Quotient of sum of exps

Average Error: 0.7 → 1.0

Time: 7.1s

Precision: binary64

Cost: 26441

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;e^{b} \leq 0.9999998 \lor \neg \left(e^{b} \leq 1\right):\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \mathbf{else}:\\ \;\;\;\;e^{a} \cdot \frac{1}{e^{a} + \left(1 + b\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (if (or (<= (exp b) 0.9999998) (not (<= (exp b) 1.0)))
   (/ 1.0 (+ 1.0 (exp b)))
   (* (exp a) (/ 1.0 (+ (exp a) (+ 1.0 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(b) <= 0.9999998) || !(exp(b) <= 1.0)) {
		tmp = 1.0 / (1.0 + exp(b));
	} else {
		tmp = exp(a) * (1.0 / (exp(a) + (1.0 + 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(b) <= 0.9999998d0) .or. (.not. (exp(b) <= 1.0d0))) then
        tmp = 1.0d0 / (1.0d0 + exp(b))
    else
        tmp = exp(a) * (1.0d0 / (exp(a) + (1.0d0 + 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(b) <= 0.9999998) || !(Math.exp(b) <= 1.0)) {
		tmp = 1.0 / (1.0 + Math.exp(b));
	} else {
		tmp = Math.exp(a) * (1.0 / (Math.exp(a) + (1.0 + 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(b) <= 0.9999998) or not (math.exp(b) <= 1.0):
		tmp = 1.0 / (1.0 + math.exp(b))
	else:
		tmp = math.exp(a) * (1.0 / (math.exp(a) + (1.0 + 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(b) <= 0.9999998) || !(exp(b) <= 1.0))
		tmp = Float64(1.0 / Float64(1.0 + exp(b)));
	else
		tmp = Float64(exp(a) * Float64(1.0 / Float64(exp(a) + Float64(1.0 + 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(b) <= 0.9999998) || ~((exp(b) <= 1.0)))
		tmp = 1.0 / (1.0 + exp(b));
	else
		tmp = exp(a) * (1.0 / (exp(a) + (1.0 + 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[Or[LessEqual[N[Exp[b], $MachinePrecision], 0.9999998], N[Not[LessEqual[N[Exp[b], $MachinePrecision], 1.0]], $MachinePrecision]], N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[a], $MachinePrecision] * N[(1.0 / N[(N[Exp[a], $MachinePrecision] + N[(1.0 + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	0.7
Target	0.0
Herbie	1.0

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 b) < 0.999999799999999994 or 1 < (exp.f64 b)

   1. Initial program 1.1

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

   2. Taylor expanded in a around 0 1.7

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

   if 0.999999799999999994 < (exp.f64 b) < 1

   1. Initial program 0.3

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

   2. Applied egg-rr 0.3

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

   3. Taylor expanded in b around 0 0.3

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

   4. Simplified 0.3

      \[\leadsto \frac{1}{\color{blue}{e^{a} + \left(1 + b\right)}} \cdot e^{a} \]
      Proof
      (*.f64 (/.f64 1 (+.f64 (exp.f64 a) (+.f64 1 b))) (exp.f64 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 1 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (exp.f64 a) 1) b))) (exp.f64 a)): 4 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 1 (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 1 (exp.f64 a))) b)) (exp.f64 a)): 0 points increase in error, 4 points decrease in error
      (*.f64 (/.f64 1 (Rewrite<= associate-+r+_binary64 (+.f64 1 (+.f64 (exp.f64 a) b)))) (exp.f64 a)): 0 points increase in error, 4 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{b} \leq 0.9999998 \lor \neg \left(e^{b} \leq 1\right):\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \mathbf{else}:\\ \;\;\;\;e^{a} \cdot \frac{1}{e^{a} + \left(1 + b\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.0
Cost	26313

\[\begin{array}{l} \mathbf{if}\;e^{b} \leq 0.9999998 \lor \neg \left(e^{b} \leq 1\right):\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{a}}{e^{a} + \left(1 + b\right)}\\ \end{array} \]

Alternative 2
Error	1.0
Cost	26185

\[\begin{array}{l} \mathbf{if}\;e^{b} \leq 0.9999998 \lor \neg \left(e^{b} \leq 1\right):\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{a}}{1 + e^{a}}\\ \end{array} \]

Alternative 3
Error	0.7
Cost	19648

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

Alternative 4
Error	0.7
Cost	19520

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

Alternative 5
Error	14.0
Cost	6861

\[\begin{array}{l} \mathbf{if}\;a \leq -2.7 \cdot 10^{-8} \lor \neg \left(a \leq -1.1 \cdot 10^{-67}\right) \land a \leq -2.9 \cdot 10^{-100}:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{1}{b + 2}\right) + -1\\ \end{array} \]

Alternative 6
Error	0.9
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a \leq -360000000:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]

Alternative 7
Error	12.8
Cost	708

\[\begin{array}{l} \mathbf{if}\;a \leq -360:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{1}{b + 2}\right) + -1\\ \end{array} \]

Alternative 8
Error	22.0
Cost	452

\[\begin{array}{l} \mathbf{if}\;a \leq -0.1:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \]

Alternative 9
Error	22.3
Cost	196

\[\begin{array}{l} \mathbf{if}\;a \leq -36:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 10
Error	39.0
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2022343 
(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))))