Quotient of sum of exps

Average Error: 0.7 → 2.0

Time: 5.9s

Precision: binary64

Cost: 19912

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;e^{b} \leq 0:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \mathbf{elif}\;e^{b} \leq 10^{+106}:\\ \;\;\;\;\frac{e^{a}}{a + \left(b + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{1}{b + 2}\right) + -1\\ \end{array} \]

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (if (<= (exp b) 0.0)
   (/ 1.0 (+ (exp b) 1.0))
   (if (<= (exp b) 1e+106)
     (/ (exp a) (+ a (+ b 2.0)))
     (+ (+ 1.0 (/ 1.0 (+ b 2.0))) -1.0))))

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) <= 0.0) {
		tmp = 1.0 / (exp(b) + 1.0);
	} else if (exp(b) <= 1e+106) {
		tmp = exp(a) / (a + (b + 2.0));
	} else {
		tmp = (1.0 + (1.0 / (b + 2.0))) + -1.0;
	}
	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) <= 0.0d0) then
        tmp = 1.0d0 / (exp(b) + 1.0d0)
    else if (exp(b) <= 1d+106) then
        tmp = exp(a) / (a + (b + 2.0d0))
    else
        tmp = (1.0d0 + (1.0d0 / (b + 2.0d0))) + (-1.0d0)
    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) <= 0.0) {
		tmp = 1.0 / (Math.exp(b) + 1.0);
	} else if (Math.exp(b) <= 1e+106) {
		tmp = Math.exp(a) / (a + (b + 2.0));
	} else {
		tmp = (1.0 + (1.0 / (b + 2.0))) + -1.0;
	}
	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) <= 0.0:
		tmp = 1.0 / (math.exp(b) + 1.0)
	elif math.exp(b) <= 1e+106:
		tmp = math.exp(a) / (a + (b + 2.0))
	else:
		tmp = (1.0 + (1.0 / (b + 2.0))) + -1.0
	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) <= 0.0)
		tmp = Float64(1.0 / Float64(exp(b) + 1.0));
	elseif (exp(b) <= 1e+106)
		tmp = Float64(exp(a) / Float64(a + Float64(b + 2.0)));
	else
		tmp = Float64(Float64(1.0 + Float64(1.0 / Float64(b + 2.0))) + -1.0);
	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) <= 0.0)
		tmp = 1.0 / (exp(b) + 1.0);
	elseif (exp(b) <= 1e+106)
		tmp = exp(a) / (a + (b + 2.0));
	else
		tmp = (1.0 + (1.0 / (b + 2.0))) + -1.0;
	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], 0.0], N[(1.0 / N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[b], $MachinePrecision], 1e+106], N[(N[Exp[a], $MachinePrecision] / N[(a + N[(b + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(1.0 / N[(b + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]

Target

Original	0.7
Target	0.0
Herbie	2.0

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

Derivation

1. Split input into 3 regimes

2. if (exp.f64 b) < 0.0

   1. Initial program 1.8

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

   2. Taylor expanded in a around 0 0.4

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

   if 0.0 < (exp.f64 b) < 1.00000000000000009e106

   1. Initial program 0.5

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

   2. Taylor expanded in b around 0 1.2

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

   3. Simplified 1.2

      \[\leadsto \frac{e^{a}}{\color{blue}{e^{a} + \left(1 + b\right)}} \]
      Proof
      (+.f64 (exp.f64 a) (+.f64 1 b)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (exp.f64 a) 1) b)): 1 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 1 (exp.f64 a))) b): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-+r+_binary64 (+.f64 1 (+.f64 (exp.f64 a) b))): 0 points increase in error, 1 points decrease in error

   4. Taylor expanded in a around 0 2.0

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

   5. Simplified 2.0

      \[\leadsto \frac{e^{a}}{\color{blue}{2 + \left(b + a\right)}} \]
      Proof
      (+.f64 2 (+.f64 b a)): 0 points increase in error, 0 points decrease in error
      (+.f64 2 (Rewrite<= +-commutative_binary64 (+.f64 a b))): 0 points increase in error, 0 points decrease in error

   6. Applied egg-rr 2.0

      \[\leadsto \color{blue}{0 + \frac{e^{a}}{a + \left(2 + b\right)}} \]

   if 1.00000000000000009e106 < (exp.f64 b)

   1. Initial program 0.3

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

   2. Taylor expanded in a around 0 0.0

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

   3. Taylor expanded in b around 0 60.5

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

   4. Simplified 60.5

      \[\leadsto \frac{1}{\color{blue}{b + 2}} \]
      Proof
      (+.f64 b 2): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 2 b)): 0 points increase in error, 0 points decrease in error

   5. Applied egg-rr 2.9

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

4. Final simplification 2.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{b} \leq 0:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \mathbf{elif}\;e^{b} \leq 10^{+106}:\\ \;\;\;\;\frac{e^{a}}{a + \left(b + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{1}{b + 2}\right) + -1\\ \end{array} \]

Alternatives

Alternative 1
Error	2.0
Cost	19912

\[\begin{array}{l} \mathbf{if}\;e^{b} \leq 0:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \mathbf{elif}\;e^{b} \leq 10^{+106}:\\ \;\;\;\;\frac{e^{a}}{2 + \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{1}{b + 2}\right) + -1\\ \end{array} \]

Alternative 2
Error	1.1
Cost	19652

\[\begin{array}{l} \mathbf{if}\;e^{a} \leq 0.999:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]

Alternative 3
Error	0.7
Cost	19520

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

Alternative 4
Error	13.1
Cost	13124

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

Alternative 5
Error	1.2
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a \leq -92000000000:\\ \;\;\;\;\frac{e^{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]

Alternative 6
Error	11.9
Cost	6724

\[\begin{array}{l} \mathbf{if}\;b \leq 243.6817491012524:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{1}{b + 2}\right) + -1\\ \end{array} \]

Alternative 7
Error	22.9
Cost	708

\[\begin{array}{l} \mathbf{if}\;b \leq -5.3292848264969346 \cdot 10^{-108}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{1}{b + 2}\right) + -1\\ \end{array} \]

Alternative 8
Error	32.9
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 0.01594305247736646:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{b}\\ \end{array} \]

Alternative 9
Error	33.1
Cost	324

\[\begin{array}{l} \mathbf{if}\;b \leq 0.01594305247736646:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{b}\\ \end{array} \]

Alternative 10
Error	38.8
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2022317 
(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))))