Quotient of sum of exps

Average Error: 0.7 → 1.2

Time: 4.6s

Precision: binary64

Cost: 13316

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;e^{a} \leq 0.999999:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.999999)
   (/ 1.0 (+ 1.0 (exp (- a))))
   (/ 1.0 (+ 1.0 (exp b)))))

C

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

↓

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.999999) {
		tmp = 1.0 / (1.0 + exp(-a));
	} else {
		tmp = 1.0 / (1.0 + exp(b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    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(a) <= 0.999999d0) then
        tmp = 1.0d0 / (1.0d0 + exp(-a))
    else
        tmp = 1.0d0 / (1.0d0 + exp(b))
    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(a) <= 0.999999) {
		tmp = 1.0 / (1.0 + Math.exp(-a));
	} else {
		tmp = 1.0 / (1.0 + Math.exp(b));
	}
	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(a) <= 0.999999:
		tmp = 1.0 / (1.0 + math.exp(-a))
	else:
		tmp = 1.0 / (1.0 + math.exp(b))
	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(a) <= 0.999999)
		tmp = Float64(1.0 / Float64(1.0 + exp(Float64(-a))));
	else
		tmp = Float64(1.0 / Float64(1.0 + exp(b)));
	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(a) <= 0.999999)
		tmp = 1.0 / (1.0 + exp(-a));
	else
		tmp = 1.0 / (1.0 + exp(b));
	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[a], $MachinePrecision], 0.999999], N[(1.0 / N[(1.0 + N[Exp[(-a)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	0.7
Target	0.0
Herbie	1.2

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.999998999999999971

   1. Initial program 0.8

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

   2. Taylor expanded in b around 0 2.1

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

   3. Applied egg-rr 2.1

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

   4. Simplified 2.1

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

      [Start] 2.1	\[ \left(-e^{a}\right) \cdot \frac{1}{\left(-e^{a}\right) + -1} \]
      associate-*r/ [=>] 2.1	\[ \color{blue}{\frac{\left(-e^{a}\right) \cdot 1}{\left(-e^{a}\right) + -1}} \]
      *-rgt-identity [=>] 2.1	\[ \frac{\color{blue}{-e^{a}}}{\left(-e^{a}\right) + -1} \]
      neg-mul-1 [=>] 2.1	\[ \frac{\color{blue}{-1 \cdot e^{a}}}{\left(-e^{a}\right) + -1} \]
      associate-/l* [=>] 2.1	\[ \color{blue}{\frac{-1}{\frac{\left(-e^{a}\right) + -1}{e^{a}}}} \]
      +-commutative [=>] 2.1	\[ \frac{-1}{\frac{\color{blue}{-1 + \left(-e^{a}\right)}}{e^{a}}} \]
      unsub-neg [=>] 2.1	\[ \frac{-1}{\frac{\color{blue}{-1 - e^{a}}}{e^{a}}} \]
      div-sub [=>] 62.8	\[ \frac{-1}{\color{blue}{\frac{-1}{e^{a}} - \frac{e^{a}}{e^{a}}}} \]
      *-inverses [=>] 2.1	\[ \frac{-1}{\frac{-1}{e^{a}} - \color{blue}{1}} \]
      sub-neg [=>] 2.1	\[ \frac{-1}{\color{blue}{\frac{-1}{e^{a}} + \left(-1\right)}} \]
      metadata-eval [=>] 2.1	\[ \frac{-1}{\frac{-1}{e^{a}} + \color{blue}{-1}} \]

   5. Taylor expanded in a around inf 2.1

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

   6. Simplified 2.1

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

      [Start] 2.1	\[ \frac{1}{1 + \frac{1}{e^{a}}} \]
      metadata-eval [<=] 2.1	\[ \frac{1}{1 + \frac{\color{blue}{--1}}{e^{a}}} \]
      distribute-neg-frac [<=] 2.1	\[ \frac{1}{1 + \color{blue}{\left(-\frac{-1}{e^{a}}\right)}} \]
      distribute-neg-frac [=>] 2.1	\[ \frac{1}{1 + \color{blue}{\frac{--1}{e^{a}}}} \]
      metadata-eval [=>] 2.1	\[ \frac{1}{1 + \frac{\color{blue}{1}}{e^{a}}} \]
      exp-neg [<=] 2.1	\[ \frac{1}{1 + \color{blue}{e^{-a}}} \]

   if 0.999998999999999971 < (exp.f64 a)

   1. Initial program 0.7

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

   2. Taylor expanded in a around 0 0.9

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

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.999999:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.0
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a \leq -720:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]

Alternative 2
Error	22.8
Cost	981

\[\begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{-97}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq -1.02 \cdot 10^{-130}:\\ \;\;\;\;0\\ \mathbf{elif}\;b \leq 2.55 \cdot 10^{-281} \lor \neg \left(b \leq 1.7 \cdot 10^{-268}\right) \land b \leq 1.08 \cdot 10^{-16}:\\ \;\;\;\;\frac{-1}{a + -2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 3
Error	23.1
Cost	724

\[\begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{-97}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq -1.02 \cdot 10^{-130}:\\ \;\;\;\;0\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{-285}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-268}:\\ \;\;\;\;0\\ \mathbf{elif}\;b \leq 1.12 \cdot 10^{-16}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 4
Error	23.1
Cost	724

\[\begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{-97}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq -8.5 \cdot 10^{-131}:\\ \;\;\;\;0\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{-287}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{-268}:\\ \;\;\;\;0\\ \mathbf{elif}\;b \leq 1.12 \cdot 10^{-16}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5
Error	13.4
Cost	708

\[\begin{array}{l} \mathbf{if}\;a \leq -350:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{1}{b + 2}\right) + -1\\ \end{array} \]

Alternative 6
Error	39.4
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023066 
(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))))