Quotient of sum of exps

Percentage Accurate: 99.0% → 100.0%

Time: 10.5s

Alternatives: 15

Speedup: 2.9×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. *-lft-identity 99.2%

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

   2. associate-*l/ 99.2%

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

   3. associate-/r/ 99.2%

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

   4. remove-double-neg 99.2%

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

   5. unsub-neg 99.2%

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

   6. div-sub 71.8%

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

   7. *-lft-identity 71.8%

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

   8. associate-*l/ 71.8%

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

   9. lft-mult-inverse 99.6%

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

   10. sub-neg 99.6%

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

   11. distribute-frac-neg 99.6%

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

   12. remove-double-neg 99.6%

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

   13. div-exp 100.0%

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

3. Simplified 100.0%

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

   1. add-exp-log 100.0%

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

   2. log-rec 100.0%

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

   3. log1p-define 100.0%

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

6. Applied egg-rr 100.0%

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

Alternative 2: 98.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -720000000:\\ \;\;\;\;\frac{e^{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -720000000.0) (/ (exp a) b) (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -720000000.0) {
		tmp = exp(a) / b;
	} 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 <= (-720000000.0d0)) then
        tmp = exp(a) / b
    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 <= -720000000.0) {
		tmp = Math.exp(a) / b;
	} else {
		tmp = 1.0 / (1.0 + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -720000000.0:
		tmp = math.exp(a) / b
	else:
		tmp = 1.0 / (1.0 + math.exp(b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -720000000.0)
		tmp = Float64(exp(a) / b);
	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 <= -720000000.0)
		tmp = exp(a) / b;
	else
		tmp = 1.0 / (1.0 + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -720000000.0], N[(N[Exp[a], $MachinePrecision] / b), $MachinePrecision], N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -7.2e8

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/r/ 100.0%

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

      4. remove-double-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. div-sub 0.0%

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

      7. *-lft-identity 0.0%

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

      8. associate-*l/ 0.0%

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

      9. lft-mult-inverse 100.0%

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

      10. sub-neg 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. remove-double-neg 100.0%

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

      13. div-exp 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around 0 67.1%

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

      1. distribute-rgt1-in 100.0%

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

      2. rec-exp 100.0%

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

      3. associate-*r/ 100.0%

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

      4. *-rgt-identity 100.0%

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

      5. +-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in b around inf 100.0%

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

   if -7.2e8 < a

   1. Initial program 98.9%

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

      1. *-lft-identity 98.9%

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

      2. associate-*l/ 98.9%

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

      3. associate-/r/ 98.9%

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

      4. remove-double-neg 98.9%

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

      5. unsub-neg 98.9%

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

      6. div-sub 98.9%

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

      7. *-lft-identity 98.9%

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

      8. associate-*l/ 98.9%

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

      9. lft-mult-inverse 99.4%

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

      10. sub-neg 99.4%

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

      11. distribute-frac-neg 99.4%

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

      12. remove-double-neg 99.4%

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

      13. div-exp 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0 98.2%

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

Alternative 3: 71.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -720000000:\\ \;\;\;\;\frac{e^{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \left(a \cdot \left(\left(b + 1\right) + a \cdot \left(\left(b \cdot 0.5 - b\right) + \left(a \cdot \left(0.16666666666666666 + \frac{\left(b \cdot -0.8333333333333334\right) \cdot \left(b \cdot -0.16666666666666666\right)}{b \cdot -0.16666666666666666}\right) - 0.5\right)\right)\right) - b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -720000000.0)
   (/ (exp a) b)
   (/
    1.0
    (-
     2.0
     (-
      (*
       a
       (+
        (+ b 1.0)
        (*
         a
         (+
          (- (* b 0.5) b)
          (-
           (*
            a
            (+
             0.16666666666666666
             (/
              (* (* b -0.8333333333333334) (* b -0.16666666666666666))
              (* b -0.16666666666666666))))
           0.5)))))
      b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -720000000.0) {
		tmp = exp(a) / b;
	} else {
		tmp = 1.0 / (2.0 - ((a * ((b + 1.0) + (a * (((b * 0.5) - b) + ((a * (0.16666666666666666 + (((b * -0.8333333333333334) * (b * -0.16666666666666666)) / (b * -0.16666666666666666)))) - 0.5))))) - 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 <= (-720000000.0d0)) then
        tmp = exp(a) / b
    else
        tmp = 1.0d0 / (2.0d0 - ((a * ((b + 1.0d0) + (a * (((b * 0.5d0) - b) + ((a * (0.16666666666666666d0 + (((b * (-0.8333333333333334d0)) * (b * (-0.16666666666666666d0))) / (b * (-0.16666666666666666d0))))) - 0.5d0))))) - b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -720000000.0) {
		tmp = Math.exp(a) / b;
	} else {
		tmp = 1.0 / (2.0 - ((a * ((b + 1.0) + (a * (((b * 0.5) - b) + ((a * (0.16666666666666666 + (((b * -0.8333333333333334) * (b * -0.16666666666666666)) / (b * -0.16666666666666666)))) - 0.5))))) - b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -720000000.0:
		tmp = math.exp(a) / b
	else:
		tmp = 1.0 / (2.0 - ((a * ((b + 1.0) + (a * (((b * 0.5) - b) + ((a * (0.16666666666666666 + (((b * -0.8333333333333334) * (b * -0.16666666666666666)) / (b * -0.16666666666666666)))) - 0.5))))) - b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -720000000.0)
		tmp = Float64(exp(a) / b);
	else
		tmp = Float64(1.0 / Float64(2.0 - Float64(Float64(a * Float64(Float64(b + 1.0) + Float64(a * Float64(Float64(Float64(b * 0.5) - b) + Float64(Float64(a * Float64(0.16666666666666666 + Float64(Float64(Float64(b * -0.8333333333333334) * Float64(b * -0.16666666666666666)) / Float64(b * -0.16666666666666666)))) - 0.5))))) - b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -720000000.0)
		tmp = exp(a) / b;
	else
		tmp = 1.0 / (2.0 - ((a * ((b + 1.0) + (a * (((b * 0.5) - b) + ((a * (0.16666666666666666 + (((b * -0.8333333333333334) * (b * -0.16666666666666666)) / (b * -0.16666666666666666)))) - 0.5))))) - b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -720000000.0], N[(N[Exp[a], $MachinePrecision] / b), $MachinePrecision], N[(1.0 / N[(2.0 - N[(N[(a * N[(N[(b + 1.0), $MachinePrecision] + N[(a * N[(N[(N[(b * 0.5), $MachinePrecision] - b), $MachinePrecision] + N[(N[(a * N[(0.16666666666666666 + N[(N[(N[(b * -0.8333333333333334), $MachinePrecision] * N[(b * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] / N[(b * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -7.2e8

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/r/ 100.0%

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

      4. remove-double-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. div-sub 0.0%

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

      7. *-lft-identity 0.0%

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

      8. associate-*l/ 0.0%

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

      9. lft-mult-inverse 100.0%

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

      10. sub-neg 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. remove-double-neg 100.0%

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

      13. div-exp 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around 0 67.1%

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

      1. distribute-rgt1-in 100.0%

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

      2. rec-exp 100.0%

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

      3. associate-*r/ 100.0%

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

      4. *-rgt-identity 100.0%

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

      5. +-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in b around inf 100.0%

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

   if -7.2e8 < a

   1. Initial program 98.9%

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

      1. *-lft-identity 98.9%

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

      2. associate-*l/ 98.9%

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

      3. associate-/r/ 98.9%

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

      4. remove-double-neg 98.9%

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

      5. unsub-neg 98.9%

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

      6. div-sub 98.9%

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

      7. *-lft-identity 98.9%

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

      8. associate-*l/ 98.9%

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

      9. lft-mult-inverse 99.4%

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

      10. sub-neg 99.4%

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

      11. distribute-frac-neg 99.4%

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

      12. remove-double-neg 99.4%

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

      13. div-exp 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around 0 51.9%

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

      1. distribute-rgt1-in 51.9%

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

      2. rec-exp 51.9%

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

      3. associate-*r/ 51.9%

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

      4. *-rgt-identity 51.9%

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

      5. +-commutative 51.9%

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

   7. Simplified 51.9%

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

   8. Taylor expanded in a around 0 50.8%

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

      1. flip-+ 49.7%

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

      2. mul-1-neg 49.7%

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

      3. distribute-rgt-out 49.7%

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

      4. metadata-eval 49.7%

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

      5. *-commutative 49.7%

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

      6. mul-1-neg 49.7%

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

      7. distribute-rgt-out 49.7%

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

      8. metadata-eval 49.7%

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

      9. *-commutative 49.7%

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

      10. sqr-neg 49.7%

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

   10. Applied egg-rr 49.7%

       \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(\left(0.5 + -1 \cdot \left(a \cdot \left(0.16666666666666666 + \color{blue}{\left(\frac{{\left(b \cdot -0.5\right)}^{2}}{b \cdot -0.5 - b \cdot -0.3333333333333333} - \frac{{\left(b \cdot -0.3333333333333333\right)}^{2}}{b \cdot -0.5 - b \cdot -0.3333333333333333}\right)}\right)\right)\right) - \left(-1 \cdot b + 0.5 \cdot b\right)\right) - \left(1 + b\right)\right)\right)} \]
   11. Step-by-step derivation

       1. div-sub 49.7%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(\left(0.5 + -1 \cdot \left(a \cdot \left(0.16666666666666666 + \color{blue}{\frac{{\left(b \cdot -0.5\right)}^{2} - {\left(b \cdot -0.3333333333333333\right)}^{2}}{b \cdot -0.5 - b \cdot -0.3333333333333333}}\right)\right)\right) - \left(-1 \cdot b + 0.5 \cdot b\right)\right) - \left(1 + b\right)\right)\right)} \]

       2. unpow2 49.7%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(\left(0.5 + -1 \cdot \left(a \cdot \left(0.16666666666666666 + \frac{\color{blue}{\left(b \cdot -0.5\right) \cdot \left(b \cdot -0.5\right)} - {\left(b \cdot -0.3333333333333333\right)}^{2}}{b \cdot -0.5 - b \cdot -0.3333333333333333}\right)\right)\right) - \left(-1 \cdot b + 0.5 \cdot b\right)\right) - \left(1 + b\right)\right)\right)} \]

       3. unpow2 49.7%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(\left(0.5 + -1 \cdot \left(a \cdot \left(0.16666666666666666 + \frac{\left(b \cdot -0.5\right) \cdot \left(b \cdot -0.5\right) - \color{blue}{\left(b \cdot -0.3333333333333333\right) \cdot \left(b \cdot -0.3333333333333333\right)}}{b \cdot -0.5 - b \cdot -0.3333333333333333}\right)\right)\right) - \left(-1 \cdot b + 0.5 \cdot b\right)\right) - \left(1 + b\right)\right)\right)} \]

       4. difference-of-squares 62.4%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(\left(0.5 + -1 \cdot \left(a \cdot \left(0.16666666666666666 + \frac{\color{blue}{\left(b \cdot -0.5 + b \cdot -0.3333333333333333\right) \cdot \left(b \cdot -0.5 - b \cdot -0.3333333333333333\right)}}{b \cdot -0.5 - b \cdot -0.3333333333333333}\right)\right)\right) - \left(-1 \cdot b + 0.5 \cdot b\right)\right) - \left(1 + b\right)\right)\right)} \]

       5. +-commutative 62.4%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(\left(0.5 + -1 \cdot \left(a \cdot \left(0.16666666666666666 + \frac{\color{blue}{\left(b \cdot -0.3333333333333333 + b \cdot -0.5\right)} \cdot \left(b \cdot -0.5 - b \cdot -0.3333333333333333\right)}{b \cdot -0.5 - b \cdot -0.3333333333333333}\right)\right)\right) - \left(-1 \cdot b + 0.5 \cdot b\right)\right) - \left(1 + b\right)\right)\right)} \]

       6. distribute-lft-out 62.4%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(\left(0.5 + -1 \cdot \left(a \cdot \left(0.16666666666666666 + \frac{\color{blue}{\left(b \cdot \left(-0.3333333333333333 + -0.5\right)\right)} \cdot \left(b \cdot -0.5 - b \cdot -0.3333333333333333\right)}{b \cdot -0.5 - b \cdot -0.3333333333333333}\right)\right)\right) - \left(-1 \cdot b + 0.5 \cdot b\right)\right) - \left(1 + b\right)\right)\right)} \]

       7. metadata-eval 62.4%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(\left(0.5 + -1 \cdot \left(a \cdot \left(0.16666666666666666 + \frac{\left(b \cdot \color{blue}{-0.8333333333333334}\right) \cdot \left(b \cdot -0.5 - b \cdot -0.3333333333333333\right)}{b \cdot -0.5 - b \cdot -0.3333333333333333}\right)\right)\right) - \left(-1 \cdot b + 0.5 \cdot b\right)\right) - \left(1 + b\right)\right)\right)} \]

       8. distribute-lft-out-- 62.4%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(\left(0.5 + -1 \cdot \left(a \cdot \left(0.16666666666666666 + \frac{\left(b \cdot -0.8333333333333334\right) \cdot \color{blue}{\left(b \cdot \left(-0.5 - -0.3333333333333333\right)\right)}}{b \cdot -0.5 - b \cdot -0.3333333333333333}\right)\right)\right) - \left(-1 \cdot b + 0.5 \cdot b\right)\right) - \left(1 + b\right)\right)\right)} \]

       9. metadata-eval 62.4%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(\left(0.5 + -1 \cdot \left(a \cdot \left(0.16666666666666666 + \frac{\left(b \cdot -0.8333333333333334\right) \cdot \left(b \cdot \color{blue}{-0.16666666666666666}\right)}{b \cdot -0.5 - b \cdot -0.3333333333333333}\right)\right)\right) - \left(-1 \cdot b + 0.5 \cdot b\right)\right) - \left(1 + b\right)\right)\right)} \]

       10. *-commutative 62.4%

           \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(\left(0.5 + -1 \cdot \left(a \cdot \left(0.16666666666666666 + \frac{\left(b \cdot -0.8333333333333334\right) \cdot \color{blue}{\left(-0.16666666666666666 \cdot b\right)}}{b \cdot -0.5 - b \cdot -0.3333333333333333}\right)\right)\right) - \left(-1 \cdot b + 0.5 \cdot b\right)\right) - \left(1 + b\right)\right)\right)} \]

       11. distribute-lft-out-- 62.4%

           \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(\left(0.5 + -1 \cdot \left(a \cdot \left(0.16666666666666666 + \frac{\left(b \cdot -0.8333333333333334\right) \cdot \left(-0.16666666666666666 \cdot b\right)}{\color{blue}{b \cdot \left(-0.5 - -0.3333333333333333\right)}}\right)\right)\right) - \left(-1 \cdot b + 0.5 \cdot b\right)\right) - \left(1 + b\right)\right)\right)} \]

       12. metadata-eval 62.4%

           \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(\left(0.5 + -1 \cdot \left(a \cdot \left(0.16666666666666666 + \frac{\left(b \cdot -0.8333333333333334\right) \cdot \left(-0.16666666666666666 \cdot b\right)}{b \cdot \color{blue}{-0.16666666666666666}}\right)\right)\right) - \left(-1 \cdot b + 0.5 \cdot b\right)\right) - \left(1 + b\right)\right)\right)} \]

       13. *-commutative 62.4%

           \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(\left(0.5 + -1 \cdot \left(a \cdot \left(0.16666666666666666 + \frac{\left(b \cdot -0.8333333333333334\right) \cdot \left(-0.16666666666666666 \cdot b\right)}{\color{blue}{-0.16666666666666666 \cdot b}}\right)\right)\right) - \left(-1 \cdot b + 0.5 \cdot b\right)\right) - \left(1 + b\right)\right)\right)} \]

   12. Simplified 62.4%

       \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(\left(0.5 + -1 \cdot \left(a \cdot \left(0.16666666666666666 + \color{blue}{\frac{\left(b \cdot -0.8333333333333334\right) \cdot \left(-0.16666666666666666 \cdot b\right)}{-0.16666666666666666 \cdot b}}\right)\right)\right) - \left(-1 \cdot b + 0.5 \cdot b\right)\right) - \left(1 + b\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -720000000:\\ \;\;\;\;\frac{e^{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \left(a \cdot \left(\left(b + 1\right) + a \cdot \left(\left(b \cdot 0.5 - b\right) + \left(a \cdot \left(0.16666666666666666 + \frac{\left(b \cdot -0.8333333333333334\right) \cdot \left(b \cdot -0.16666666666666666\right)}{b \cdot -0.16666666666666666}\right) - 0.5\right)\right)\right) - b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 100.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. *-lft-identity 99.2%

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

   2. associate-*l/ 99.2%

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

   3. associate-/r/ 99.2%

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

   4. remove-double-neg 99.2%

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

   5. unsub-neg 99.2%

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

   6. div-sub 71.8%

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

   7. *-lft-identity 71.8%

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

   8. associate-*l/ 71.8%

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

   9. lft-mult-inverse 99.6%

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

   10. sub-neg 99.6%

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

   11. distribute-frac-neg 99.6%

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

   12. remove-double-neg 99.6%

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

   13. div-exp 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 5: 68.4% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.65:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \left(a \cdot \left(\left(b + 1\right) + a \cdot \left(\left(b \cdot 0.5 - b\right) + \left(a \cdot \left(0.16666666666666666 + \frac{\left(b \cdot -0.8333333333333334\right) \cdot \left(b \cdot -0.16666666666666666\right)}{b \cdot -0.16666666666666666}\right) - 0.5\right)\right)\right) - b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -1.65)
   0.5
   (/
    1.0
    (-
     2.0
     (-
      (*
       a
       (+
        (+ b 1.0)
        (*
         a
         (+
          (- (* b 0.5) b)
          (-
           (*
            a
            (+
             0.16666666666666666
             (/
              (* (* b -0.8333333333333334) (* b -0.16666666666666666))
              (* b -0.16666666666666666))))
           0.5)))))
      b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -1.65) {
		tmp = 0.5;
	} else {
		tmp = 1.0 / (2.0 - ((a * ((b + 1.0) + (a * (((b * 0.5) - b) + ((a * (0.16666666666666666 + (((b * -0.8333333333333334) * (b * -0.16666666666666666)) / (b * -0.16666666666666666)))) - 0.5))))) - 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 <= (-1.65d0)) then
        tmp = 0.5d0
    else
        tmp = 1.0d0 / (2.0d0 - ((a * ((b + 1.0d0) + (a * (((b * 0.5d0) - b) + ((a * (0.16666666666666666d0 + (((b * (-0.8333333333333334d0)) * (b * (-0.16666666666666666d0))) / (b * (-0.16666666666666666d0))))) - 0.5d0))))) - b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -1.65) {
		tmp = 0.5;
	} else {
		tmp = 1.0 / (2.0 - ((a * ((b + 1.0) + (a * (((b * 0.5) - b) + ((a * (0.16666666666666666 + (((b * -0.8333333333333334) * (b * -0.16666666666666666)) / (b * -0.16666666666666666)))) - 0.5))))) - b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -1.65:
		tmp = 0.5
	else:
		tmp = 1.0 / (2.0 - ((a * ((b + 1.0) + (a * (((b * 0.5) - b) + ((a * (0.16666666666666666 + (((b * -0.8333333333333334) * (b * -0.16666666666666666)) / (b * -0.16666666666666666)))) - 0.5))))) - b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -1.65)
		tmp = 0.5;
	else
		tmp = Float64(1.0 / Float64(2.0 - Float64(Float64(a * Float64(Float64(b + 1.0) + Float64(a * Float64(Float64(Float64(b * 0.5) - b) + Float64(Float64(a * Float64(0.16666666666666666 + Float64(Float64(Float64(b * -0.8333333333333334) * Float64(b * -0.16666666666666666)) / Float64(b * -0.16666666666666666)))) - 0.5))))) - b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -1.65)
		tmp = 0.5;
	else
		tmp = 1.0 / (2.0 - ((a * ((b + 1.0) + (a * (((b * 0.5) - b) + ((a * (0.16666666666666666 + (((b * -0.8333333333333334) * (b * -0.16666666666666666)) / (b * -0.16666666666666666)))) - 0.5))))) - b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -1.65], 0.5, N[(1.0 / N[(2.0 - N[(N[(a * N[(N[(b + 1.0), $MachinePrecision] + N[(a * N[(N[(N[(b * 0.5), $MachinePrecision] - b), $MachinePrecision] + N[(N[(a * N[(0.16666666666666666 + N[(N[(N[(b * -0.8333333333333334), $MachinePrecision] * N[(b * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] / N[(b * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.6499999999999999

   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%

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

      3. associate-/r/ 97.8%

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

      4. remove-double-neg 97.8%

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

      5. unsub-neg 97.8%

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

      6. div-sub 97.8%

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

      7. *-lft-identity 97.8%

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

      8. associate-*l/ 97.8%

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

      9. lft-mult-inverse 97.8%

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

      10. sub-neg 97.8%

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

      11. distribute-frac-neg 97.8%

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

      12. remove-double-neg 97.8%

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

      13. div-exp 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0 99.0%

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

   6. Taylor expanded in b around 0 18.8%

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

   if -1.6499999999999999 < b

   1. Initial program 99.5%

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

      1. *-lft-identity 99.5%

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

      2. associate-*l/ 99.5%

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

      3. associate-/r/ 99.5%

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

      4. remove-double-neg 99.5%

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

      5. unsub-neg 99.5%

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

      6. div-sub 66.2%

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

      7. *-lft-identity 66.2%

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

      8. associate-*l/ 66.2%

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

      9. lft-mult-inverse 100.0%

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

      10. sub-neg 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. remove-double-neg 100.0%

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

      13. div-exp 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around 0 67.8%

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

      1. distribute-rgt1-in 78.7%

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

      2. rec-exp 78.7%

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

      3. associate-*r/ 78.7%

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

      4. *-rgt-identity 78.7%

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

      5. +-commutative 78.7%

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

   7. Simplified 78.7%

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

   8. Taylor expanded in a around 0 68.4%

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

      1. flip-+ 61.5%

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

      2. mul-1-neg 61.5%

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

      3. distribute-rgt-out 61.5%

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

      4. metadata-eval 61.5%

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

      5. *-commutative 61.5%

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

      6. mul-1-neg 61.5%

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

      7. distribute-rgt-out 61.5%

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

      8. metadata-eval 61.5%

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

      9. *-commutative 61.5%

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

      10. sqr-neg 61.5%

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

   10. Applied egg-rr 61.5%

       \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(\left(0.5 + -1 \cdot \left(a \cdot \left(0.16666666666666666 + \color{blue}{\left(\frac{{\left(b \cdot -0.5\right)}^{2}}{b \cdot -0.5 - b \cdot -0.3333333333333333} - \frac{{\left(b \cdot -0.3333333333333333\right)}^{2}}{b \cdot -0.5 - b \cdot -0.3333333333333333}\right)}\right)\right)\right) - \left(-1 \cdot b + 0.5 \cdot b\right)\right) - \left(1 + b\right)\right)\right)} \]
   11. Step-by-step derivation

       1. div-sub 61.5%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(\left(0.5 + -1 \cdot \left(a \cdot \left(0.16666666666666666 + \color{blue}{\frac{{\left(b \cdot -0.5\right)}^{2} - {\left(b \cdot -0.3333333333333333\right)}^{2}}{b \cdot -0.5 - b \cdot -0.3333333333333333}}\right)\right)\right) - \left(-1 \cdot b + 0.5 \cdot b\right)\right) - \left(1 + b\right)\right)\right)} \]

       2. unpow2 61.5%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(\left(0.5 + -1 \cdot \left(a \cdot \left(0.16666666666666666 + \frac{\color{blue}{\left(b \cdot -0.5\right) \cdot \left(b \cdot -0.5\right)} - {\left(b \cdot -0.3333333333333333\right)}^{2}}{b \cdot -0.5 - b \cdot -0.3333333333333333}\right)\right)\right) - \left(-1 \cdot b + 0.5 \cdot b\right)\right) - \left(1 + b\right)\right)\right)} \]

       3. unpow2 61.5%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(\left(0.5 + -1 \cdot \left(a \cdot \left(0.16666666666666666 + \frac{\left(b \cdot -0.5\right) \cdot \left(b \cdot -0.5\right) - \color{blue}{\left(b \cdot -0.3333333333333333\right) \cdot \left(b \cdot -0.3333333333333333\right)}}{b \cdot -0.5 - b \cdot -0.3333333333333333}\right)\right)\right) - \left(-1 \cdot b + 0.5 \cdot b\right)\right) - \left(1 + b\right)\right)\right)} \]

       4. difference-of-squares 78.6%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(\left(0.5 + -1 \cdot \left(a \cdot \left(0.16666666666666666 + \frac{\color{blue}{\left(b \cdot -0.5 + b \cdot -0.3333333333333333\right) \cdot \left(b \cdot -0.5 - b \cdot -0.3333333333333333\right)}}{b \cdot -0.5 - b \cdot -0.3333333333333333}\right)\right)\right) - \left(-1 \cdot b + 0.5 \cdot b\right)\right) - \left(1 + b\right)\right)\right)} \]

       5. +-commutative 78.6%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(\left(0.5 + -1 \cdot \left(a \cdot \left(0.16666666666666666 + \frac{\color{blue}{\left(b \cdot -0.3333333333333333 + b \cdot -0.5\right)} \cdot \left(b \cdot -0.5 - b \cdot -0.3333333333333333\right)}{b \cdot -0.5 - b \cdot -0.3333333333333333}\right)\right)\right) - \left(-1 \cdot b + 0.5 \cdot b\right)\right) - \left(1 + b\right)\right)\right)} \]

       6. distribute-lft-out 78.6%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(\left(0.5 + -1 \cdot \left(a \cdot \left(0.16666666666666666 + \frac{\color{blue}{\left(b \cdot \left(-0.3333333333333333 + -0.5\right)\right)} \cdot \left(b \cdot -0.5 - b \cdot -0.3333333333333333\right)}{b \cdot -0.5 - b \cdot -0.3333333333333333}\right)\right)\right) - \left(-1 \cdot b + 0.5 \cdot b\right)\right) - \left(1 + b\right)\right)\right)} \]

       7. metadata-eval 78.6%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(\left(0.5 + -1 \cdot \left(a \cdot \left(0.16666666666666666 + \frac{\left(b \cdot \color{blue}{-0.8333333333333334}\right) \cdot \left(b \cdot -0.5 - b \cdot -0.3333333333333333\right)}{b \cdot -0.5 - b \cdot -0.3333333333333333}\right)\right)\right) - \left(-1 \cdot b + 0.5 \cdot b\right)\right) - \left(1 + b\right)\right)\right)} \]

       8. distribute-lft-out-- 78.6%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(\left(0.5 + -1 \cdot \left(a \cdot \left(0.16666666666666666 + \frac{\left(b \cdot -0.8333333333333334\right) \cdot \color{blue}{\left(b \cdot \left(-0.5 - -0.3333333333333333\right)\right)}}{b \cdot -0.5 - b \cdot -0.3333333333333333}\right)\right)\right) - \left(-1 \cdot b + 0.5 \cdot b\right)\right) - \left(1 + b\right)\right)\right)} \]

       9. metadata-eval 78.6%

          \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(\left(0.5 + -1 \cdot \left(a \cdot \left(0.16666666666666666 + \frac{\left(b \cdot -0.8333333333333334\right) \cdot \left(b \cdot \color{blue}{-0.16666666666666666}\right)}{b \cdot -0.5 - b \cdot -0.3333333333333333}\right)\right)\right) - \left(-1 \cdot b + 0.5 \cdot b\right)\right) - \left(1 + b\right)\right)\right)} \]

       10. *-commutative 78.6%

           \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(\left(0.5 + -1 \cdot \left(a \cdot \left(0.16666666666666666 + \frac{\left(b \cdot -0.8333333333333334\right) \cdot \color{blue}{\left(-0.16666666666666666 \cdot b\right)}}{b \cdot -0.5 - b \cdot -0.3333333333333333}\right)\right)\right) - \left(-1 \cdot b + 0.5 \cdot b\right)\right) - \left(1 + b\right)\right)\right)} \]

       11. distribute-lft-out-- 78.6%

           \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(\left(0.5 + -1 \cdot \left(a \cdot \left(0.16666666666666666 + \frac{\left(b \cdot -0.8333333333333334\right) \cdot \left(-0.16666666666666666 \cdot b\right)}{\color{blue}{b \cdot \left(-0.5 - -0.3333333333333333\right)}}\right)\right)\right) - \left(-1 \cdot b + 0.5 \cdot b\right)\right) - \left(1 + b\right)\right)\right)} \]

       12. metadata-eval 78.6%

           \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(\left(0.5 + -1 \cdot \left(a \cdot \left(0.16666666666666666 + \frac{\left(b \cdot -0.8333333333333334\right) \cdot \left(-0.16666666666666666 \cdot b\right)}{b \cdot \color{blue}{-0.16666666666666666}}\right)\right)\right) - \left(-1 \cdot b + 0.5 \cdot b\right)\right) - \left(1 + b\right)\right)\right)} \]

       13. *-commutative 78.6%

           \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(\left(0.5 + -1 \cdot \left(a \cdot \left(0.16666666666666666 + \frac{\left(b \cdot -0.8333333333333334\right) \cdot \left(-0.16666666666666666 \cdot b\right)}{\color{blue}{-0.16666666666666666 \cdot b}}\right)\right)\right) - \left(-1 \cdot b + 0.5 \cdot b\right)\right) - \left(1 + b\right)\right)\right)} \]

   12. Simplified 78.6%

       \[\leadsto \frac{1}{2 + \left(b + a \cdot \left(a \cdot \left(\left(0.5 + -1 \cdot \left(a \cdot \left(0.16666666666666666 + \color{blue}{\frac{\left(b \cdot -0.8333333333333334\right) \cdot \left(-0.16666666666666666 \cdot b\right)}{-0.16666666666666666 \cdot b}}\right)\right)\right) - \left(-1 \cdot b + 0.5 \cdot b\right)\right) - \left(1 + b\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.65:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \left(a \cdot \left(\left(b + 1\right) + a \cdot \left(\left(b \cdot 0.5 - b\right) + \left(a \cdot \left(0.16666666666666666 + \frac{\left(b \cdot -0.8333333333333334\right) \cdot \left(b \cdot -0.16666666666666666\right)}{b \cdot -0.16666666666666666}\right) - 0.5\right)\right)\right) - b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.0% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)\\ \mathbf{if}\;b \leq -1.12 \cdot 10^{-296}:\\ \;\;\;\;\frac{1}{2 + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(b + b \cdot \left(t\_0 + \frac{a \cdot \left(-1 + a \cdot 0.5\right)}{b}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (+ -1.0 (* a (+ 0.5 (* a -0.16666666666666666)))))))
   (if (<= b -1.12e-296)
     (/ 1.0 (+ 2.0 t_0))
     (/ 1.0 (+ 2.0 (+ b (* b (+ t_0 (/ (* a (+ -1.0 (* a 0.5))) b)))))))))

C

double code(double a, double b) {
	double t_0 = a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))));
	double tmp;
	if (b <= -1.12e-296) {
		tmp = 1.0 / (2.0 + t_0);
	} else {
		tmp = 1.0 / (2.0 + (b + (b * (t_0 + ((a * (-1.0 + (a * 0.5))) / 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 = a * ((-1.0d0) + (a * (0.5d0 + (a * (-0.16666666666666666d0)))))
    if (b <= (-1.12d-296)) then
        tmp = 1.0d0 / (2.0d0 + t_0)
    else
        tmp = 1.0d0 / (2.0d0 + (b + (b * (t_0 + ((a * ((-1.0d0) + (a * 0.5d0))) / b)))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))));
	double tmp;
	if (b <= -1.12e-296) {
		tmp = 1.0 / (2.0 + t_0);
	} else {
		tmp = 1.0 / (2.0 + (b + (b * (t_0 + ((a * (-1.0 + (a * 0.5))) / b)))));
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))
	tmp = 0
	if b <= -1.12e-296:
		tmp = 1.0 / (2.0 + t_0)
	else:
		tmp = 1.0 / (2.0 + (b + (b * (t_0 + ((a * (-1.0 + (a * 0.5))) / b)))))
	return tmp

Julia

function code(a, b)
	t_0 = Float64(a * Float64(-1.0 + Float64(a * Float64(0.5 + Float64(a * -0.16666666666666666)))))
	tmp = 0.0
	if (b <= -1.12e-296)
		tmp = Float64(1.0 / Float64(2.0 + t_0));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b + Float64(b * Float64(t_0 + Float64(Float64(a * Float64(-1.0 + Float64(a * 0.5))) / b))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))));
	tmp = 0.0;
	if (b <= -1.12e-296)
		tmp = 1.0 / (2.0 + t_0);
	else
		tmp = 1.0 / (2.0 + (b + (b * (t_0 + ((a * (-1.0 + (a * 0.5))) / b)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * N[(-1.0 + N[(a * N[(0.5 + N[(a * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.12e-296], N[(1.0 / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b + N[(b * N[(t$95$0 + N[(N[(a * N[(-1.0 + N[(a * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.11999999999999994e-296

   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%

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

      3. associate-/r/ 98.2%

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

      4. remove-double-neg 98.2%

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

      5. unsub-neg 98.2%

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

      6. div-sub 79.6%

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

      7. *-lft-identity 79.6%

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

      8. associate-*l/ 79.6%

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

      9. lft-mult-inverse 99.1%

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

      10. sub-neg 99.1%

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

      11. distribute-frac-neg 99.1%

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

      12. remove-double-neg 99.1%

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

      13. div-exp 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around 0 67.9%

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

   6. Taylor expanded in a around 0 63.1%

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

   if -1.11999999999999994e-296 < b

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/r/ 100.0%

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

      4. remove-double-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. div-sub 65.2%

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

      7. *-lft-identity 65.2%

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

      8. associate-*l/ 65.2%

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

      9. lft-mult-inverse 100.0%

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

      10. sub-neg 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. remove-double-neg 100.0%

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

      13. div-exp 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around 0 66.9%

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

      1. distribute-rgt1-in 67.6%

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

      2. rec-exp 67.6%

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

      3. associate-*r/ 67.6%

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

      4. *-rgt-identity 67.6%

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

      5. +-commutative 67.6%

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

   7. Simplified 67.6%

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

   8. Taylor expanded in a around 0 56.1%

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

   9. Taylor expanded in b around inf 53.3%

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

   10. Taylor expanded in b around inf 58.8%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.12 \cdot 10^{-296}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(b + b \cdot \left(a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right) + \frac{a \cdot \left(-1 + a \cdot 0.5\right)}{b}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.3% accurate, 16.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -1

   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%

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

      3. associate-/r/ 97.8%

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

      4. remove-double-neg 97.8%

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

      5. unsub-neg 97.8%

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

      6. div-sub 97.8%

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

      7. *-lft-identity 97.8%

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

      8. associate-*l/ 97.8%

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

      9. lft-mult-inverse 97.8%

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

      10. sub-neg 97.8%

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

      11. distribute-frac-neg 97.8%

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

      12. remove-double-neg 97.8%

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

      13. div-exp 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0 99.0%

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

   6. Taylor expanded in b around 0 18.8%

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

   if -1 < b

   1. Initial program 99.5%

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

      1. *-lft-identity 99.5%

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

      2. associate-*l/ 99.5%

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

      3. associate-/r/ 99.5%

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

      4. remove-double-neg 99.5%

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

      5. unsub-neg 99.5%

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

      6. div-sub 66.2%

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

      7. *-lft-identity 66.2%

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

      8. associate-*l/ 66.2%

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

      9. lft-mult-inverse 100.0%

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

      10. sub-neg 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. remove-double-neg 100.0%

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

      13. div-exp 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around 0 67.8%

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

      1. distribute-rgt1-in 78.7%

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

      2. rec-exp 78.7%

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

      3. associate-*r/ 78.7%

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

      4. *-rgt-identity 78.7%

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

      5. +-commutative 78.7%

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

   7. Simplified 78.7%

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

   8. Taylor expanded in a around 0 51.2%

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

      1. associate-+r+ 51.2%

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

      2. +-commutative 51.2%

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

      3. mul-1-neg 51.2%

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

      4. distribute-rgt-neg-in 51.2%

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

      5. neg-sub0 51.2%

         \[\leadsto \frac{1}{\left(b + 2\right) + a \cdot \color{blue}{\left(0 - \left(1 + b\right)\right)}} \]

      6. associate--r+ 51.2%

         \[\leadsto \frac{1}{\left(b + 2\right) + a \cdot \color{blue}{\left(\left(0 - 1\right) - b\right)}} \]

      7. metadata-eval 51.2%

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

   10. Simplified 51.2%

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

   11. Taylor expanded in b around inf 62.4%

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

       1. mul-1-neg 62.4%

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

       2. unsub-neg 62.4%

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

       3. mul-1-neg 62.4%

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

   13. Simplified 62.4%

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

4. Final simplification 54.6%

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

Alternative 8: 57.1% accurate, 20.3× speedup

Math

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

FPCore

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

C

double code(double a, double b) {
	return 1.0 / (2.0 + (a * (-1.0 + (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 * ((-1.0d0) + (a * (0.5d0 + (a * (-0.16666666666666666d0)))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. *-lft-identity 99.2%

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

   2. associate-*l/ 99.2%

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

   3. associate-/r/ 99.2%

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

   4. remove-double-neg 99.2%

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

   5. unsub-neg 99.2%

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

   6. div-sub 71.8%

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

   7. *-lft-identity 71.8%

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

   8. associate-*l/ 71.8%

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

   9. lft-mult-inverse 99.6%

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

   10. sub-neg 99.6%

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

   11. distribute-frac-neg 99.6%

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

   12. remove-double-neg 99.6%

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

   13. div-exp 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in b around 0 67.0%

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

6. Taylor expanded in a around 0 57.8%

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

7. Final simplification 57.8%

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

Alternative 9: 52.5% accurate, 21.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{-1}{b \cdot \left(a + \frac{a}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.1e-6) (/ -1.0 (* b (+ a (/ a b)))) (+ 0.5 (* a 0.25))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.1e-6) {
		tmp = -1.0 / (b * (a + (a / b)));
	} else {
		tmp = 0.5 + (a * 0.25);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-1.1d-6)) then
        tmp = (-1.0d0) / (b * (a + (a / b)))
    else
        tmp = 0.5d0 + (a * 0.25d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1.1e-6) {
		tmp = -1.0 / (b * (a + (a / b)));
	} else {
		tmp = 0.5 + (a * 0.25);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.1e-6:
		tmp = -1.0 / (b * (a + (a / b)))
	else:
		tmp = 0.5 + (a * 0.25)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.1e-6)
		tmp = Float64(-1.0 / Float64(b * Float64(a + Float64(a / b))));
	else
		tmp = Float64(0.5 + Float64(a * 0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.1e-6)
		tmp = -1.0 / (b * (a + (a / b)));
	else
		tmp = 0.5 + (a * 0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.1e-6], N[(-1.0 / N[(b * N[(a + N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.1000000000000001e-6

   1. Initial program 98.7%

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

      1. *-lft-identity 98.7%

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

      2. associate-*l/ 98.7%

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

      3. associate-/r/ 98.7%

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

      4. remove-double-neg 98.7%

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

      5. unsub-neg 98.7%

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

      6. div-sub 5.3%

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

      7. *-lft-identity 5.3%

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

      8. associate-*l/ 5.3%

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

      9. lft-mult-inverse 98.7%

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

      10. sub-neg 98.7%

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

      11. distribute-frac-neg 98.7%

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

      12. remove-double-neg 98.7%

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

      13. div-exp 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around 0 63.1%

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

      1. distribute-rgt1-in 93.8%

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

      2. rec-exp 93.8%

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

      3. associate-*r/ 93.8%

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

      4. *-rgt-identity 93.8%

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

      5. +-commutative 93.8%

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

   7. Simplified 93.8%

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

   8. Taylor expanded in a around 0 19.4%

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

      1. associate-+r+ 19.4%

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

      2. +-commutative 19.4%

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

      3. mul-1-neg 19.4%

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

      4. distribute-rgt-neg-in 19.4%

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

      5. neg-sub0 19.4%

         \[\leadsto \frac{1}{\left(b + 2\right) + a \cdot \color{blue}{\left(0 - \left(1 + b\right)\right)}} \]

      6. associate--r+ 19.4%

         \[\leadsto \frac{1}{\left(b + 2\right) + a \cdot \color{blue}{\left(\left(0 - 1\right) - b\right)}} \]

      7. metadata-eval 19.4%

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

   10. Simplified 19.4%

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

   11. Taylor expanded in a around inf 19.4%

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

   12. Taylor expanded in b around inf 50.7%

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

   if -1.1000000000000001e-6 < a

   1. Initial program 99.4%

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

      1. *-lft-identity 99.4%

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

      2. associate-*l/ 99.4%

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

      3. associate-/r/ 99.4%

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

      4. remove-double-neg 99.4%

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

      5. unsub-neg 99.4%

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

      6. div-sub 99.4%

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

      7. *-lft-identity 99.4%

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

      8. associate-*l/ 99.4%

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

      9. lft-mult-inverse 100.0%

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

      10. sub-neg 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. remove-double-neg 100.0%

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

      13. div-exp 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around 0 56.0%

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

   6. Taylor expanded in a around 0 55.1%

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

      1. *-commutative 55.1%

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

   8. Simplified 55.1%

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

Alternative 10: 44.8% accurate, 25.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < 2

   1. Initial program 98.9%

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

      1. *-lft-identity 98.9%

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

      2. associate-*l/ 98.9%

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

      3. associate-/r/ 98.9%

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

      4. remove-double-neg 98.9%

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

      5. unsub-neg 98.9%

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

      6. div-sub 74.6%

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

      7. *-lft-identity 74.6%

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

      8. associate-*l/ 74.6%

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

      9. lft-mult-inverse 99.4%

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

      10. sub-neg 99.4%

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

      11. distribute-frac-neg 99.4%

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

      12. remove-double-neg 99.4%

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

      13. div-exp 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around 0 78.4%

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

   6. Taylor expanded in a around 0 54.3%

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

      1. neg-mul-1 54.3%

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

      2. unsub-neg 54.3%

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

   8. Simplified 54.3%

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

   if 2 < b

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/r/ 100.0%

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

      4. remove-double-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. div-sub 64.8%

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

      7. *-lft-identity 64.8%

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

      8. associate-*l/ 64.8%

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

      9. lft-mult-inverse 100.0%

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

      10. sub-neg 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. remove-double-neg 100.0%

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

      13. div-exp 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around 0 38.7%

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

      1. distribute-rgt1-in 38.7%

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

      2. rec-exp 38.7%

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

      3. associate-*r/ 38.7%

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

      4. *-rgt-identity 38.7%

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

      5. +-commutative 38.7%

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

   7. Simplified 38.7%

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

   8. Taylor expanded in a around 0 20.2%

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

      1. associate-+r+ 20.2%

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

      2. +-commutative 20.2%

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

      3. mul-1-neg 20.2%

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

      4. distribute-rgt-neg-in 20.2%

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

      5. neg-sub0 20.2%

         \[\leadsto \frac{1}{\left(b + 2\right) + a \cdot \color{blue}{\left(0 - \left(1 + b\right)\right)}} \]

      6. associate--r+ 20.2%

         \[\leadsto \frac{1}{\left(b + 2\right) + a \cdot \color{blue}{\left(\left(0 - 1\right) - b\right)}} \]

      7. metadata-eval 20.2%

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

   10. Simplified 20.2%

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

   11. Taylor expanded in b around inf 20.2%

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

       1. mul-1-neg 20.2%

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

       2. unsub-neg 20.2%

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

   13. Simplified 20.2%

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

4. Final simplification 44.9%

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

Alternative 11: 44.4% accurate, 25.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{-1}{a \cdot \left(b + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.1e-6) (/ -1.0 (* a (+ b 1.0))) (+ 0.5 (* a 0.25))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.1e-6) {
		tmp = -1.0 / (a * (b + 1.0));
	} 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 <= (-1.1d-6)) then
        tmp = (-1.0d0) / (a * (b + 1.0d0))
    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 <= -1.1e-6) {
		tmp = -1.0 / (a * (b + 1.0));
	} else {
		tmp = 0.5 + (a * 0.25);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.1e-6:
		tmp = -1.0 / (a * (b + 1.0))
	else:
		tmp = 0.5 + (a * 0.25)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.1e-6)
		tmp = Float64(-1.0 / Float64(a * Float64(b + 1.0)));
	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 <= -1.1e-6)
		tmp = -1.0 / (a * (b + 1.0));
	else
		tmp = 0.5 + (a * 0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.1e-6], N[(-1.0 / N[(a * N[(b + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.1000000000000001e-6

   1. Initial program 98.7%

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

      1. *-lft-identity 98.7%

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

      2. associate-*l/ 98.7%

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

      3. associate-/r/ 98.7%

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

      4. remove-double-neg 98.7%

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

      5. unsub-neg 98.7%

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

      6. div-sub 5.3%

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

      7. *-lft-identity 5.3%

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

      8. associate-*l/ 5.3%

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

      9. lft-mult-inverse 98.7%

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

      10. sub-neg 98.7%

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

      11. distribute-frac-neg 98.7%

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

      12. remove-double-neg 98.7%

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

      13. div-exp 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around 0 63.1%

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

      1. distribute-rgt1-in 93.8%

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

      2. rec-exp 93.8%

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

      3. associate-*r/ 93.8%

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

      4. *-rgt-identity 93.8%

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

      5. +-commutative 93.8%

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

   7. Simplified 93.8%

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

   8. Taylor expanded in a around 0 19.4%

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

      1. associate-+r+ 19.4%

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

      2. +-commutative 19.4%

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

      3. mul-1-neg 19.4%

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

      4. distribute-rgt-neg-in 19.4%

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

      5. neg-sub0 19.4%

         \[\leadsto \frac{1}{\left(b + 2\right) + a \cdot \color{blue}{\left(0 - \left(1 + b\right)\right)}} \]

      6. associate--r+ 19.4%

         \[\leadsto \frac{1}{\left(b + 2\right) + a \cdot \color{blue}{\left(\left(0 - 1\right) - b\right)}} \]

      7. metadata-eval 19.4%

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

   10. Simplified 19.4%

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

   11. Taylor expanded in a around inf 19.4%

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

   if -1.1000000000000001e-6 < a

   1. Initial program 99.4%

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

      1. *-lft-identity 99.4%

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

      2. associate-*l/ 99.4%

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

      3. associate-/r/ 99.4%

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

      4. remove-double-neg 99.4%

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

      5. unsub-neg 99.4%

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

      6. div-sub 99.4%

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

      7. *-lft-identity 99.4%

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

      8. associate-*l/ 99.4%

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

      9. lft-mult-inverse 100.0%

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

      10. sub-neg 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. remove-double-neg 100.0%

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

      13. div-exp 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around 0 56.0%

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

   6. Taylor expanded in a around 0 55.1%

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

      1. *-commutative 55.1%

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

   8. Simplified 55.1%

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

4. Final simplification 44.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{-1}{a \cdot \left(b + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 44.1% accurate, 30.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6 \cdot 10^{+87}:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{b \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 6e+87) (/ 1.0 (- 2.0 a)) (/ -1.0 (* b a))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 6e+87) {
		tmp = 1.0 / (2.0 - a);
	} else {
		tmp = -1.0 / (b * 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 <= 6d+87) then
        tmp = 1.0d0 / (2.0d0 - a)
    else
        tmp = (-1.0d0) / (b * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 6e+87) {
		tmp = 1.0 / (2.0 - a);
	} else {
		tmp = -1.0 / (b * a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 6e+87:
		tmp = 1.0 / (2.0 - a)
	else:
		tmp = -1.0 / (b * a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 6e+87)
		tmp = Float64(1.0 / Float64(2.0 - a));
	else
		tmp = Float64(-1.0 / Float64(b * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 6e+87)
		tmp = 1.0 / (2.0 - a);
	else
		tmp = -1.0 / (b * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 6e+87], N[(1.0 / N[(2.0 - a), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(b * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.9999999999999998e87

   1. Initial program 99.0%

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

      1. *-lft-identity 99.0%

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

      2. associate-*l/ 99.0%

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

      3. associate-/r/ 99.0%

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

      4. remove-double-neg 99.0%

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

      5. unsub-neg 99.0%

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

      6. div-sub 74.7%

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

      7. *-lft-identity 74.7%

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

      8. associate-*l/ 74.7%

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

      9. lft-mult-inverse 99.5%

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

      10. sub-neg 99.5%

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

      11. distribute-frac-neg 99.5%

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

      12. remove-double-neg 99.5%

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

      13. div-exp 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around 0 73.1%

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

   6. Taylor expanded in a around 0 49.1%

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

      1. neg-mul-1 49.1%

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

      2. unsub-neg 49.1%

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

   8. Simplified 49.1%

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

   if 5.9999999999999998e87 < b

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/r/ 100.0%

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

      4. remove-double-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. div-sub 60.0%

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

      7. *-lft-identity 60.0%

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

      8. associate-*l/ 60.0%

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

      9. lft-mult-inverse 100.0%

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

      10. sub-neg 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. remove-double-neg 100.0%

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

      13. div-exp 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around 0 43.7%

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

      1. distribute-rgt1-in 43.7%

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

      2. rec-exp 43.7%

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

      3. associate-*r/ 43.7%

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

      4. *-rgt-identity 43.7%

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

      5. +-commutative 43.7%

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

   7. Simplified 43.7%

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

   8. Taylor expanded in a around 0 27.1%

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

      1. associate-+r+ 27.1%

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

      2. +-commutative 27.1%

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

      3. mul-1-neg 27.1%

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

      4. distribute-rgt-neg-in 27.1%

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

      5. neg-sub0 27.1%

         \[\leadsto \frac{1}{\left(b + 2\right) + a \cdot \color{blue}{\left(0 - \left(1 + b\right)\right)}} \]

      6. associate--r+ 27.1%

         \[\leadsto \frac{1}{\left(b + 2\right) + a \cdot \color{blue}{\left(\left(0 - 1\right) - b\right)}} \]

      7. metadata-eval 27.1%

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

   10. Simplified 27.1%

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

   11. Taylor expanded in a around inf 25.8%

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

   12. Taylor expanded in b around inf 25.8%

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

       1. *-commutative 25.8%

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

   14. Simplified 25.8%

       \[\leadsto \color{blue}{\frac{-1}{b \cdot a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 44.0% accurate, 30.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{-1}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.1e-6) (/ -1.0 (* b a)) (+ 0.5 (* a 0.25))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.1e-6) {
		tmp = -1.0 / (b * 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 <= (-1.1d-6)) then
        tmp = (-1.0d0) / (b * 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 <= -1.1e-6) {
		tmp = -1.0 / (b * a);
	} else {
		tmp = 0.5 + (a * 0.25);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.1e-6:
		tmp = -1.0 / (b * a)
	else:
		tmp = 0.5 + (a * 0.25)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.1e-6)
		tmp = Float64(-1.0 / Float64(b * 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 <= -1.1e-6)
		tmp = -1.0 / (b * a);
	else
		tmp = 0.5 + (a * 0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.1e-6], N[(-1.0 / N[(b * a), $MachinePrecision]), $MachinePrecision], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.1000000000000001e-6

   1. Initial program 98.7%

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

      1. *-lft-identity 98.7%

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

      2. associate-*l/ 98.7%

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

      3. associate-/r/ 98.7%

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

      4. remove-double-neg 98.7%

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

      5. unsub-neg 98.7%

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

      6. div-sub 5.3%

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

      7. *-lft-identity 5.3%

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

      8. associate-*l/ 5.3%

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

      9. lft-mult-inverse 98.7%

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

      10. sub-neg 98.7%

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

      11. distribute-frac-neg 98.7%

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

      12. remove-double-neg 98.7%

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

      13. div-exp 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around 0 63.1%

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

      1. distribute-rgt1-in 93.8%

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

      2. rec-exp 93.8%

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

      3. associate-*r/ 93.8%

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

      4. *-rgt-identity 93.8%

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

      5. +-commutative 93.8%

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

   7. Simplified 93.8%

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

   8. Taylor expanded in a around 0 19.4%

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

      1. associate-+r+ 19.4%

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

      2. +-commutative 19.4%

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

      3. mul-1-neg 19.4%

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

      4. distribute-rgt-neg-in 19.4%

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

      5. neg-sub0 19.4%

         \[\leadsto \frac{1}{\left(b + 2\right) + a \cdot \color{blue}{\left(0 - \left(1 + b\right)\right)}} \]

      6. associate--r+ 19.4%

         \[\leadsto \frac{1}{\left(b + 2\right) + a \cdot \color{blue}{\left(\left(0 - 1\right) - b\right)}} \]

      7. metadata-eval 19.4%

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

   10. Simplified 19.4%

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

   11. Taylor expanded in a around inf 19.4%

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

   12. Taylor expanded in b around inf 18.2%

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

       1. *-commutative 18.2%

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

   14. Simplified 18.2%

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

   if -1.1000000000000001e-6 < a

   1. Initial program 99.4%

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

      1. *-lft-identity 99.4%

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

      2. associate-*l/ 99.4%

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

      3. associate-/r/ 99.4%

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

      4. remove-double-neg 99.4%

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

      5. unsub-neg 99.4%

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

      6. div-sub 99.4%

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

      7. *-lft-identity 99.4%

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

      8. associate-*l/ 99.4%

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

      9. lft-mult-inverse 100.0%

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

      10. sub-neg 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. remove-double-neg 100.0%

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

      13. div-exp 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around 0 56.0%

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

   6. Taylor expanded in a around 0 55.1%

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

      1. *-commutative 55.1%

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

   8. Simplified 55.1%

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

Alternative 14: 39.5% accurate, 61.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. *-lft-identity 99.2%

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

   2. associate-*l/ 99.2%

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

   3. associate-/r/ 99.2%

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

   4. remove-double-neg 99.2%

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

   5. unsub-neg 99.2%

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

   6. div-sub 71.8%

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

   7. *-lft-identity 71.8%

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

   8. associate-*l/ 71.8%

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

   9. lft-mult-inverse 99.6%

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

   10. sub-neg 99.6%

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

   11. distribute-frac-neg 99.6%

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

   12. remove-double-neg 99.6%

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

   13. div-exp 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in b around 0 67.0%

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

6. Taylor expanded in a around 0 39.6%

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

   1. *-commutative 39.6%

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

8. Simplified 39.6%

   \[\leadsto \color{blue}{0.5 + a \cdot 0.25} \]
9. Add Preprocessing

Alternative 15: 39.3% accurate, 305.0× speedup

Math

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

FPCore

(FPCore (a b) :precision binary64 0.5)

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return 0.5

Julia

function code(a, b)
	return 0.5
end

MATLAB

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

Wolfram

code[a_, b_] := 0.5

Derivation

1. Initial program 99.2%

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

   1. *-lft-identity 99.2%

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

   2. associate-*l/ 99.2%

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

   3. associate-/r/ 99.2%

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

   4. remove-double-neg 99.2%

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

   5. unsub-neg 99.2%

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

   6. div-sub 71.8%

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

   7. *-lft-identity 71.8%

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

   8. associate-*l/ 71.8%

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

   9. lft-mult-inverse 99.6%

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

   10. sub-neg 99.6%

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

   11. distribute-frac-neg 99.6%

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

   12. remove-double-neg 99.6%

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

   13. div-exp 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in a around 0 81.7%

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

6. Taylor expanded in b around 0 39.5%

   \[\leadsto \color{blue}{0.5} \]
7. Add Preprocessing

Developer target: 100.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024088 
(FPCore (a b)
  :name "Quotient of sum of exps"
  :precision binary64

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

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