Quotient of sum of exps

Average Error: 1.02% → 1.4%

Time: 8.8s

Precision: binary64

Cost: 6980

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \leq -13000:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-1 + \left(e^{b} + 2\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (if (<= a -13000.0) 0.0 (/ 1.0 (+ -1.0 (+ (exp b) 2.0)))))

C

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

↓

double code(double a, double b) {
	double tmp;
	if (a <= -13000.0) {
		tmp = 0.0;
	} else {
		tmp = 1.0 / (-1.0 + (exp(b) + 2.0));
	}
	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 (a <= (-13000.0d0)) then
        tmp = 0.0d0
    else
        tmp = 1.0d0 / ((-1.0d0) + (exp(b) + 2.0d0))
    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 (a <= -13000.0) {
		tmp = 0.0;
	} else {
		tmp = 1.0 / (-1.0 + (Math.exp(b) + 2.0));
	}
	return tmp;
}

Python

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

↓

def code(a, b):
	tmp = 0
	if a <= -13000.0:
		tmp = 0.0
	else:
		tmp = 1.0 / (-1.0 + (math.exp(b) + 2.0))
	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 (a <= -13000.0)
		tmp = 0.0;
	else
		tmp = Float64(1.0 / Float64(-1.0 + Float64(exp(b) + 2.0)));
	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 (a <= -13000.0)
		tmp = 0.0;
	else
		tmp = 1.0 / (-1.0 + (exp(b) + 2.0));
	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[a, -13000.0], 0.0, N[(1.0 / N[(-1.0 + N[(N[Exp[b], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	1.02%
Target	0.02%
Herbie	1.4%

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

Derivation

1. Split input into 2 regimes

2. if a < -13000

   1. Initial program 1.14

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

   2. Taylor expanded in a around 0 64.79

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

   3. Taylor expanded in b around 0 96.07

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

   4. Simplified 96.07

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

      [Start] 96.07	\[ \frac{1}{2 + b} \]
      +-commutative [=>] 96.07	\[ \frac{1}{\color{blue}{b + 2}} \]

   5. Applied egg-rr 66.58

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

   6. Simplified 66.58

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

      [Start] 66.58	\[ \left(1 + \frac{1}{b + 2}\right) - 1 \]
      associate--l+ [=>] 66.58	\[ \color{blue}{1 + \left(\frac{1}{b + 2} - 1\right)} \]

   7. Taylor expanded in b around inf 0.55

      \[\leadsto 1 + \color{blue}{-1} \]

   if -13000 < a

   1. Initial program 0.98

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

   2. Taylor expanded in a around 0 1.7

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

   3. Applied egg-rr 1.7

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

   4. Simplified 1.7

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

      [Start] 1.7	\[ \frac{1}{\left(1 + \left(1 + e^{b}\right)\right) - 1} \]
      sub-neg [=>] 1.7	\[ \frac{1}{\color{blue}{\left(1 + \left(1 + e^{b}\right)\right) + \left(-1\right)}} \]
      associate-+r+ [=>] 1.7	\[ \frac{1}{\color{blue}{\left(\left(1 + 1\right) + e^{b}\right)} + \left(-1\right)} \]
      metadata-eval [=>] 1.7	\[ \frac{1}{\left(\color{blue}{2} + e^{b}\right) + \left(-1\right)} \]
      metadata-eval [=>] 1.7	\[ \frac{1}{\left(2 + e^{b}\right) + \color{blue}{-1}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -13000:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-1 + \left(e^{b} + 2\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.1%
Cost	32448

\[\frac{e^{a}}{{\left({\left(e^{a} + e^{b}\right)}^{3}\right)}^{0.3333333333333333}} \]

Alternative 2
Error	1.02%
Cost	19520

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

Alternative 3
Error	1.39%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a \leq -13000:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]

Alternative 4
Error	17.22%
Cost	6724

\[\begin{array}{l} \mathbf{if}\;b \leq 340:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5
Error	36.75%
Cost	717

\[\begin{array}{l} \mathbf{if}\;a \leq -0.000108:\\ \;\;\;\;0\\ \mathbf{elif}\;a \leq -1.06 \cdot 10^{-150} \lor \neg \left(a \leq -1.75 \cdot 10^{-218}\right):\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 6
Error	20.38%
Cost	708

\[\begin{array}{l} \mathbf{if}\;a \leq -7.5 \cdot 10^{-5}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-1 + \frac{1}{b + 2}\right)\\ \end{array} \]

Alternative 7
Error	37.11%
Cost	460

\[\begin{array}{l} \mathbf{if}\;a \leq -8 \cdot 10^{-5}:\\ \;\;\;\;0\\ \mathbf{elif}\;a \leq -4 \cdot 10^{-151}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;a \leq -2.15 \cdot 10^{-218}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 8
Error	60.57%
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023102 
(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))))