Quotient of sum of exps

Percentage Accurate: 98.6% → 100.0%

Time: 8.2s

Alternatives: 13

Speedup: 2.9×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

   1. *-lft-identity 98.8%

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

   2. associate-*l/ 98.8%

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

   3. associate-/r/ 98.8%

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

   4. remove-double-neg 98.8%

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

   5. unsub-neg 98.8%

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

   6. div-sub 71.1%

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

   7. *-lft-identity 71.1%

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

   8. associate-*l/ 71.1%

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

   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. div-exp 99.6%

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

   2. +-commutative 99.6%

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

   3. metadata-eval 99.6%

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

   4. sub-neg 99.6%

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

   5. add-exp-log 99.6%

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

   6. rec-exp 99.6%

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

   7. sub-neg 99.6%

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

   8. metadata-eval 99.6%

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

   9. +-commutative 99.6%

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

   10. log1p-define 99.6%

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

   11. div-exp 100.0%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 86.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -860:\\ \;\;\;\;1 + e^{b}\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{+96}:\\ \;\;\;\;\frac{1}{\left(1 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)\right) - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -860.0)
   (+ 1.0 (exp b))
   (if (<= b 1.95e+96)
     (/
      1.0
      (- (+ 1.0 (* a (+ -1.0 (* a (+ 0.5 (* a -0.16666666666666666)))))) -1.0))
     (/ 6.0 (pow b 3.0)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -860.0) {
		tmp = 1.0 + exp(b);
	} else if (b <= 1.95e+96) {
		tmp = 1.0 / ((1.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666)))))) - -1.0);
	} else {
		tmp = 6.0 / pow(b, 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-860.0d0)) then
        tmp = 1.0d0 + exp(b)
    else if (b <= 1.95d+96) then
        tmp = 1.0d0 / ((1.0d0 + (a * ((-1.0d0) + (a * (0.5d0 + (a * (-0.16666666666666666d0))))))) - (-1.0d0))
    else
        tmp = 6.0d0 / (b ** 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -860.0) {
		tmp = 1.0 + Math.exp(b);
	} else if (b <= 1.95e+96) {
		tmp = 1.0 / ((1.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666)))))) - -1.0);
	} else {
		tmp = 6.0 / Math.pow(b, 3.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -860.0:
		tmp = 1.0 + math.exp(b)
	elif b <= 1.95e+96:
		tmp = 1.0 / ((1.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666)))))) - -1.0)
	else:
		tmp = 6.0 / math.pow(b, 3.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -860.0)
		tmp = Float64(1.0 + exp(b));
	elseif (b <= 1.95e+96)
		tmp = Float64(1.0 / Float64(Float64(1.0 + Float64(a * Float64(-1.0 + Float64(a * Float64(0.5 + Float64(a * -0.16666666666666666)))))) - -1.0));
	else
		tmp = Float64(6.0 / (b ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -860.0)
		tmp = 1.0 + exp(b);
	elseif (b <= 1.95e+96)
		tmp = 1.0 / ((1.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666)))))) - -1.0);
	else
		tmp = 6.0 / (b ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -860.0], N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.95e+96], N[(1.0 / N[(N[(1.0 + N[(a * N[(-1.0 + N[(a * N[(0.5 + N[(a * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], N[(6.0 / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -860

   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 100.0%

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

      7. *-lft-identity 100.0%

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

      8. associate-*l/ 100.0%

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

      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. Step-by-step derivation

      1. div-exp 100.0%

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

      2. +-commutative 100.0%

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

      3. metadata-eval 100.0%

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

      4. sub-neg 100.0%

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

      5. add-exp-log 100.0%

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

      6. rec-exp 100.0%

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

      7. sub-neg 100.0%

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

      8. metadata-eval 100.0%

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

      9. +-commutative 100.0%

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

      10. log1p-define 100.0%

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

      11. div-exp 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in a around 0 100.0%

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

      1. log1p-define 100.0%

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

   9. Simplified 100.0%

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

       1. add-sqr-sqrt 100.0%

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

       2. sqrt-unprod 100.0%

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

       3. sqr-neg 100.0%

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

       4. sqrt-unprod 100.0%

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

       5. add-sqr-sqrt 100.0%

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

       6. log1p-undefine 100.0%

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

       7. rem-exp-log 100.0%

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

       8. +-commutative 100.0%

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

   11. Applied egg-rr 100.0%

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

   if -860 < b < 1.95e96

   1. Initial program 98.1%

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

      1. *-lft-identity 98.1%

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

      2. associate-*l/ 98.1%

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

      3. associate-/r/ 98.1%

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

      4. +-commutative 98.1%

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

      5. remove-double-neg 98.1%

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

      6. sub-neg 98.1%

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

      7. div-sub 63.5%

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

      8. neg-mul-1 63.5%

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

      9. *-commutative 63.5%

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

      10. associate-*r/ 63.5%

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

      11. metadata-eval 63.5%

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

      12. distribute-neg-frac 63.5%

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

      13. exp-neg 63.5%

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

      14. distribute-rgt-neg-out 63.5%

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

      15. exp-neg 63.5%

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

      16. rgt-mult-inverse 99.3%

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

      17. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in b around 0 89.9%

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

   6. Taylor expanded in a around 0 75.6%

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

   if 1.95e96 < 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. +-commutative 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. div-sub 64.3%

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

      8. neg-mul-1 64.3%

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

      9. *-commutative 64.3%

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

      10. associate-*r/ 64.3%

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

      11. metadata-eval 64.3%

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

      12. distribute-neg-frac 64.3%

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

      13. exp-neg 64.3%

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

      14. distribute-rgt-neg-out 64.3%

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

      15. exp-neg 64.3%

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

      16. rgt-mult-inverse 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0 100.0%

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

   6. Taylor expanded in b around 0 97.9%

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

      1. *-commutative 97.9%

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

   8. Simplified 97.9%

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

   9. Taylor expanded in b around inf 97.9%

      \[\leadsto \color{blue}{\frac{6}{{b}^{3}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -860:\\ \;\;\;\;1 + e^{b}\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{+96}:\\ \;\;\;\;\frac{1}{\left(1 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)\right) - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{{b}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -114:\\ \;\;\;\;\frac{1}{e^{-a} - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -114.0) (/ 1.0 (- (exp (- a)) -1.0)) (/ 1.0 (- (exp b) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -114.0) {
		tmp = 1.0 / (exp(-a) - -1.0);
	} else {
		tmp = 1.0 / (exp(b) - -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 (a <= (-114.0d0)) then
        tmp = 1.0d0 / (exp(-a) - (-1.0d0))
    else
        tmp = 1.0d0 / (exp(b) - (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -114.0) {
		tmp = 1.0 / (Math.exp(-a) - -1.0);
	} else {
		tmp = 1.0 / (Math.exp(b) - -1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -114.0:
		tmp = 1.0 / (math.exp(-a) - -1.0)
	else:
		tmp = 1.0 / (math.exp(b) - -1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -114.0)
		tmp = Float64(1.0 / Float64(exp(Float64(-a)) - -1.0));
	else
		tmp = Float64(1.0 / Float64(exp(b) - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -114.0)
		tmp = 1.0 / (exp(-a) - -1.0);
	else
		tmp = 1.0 / (exp(b) - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -114.0], N[(1.0 / N[(N[Exp[(-a)], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Exp[b], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -114

   1. Initial program 98.6%

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

      1. *-lft-identity 98.6%

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

      2. associate-*l/ 98.6%

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

      3. associate-/r/ 98.6%

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

      4. +-commutative 98.6%

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

      5. remove-double-neg 98.6%

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

      6. sub-neg 98.6%

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

      7. div-sub 2.7%

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

      8. neg-mul-1 2.7%

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

      9. *-commutative 2.7%

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

      10. associate-*r/ 2.7%

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

      11. metadata-eval 2.7%

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

      12. distribute-neg-frac 2.7%

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

      13. exp-neg 2.7%

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

      14. distribute-rgt-neg-out 2.7%

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

      15. exp-neg 2.7%

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

      16. rgt-mult-inverse 98.6%

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

      17. metadata-eval 98.6%

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

   3. Simplified 98.6%

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

   5. Taylor expanded in b around 0 100.0%

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

      1. rec-exp 100.0%

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

   7. Simplified 100.0%

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

   if -114 < a

   1. Initial program 98.8%

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

      1. *-lft-identity 98.8%

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

      2. associate-*l/ 98.8%

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

      3. associate-/r/ 98.8%

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

      4. +-commutative 98.8%

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

      5. remove-double-neg 98.8%

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

      6. sub-neg 98.8%

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

      7. div-sub 98.8%

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

      8. neg-mul-1 98.8%

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

      9. *-commutative 98.8%

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

      10. associate-*r/ 98.8%

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

      11. metadata-eval 98.8%

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

      12. distribute-neg-frac 98.8%

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

      13. exp-neg 98.8%

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

      14. distribute-rgt-neg-out 98.8%

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

      15. exp-neg 98.8%

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

      16. rgt-mult-inverse 99.9%

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

      17. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in a around 0 97.9%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -114:\\ \;\;\;\;\frac{1}{e^{-a} - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -475000000:\\ \;\;\;\;\frac{e^{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -475000000.0) (/ (exp a) a) (/ 1.0 (- (exp b) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -475000000.0) {
		tmp = exp(a) / a;
	} else {
		tmp = 1.0 / (exp(b) - -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 (a <= (-475000000.0d0)) then
        tmp = exp(a) / a
    else
        tmp = 1.0d0 / (exp(b) - (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -475000000.0) {
		tmp = Math.exp(a) / a;
	} else {
		tmp = 1.0 / (Math.exp(b) - -1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -475000000.0:
		tmp = math.exp(a) / a
	else:
		tmp = 1.0 / (math.exp(b) - -1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -475000000.0)
		tmp = Float64(exp(a) / a);
	else
		tmp = Float64(1.0 / Float64(exp(b) - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -475000000.0)
		tmp = exp(a) / a;
	else
		tmp = 1.0 / (exp(b) - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -475000000.0], N[(N[Exp[a], $MachinePrecision] / a), $MachinePrecision], N[(1.0 / N[(N[Exp[b], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.75e8

   1. Initial program 98.6%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0 100.0%

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

   4. Taylor expanded in a around 0 100.0%

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

      1. +-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in a around inf 100.0%

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

   if -4.75e8 < 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. +-commutative 98.9%

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

      5. remove-double-neg 98.9%

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

      6. sub-neg 98.9%

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

      7. div-sub 98.3%

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

      8. neg-mul-1 98.3%

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

      9. *-commutative 98.3%

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

      10. associate-*r/ 98.3%

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

      11. metadata-eval 98.3%

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

      12. distribute-neg-frac 98.3%

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

      13. exp-neg 98.3%

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

      14. distribute-rgt-neg-out 98.3%

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

      15. exp-neg 98.3%

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

      16. rgt-mult-inverse 99.9%

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

      17. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in a around 0 96.9%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -475000000:\\ \;\;\;\;\frac{e^{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -860:\\ \;\;\;\;1 + e^{b}\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{+98}:\\ \;\;\;\;\frac{1}{\left(1 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)\right) - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -860.0)
   (+ 1.0 (exp b))
   (if (<= b 1.3e+98)
     (/
      1.0
      (- (+ 1.0 (* a (+ -1.0 (* a (+ 0.5 (* a -0.16666666666666666)))))) -1.0))
     (/ 1.0 (+ 2.0 (* b (+ 1.0 (* b (+ 0.5 (* b 0.16666666666666666))))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -860.0) {
		tmp = 1.0 + exp(b);
	} else if (b <= 1.3e+98) {
		tmp = 1.0 / ((1.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666)))))) - -1.0);
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-860.0d0)) then
        tmp = 1.0d0 + exp(b)
    else if (b <= 1.3d+98) then
        tmp = 1.0d0 / ((1.0d0 + (a * ((-1.0d0) + (a * (0.5d0 + (a * (-0.16666666666666666d0))))))) - (-1.0d0))
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (b * (0.5d0 + (b * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -860.0) {
		tmp = 1.0 + Math.exp(b);
	} else if (b <= 1.3e+98) {
		tmp = 1.0 / ((1.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666)))))) - -1.0);
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -860.0:
		tmp = 1.0 + math.exp(b)
	elif b <= 1.3e+98:
		tmp = 1.0 / ((1.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666)))))) - -1.0)
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -860.0)
		tmp = Float64(1.0 + exp(b));
	elseif (b <= 1.3e+98)
		tmp = Float64(1.0 / Float64(Float64(1.0 + Float64(a * Float64(-1.0 + Float64(a * Float64(0.5 + Float64(a * -0.16666666666666666)))))) - -1.0));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(b * Float64(0.5 + Float64(b * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -860.0)
		tmp = 1.0 + exp(b);
	elseif (b <= 1.3e+98)
		tmp = 1.0 / ((1.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666)))))) - -1.0);
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -860.0], N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.3e+98], N[(1.0 / N[(N[(1.0 + N[(a * N[(-1.0 + N[(a * N[(0.5 + N[(a * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(b * N[(0.5 + N[(b * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -860

   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 100.0%

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

      7. *-lft-identity 100.0%

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

      8. associate-*l/ 100.0%

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

      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. Step-by-step derivation

      1. div-exp 100.0%

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

      2. +-commutative 100.0%

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

      3. metadata-eval 100.0%

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

      4. sub-neg 100.0%

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

      5. add-exp-log 100.0%

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

      6. rec-exp 100.0%

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

      7. sub-neg 100.0%

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

      8. metadata-eval 100.0%

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

      9. +-commutative 100.0%

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

      10. log1p-define 100.0%

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

      11. div-exp 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in a around 0 100.0%

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

      1. log1p-define 100.0%

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

   9. Simplified 100.0%

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

       1. add-sqr-sqrt 100.0%

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

       2. sqrt-unprod 100.0%

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

       3. sqr-neg 100.0%

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

       4. sqrt-unprod 100.0%

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

       5. add-sqr-sqrt 100.0%

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

       6. log1p-undefine 100.0%

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

       7. rem-exp-log 100.0%

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

       8. +-commutative 100.0%

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

   11. Applied egg-rr 100.0%

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

   if -860 < b < 1.3e98

   1. Initial program 98.1%

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

      1. *-lft-identity 98.1%

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

      2. associate-*l/ 98.1%

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

      3. associate-/r/ 98.1%

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

      4. +-commutative 98.1%

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

      5. remove-double-neg 98.1%

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

      6. sub-neg 98.1%

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

      7. div-sub 63.5%

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

      8. neg-mul-1 63.5%

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

      9. *-commutative 63.5%

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

      10. associate-*r/ 63.5%

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

      11. metadata-eval 63.5%

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

      12. distribute-neg-frac 63.5%

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

      13. exp-neg 63.5%

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

      14. distribute-rgt-neg-out 63.5%

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

      15. exp-neg 63.5%

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

      16. rgt-mult-inverse 99.3%

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

      17. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in b around 0 89.9%

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

   6. Taylor expanded in a around 0 75.6%

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

   if 1.3e98 < 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. +-commutative 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. div-sub 64.3%

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

      8. neg-mul-1 64.3%

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

      9. *-commutative 64.3%

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

      10. associate-*r/ 64.3%

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

      11. metadata-eval 64.3%

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

      12. distribute-neg-frac 64.3%

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

      13. exp-neg 64.3%

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

      14. distribute-rgt-neg-out 64.3%

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

      15. exp-neg 64.3%

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

      16. rgt-mult-inverse 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0 100.0%

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

   6. Taylor expanded in b around 0 97.9%

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

      1. *-commutative 97.9%

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

   8. Simplified 97.9%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -860:\\ \;\;\;\;1 + e^{b}\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{+98}:\\ \;\;\;\;\frac{1}{\left(1 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)\right) - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 100.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

   1. *-lft-identity 98.8%

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

   2. associate-*l/ 98.8%

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

   3. associate-/r/ 98.8%

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

   4. remove-double-neg 98.8%

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

   5. unsub-neg 98.8%

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

   6. div-sub 71.1%

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

   7. *-lft-identity 71.1%

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

   8. associate-*l/ 71.1%

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

   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 7: 70.9% accurate, 13.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.5 \cdot 10^{+99}:\\ \;\;\;\;\frac{1}{\left(1 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)\right) - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 6.5e+99)
   (/
    1.0
    (- (+ 1.0 (* a (+ -1.0 (* a (+ 0.5 (* a -0.16666666666666666)))))) -1.0))
   (/ 1.0 (+ 2.0 (* b (+ 1.0 (* b (+ 0.5 (* b 0.16666666666666666)))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 6.5e+99) {
		tmp = 1.0 / ((1.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666)))))) - -1.0);
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 6.5d+99) then
        tmp = 1.0d0 / ((1.0d0 + (a * ((-1.0d0) + (a * (0.5d0 + (a * (-0.16666666666666666d0))))))) - (-1.0d0))
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (b * (0.5d0 + (b * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 6.5e+99) {
		tmp = 1.0 / ((1.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666)))))) - -1.0);
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 6.5e+99:
		tmp = 1.0 / ((1.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666)))))) - -1.0)
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 6.5e+99)
		tmp = Float64(1.0 / Float64(Float64(1.0 + Float64(a * Float64(-1.0 + Float64(a * Float64(0.5 + Float64(a * -0.16666666666666666)))))) - -1.0));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(b * Float64(0.5 + Float64(b * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 6.5e+99)
		tmp = 1.0 / ((1.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666)))))) - -1.0);
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 6.5e+99], N[(1.0 / N[(N[(1.0 + N[(a * N[(-1.0 + N[(a * N[(0.5 + N[(a * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(b * N[(0.5 + N[(b * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.5000000000000004e99

   1. Initial program 98.6%

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

      1. *-lft-identity 98.6%

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

      2. associate-*l/ 98.5%

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

      3. associate-/r/ 98.5%

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

      4. +-commutative 98.5%

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

      5. remove-double-neg 98.5%

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

      6. sub-neg 98.5%

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

      7. div-sub 72.4%

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

      8. neg-mul-1 72.4%

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

      9. *-commutative 72.4%

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

      10. associate-*r/ 72.4%

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

      11. metadata-eval 72.4%

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

      12. distribute-neg-frac 72.4%

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

      13. exp-neg 72.4%

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

      14. distribute-rgt-neg-out 72.4%

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

      15. exp-neg 72.4%

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

      16. rgt-mult-inverse 99.5%

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

      17. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in b around 0 72.6%

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

   6. Taylor expanded in a around 0 61.8%

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

   if 6.5000000000000004e99 < 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. +-commutative 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. div-sub 64.3%

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

      8. neg-mul-1 64.3%

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

      9. *-commutative 64.3%

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

      10. associate-*r/ 64.3%

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

      11. metadata-eval 64.3%

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

      12. distribute-neg-frac 64.3%

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

      13. exp-neg 64.3%

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

      14. distribute-rgt-neg-out 64.3%

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

      15. exp-neg 64.3%

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

      16. rgt-mult-inverse 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0 100.0%

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

   6. Taylor expanded in b around 0 97.9%

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

      1. *-commutative 97.9%

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

   8. Simplified 97.9%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.5 \cdot 10^{+99}:\\ \;\;\;\;\frac{1}{\left(1 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)\right) - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.3% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.5 \cdot 10^{+145}:\\ \;\;\;\;\frac{1}{a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 6.5e+145)
   (/ 1.0 (+ (* a (+ -1.0 (* a (+ 0.5 (* a -0.16666666666666666))))) 2.0))
   (/ 1.0 (+ 2.0 (* b (+ 1.0 (* b 0.5)))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 6.5e+145) {
		tmp = 1.0 / ((a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))) + 2.0);
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 6.5d+145) then
        tmp = 1.0d0 / ((a * ((-1.0d0) + (a * (0.5d0 + (a * (-0.16666666666666666d0)))))) + 2.0d0)
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (b * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 6.5e+145) {
		tmp = 1.0 / ((a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))) + 2.0);
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 6.5e+145:
		tmp = 1.0 / ((a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))) + 2.0)
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 6.5e+145)
		tmp = Float64(1.0 / Float64(Float64(a * Float64(-1.0 + Float64(a * Float64(0.5 + Float64(a * -0.16666666666666666))))) + 2.0));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(b * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 6.5e+145)
		tmp = 1.0 / ((a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))) + 2.0);
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 6.5e+145], N[(1.0 / N[(N[(a * N[(-1.0 + N[(a * N[(0.5 + N[(a * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(b * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.50000000000000034e145

   1. Initial program 98.6%

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

      1. *-lft-identity 98.6%

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

      2. associate-*l/ 98.6%

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

      3. associate-/r/ 98.6%

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

      4. +-commutative 98.6%

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

      5. remove-double-neg 98.6%

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

      6. sub-neg 98.6%

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

      7. div-sub 72.2%

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

      8. neg-mul-1 72.2%

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

      9. *-commutative 72.2%

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

      10. associate-*r/ 72.1%

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

      11. metadata-eval 72.1%

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

      12. distribute-neg-frac 72.1%

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

      13. exp-neg 72.1%

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

      14. distribute-rgt-neg-out 72.1%

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

      15. exp-neg 72.1%

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

      16. rgt-mult-inverse 99.5%

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

      17. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in b around 0 71.1%

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

      1. rec-exp 71.1%

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

   7. Simplified 71.1%

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

   8. Taylor expanded in a around 0 60.7%

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

   if 6.50000000000000034e145 < 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. +-commutative 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. div-sub 63.6%

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

      8. neg-mul-1 63.6%

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

      9. *-commutative 63.6%

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

      10. associate-*r/ 63.6%

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

      11. metadata-eval 63.6%

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

      12. distribute-neg-frac 63.6%

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

      13. exp-neg 63.6%

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

      14. distribute-rgt-neg-out 63.6%

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

      15. exp-neg 63.6%

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

      16. rgt-mult-inverse 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0 100.0%

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

   6. Taylor expanded in b around 0 91.8%

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

      1. *-commutative 91.8%

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

   8. Simplified 91.8%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.5 \cdot 10^{+145}:\\ \;\;\;\;\frac{1}{a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.8% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.5 \cdot 10^{+96}:\\ \;\;\;\;\frac{1}{a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.5e+96)
   (/ 1.0 (+ (* a (+ -1.0 (* a (+ 0.5 (* a -0.16666666666666666))))) 2.0))
   (/ 1.0 (+ 2.0 (* b (+ 1.0 (* b (+ 0.5 (* b 0.16666666666666666)))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.5e+96) {
		tmp = 1.0 / ((a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))) + 2.0);
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 2.5d+96) then
        tmp = 1.0d0 / ((a * ((-1.0d0) + (a * (0.5d0 + (a * (-0.16666666666666666d0)))))) + 2.0d0)
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (b * (0.5d0 + (b * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 2.5e+96) {
		tmp = 1.0 / ((a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))) + 2.0);
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2.5e+96:
		tmp = 1.0 / ((a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))) + 2.0)
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.5e+96)
		tmp = Float64(1.0 / Float64(Float64(a * Float64(-1.0 + Float64(a * Float64(0.5 + Float64(a * -0.16666666666666666))))) + 2.0));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(b * Float64(0.5 + Float64(b * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2.5e+96)
		tmp = 1.0 / ((a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))) + 2.0);
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.5e+96], N[(1.0 / N[(N[(a * N[(-1.0 + N[(a * N[(0.5 + N[(a * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(b * N[(0.5 + N[(b * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.5000000000000002e96

   1. Initial program 98.6%

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

      1. *-lft-identity 98.6%

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

      2. associate-*l/ 98.5%

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

      3. associate-/r/ 98.5%

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

      4. +-commutative 98.5%

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

      5. remove-double-neg 98.5%

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

      6. sub-neg 98.5%

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

      7. div-sub 72.4%

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

      8. neg-mul-1 72.4%

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

      9. *-commutative 72.4%

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

      10. associate-*r/ 72.4%

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

      11. metadata-eval 72.4%

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

      12. distribute-neg-frac 72.4%

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

      13. exp-neg 72.4%

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

      14. distribute-rgt-neg-out 72.4%

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

      15. exp-neg 72.4%

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

      16. rgt-mult-inverse 99.5%

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

      17. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in b around 0 72.6%

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

      1. rec-exp 72.6%

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

   7. Simplified 72.6%

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

   8. Taylor expanded in a around 0 61.8%

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

   if 2.5000000000000002e96 < 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. +-commutative 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. div-sub 64.3%

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

      8. neg-mul-1 64.3%

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

      9. *-commutative 64.3%

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

      10. associate-*r/ 64.3%

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

      11. metadata-eval 64.3%

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

      12. distribute-neg-frac 64.3%

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

      13. exp-neg 64.3%

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

      14. distribute-rgt-neg-out 64.3%

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

      15. exp-neg 64.3%

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

      16. rgt-mult-inverse 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0 100.0%

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

   6. Taylor expanded in b around 0 97.9%

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

      1. *-commutative 97.9%

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

   8. Simplified 97.9%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.5 \cdot 10^{+96}:\\ \;\;\;\;\frac{1}{a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.6% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.35 \cdot 10^{+129}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.35e+129)
   (/ 1.0 (+ 2.0 (* a (+ -1.0 (* a 0.5)))))
   (/ 1.0 (+ 2.0 (* b (+ 1.0 (* b 0.5)))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.35e+129) {
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * 0.5))));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 1.35d+129) then
        tmp = 1.0d0 / (2.0d0 + (a * ((-1.0d0) + (a * 0.5d0))))
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (b * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.35e+129) {
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * 0.5))));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.35e+129:
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * 0.5))))
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.35e+129)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(-1.0 + Float64(a * 0.5)))));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(b * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.35e+129)
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * 0.5))));
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.35e+129], N[(1.0 / N[(2.0 + N[(a * N[(-1.0 + N[(a * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(b * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.35e129

   1. Initial program 98.6%

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

      1. *-lft-identity 98.6%

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

      2. associate-*l/ 98.6%

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

      3. associate-/r/ 98.6%

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

      4. +-commutative 98.6%

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

      5. remove-double-neg 98.6%

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

      6. sub-neg 98.6%

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

      7. div-sub 72.0%

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

      8. neg-mul-1 72.0%

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

      9. *-commutative 72.0%

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

      10. associate-*r/ 72.0%

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

      11. metadata-eval 72.0%

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

      12. distribute-neg-frac 72.0%

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

      13. exp-neg 72.0%

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

      14. distribute-rgt-neg-out 72.0%

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

      15. exp-neg 72.0%

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

      16. rgt-mult-inverse 99.5%

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

      17. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in b around 0 72.2%

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

      1. rec-exp 72.2%

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

   7. Simplified 72.2%

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

   8. Taylor expanded in a around 0 57.8%

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

   if 1.35e129 < 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. +-commutative 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. div-sub 65.8%

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

      8. neg-mul-1 65.8%

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

      9. *-commutative 65.8%

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

      10. associate-*r/ 65.8%

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

      11. metadata-eval 65.8%

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

      12. distribute-neg-frac 65.8%

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

      13. exp-neg 65.8%

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

      14. distribute-rgt-neg-out 65.8%

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

      15. exp-neg 65.8%

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

      16. rgt-mult-inverse 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0 100.0%

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

   6. Taylor expanded in b around 0 80.8%

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

      1. *-commutative 80.8%

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

   8. Simplified 80.8%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.35 \cdot 10^{+129}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 53.3% accurate, 27.7× speedup

Math

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

FPCore

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

C

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

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))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

   1. *-lft-identity 98.8%

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

   2. associate-*l/ 98.8%

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

   3. associate-/r/ 98.8%

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

   4. +-commutative 98.8%

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

   5. remove-double-neg 98.8%

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

   6. sub-neg 98.8%

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

   7. div-sub 71.1%

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

   8. neg-mul-1 71.1%

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

   9. *-commutative 71.1%

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

   10. associate-*r/ 71.1%

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

   11. metadata-eval 71.1%

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

   12. distribute-neg-frac 71.1%

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

   13. exp-neg 71.0%

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

   14. distribute-rgt-neg-out 71.0%

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

   15. exp-neg 71.1%

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

   16. rgt-mult-inverse 99.6%

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

   17. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in b around 0 66.9%

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

   1. rec-exp 66.9%

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

7. Simplified 66.9%

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

8. Taylor expanded in a around 0 52.4%

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

9. Final simplification 52.4%

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

Alternative 12: 40.0% accurate, 61.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{2 - a} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

   1. *-lft-identity 98.8%

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

   2. associate-*l/ 98.8%

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

   3. associate-/r/ 98.8%

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

   4. +-commutative 98.8%

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

   5. remove-double-neg 98.8%

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

   6. sub-neg 98.8%

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

   7. div-sub 71.1%

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

   8. neg-mul-1 71.1%

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

   9. *-commutative 71.1%

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

   10. associate-*r/ 71.1%

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

   11. metadata-eval 71.1%

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

   12. distribute-neg-frac 71.1%

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

   13. exp-neg 71.0%

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

   14. distribute-rgt-neg-out 71.0%

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

   15. exp-neg 71.1%

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

   16. rgt-mult-inverse 99.6%

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

   17. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in b around 0 66.9%

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

   1. rec-exp 66.9%

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

7. Simplified 66.9%

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

8. Taylor expanded in a around 0 38.1%

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

   1. neg-mul-1 38.1%

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

   2. unsub-neg 38.1%

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

10. Simplified 38.1%

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

11. Final simplification 38.1%

    \[\leadsto \frac{1}{2 - a} \]
12. Add Preprocessing

Alternative 13: 39.2% accurate, 305.0× speedup

Math

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

FPCore

(FPCore (a b) :precision binary64 0.5)

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return 0.5

Julia

function code(a, b)
	return 0.5
end

MATLAB

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

Wolfram

code[a_, b_] := 0.5

Derivation

1. Initial program 98.8%

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

   1. *-lft-identity 98.8%

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

   2. associate-*l/ 98.8%

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

   3. associate-/r/ 98.8%

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

   4. +-commutative 98.8%

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

   5. remove-double-neg 98.8%

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

   6. sub-neg 98.8%

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

   7. div-sub 71.1%

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

   8. neg-mul-1 71.1%

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

   9. *-commutative 71.1%

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

   10. associate-*r/ 71.1%

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

   11. metadata-eval 71.1%

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

   12. distribute-neg-frac 71.1%

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

   13. exp-neg 71.0%

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

   14. distribute-rgt-neg-out 71.0%

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

   15. exp-neg 71.1%

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

   16. rgt-mult-inverse 99.6%

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

   17. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in a around 0 81.1%

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

6. Taylor expanded in b around 0 37.4%

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

7. Final simplification 37.4%

   \[\leadsto 0.5 \]
8. 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 2024077 
(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))))