Quotient of sum of exps

Percentage Accurate: 98.9% → 100.0%

Time: 8.6s

Alternatives: 12

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: 98.9% 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 98.4%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
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 70.3%

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

   9. exp-neg 70.3%

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

   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. add-log-exp 99.8%

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

   2. *-un-lft-identity 99.8%

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

   3. log-prod 99.8%

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

   4. metadata-eval 99.8%

      \[\leadsto \color{blue}{0} + \log \left(e^{\frac{1}{1 + e^{b - a}}}\right) \]

   5. add-log-exp 100.0%

      \[\leadsto 0 + \color{blue}{\frac{1}{1 + e^{b - a}}} \]

   6. add-exp-log 100.0%

      \[\leadsto 0 + \color{blue}{e^{\log \left(\frac{1}{1 + e^{b - a}}\right)}} \]

   7. log-rec 100.0%

      \[\leadsto 0 + e^{\color{blue}{-\log \left(1 + e^{b - a}\right)}} \]

   8. log1p-udef 100.0%

      \[\leadsto 0 + e^{-\color{blue}{\mathsf{log1p}\left(e^{b - a}\right)}} \]

5. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{0 + e^{-\mathsf{log1p}\left(e^{b - a}\right)}} \]
6. Step-by-step derivation

   1. +-lft-identity 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

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

Alternative 2: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{b} \leq 0.9999 \lor \neg \left(e^{b} \leq 2\right):\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= (exp b) 0.9999) (not (<= (exp b) 2.0)))
   (/ 1.0 (+ 1.0 (exp b)))
   (/ 1.0 (+ 1.0 (exp (- a))))))

C

double code(double a, double b) {
	double tmp;
	if ((exp(b) <= 0.9999) || !(exp(b) <= 2.0)) {
		tmp = 1.0 / (1.0 + exp(b));
	} 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) :: tmp
    if ((exp(b) <= 0.9999d0) .or. (.not. (exp(b) <= 2.0d0))) then
        tmp = 1.0d0 / (1.0d0 + exp(b))
    else
        tmp = 1.0d0 / (1.0d0 + exp(-a))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((Math.exp(b) <= 0.9999) || !(Math.exp(b) <= 2.0)) {
		tmp = 1.0 / (1.0 + Math.exp(b));
	} else {
		tmp = 1.0 / (1.0 + Math.exp(-a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (math.exp(b) <= 0.9999) or not (math.exp(b) <= 2.0):
		tmp = 1.0 / (1.0 + math.exp(b))
	else:
		tmp = 1.0 / (1.0 + math.exp(-a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((exp(b) <= 0.9999) || !(exp(b) <= 2.0))
		tmp = Float64(1.0 / Float64(1.0 + exp(b)));
	else
		tmp = Float64(1.0 / Float64(1.0 + exp(Float64(-a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((exp(b) <= 0.9999) || ~((exp(b) <= 2.0)))
		tmp = 1.0 / (1.0 + exp(b));
	else
		tmp = 1.0 / (1.0 + exp(-a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[N[Exp[b], $MachinePrecision], 0.9999], N[Not[LessEqual[N[Exp[b], $MachinePrecision], 2.0]], $MachinePrecision]], N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 + N[Exp[(-a)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 b) < 0.99990000000000001 or 2 < (exp.f64 b)

   1. Initial program 98.5%

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

      1. *-lft-identity 98.5%

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

      2. associate-/l* 98.5%

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

      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 79.4%

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

      9. exp-neg 79.4%

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

      10. rgt-mult-inverse 98.5%

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

      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%

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

   if 0.99990000000000001 < (exp.f64 b) < 2

   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 60.8%

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

      9. exp-neg 60.8%

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

      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 99.3%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{b} \leq 0.9999 \lor \neg \left(e^{b} \leq 2\right):\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \end{array} \]

Alternative 3: 98.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3200000:\\ \;\;\;\;\frac{e^{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3200000.0) (/ (exp a) a) (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -3200000.0) {
		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 <= (-3200000.0d0)) 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 <= -3200000.0) {
		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 <= -3200000.0:
		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 <= -3200000.0)
		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 <= -3200000.0)
		tmp = exp(a) / a;
	else
		tmp = 1.0 / (1.0 + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -3200000.0], 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 < -3.2e6

   1. Initial program 100.0%

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

   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 -3.2e6 < a

   1. Initial program 97.8%

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

      1. *-lft-identity 97.8%

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

      2. associate-/l* 97.8%

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

      3. remove-double-div 97.8%

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

      4. exp-neg 97.8%

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

      5. associate-/r/ 97.8%

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

      6. /-rgt-identity 97.8%

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

      7. *-commutative 97.8%

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

      8. distribute-rgt-in 97.8%

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

      9. exp-neg 97.8%

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

      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 a around 0 97.3%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3200000:\\ \;\;\;\;\frac{e^{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]

Alternative 4: 100.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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 70.3%

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

   9. exp-neg 70.3%

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

   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 5: 71.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1150000:\\ \;\;\;\;\frac{e^{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(b + 2\right) + 0.5 \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1150000.0) (/ (exp a) a) (/ 1.0 (+ (+ b 2.0) (* 0.5 (* b b))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1150000.0) {
		tmp = exp(a) / a;
	} else {
		tmp = 1.0 / ((b + 2.0) + (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 <= (-1150000.0d0)) then
        tmp = exp(a) / a
    else
        tmp = 1.0d0 / ((b + 2.0d0) + (0.5d0 * (b * b)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1150000.0) {
		tmp = Math.exp(a) / a;
	} else {
		tmp = 1.0 / ((b + 2.0) + (0.5 * (b * b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1150000.0:
		tmp = math.exp(a) / a
	else:
		tmp = 1.0 / ((b + 2.0) + (0.5 * (b * b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1150000.0)
		tmp = Float64(exp(a) / a);
	else
		tmp = Float64(1.0 / Float64(Float64(b + 2.0) + Float64(0.5 * Float64(b * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1150000.0)
		tmp = exp(a) / a;
	else
		tmp = 1.0 / ((b + 2.0) + (0.5 * (b * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1150000.0], N[(N[Exp[a], $MachinePrecision] / a), $MachinePrecision], N[(1.0 / N[(N[(b + 2.0), $MachinePrecision] + N[(0.5 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.15e6

   1. Initial program 100.0%

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

   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 -1.15e6 < a

   1. Initial program 97.8%

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

      1. *-lft-identity 97.8%

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

      2. associate-/l* 97.8%

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

      3. remove-double-div 97.8%

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

      4. exp-neg 97.8%

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

      5. associate-/r/ 97.8%

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

      6. /-rgt-identity 97.8%

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

      7. *-commutative 97.8%

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

      8. distribute-rgt-in 97.8%

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

      9. exp-neg 97.8%

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

      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 a around 0 97.3%

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

   5. Taylor expanded in b around 0 59.6%

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

      1. associate-+r+ 59.6%

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

      2. +-commutative 59.6%

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

      3. unpow2 59.6%

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

   7. Simplified 59.6%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1150000:\\ \;\;\;\;\frac{e^{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(b + 2\right) + 0.5 \cdot \left(b \cdot b\right)}\\ \end{array} \]

Alternative 6: 69.1% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + b \cdot \left(b \cdot 0.5\right)\\ \mathbf{if}\;b \leq 9.5 \cdot 10^{+54}:\\ \;\;\;\;\frac{1}{\left(2 - a\right) + a \cdot \left(a \cdot 0.5\right)}\\ \mathbf{elif}\;b \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\frac{b \cdot b - t_0 \cdot t_0}{b - t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (+ 2.0 (* b (* b 0.5)))))
   (if (<= b 9.5e+54)
     (/ 1.0 (+ (- 2.0 a) (* a (* a 0.5))))
     (if (<= b 1.32e+154)
       (/ 1.0 (/ (- (* b b) (* t_0 t_0)) (- b t_0)))
       (/ 2.0 (* b b))))))

C

double code(double a, double b) {
	double t_0 = 2.0 + (b * (b * 0.5));
	double tmp;
	if (b <= 9.5e+54) {
		tmp = 1.0 / ((2.0 - a) + (a * (a * 0.5)));
	} else if (b <= 1.32e+154) {
		tmp = 1.0 / (((b * b) - (t_0 * t_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 = 2.0d0 + (b * (b * 0.5d0))
    if (b <= 9.5d+54) then
        tmp = 1.0d0 / ((2.0d0 - a) + (a * (a * 0.5d0)))
    else if (b <= 1.32d+154) then
        tmp = 1.0d0 / (((b * b) - (t_0 * t_0)) / (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 = 2.0 + (b * (b * 0.5));
	double tmp;
	if (b <= 9.5e+54) {
		tmp = 1.0 / ((2.0 - a) + (a * (a * 0.5)));
	} else if (b <= 1.32e+154) {
		tmp = 1.0 / (((b * b) - (t_0 * t_0)) / (b - t_0));
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = 2.0 + (b * (b * 0.5))
	tmp = 0
	if b <= 9.5e+54:
		tmp = 1.0 / ((2.0 - a) + (a * (a * 0.5)))
	elif b <= 1.32e+154:
		tmp = 1.0 / (((b * b) - (t_0 * t_0)) / (b - t_0))
	else:
		tmp = 2.0 / (b * b)
	return tmp

Julia

function code(a, b)
	t_0 = Float64(2.0 + Float64(b * Float64(b * 0.5)))
	tmp = 0.0
	if (b <= 9.5e+54)
		tmp = Float64(1.0 / Float64(Float64(2.0 - a) + Float64(a * Float64(a * 0.5))));
	elseif (b <= 1.32e+154)
		tmp = Float64(1.0 / Float64(Float64(Float64(b * b) - Float64(t_0 * t_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 = 2.0 + (b * (b * 0.5));
	tmp = 0.0;
	if (b <= 9.5e+54)
		tmp = 1.0 / ((2.0 - a) + (a * (a * 0.5)));
	elseif (b <= 1.32e+154)
		tmp = 1.0 / (((b * b) - (t_0 * t_0)) / (b - t_0));
	else
		tmp = 2.0 / (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(2.0 + N[(b * N[(b * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 9.5e+54], N[(1.0 / N[(N[(2.0 - a), $MachinePrecision] + N[(a * N[(a * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.32e+154], N[(1.0 / N[(N[(N[(b * b), $MachinePrecision] - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(b - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < 9.4999999999999999e54

   1. Initial program 97.8%

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

      1. *-lft-identity 97.8%

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

      2. associate-/l* 97.8%

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

      3. remove-double-div 97.8%

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

      4. exp-neg 97.8%

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

      5. associate-/r/ 97.8%

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

      6. /-rgt-identity 97.8%

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

      7. *-commutative 97.8%

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

      8. distribute-rgt-in 69.9%

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

      9. exp-neg 69.9%

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

      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 74.7%

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

   5. Taylor expanded in a around 0 57.4%

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

      1. associate-+r+ 57.4%

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

      2. neg-mul-1 57.4%

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

      3. unsub-neg 57.4%

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

      4. *-commutative 57.4%

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

      5. unpow2 57.4%

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

      6. associate-*l* 57.4%

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

   7. Simplified 57.4%

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

   if 9.4999999999999999e54 < b < 1.31999999999999998e154

   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 78.3%

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

      9. exp-neg 78.3%

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

      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 7.6%

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

      1. associate-+r+ 7.6%

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

      2. +-commutative 7.6%

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

      3. unpow2 7.6%

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

   7. Simplified 7.6%

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

      1. associate-+l+ 7.6%

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

      2. flip-+ 70.9%

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

      3. *-commutative 70.9%

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

      4. associate-*l* 70.9%

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

      5. *-commutative 70.9%

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

      6. associate-*l* 70.9%

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

      7. *-commutative 70.9%

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

      8. associate-*l* 70.9%

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

   9. Applied egg-rr 70.9%

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

   if 1.31999999999999998e154 < 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 68.1%

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

      9. exp-neg 68.1%

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

      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 0 100.0%

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

      1. associate-+r+ 100.0%

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

      2. +-commutative 100.0%

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

      3. unpow2 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in b around inf 100.0%

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

      1. unpow2 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9.5 \cdot 10^{+54}:\\ \;\;\;\;\frac{1}{\left(2 - a\right) + a \cdot \left(a \cdot 0.5\right)}\\ \mathbf{elif}\;b \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\frac{b \cdot b - \left(2 + b \cdot \left(b \cdot 0.5\right)\right) \cdot \left(2 + b \cdot \left(b \cdot 0.5\right)\right)}{b - \left(2 + b \cdot \left(b \cdot 0.5\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \]

Alternative 7: 60.6% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.62 \cdot 10^{+97}:\\ \;\;\;\;\frac{2}{a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(b + 2\right) + 0.5 \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.62e+97) (/ 2.0 (* a a)) (/ 1.0 (+ (+ b 2.0) (* 0.5 (* b b))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.62e+97) {
		tmp = 2.0 / (a * a);
	} else {
		tmp = 1.0 / ((b + 2.0) + (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 <= (-1.62d+97)) then
        tmp = 2.0d0 / (a * a)
    else
        tmp = 1.0d0 / ((b + 2.0d0) + (0.5d0 * (b * b)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1.62e+97) {
		tmp = 2.0 / (a * a);
	} else {
		tmp = 1.0 / ((b + 2.0) + (0.5 * (b * b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.62e+97:
		tmp = 2.0 / (a * a)
	else:
		tmp = 1.0 / ((b + 2.0) + (0.5 * (b * b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.62e+97)
		tmp = Float64(2.0 / Float64(a * a));
	else
		tmp = Float64(1.0 / Float64(Float64(b + 2.0) + Float64(0.5 * Float64(b * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.62e+97)
		tmp = 2.0 / (a * a);
	else
		tmp = 1.0 / ((b + 2.0) + (0.5 * (b * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.62e+97], N[(2.0 / N[(a * a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(b + 2.0), $MachinePrecision] + N[(0.5 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.62e97

   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 71.7%

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

      1. associate-+r+ 71.7%

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

      2. neg-mul-1 71.7%

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

      3. unsub-neg 71.7%

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

      4. *-commutative 71.7%

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

      5. unpow2 71.7%

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

      6. associate-*l* 71.7%

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

   7. Simplified 71.7%

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

   8. Taylor expanded in a around inf 71.7%

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

      1. unpow2 71.7%

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

   10. Simplified 71.7%

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

   if -1.62e97 < a

   1. Initial program 98.1%

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

      1. *-lft-identity 98.1%

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

      2. associate-/l* 98.1%

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

      3. remove-double-div 98.0%

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

      4. exp-neg 98.0%

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

      5. associate-/r/ 98.1%

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

      6. /-rgt-identity 98.1%

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

      7. *-commutative 98.1%

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

      8. distribute-rgt-in 85.7%

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

      9. exp-neg 85.7%

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

      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 a around 0 90.7%

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

   5. Taylor expanded in b around 0 55.9%

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

      1. associate-+r+ 55.9%

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

      2. +-commutative 55.9%

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

      3. unpow2 55.9%

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

   7. Simplified 55.9%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.62 \cdot 10^{+97}:\\ \;\;\;\;\frac{2}{a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(b + 2\right) + 0.5 \cdot \left(b \cdot b\right)}\\ \end{array} \]

Alternative 8: 63.4% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 9.2 \cdot 10^{+113}:\\ \;\;\;\;\frac{1}{\left(2 - a\right) + a \cdot \left(a \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(b + 2\right) + 0.5 \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 9.2e+113)
   (/ 1.0 (+ (- 2.0 a) (* a (* a 0.5))))
   (/ 1.0 (+ (+ b 2.0) (* 0.5 (* b b))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 9.2e+113) {
		tmp = 1.0 / ((2.0 - a) + (a * (a * 0.5)));
	} else {
		tmp = 1.0 / ((b + 2.0) + (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 <= 9.2d+113) then
        tmp = 1.0d0 / ((2.0d0 - a) + (a * (a * 0.5d0)))
    else
        tmp = 1.0d0 / ((b + 2.0d0) + (0.5d0 * (b * b)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 9.2e+113) {
		tmp = 1.0 / ((2.0 - a) + (a * (a * 0.5)));
	} else {
		tmp = 1.0 / ((b + 2.0) + (0.5 * (b * b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 9.2e+113:
		tmp = 1.0 / ((2.0 - a) + (a * (a * 0.5)))
	else:
		tmp = 1.0 / ((b + 2.0) + (0.5 * (b * b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 9.2e+113)
		tmp = Float64(1.0 / Float64(Float64(2.0 - a) + Float64(a * Float64(a * 0.5))));
	else
		tmp = Float64(1.0 / Float64(Float64(b + 2.0) + Float64(0.5 * Float64(b * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 9.2e+113)
		tmp = 1.0 / ((2.0 - a) + (a * (a * 0.5)));
	else
		tmp = 1.0 / ((b + 2.0) + (0.5 * (b * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 9.2e+113], N[(1.0 / N[(N[(2.0 - a), $MachinePrecision] + N[(a * N[(a * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(b + 2.0), $MachinePrecision] + N[(0.5 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9.19999999999999987e113

   1. Initial program 98.0%

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

      1. *-lft-identity 98.0%

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

      2. associate-/l* 98.0%

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

      3. remove-double-div 97.9%

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

      4. exp-neg 97.9%

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

      5. associate-/r/ 98.0%

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

      6. /-rgt-identity 98.0%

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

      7. *-commutative 98.0%

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

      8. distribute-rgt-in 70.3%

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

      9. exp-neg 70.3%

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

      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 71.4%

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

   5. Taylor expanded in a around 0 54.4%

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

      1. associate-+r+ 54.4%

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

      2. neg-mul-1 54.4%

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

      3. unsub-neg 54.4%

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

      4. *-commutative 54.4%

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

      5. unpow2 54.4%

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

      6. associate-*l* 54.4%

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

   7. Simplified 54.4%

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

   if 9.19999999999999987e113 < 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 70.2%

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

      9. exp-neg 70.2%

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

      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 84.4%

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

      1. associate-+r+ 84.4%

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

      2. +-commutative 84.4%

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

      3. unpow2 84.4%

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

   7. Simplified 84.4%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9.2 \cdot 10^{+113}:\\ \;\;\;\;\frac{1}{\left(2 - a\right) + a \cdot \left(a \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(b + 2\right) + 0.5 \cdot \left(b \cdot b\right)}\\ \end{array} \]

Alternative 9: 53.2% accurate, 43.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.35:\\ \;\;\;\;\frac{2}{a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.35) (/ 2.0 (* a a)) (+ 0.5 (* a 0.25))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.35) {
		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 <= (-1.35d0)) 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 <= -1.35) {
		tmp = 2.0 / (a * a);
	} else {
		tmp = 0.5 + (a * 0.25);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.35:
		tmp = 2.0 / (a * a)
	else:
		tmp = 0.5 + (a * 0.25)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.35)
		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 <= -1.35)
		tmp = 2.0 / (a * a);
	else
		tmp = 0.5 + (a * 0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.35], 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 < -1.3500000000000001

   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 98.7%

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

      4. exp-neg 98.7%

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

      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 2.7%

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

      9. exp-neg 2.7%

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

      10. rgt-mult-inverse 98.7%

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

      11. prod-exp 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 97.4%

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

   5. Taylor expanded in a around 0 45.7%

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

      1. associate-+r+ 45.7%

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

      2. neg-mul-1 45.7%

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

      3. unsub-neg 45.7%

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

      4. *-commutative 45.7%

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

      5. unpow2 45.7%

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

      6. associate-*l* 45.7%

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

   7. Simplified 45.7%

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

   8. Taylor expanded in a around inf 45.7%

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

      1. unpow2 45.7%

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

   10. Simplified 45.7%

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

   if -1.3500000000000001 < a

   1. Initial program 98.3%

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

      1. *-lft-identity 98.3%

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

      2. associate-/l* 98.3%

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

      3. remove-double-div 98.3%

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

      4. exp-neg 98.3%

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

      5. associate-/r/ 98.3%

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

      6. /-rgt-identity 98.3%

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

      7. *-commutative 98.3%

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

      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 b around 0 48.3%

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

   5. Taylor expanded in a around 0 46.2%

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

      1. *-commutative 46.2%

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

   7. Simplified 46.2%

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

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.35:\\ \;\;\;\;\frac{2}{a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \]

Alternative 10: 53.1% accurate, 43.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.28 \cdot 10^{+41}:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.28e+41) (/ 1.0 (- 2.0 a)) (/ 2.0 (* b b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.28e+41) {
		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 <= 1.28d+41) 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 <= 1.28e+41) {
		tmp = 1.0 / (2.0 - a);
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.28e+41:
		tmp = 1.0 / (2.0 - a)
	else:
		tmp = 2.0 / (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.28e+41)
		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 <= 1.28e+41)
		tmp = 1.0 / (2.0 - a);
	else
		tmp = 2.0 / (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.28e+41], 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 < 1.27999999999999992e41

   1. Initial program 97.8%

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

      1. *-lft-identity 97.8%

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

      2. associate-/l* 97.8%

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

      3. remove-double-div 97.7%

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

      4. exp-neg 97.7%

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

      5. associate-/r/ 97.7%

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

      6. /-rgt-identity 97.7%

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

      7. *-commutative 97.7%

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

      8. distribute-rgt-in 70.1%

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

      9. exp-neg 70.1%

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

      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 75.6%

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

   5. Taylor expanded in a around 0 46.5%

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

      1. neg-mul-1 46.5%

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

      2. unsub-neg 46.5%

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

   7. Simplified 46.5%

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

   if 1.27999999999999992e41 < 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 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 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 65.3%

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

      1. associate-+r+ 65.3%

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

      2. +-commutative 65.3%

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

      3. unpow2 65.3%

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

   7. Simplified 65.3%

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

   8. Taylor expanded in b around inf 65.3%

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

      1. unpow2 65.3%

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

   10. Simplified 65.3%

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

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.28 \cdot 10^{+41}:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \]

Alternative 11: 40.5% 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 98.4%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
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 70.3%

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

   9. exp-neg 70.3%

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

   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 62.7%

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

5. Taylor expanded in a around 0 34.0%

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

   1. neg-mul-1 34.0%

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

   2. unsub-neg 34.0%

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

7. Simplified 34.0%

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

8. Final simplification 34.0%

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

Alternative 12: 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 98.4%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
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 70.3%

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

   9. exp-neg 70.3%

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

   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 80.3%

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

5. Taylor expanded in b around 0 33.4%

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

6. Final simplification 33.4%

   \[\leadsto 0.5 \]

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 2023283 
(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))))