Quotient of sum of exps

Average Error: 0.5 → 1.0

Time: 4.2s

Precision: binary64

Cost: 6852

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \leq -740:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (if (<= a -740.0) 0.0 (/ 1.0 (+ (exp b) 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 (a <= -740.0) {
		tmp = 0.0;
	} else {
		tmp = 1.0 / (exp(b) + 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 (a <= (-740.0d0)) then
        tmp = 0.0d0
    else
        tmp = 1.0d0 / (exp(b) + 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 (a <= -740.0) {
		tmp = 0.0;
	} else {
		tmp = 1.0 / (Math.exp(b) + 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 a <= -740.0:
		tmp = 0.0
	else:
		tmp = 1.0 / (math.exp(b) + 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 (a <= -740.0)
		tmp = 0.0;
	else
		tmp = Float64(1.0 / Float64(exp(b) + 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 (a <= -740.0)
		tmp = 0.0;
	else
		tmp = 1.0 / (exp(b) + 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[a, -740.0], 0.0, N[(1.0 / N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Target

Original	0.5
Target	0.0
Herbie	1.0

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

Derivation

1. Split input into 2 regimes

2. if a < -740

   1. Initial program 0.9

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

   2. Taylor expanded in a around 0 42.0

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

   3. Taylor expanded in b around 0 61.6

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

   4. Simplified 61.6

      \[\leadsto \frac{1}{\color{blue}{b + 2}} \]
      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 42.9

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

   6. Taylor expanded in b around inf 0.3

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

   if -740 < a

   1. Initial program 0.4

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

   2. Taylor expanded in a around 0 1.2

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

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -740:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]

Alternatives

Alternative 1
Error	12.0
Cost	19848

\[\begin{array}{l} t_0 := \frac{1}{e^{b} + 1}\\ \mathbf{if}\;e^{b} \leq 1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;e^{b} \leq 1.000000001:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.5
Cost	19520

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

Alternative 3
Error	24.4
Cost	980

\[\begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{-193}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq -1.6 \cdot 10^{-284}:\\ \;\;\;\;0\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-184}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-151}:\\ \;\;\;\;0\\ \mathbf{elif}\;b \leq 0.54:\\ \;\;\;\;0.5 + b \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 4
Error	24.4
Cost	980

\[\begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{-193}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq -1.6 \cdot 10^{-284}:\\ \;\;\;\;0\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{-184}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{-151}:\\ \;\;\;\;0\\ \mathbf{elif}\;b \leq 0.27:\\ \;\;\;\;0.5 + b \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5
Error	24.5
Cost	724

\[\begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{-193}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq -1.6 \cdot 10^{-284}:\\ \;\;\;\;0\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{-184}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{-151}:\\ \;\;\;\;0\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{-11}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 6
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 7
Error	38.6
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2022331 
(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))))