Quotient of sum of exps

Average Accuracy: 99.1% → 98.2%

Time: 6.6s

Precision: binary64

Cost: 26184

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;e^{b} \leq 0.99999999999999:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \mathbf{elif}\;e^{b} \leq 1:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (if (<= (exp b) 0.99999999999999)
   (/ 1.0 (+ (exp b) 1.0))
   (if (<= (exp b) 1.0) (/ (exp a) (+ (exp a) 1.0)) 0.0)))

C

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

↓

double code(double a, double b) {
	double tmp;
	if (exp(b) <= 0.99999999999999) {
		tmp = 1.0 / (exp(b) + 1.0);
	} else if (exp(b) <= 1.0) {
		tmp = exp(a) / (exp(a) + 1.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

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

↓

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(b) <= 0.99999999999999d0) then
        tmp = 1.0d0 / (exp(b) + 1.0d0)
    else if (exp(b) <= 1.0d0) then
        tmp = exp(a) / (exp(a) + 1.0d0)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

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

↓

public static double code(double a, double b) {
	double tmp;
	if (Math.exp(b) <= 0.99999999999999) {
		tmp = 1.0 / (Math.exp(b) + 1.0);
	} else if (Math.exp(b) <= 1.0) {
		tmp = Math.exp(a) / (Math.exp(a) + 1.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

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

↓

def code(a, b):
	tmp = 0
	if math.exp(b) <= 0.99999999999999:
		tmp = 1.0 / (math.exp(b) + 1.0)
	elif math.exp(b) <= 1.0:
		tmp = math.exp(a) / (math.exp(a) + 1.0)
	else:
		tmp = 0.0
	return tmp

Julia

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

↓

function code(a, b)
	tmp = 0.0
	if (exp(b) <= 0.99999999999999)
		tmp = Float64(1.0 / Float64(exp(b) + 1.0));
	elseif (exp(b) <= 1.0)
		tmp = Float64(exp(a) / Float64(exp(a) + 1.0));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(b) <= 0.99999999999999)
		tmp = 1.0 / (exp(b) + 1.0);
	elseif (exp(b) <= 1.0)
		tmp = exp(a) / (exp(a) + 1.0);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[a_, b_] := If[LessEqual[N[Exp[b], $MachinePrecision], 0.99999999999999], N[(1.0 / N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[b], $MachinePrecision], 1.0], N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], 0.0]]

Target

Original	99.1%
Target	100.0%
Herbie	98.2%

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

Derivation

1. Split input into 3 regimes

2. if (exp.f64 b) < 0.99999999999999001

   1. Initial program 98.1%

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

   2. Taylor expanded in a around 0 97.1%

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

   if 0.99999999999999001 < (exp.f64 b) < 1

   1. Initial program 99.4%

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

   2. Taylor expanded in b around 0 99.4%

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

   if 1 < (exp.f64 b)

   1. Initial program 99.4%

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

   2. Taylor expanded in a around 0 98.1%

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

   3. Taylor expanded in b around 0 7.4%

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

   4. Simplified 7.4%

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

      [Start] 7.4	\[ \frac{1}{2 + b} \]
      +-commutative [=>] 7.4	\[ \frac{1}{\color{blue}{b + 2}} \]

   5. Applied egg-rr 92.9%

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

   6. Taylor expanded in b around inf 96.6%

      \[\leadsto \color{blue}{1} - 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{b} \leq 0.99999999999999:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \mathbf{elif}\;e^{b} \leq 1:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.1%
Cost	32320

\[\frac{e^{a}}{e^{a} + \mathsf{expm1}\left(\mathsf{log1p}\left(e^{b}\right)\right)} \]

Alternative 2
Accuracy	99.1%
Cost	19520

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

Alternative 3
Accuracy	98.0%
Cost	13252

\[\begin{array}{l} \mathbf{if}\;e^{a} \leq 0.99999995:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]

Alternative 4
Accuracy	81.9%
Cost	6724

\[\begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{-26}:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5
Accuracy	79.0%
Cost	708

\[\begin{array}{l} \mathbf{if}\;a \leq -2100000000000:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{1}{b + 2}\right) + -1\\ \end{array} \]

Alternative 6
Accuracy	63.2%
Cost	460

\[\begin{array}{l} \mathbf{if}\;b \leq 7.5 \cdot 10^{-269}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq 10^{-187}:\\ \;\;\;\;0\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-26}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7
Accuracy	63.4%
Cost	460

\[\begin{array}{l} \mathbf{if}\;b \leq 7.5 \cdot 10^{-269}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{elif}\;b \leq 9 \cdot 10^{-188}:\\ \;\;\;\;0\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{-26}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 8
Accuracy	39.3%
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023129 
(FPCore (a b)
  :name "Quotient of sum of exps"
  :precision binary64

  :herbie-target
  (/ 1.0 (+ 1.0 (exp (- b a))))

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