Quotient of sum of exps

Percentage Accurate: 98.8% → 100.0%

Time: 6.4s

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: 98.8% 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 69.1%

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

   9. exp-neg 69.1%

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

   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 99.5%

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

   2. *-un-lft-identity 99.5%

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

   3. log-prod 99.5%

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

   4. metadata-eval 99.5%

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

   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: 98.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1700 \lor \neg \left(b \leq 2.6 \cdot 10^{-18}\right):\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= b -1700.0) (not (<= b 2.6e-18)))
   (/ 1.0 (+ 1.0 (exp b)))
   (/ 1.0 (+ 1.0 (exp (- a))))))

C

double code(double a, double b) {
	double tmp;
	if ((b <= -1700.0) || !(b <= 2.6e-18)) {
		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 ((b <= (-1700.0d0)) .or. (.not. (b <= 2.6d-18))) 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 ((b <= -1700.0) || !(b <= 2.6e-18)) {
		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 (b <= -1700.0) or not (b <= 2.6e-18):
		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 ((b <= -1700.0) || !(b <= 2.6e-18))
		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 ((b <= -1700.0) || ~((b <= 2.6e-18)))
		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[b, -1700.0], N[Not[LessEqual[b, 2.6e-18]], $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 b < -1700 or 2.6e-18 < b

   1. Initial program 99.1%

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

      1. *-lft-identity 99.1%

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

      2. associate-/l* 99.1%

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

      3. remove-double-div 99.1%

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

      4. exp-neg 99.1%

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

      5. associate-/r/ 99.1%

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

      6. /-rgt-identity 99.1%

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

      7. *-commutative 99.1%

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

      8. distribute-rgt-in 77.8%

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

      9. exp-neg 77.8%

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

      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 a around 0 100.0%

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

   if -1700 < b < 2.6e-18

   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 61.8%

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

      9. exp-neg 61.8%

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

      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 0 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1700 \lor \neg \left(b \leq 2.6 \cdot 10^{-18}\right):\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \end{array} \]

Alternative 3: 98.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -720:\\ \;\;\;\;\frac{e^{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -720.0) (/ (exp a) a) (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -720.0) {
		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 <= (-720.0d0)) 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 <= -720.0) {
		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 <= -720.0:
		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 <= -720.0)
		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 <= -720.0)
		tmp = exp(a) / a;
	else
		tmp = 1.0 / (1.0 + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -720.0], 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 < -720

   1. Initial program 98.7%

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

   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 -720 < 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 a around 0 96.7%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -720:\\ \;\;\;\;\frac{e^{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]

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 69.1%

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

   9. exp-neg 69.1%

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

   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: 66.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -465:\\ \;\;\;\;\frac{e^{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(2 - a\right) + \frac{\left(0.25 - a \cdot \left(a \cdot 0.027777777777777776\right)\right) \cdot \left(a \cdot a\right)}{0.5 + a \cdot 0.16666666666666666}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -465.0)
   (/ (exp a) a)
   (/
    1.0
    (+
     (- 2.0 a)
     (/
      (* (- 0.25 (* a (* a 0.027777777777777776))) (* a a))
      (+ 0.5 (* a 0.16666666666666666)))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -465.0) {
		tmp = exp(a) / a;
	} else {
		tmp = 1.0 / ((2.0 - a) + (((0.25 - (a * (a * 0.027777777777777776))) * (a * a)) / (0.5 + (a * 0.16666666666666666))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-465.0d0)) then
        tmp = exp(a) / a
    else
        tmp = 1.0d0 / ((2.0d0 - a) + (((0.25d0 - (a * (a * 0.027777777777777776d0))) * (a * a)) / (0.5d0 + (a * 0.16666666666666666d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -465.0) {
		tmp = Math.exp(a) / a;
	} else {
		tmp = 1.0 / ((2.0 - a) + (((0.25 - (a * (a * 0.027777777777777776))) * (a * a)) / (0.5 + (a * 0.16666666666666666))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -465.0:
		tmp = math.exp(a) / a
	else:
		tmp = 1.0 / ((2.0 - a) + (((0.25 - (a * (a * 0.027777777777777776))) * (a * a)) / (0.5 + (a * 0.16666666666666666))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -465.0)
		tmp = Float64(exp(a) / a);
	else
		tmp = Float64(1.0 / Float64(Float64(2.0 - a) + Float64(Float64(Float64(0.25 - Float64(a * Float64(a * 0.027777777777777776))) * Float64(a * a)) / Float64(0.5 + Float64(a * 0.16666666666666666)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -465.0)
		tmp = exp(a) / a;
	else
		tmp = 1.0 / ((2.0 - a) + (((0.25 - (a * (a * 0.027777777777777776))) * (a * a)) / (0.5 + (a * 0.16666666666666666))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -465.0], N[(N[Exp[a], $MachinePrecision] / a), $MachinePrecision], N[(1.0 / N[(N[(2.0 - a), $MachinePrecision] + N[(N[(N[(0.25 - N[(a * N[(a * 0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[(0.5 + N[(a * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -465

   1. Initial program 98.7%

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

   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 -465 < 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 55.7%

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

   5. Taylor expanded in a around 0 53.7%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(-1 \cdot a + \left(-0.16666666666666666 \cdot {a}^{3} + 0.5 \cdot {a}^{2}\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-+r+ 53.7%

         \[\leadsto \frac{1}{\color{blue}{\left(2 + -1 \cdot a\right) + \left(-0.16666666666666666 \cdot {a}^{3} + 0.5 \cdot {a}^{2}\right)}} \]

      2. neg-mul-1 53.7%

         \[\leadsto \frac{1}{\left(2 + \color{blue}{\left(-a\right)}\right) + \left(-0.16666666666666666 \cdot {a}^{3} + 0.5 \cdot {a}^{2}\right)} \]

      3. unsub-neg 53.7%

         \[\leadsto \frac{1}{\color{blue}{\left(2 - a\right)} + \left(-0.16666666666666666 \cdot {a}^{3} + 0.5 \cdot {a}^{2}\right)} \]

      4. +-commutative 53.7%

         \[\leadsto \frac{1}{\left(2 - a\right) + \color{blue}{\left(0.5 \cdot {a}^{2} + -0.16666666666666666 \cdot {a}^{3}\right)}} \]

      5. *-commutative 53.7%

         \[\leadsto \frac{1}{\left(2 - a\right) + \left(\color{blue}{{a}^{2} \cdot 0.5} + -0.16666666666666666 \cdot {a}^{3}\right)} \]

      6. *-commutative 53.7%

         \[\leadsto \frac{1}{\left(2 - a\right) + \left({a}^{2} \cdot 0.5 + \color{blue}{{a}^{3} \cdot -0.16666666666666666}\right)} \]

      7. unpow3 53.7%

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

      8. unpow2 53.7%

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

      9. associate-*l* 53.7%

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

      10. distribute-lft-out 53.7%

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

      11. unpow2 53.7%

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

   7. Simplified 53.7%

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

      1. *-commutative 53.7%

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

      2. flip-+ 53.7%

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

      3. associate-*l/ 53.7%

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

      4. metadata-eval 53.7%

         \[\leadsto \frac{1}{\left(2 - a\right) + \frac{\left(\color{blue}{0.25} - \left(a \cdot -0.16666666666666666\right) \cdot \left(a \cdot -0.16666666666666666\right)\right) \cdot \left(a \cdot a\right)}{0.5 - a \cdot -0.16666666666666666}} \]

      5. swap-sqr 53.7%

         \[\leadsto \frac{1}{\left(2 - a\right) + \frac{\left(0.25 - \color{blue}{\left(a \cdot a\right) \cdot \left(-0.16666666666666666 \cdot -0.16666666666666666\right)}\right) \cdot \left(a \cdot a\right)}{0.5 - a \cdot -0.16666666666666666}} \]

      6. metadata-eval 53.7%

         \[\leadsto \frac{1}{\left(2 - a\right) + \frac{\left(0.25 - \left(a \cdot a\right) \cdot \color{blue}{0.027777777777777776}\right) \cdot \left(a \cdot a\right)}{0.5 - a \cdot -0.16666666666666666}} \]

      7. *-commutative 53.7%

         \[\leadsto \frac{1}{\left(2 - a\right) + \frac{\left(0.25 - \left(a \cdot a\right) \cdot 0.027777777777777776\right) \cdot \left(a \cdot a\right)}{0.5 - \color{blue}{-0.16666666666666666 \cdot a}}} \]

      8. cancel-sign-sub-inv 53.7%

         \[\leadsto \frac{1}{\left(2 - a\right) + \frac{\left(0.25 - \left(a \cdot a\right) \cdot 0.027777777777777776\right) \cdot \left(a \cdot a\right)}{\color{blue}{0.5 + \left(--0.16666666666666666\right) \cdot a}}} \]

      9. metadata-eval 53.7%

         \[\leadsto \frac{1}{\left(2 - a\right) + \frac{\left(0.25 - \left(a \cdot a\right) \cdot 0.027777777777777776\right) \cdot \left(a \cdot a\right)}{0.5 + \color{blue}{0.16666666666666666} \cdot a}} \]

   9. Applied egg-rr 53.7%

      \[\leadsto \frac{1}{\left(2 - a\right) + \color{blue}{\frac{\left(0.25 - \left(a \cdot a\right) \cdot 0.027777777777777776\right) \cdot \left(a \cdot a\right)}{0.5 + 0.16666666666666666 \cdot a}}} \]

   10. Taylor expanded in a around 0 53.7%

       \[\leadsto \frac{1}{\left(2 - a\right) + \frac{\left(0.25 - \color{blue}{0.027777777777777776 \cdot {a}^{2}}\right) \cdot \left(a \cdot a\right)}{0.5 + 0.16666666666666666 \cdot a}} \]
   11. Step-by-step derivation

       1. *-commutative 53.7%

          \[\leadsto \frac{1}{\left(2 - a\right) + \frac{\left(0.25 - \color{blue}{{a}^{2} \cdot 0.027777777777777776}\right) \cdot \left(a \cdot a\right)}{0.5 + 0.16666666666666666 \cdot a}} \]

       2. unpow2 53.7%

          \[\leadsto \frac{1}{\left(2 - a\right) + \frac{\left(0.25 - \color{blue}{\left(a \cdot a\right)} \cdot 0.027777777777777776\right) \cdot \left(a \cdot a\right)}{0.5 + 0.16666666666666666 \cdot a}} \]

       3. associate-*r* 53.7%

          \[\leadsto \frac{1}{\left(2 - a\right) + \frac{\left(0.25 - \color{blue}{a \cdot \left(a \cdot 0.027777777777777776\right)}\right) \cdot \left(a \cdot a\right)}{0.5 + 0.16666666666666666 \cdot a}} \]

   12. Simplified 53.7%

       \[\leadsto \frac{1}{\left(2 - a\right) + \frac{\left(0.25 - \color{blue}{a \cdot \left(a \cdot 0.027777777777777776\right)}\right) \cdot \left(a \cdot a\right)}{0.5 + 0.16666666666666666 \cdot a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -465:\\ \;\;\;\;\frac{e^{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(2 - a\right) + \frac{\left(0.25 - a \cdot \left(a \cdot 0.027777777777777776\right)\right) \cdot \left(a \cdot a\right)}{0.5 + a \cdot 0.16666666666666666}}\\ \end{array} \]

Alternative 6: 59.7% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(2 - a\right) + \frac{\left(0.25 - a \cdot \left(a \cdot 0.027777777777777776\right)\right) \cdot \left(a \cdot a\right)}{0.5 + a \cdot 0.16666666666666666}} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (/
  1.0
  (+
   (- 2.0 a)
   (/
    (* (- 0.25 (* a (* a 0.027777777777777776))) (* a a))
    (+ 0.5 (* a 0.16666666666666666))))))

C

double code(double a, double b) {
	return 1.0 / ((2.0 - a) + (((0.25 - (a * (a * 0.027777777777777776))) * (a * a)) / (0.5 + (a * 0.16666666666666666))));
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 1.0d0 / ((2.0d0 - a) + (((0.25d0 - (a * (a * 0.027777777777777776d0))) * (a * a)) / (0.5d0 + (a * 0.16666666666666666d0))))
end function

Java

public static double code(double a, double b) {
	return 1.0 / ((2.0 - a) + (((0.25 - (a * (a * 0.027777777777777776))) * (a * a)) / (0.5 + (a * 0.16666666666666666))));
}

Python

def code(a, b):
	return 1.0 / ((2.0 - a) + (((0.25 - (a * (a * 0.027777777777777776))) * (a * a)) / (0.5 + (a * 0.16666666666666666))))

Julia

function code(a, b)
	return Float64(1.0 / Float64(Float64(2.0 - a) + Float64(Float64(Float64(0.25 - Float64(a * Float64(a * 0.027777777777777776))) * Float64(a * a)) / Float64(0.5 + Float64(a * 0.16666666666666666)))))
end

MATLAB

function tmp = code(a, b)
	tmp = 1.0 / ((2.0 - a) + (((0.25 - (a * (a * 0.027777777777777776))) * (a * a)) / (0.5 + (a * 0.16666666666666666))));
end

Wolfram

code[a_, b_] := N[(1.0 / N[(N[(2.0 - a), $MachinePrecision] + N[(N[(N[(0.25 - N[(a * N[(a * 0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[(0.5 + N[(a * 0.16666666666666666), $MachinePrecision]), $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 69.1%

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

   9. exp-neg 69.1%

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

   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 68.9%

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

5. Taylor expanded in a around 0 57.9%

   \[\leadsto \frac{1}{\color{blue}{2 + \left(-1 \cdot a + \left(-0.16666666666666666 \cdot {a}^{3} + 0.5 \cdot {a}^{2}\right)\right)}} \]
6. Step-by-step derivation

   1. associate-+r+ 57.9%

      \[\leadsto \frac{1}{\color{blue}{\left(2 + -1 \cdot a\right) + \left(-0.16666666666666666 \cdot {a}^{3} + 0.5 \cdot {a}^{2}\right)}} \]

   2. neg-mul-1 57.9%

      \[\leadsto \frac{1}{\left(2 + \color{blue}{\left(-a\right)}\right) + \left(-0.16666666666666666 \cdot {a}^{3} + 0.5 \cdot {a}^{2}\right)} \]

   3. unsub-neg 57.9%

      \[\leadsto \frac{1}{\color{blue}{\left(2 - a\right)} + \left(-0.16666666666666666 \cdot {a}^{3} + 0.5 \cdot {a}^{2}\right)} \]

   4. +-commutative 57.9%

      \[\leadsto \frac{1}{\left(2 - a\right) + \color{blue}{\left(0.5 \cdot {a}^{2} + -0.16666666666666666 \cdot {a}^{3}\right)}} \]

   5. *-commutative 57.9%

      \[\leadsto \frac{1}{\left(2 - a\right) + \left(\color{blue}{{a}^{2} \cdot 0.5} + -0.16666666666666666 \cdot {a}^{3}\right)} \]

   6. *-commutative 57.9%

      \[\leadsto \frac{1}{\left(2 - a\right) + \left({a}^{2} \cdot 0.5 + \color{blue}{{a}^{3} \cdot -0.16666666666666666}\right)} \]

   7. unpow3 57.9%

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

   8. unpow2 57.9%

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

   9. associate-*l* 57.9%

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

   10. distribute-lft-out 57.9%

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

   11. unpow2 57.9%

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

7. Simplified 57.9%

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

   1. *-commutative 57.9%

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

   2. flip-+ 57.9%

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

   3. associate-*l/ 61.1%

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

   4. metadata-eval 61.1%

      \[\leadsto \frac{1}{\left(2 - a\right) + \frac{\left(\color{blue}{0.25} - \left(a \cdot -0.16666666666666666\right) \cdot \left(a \cdot -0.16666666666666666\right)\right) \cdot \left(a \cdot a\right)}{0.5 - a \cdot -0.16666666666666666}} \]

   5. swap-sqr 61.1%

      \[\leadsto \frac{1}{\left(2 - a\right) + \frac{\left(0.25 - \color{blue}{\left(a \cdot a\right) \cdot \left(-0.16666666666666666 \cdot -0.16666666666666666\right)}\right) \cdot \left(a \cdot a\right)}{0.5 - a \cdot -0.16666666666666666}} \]

   6. metadata-eval 61.1%

      \[\leadsto \frac{1}{\left(2 - a\right) + \frac{\left(0.25 - \left(a \cdot a\right) \cdot \color{blue}{0.027777777777777776}\right) \cdot \left(a \cdot a\right)}{0.5 - a \cdot -0.16666666666666666}} \]

   7. *-commutative 61.1%

      \[\leadsto \frac{1}{\left(2 - a\right) + \frac{\left(0.25 - \left(a \cdot a\right) \cdot 0.027777777777777776\right) \cdot \left(a \cdot a\right)}{0.5 - \color{blue}{-0.16666666666666666 \cdot a}}} \]

   8. cancel-sign-sub-inv 61.1%

      \[\leadsto \frac{1}{\left(2 - a\right) + \frac{\left(0.25 - \left(a \cdot a\right) \cdot 0.027777777777777776\right) \cdot \left(a \cdot a\right)}{\color{blue}{0.5 + \left(--0.16666666666666666\right) \cdot a}}} \]

   9. metadata-eval 61.1%

      \[\leadsto \frac{1}{\left(2 - a\right) + \frac{\left(0.25 - \left(a \cdot a\right) \cdot 0.027777777777777776\right) \cdot \left(a \cdot a\right)}{0.5 + \color{blue}{0.16666666666666666} \cdot a}} \]

9. Applied egg-rr 61.1%

   \[\leadsto \frac{1}{\left(2 - a\right) + \color{blue}{\frac{\left(0.25 - \left(a \cdot a\right) \cdot 0.027777777777777776\right) \cdot \left(a \cdot a\right)}{0.5 + 0.16666666666666666 \cdot a}}} \]

10. Taylor expanded in a around 0 61.1%

    \[\leadsto \frac{1}{\left(2 - a\right) + \frac{\left(0.25 - \color{blue}{0.027777777777777776 \cdot {a}^{2}}\right) \cdot \left(a \cdot a\right)}{0.5 + 0.16666666666666666 \cdot a}} \]
11. Step-by-step derivation

    1. *-commutative 61.1%

       \[\leadsto \frac{1}{\left(2 - a\right) + \frac{\left(0.25 - \color{blue}{{a}^{2} \cdot 0.027777777777777776}\right) \cdot \left(a \cdot a\right)}{0.5 + 0.16666666666666666 \cdot a}} \]

    2. unpow2 61.1%

       \[\leadsto \frac{1}{\left(2 - a\right) + \frac{\left(0.25 - \color{blue}{\left(a \cdot a\right)} \cdot 0.027777777777777776\right) \cdot \left(a \cdot a\right)}{0.5 + 0.16666666666666666 \cdot a}} \]

    3. associate-*r* 61.1%

       \[\leadsto \frac{1}{\left(2 - a\right) + \frac{\left(0.25 - \color{blue}{a \cdot \left(a \cdot 0.027777777777777776\right)}\right) \cdot \left(a \cdot a\right)}{0.5 + 0.16666666666666666 \cdot a}} \]

12. Simplified 61.1%

    \[\leadsto \frac{1}{\left(2 - a\right) + \frac{\left(0.25 - \color{blue}{a \cdot \left(a \cdot 0.027777777777777776\right)}\right) \cdot \left(a \cdot a\right)}{0.5 + 0.16666666666666666 \cdot a}} \]

13. Final simplification 61.1%

    \[\leadsto \frac{1}{\left(2 - a\right) + \frac{\left(0.25 - a \cdot \left(a \cdot 0.027777777777777776\right)\right) \cdot \left(a \cdot a\right)}{0.5 + a \cdot 0.16666666666666666}} \]

Alternative 7: 57.4% accurate, 20.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(2 - a\right) + \left(a \cdot a\right) \cdot \left(0.5 + a \cdot -0.16666666666666666\right)} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (/ 1.0 (+ (- 2.0 a) (* (* a a) (+ 0.5 (* a -0.16666666666666666))))))

C

double code(double a, double b) {
	return 1.0 / ((2.0 - a) + ((a * a) * (0.5 + (a * -0.16666666666666666))));
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 1.0d0 / ((2.0d0 - a) + ((a * a) * (0.5d0 + (a * (-0.16666666666666666d0)))))
end function

Java

public static double code(double a, double b) {
	return 1.0 / ((2.0 - a) + ((a * a) * (0.5 + (a * -0.16666666666666666))));
}

Python

def code(a, b):
	return 1.0 / ((2.0 - a) + ((a * a) * (0.5 + (a * -0.16666666666666666))))

Julia

function code(a, b)
	return Float64(1.0 / Float64(Float64(2.0 - a) + Float64(Float64(a * a) * Float64(0.5 + Float64(a * -0.16666666666666666)))))
end

MATLAB

function tmp = code(a, b)
	tmp = 1.0 / ((2.0 - a) + ((a * a) * (0.5 + (a * -0.16666666666666666))));
end

Wolfram

code[a_, b_] := N[(1.0 / N[(N[(2.0 - a), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * N[(0.5 + N[(a * -0.16666666666666666), $MachinePrecision]), $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 69.1%

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

   9. exp-neg 69.1%

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

   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 68.9%

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

5. Taylor expanded in a around 0 57.9%

   \[\leadsto \frac{1}{\color{blue}{2 + \left(-1 \cdot a + \left(-0.16666666666666666 \cdot {a}^{3} + 0.5 \cdot {a}^{2}\right)\right)}} \]
6. Step-by-step derivation

   1. associate-+r+ 57.9%

      \[\leadsto \frac{1}{\color{blue}{\left(2 + -1 \cdot a\right) + \left(-0.16666666666666666 \cdot {a}^{3} + 0.5 \cdot {a}^{2}\right)}} \]

   2. neg-mul-1 57.9%

      \[\leadsto \frac{1}{\left(2 + \color{blue}{\left(-a\right)}\right) + \left(-0.16666666666666666 \cdot {a}^{3} + 0.5 \cdot {a}^{2}\right)} \]

   3. unsub-neg 57.9%

      \[\leadsto \frac{1}{\color{blue}{\left(2 - a\right)} + \left(-0.16666666666666666 \cdot {a}^{3} + 0.5 \cdot {a}^{2}\right)} \]

   4. +-commutative 57.9%

      \[\leadsto \frac{1}{\left(2 - a\right) + \color{blue}{\left(0.5 \cdot {a}^{2} + -0.16666666666666666 \cdot {a}^{3}\right)}} \]

   5. *-commutative 57.9%

      \[\leadsto \frac{1}{\left(2 - a\right) + \left(\color{blue}{{a}^{2} \cdot 0.5} + -0.16666666666666666 \cdot {a}^{3}\right)} \]

   6. *-commutative 57.9%

      \[\leadsto \frac{1}{\left(2 - a\right) + \left({a}^{2} \cdot 0.5 + \color{blue}{{a}^{3} \cdot -0.16666666666666666}\right)} \]

   7. unpow3 57.9%

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

   8. unpow2 57.9%

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

   9. associate-*l* 57.9%

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

   10. distribute-lft-out 57.9%

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

   11. unpow2 57.9%

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

7. Simplified 57.9%

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

8. Final simplification 57.9%

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

Alternative 8: 53.2% accurate, 27.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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 69.1%

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

   9. exp-neg 69.1%

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

   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 68.9%

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

5. Taylor expanded in a around 0 52.9%

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

   1. +-commutative 52.9%

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

   2. neg-mul-1 52.9%

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

   3. unsub-neg 52.9%

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

   4. unpow2 52.9%

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

7. Simplified 52.9%

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

   1. associate-*r* 52.9%

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

   2. *-un-lft-identity 52.9%

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

   3. distribute-rgt-out-- 52.9%

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

   4. *-commutative 52.9%

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

9. Applied egg-rr 52.9%

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

10. Final simplification 52.9%

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

Alternative 9: 53.3% accurate, 43.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.75:\\ \;\;\;\;\frac{2}{a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.75) (/ 2.0 (* a a)) (+ 0.5 (* a 0.25))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.75) {
		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 <= (-1.75d0)) 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 <= -1.75) {
		tmp = 2.0 / (a * a);
	} else {
		tmp = 0.5 + (a * 0.25);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.75:
		tmp = 2.0 / (a * a)
	else:
		tmp = 0.5 + (a * 0.25)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.75)
		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 <= -1.75)
		tmp = 2.0 / (a * a);
	else
		tmp = 0.5 + (a * 0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.75], 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 < -1.75

   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 3.8%

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

      9. exp-neg 3.8%

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

      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 97.6%

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

   5. Taylor expanded in a around 0 49.6%

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

      1. +-commutative 49.6%

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

      2. neg-mul-1 49.6%

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

      3. unsub-neg 49.6%

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

      4. unpow2 49.6%

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

   7. Simplified 49.6%

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

   8. Taylor expanded in a around inf 49.6%

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

      1. unpow2 49.6%

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

   10. Simplified 49.6%

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

   if -1.75 < 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 56.1%

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

   5. Taylor expanded in a around 0 54.4%

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

      1. *-commutative 54.4%

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

   7. Simplified 54.4%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.75:\\ \;\;\;\;\frac{2}{a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \]

Alternative 10: 39.1% 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 98.4%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
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 69.1%

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

   9. exp-neg 69.1%

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

   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 68.9%

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

5. Taylor expanded in a around 0 38.3%

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

   1. *-commutative 38.3%

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

7. Simplified 38.3%

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

8. Final simplification 38.3%

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

Alternative 11: 39.8% 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 69.1%

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

   9. exp-neg 69.1%

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

   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 68.9%

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

5. Taylor expanded in a around 0 38.8%

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

   1. neg-mul-1 38.8%

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

   2. unsub-neg 38.8%

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

7. Simplified 38.8%

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

8. Final simplification 38.8%

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

Alternative 12: 38.9% 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 69.1%

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

   9. exp-neg 69.1%

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

   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 68.9%

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

5. Taylor expanded in a around 0 37.8%

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

6. Final simplification 37.8%

   \[\leadsto 0.5 \]

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 2023293 
(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))))