0 - b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -700.0)
		tmp = exp(a) / a;
	else
		tmp = (1.0 / (4.0 - (b * b))) * (2.0 - b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -700.0], N[(N[Exp[a], $MachinePrecision] / a), $MachinePrecision], N[(N[(1.0 / N[(4.0 - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 - b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -700

   1. Initial program 98.6%

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

   2. Taylor expanded in b around 0 98.7%

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

   3. Taylor expanded in a around 0 98.7%

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

      1. +-commutative 98.7%

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

   5. Simplified 98.7%

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

   6. Taylor expanded in a around inf 98.7%

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

   if -700 < a

   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%

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

      3. remove-double-div 98.9%

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

      4. exp-neg 98.9%

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

      5. associate-/r/ 98.9%

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

      6. /-rgt-identity 98.9%

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

      7. *-commutative 98.9%

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

      8. distribute-rgt-in 98.9%

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

      9. exp-neg 98.9%

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

      10. rgt-mult-inverse 99.4%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 97.4%

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

   5. Taylor expanded in b around 0 49.9%

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

      1. +-commutative 49.9%

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

   7. Simplified 49.9%

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

      1. +-commutative 49.9%

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

      2. flip-+ 58.0%

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

      3. associate-/r/ 58.0%

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

      4. metadata-eval 58.0%

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

   9. Applied egg-rr 58.0%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -700:\\ \;\;\;\;\frac{e^{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{4 - b \cdot b} \cdot \left(2 - b\right)\\ \end{array} \]

Alternative 7: 67.5% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 - b \cdot b\\ \mathbf{if}\;b \leq 1.7 \cdot 10^{+91}:\\ \;\;\;\;\frac{1}{\left(2 - a\right) + 0.5 \cdot \left(a \cdot a\right)}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{t_0}{t_0 \cdot \left(b + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(2 - b\right) \cdot \frac{-1}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (- 4.0 (* b b))))
   (if (<= b 1.7e+91)
     (/ 1.0 (+ (- 2.0 a) (* 0.5 (* a a))))
     (if (<= b 1.35e+154)
       (/ t_0 (* t_0 (+ b 2.0)))
       (* (- 2.0 b) (/ -1.0 (* b b)))))))

C

double code(double a, double b) {
	double t_0 = 4.0 - (b * b);
	double tmp;
	if (b <= 1.7e+91) {
		tmp = 1.0 / ((2.0 - a) + (0.5 * (a * a)));
	} else if (b <= 1.35e+154) {
		tmp = t_0 / (t_0 * (b + 2.0));
	} else {
		tmp = (2.0 - b) * (-1.0 / (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 = 4.0d0 - (b * b)
    if (b <= 1.7d+91) then
        tmp = 1.0d0 / ((2.0d0 - a) + (0.5d0 * (a * a)))
    else if (b <= 1.35d+154) then
        tmp = t_0 / (t_0 * (b + 2.0d0))
    else
        tmp = (2.0d0 - b) * ((-1.0d0) / (b * b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = 4.0 - (b * b);
	double tmp;
	if (b <= 1.7e+91) {
		tmp = 1.0 / ((2.0 - a) + (0.5 * (a * a)));
	} else if (b <= 1.35e+154) {
		tmp = t_0 / (t_0 * (b + 2.0));
	} else {
		tmp = (2.0 - b) * (-1.0 / (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = 4.0 - (b * b)
	tmp = 0
	if b <= 1.7e+91:
		tmp = 1.0 / ((2.0 - a) + (0.5 * (a * a)))
	elif b <= 1.35e+154:
		tmp = t_0 / (t_0 * (b + 2.0))
	else:
		tmp = (2.0 - b) * (-1.0 / (b * b))
	return tmp

Julia

function code(a, b)
	t_0 = Float64(4.0 - Float64(b * b))
	tmp = 0.0
	if (b <= 1.7e+91)
		tmp = Float64(1.0 / Float64(Float64(2.0 - a) + Float64(0.5 * Float64(a * a))));
	elseif (b <= 1.35e+154)
		tmp = Float64(t_0 / Float64(t_0 * Float64(b + 2.0)));
	else
		tmp = Float64(Float64(2.0 - b) * Float64(-1.0 / Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = 4.0 - (b * b);
	tmp = 0.0;
	if (b <= 1.7e+91)
		tmp = 1.0 / ((2.0 - a) + (0.5 * (a * a)));
	elseif (b <= 1.35e+154)
		tmp = t_0 / (t_0 * (b + 2.0));
	else
		tmp = (2.0 - b) * (-1.0 / (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(4.0 - N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.7e+91], N[(1.0 / N[(N[(2.0 - a), $MachinePrecision] + N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.35e+154], N[(t$95$0 / N[(t$95$0 * N[(b + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 - b), $MachinePrecision] * N[(-1.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.7e91

   1. Initial program 98.6%

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

      1. *-lft-identity 98.6%

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

      2. associate-/l* 98.6%

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

      3. remove-double-div 98.6%

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

      4. exp-neg 98.6%

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

      5. associate-/r/ 98.6%

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

      6. /-rgt-identity 98.6%

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

      7. *-commutative 98.6%

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

      8. distribute-rgt-in 70.7%

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

      9. exp-neg 70.7%

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

      10. rgt-mult-inverse 99.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around 0 73.9%

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

   5. Taylor expanded in a around 0 58.6%

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

      1. associate-+r+ 58.6%

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

      2. mul-1-neg 58.6%

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

      3. unsub-neg 58.6%

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

      4. unpow2 58.6%

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

   7. Simplified 58.6%

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

   if 1.7e91 < b < 1.35000000000000003e154

   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%

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

      3. remove-double-div 100.0%

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

      4. exp-neg 100.0%

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

      5. associate-/r/ 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. *-commutative 100.0%

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

      8. distribute-rgt-in 92.9%

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

      9. exp-neg 92.9%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 100.0%

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

   5. Taylor expanded in b around 0 4.2%

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

      1. +-commutative 4.2%

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

   7. Simplified 4.2%

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

      1. +-commutative 4.2%

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

      2. flip-+ 4.2%

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

      3. associate-/r/ 4.2%

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

      4. metadata-eval 4.2%

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

   9. Applied egg-rr 4.2%

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

       1. flip-- 4.2%

          \[\leadsto \frac{1}{4 - b \cdot b} \cdot \color{blue}{\frac{2 \cdot 2 - b \cdot b}{2 + b}} \]

       2. metadata-eval 4.2%

          \[\leadsto \frac{1}{4 - b \cdot b} \cdot \frac{\color{blue}{4} - b \cdot b}{2 + b} \]

       3. frac-times 79.4%

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

       4. *-lft-identity 79.4%

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

   11. Applied egg-rr 79.4%

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

   if 1.35000000000000003e154 < 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%

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

      3. remove-double-div 100.0%

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

      4. exp-neg 100.0%

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

      5. associate-/r/ 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. *-commutative 100.0%

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

      8. distribute-rgt-in 59.3%

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

      9. exp-neg 59.3%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 100.0%

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

   5. Taylor expanded in b around 0 7.2%

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

      1. +-commutative 7.2%

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

   7. Simplified 7.2%

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

      1. +-commutative 7.2%

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

      2. flip-+ 100.0%

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

      3. associate-/r/ 100.0%

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

      4. metadata-eval 100.0%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in b around inf 100.0%

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

       1. unpow2 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.7 \cdot 10^{+91}:\\ \;\;\;\;\frac{1}{\left(2 - a\right) + 0.5 \cdot \left(a \cdot a\right)}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{4 - b \cdot b}{\left(4 - b \cdot b\right) \cdot \left(b + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(2 - b\right) \cdot \frac{-1}{b \cdot b}\\ \end{array} \]

Alternative 8: 54.5% accurate, 23.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.5 \cdot 10^{+139}:\\ \;\;\;\;\frac{2}{a \cdot a}\\ \mathbf{elif}\;a \leq -3 \cdot 10^{+16}:\\ \;\;\;\;\left(2 - b\right) \cdot \frac{-1}{b \cdot b}\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.5e+139)
   (/ 2.0 (* a a))
   (if (<= a -3e+16) (* (- 2.0 b) (/ -1.0 (* b b))) (+ 0.5 (* a 0.25)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.5e+139) {
		tmp = 2.0 / (a * a);
	} else if (a <= -3e+16) {
		tmp = (2.0 - b) * (-1.0 / (b * b));
	} else {
		tmp = 0.5 + (a * 0.25);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-1.5d+139)) then
        tmp = 2.0d0 / (a * a)
    else if (a <= (-3d+16)) then
        tmp = (2.0d0 - b) * ((-1.0d0) / (b * b))
    else
        tmp = 0.5d0 + (a * 0.25d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1.5e+139) {
		tmp = 2.0 / (a * a);
	} else if (a <= -3e+16) {
		tmp = (2.0 - b) * (-1.0 / (b * b));
	} else {
		tmp = 0.5 + (a * 0.25);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.5e+139:
		tmp = 2.0 / (a * a)
	elif a <= -3e+16:
		tmp = (2.0 - b) * (-1.0 / (b * b))
	else:
		tmp = 0.5 + (a * 0.25)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.5e+139)
		tmp = Float64(2.0 / Float64(a * a));
	elseif (a <= -3e+16)
		tmp = Float64(Float64(2.0 - b) * Float64(-1.0 / Float64(b * b)));
	else
		tmp = Float64(0.5 + Float64(a * 0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.5e+139)
		tmp = 2.0 / (a * a);
	elseif (a <= -3e+16)
		tmp = (2.0 - b) * (-1.0 / (b * b));
	else
		tmp = 0.5 + (a * 0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.5e+139], N[(2.0 / N[(a * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -3e+16], N[(N[(2.0 - b), $MachinePrecision] * N[(-1.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.5e139

   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%

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

      3. remove-double-div 100.0%

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

      4. exp-neg 100.0%

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

      5. associate-/r/ 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. *-commutative 100.0%

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

      8. distribute-rgt-in 0.0%

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

      9. exp-neg 0.0%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around 0 100.0%

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

   5. Taylor expanded in a around 0 94.8%

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

      1. associate-+r+ 94.8%

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

      2. mul-1-neg 94.8%

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

      3. unsub-neg 94.8%

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

      4. unpow2 94.8%

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

   7. Simplified 94.8%

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

   8. Taylor expanded in a around inf 94.8%

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

      1. unpow2 94.8%

         \[\leadsto \frac{2}{\color{blue}{a \cdot a}} \]

   10. Simplified 94.8%

       \[\leadsto \color{blue}{\frac{2}{a \cdot a}} \]

   if -1.5e139 < a < -3e16

   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%

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

      3. remove-double-div 100.0%

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

      4. exp-neg 100.0%

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

      5. associate-/r/ 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. *-commutative 100.0%

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

      8. distribute-rgt-in 0.0%

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

      9. exp-neg 0.0%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 45.1%

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

   5. Taylor expanded in b around 0 4.1%

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

      1. +-commutative 4.1%

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

   7. Simplified 4.1%

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

      1. +-commutative 4.1%

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

      2. flip-+ 25.8%

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

      3. associate-/r/ 25.8%

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

      4. metadata-eval 25.8%

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

   9. Applied egg-rr 25.8%

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

   10. Taylor expanded in b around inf 25.2%

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

       1. unpow2 25.2%

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

   12. Simplified 25.2%

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

   if -3e16 < a

   1. Initial program 98.4%

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

      1. *-lft-identity 98.4%

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

      2. associate-/l* 98.4%

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

      3. remove-double-div 98.4%

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

      4. exp-neg 98.4%

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

      5. associate-/r/ 98.4%

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

      6. /-rgt-identity 98.4%

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

      7. *-commutative 98.4%

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

      8. distribute-rgt-in 94.7%

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

      9. exp-neg 94.7%

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

      10. rgt-mult-inverse 98.9%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around 0 55.9%

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

   5. Taylor expanded in a around 0 51.0%

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

      1. *-commutative 51.0%

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

   7. Simplified 51.0%

      \[\leadsto \color{blue}{0.5 + a \cdot 0.25} \]
3. Recombined 3 regimes into one program.

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.5 \cdot 10^{+139}:\\ \;\;\;\;\frac{2}{a \cdot a}\\ \mathbf{elif}\;a \leq -3 \cdot 10^{+16}:\\ \;\;\;\;\left(2 - b\right) \cdot \frac{-1}{b \cdot b}\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \]

Alternative 9: 60.6% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.45 \cdot 10^{+139}:\\ \;\;\;\;\frac{2}{a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{4 - b \cdot b} \cdot \left(2 - b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.45e+139) (/ 2.0 (* a a)) (* (/ 1.0 (- 4.0 (* b b))) (- 2.0 b))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.45e+139) {
		tmp = 2.0 / (a * a);
	} else {
		tmp = (1.0 / (4.0 - (b * b))) * (2.0 - 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 <= (-1.45d+139)) then
        tmp = 2.0d0 / (a * a)
    else
        tmp = (1.0d0 / (4.0d0 - (b * b))) * (2.0d0 - b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1.45e+139) {
		tmp = 2.0 / (a * a);
	} else {
		tmp = (1.0 / (4.0 - (b * b))) * (2.0 - b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.45e+139:
		tmp = 2.0 / (a * a)
	else:
		tmp = (1.0 / (4.0 - (b * b))) * (2.0 - b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.45e+139)
		tmp = Float64(2.0 / Float64(a * a));
	else
		tmp = Float64(Float64(1.0 / Float64(4.0 - Float64(b * b))) * Float64(2.0 - b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.45e+139)
		tmp = 2.0 / (a * a);
	else
		tmp = (1.0 / (4.0 - (b * b))) * (2.0 - b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.45e+139], N[(2.0 / N[(a * a), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(4.0 - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 - b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.4499999999999999e139

   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%

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

      3. remove-double-div 100.0%

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

      4. exp-neg 100.0%

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

      5. associate-/r/ 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. *-commutative 100.0%

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

      8. distribute-rgt-in 0.0%

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

      9. exp-neg 0.0%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around 0 100.0%

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

   5. Taylor expanded in a around 0 94.8%

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

      1. associate-+r+ 94.8%

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

      2. mul-1-neg 94.8%

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

      3. unsub-neg 94.8%

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

      4. unpow2 94.8%

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

   7. Simplified 94.8%

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

   8. Taylor expanded in a around inf 94.8%

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

      1. unpow2 94.8%

         \[\leadsto \frac{2}{\color{blue}{a \cdot a}} \]

   10. Simplified 94.8%

       \[\leadsto \color{blue}{\frac{2}{a \cdot a}} \]

   if -1.4499999999999999e139 < a

   1. Initial program 98.6%

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

      1. *-lft-identity 98.6%

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

      2. associate-/l* 98.6%

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

      3. remove-double-div 98.6%

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

      4. exp-neg 98.6%

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

      5. associate-/r/ 98.6%

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

      6. /-rgt-identity 98.6%

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

      7. *-commutative 98.6%

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

      8. distribute-rgt-in 81.9%

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

      9. exp-neg 81.9%

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

      10. rgt-mult-inverse 99.1%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 87.8%

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

   5. Taylor expanded in b around 0 42.0%

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

      1. +-commutative 42.0%

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

   7. Simplified 42.0%

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

      1. +-commutative 42.0%

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

      2. flip-+ 51.7%

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

      3. associate-/r/ 51.7%

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

      4. metadata-eval 51.7%

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

   9. Applied egg-rr 51.7%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.45 \cdot 10^{+139}:\\ \;\;\;\;\frac{2}{a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{4 - b \cdot b} \cdot \left(2 - b\right)\\ \end{array} \]

Alternative 10: 64.0% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.85 \cdot 10^{+144}:\\ \;\;\;\;\frac{1}{\left(2 - a\right) + 0.5 \cdot \left(a \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(2 - b\right) \cdot \frac{-1}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.85e+144)
   (/ 1.0 (+ (- 2.0 a) (* 0.5 (* a a))))
   (* (- 2.0 b) (/ -1.0 (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.85e+144) {
		tmp = 1.0 / ((2.0 - a) + (0.5 * (a * a)));
	} else {
		tmp = (2.0 - b) * (-1.0 / (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 <= 1.85d+144) then
        tmp = 1.0d0 / ((2.0d0 - a) + (0.5d0 * (a * a)))
    else
        tmp = (2.0d0 - b) * ((-1.0d0) / (b * b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.85e+144) {
		tmp = 1.0 / ((2.0 - a) + (0.5 * (a * a)));
	} else {
		tmp = (2.0 - b) * (-1.0 / (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.85e+144:
		tmp = 1.0 / ((2.0 - a) + (0.5 * (a * a)))
	else:
		tmp = (2.0 - b) * (-1.0 / (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.85e+144)
		tmp = Float64(1.0 / Float64(Float64(2.0 - a) + Float64(0.5 * Float64(a * a))));
	else
		tmp = Float64(Float64(2.0 - b) * Float64(-1.0 / Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.85e+144)
		tmp = 1.0 / ((2.0 - a) + (0.5 * (a * a)));
	else
		tmp = (2.0 - b) * (-1.0 / (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.85e+144], N[(1.0 / N[(N[(2.0 - a), $MachinePrecision] + N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 - b), $MachinePrecision] * N[(-1.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.8499999999999998e144

   1. Initial program 98.7%

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

      1. *-lft-identity 98.7%

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

      2. associate-/l* 98.7%

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

      3. remove-double-div 98.6%

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

      4. exp-neg 98.6%

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

      5. associate-/r/ 98.7%

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

      6. /-rgt-identity 98.7%

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

      7. *-commutative 98.7%

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

      8. distribute-rgt-in 71.8%

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

      9. exp-neg 71.8%

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

      10. rgt-mult-inverse 99.1%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around 0 70.6%

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

   5. Taylor expanded in a around 0 56.1%

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

      1. associate-+r+ 56.1%

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

      2. mul-1-neg 56.1%

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

      3. unsub-neg 56.1%

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

      4. unpow2 56.1%

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

   7. Simplified 56.1%

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

   if 1.8499999999999998e144 < 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%

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

      3. remove-double-div 100.0%

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

      4. exp-neg 100.0%

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

      5. associate-/r/ 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. *-commutative 100.0%

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

      8. distribute-rgt-in 62.1%

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

      9. exp-neg 62.1%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 100.0%

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

   5. Taylor expanded in b around 0 7.1%

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

      1. +-commutative 7.1%

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

   7. Simplified 7.1%

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

      1. +-commutative 7.1%

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

      2. flip-+ 93.4%

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

      3. associate-/r/ 93.4%

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

      4. metadata-eval 93.4%

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

   9. Applied egg-rr 93.4%

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

   10. Taylor expanded in b around inf 93.4%

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

       1. unpow2 93.4%

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

   12. Simplified 93.4%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.85 \cdot 10^{+144}:\\ \;\;\;\;\frac{1}{\left(2 - a\right) + 0.5 \cdot \left(a \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(2 - b\right) \cdot \frac{-1}{b \cdot b}\\ \end{array} \]

Alternative 11: 53.1% accurate, 43.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.7:\\ \;\;\;\;\frac{2}{a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.7) (/ 2.0 (* a a)) (+ 0.5 (* a 0.25))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.7) {
		tmp = 2.0 / (a * a);
	} else {
		tmp = 0.5 + (a * 0.25);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-1.7d0)) then
        tmp = 2.0d0 / (a * a)
    else
        tmp = 0.5d0 + (a * 0.25d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1.7) {
		tmp = 2.0 / (a * a);
	} else {
		tmp = 0.5 + (a * 0.25);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.7:
		tmp = 2.0 / (a * a)
	else:
		tmp = 0.5 + (a * 0.25)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.7)
		tmp = Float64(2.0 / Float64(a * a));
	else
		tmp = Float64(0.5 + Float64(a * 0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.7)
		tmp = 2.0 / (a * a);
	else
		tmp = 0.5 + (a * 0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.7], N[(2.0 / N[(a * a), $MachinePrecision]), $MachinePrecision], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.69999999999999996

   1. Initial program 98.6%

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

      1. *-lft-identity 98.6%

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

      2. associate-/l* 98.6%

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

      3. remove-double-div 98.6%

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

      4. exp-neg 98.6%

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

      5. associate-/r/ 98.6%

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

      6. /-rgt-identity 98.6%

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

      7. *-commutative 98.6%

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

      8. distribute-rgt-in 1.3%

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

      9. exp-neg 1.3%

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

      10. rgt-mult-inverse 98.6%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around 0 98.7%

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

   5. Taylor expanded in a around 0 47.7%

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

      1. associate-+r+ 47.7%

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

      2. mul-1-neg 47.7%

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

      3. unsub-neg 47.7%

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

      4. unpow2 47.7%

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

   7. Simplified 47.7%

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

   8. Taylor expanded in a around inf 47.6%

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

      1. unpow2 47.6%

         \[\leadsto \frac{2}{\color{blue}{a \cdot a}} \]

   10. Simplified 47.6%

       \[\leadsto \color{blue}{\frac{2}{a \cdot a}} \]

   if -1.69999999999999996 < a

   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%

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

      3. remove-double-div 98.9%

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

      4. exp-neg 98.9%

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

      5. associate-/r/ 98.9%

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

      6. /-rgt-identity 98.9%

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

      7. *-commutative 98.9%

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

      8. distribute-rgt-in 98.9%

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

      9. exp-neg 98.9%

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

      10. rgt-mult-inverse 99.4%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around 0 54.3%

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

   5. Taylor expanded in a around 0 53.4%

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

      1. *-commutative 53.4%

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

   7. Simplified 53.4%

      \[\leadsto \color{blue}{0.5 + a \cdot 0.25} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.7:\\ \;\;\;\;\frac{2}{a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \]

Alternative 12: 39.9% 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 98.8%

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

   1. *-lft-identity 98.8%

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

   2. associate-/l* 98.8%

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

   3. remove-double-div 98.8%

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

   4. exp-neg 98.8%

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

   5. associate-/r/ 98.8%

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

   6. /-rgt-identity 98.8%

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

   7. *-commutative 98.8%

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

   8. distribute-rgt-in 70.7%

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

   9. exp-neg 70.7%

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

   10. rgt-mult-inverse 99.2%

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

   11. prod-exp 100.0%

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

   12. unsub-neg 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in b around 0 67.1%

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

5. Taylor expanded in a around 0 38.6%

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

   1. *-commutative 38.6%

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

7. Simplified 38.6%

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

8. Final simplification 38.6%

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

Alternative 13: 40.6% 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 98.8%

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

   1. *-lft-identity 98.8%

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

   2. associate-/l* 98.8%

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

   3. remove-double-div 98.8%

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

   4. exp-neg 98.8%

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

   5. associate-/r/ 98.8%

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

   6. /-rgt-identity 98.8%

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

   7. *-commutative 98.8%

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

   8. distribute-rgt-in 70.7%

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

   9. exp-neg 70.7%

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

   10. rgt-mult-inverse 99.2%

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

   11. prod-exp 100.0%

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

   12. unsub-neg 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in b around 0 67.1%

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

5. Taylor expanded in a around 0 39.3%

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

   1. mul-1-neg 39.3%

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

   2. unsub-neg 39.3%

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

7. Simplified 39.3%

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

8. Final simplification 39.3%

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

Alternative 14: 39.7% 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 98.8%

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

   1. *-lft-identity 98.8%

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

   2. associate-/l* 98.8%

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

   3. remove-double-div 98.8%

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

   4. exp-neg 98.8%

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

   5. associate-/r/ 98.8%

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

   6. /-rgt-identity 98.8%

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

   7. *-commutative 98.8%

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

   8. distribute-rgt-in 70.7%

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

   9. exp-neg 70.7%

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

   10. rgt-mult-inverse 99.2%

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

   11. prod-exp 100.0%

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

   12. unsub-neg 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in a around 0 80.0%

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

5. Taylor expanded in b around 0 38.3%

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

6. Final simplification 38.3%

   \[\leadsto 0.5 \]

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 2023297 
(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))))