Quotient of sum of exps

Average Error: 0.7 → 1.1

Time: 6.3s

Precision: binary64

Cost: 32520

Math

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

↓

\[\begin{array}{l} t_0 := e^{b} + 1\\ \mathbf{if}\;e^{b} \leq 0.98:\\ \;\;\;\;\frac{1}{t_0}\\ \mathbf{elif}\;e^{b} \leq 1:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;e^{-\log t_0}\\ \end{array} \]

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (+ (exp b) 1.0)))
   (if (<= (exp b) 0.98)
     (/ 1.0 t_0)
     (if (<= (exp b) 1.0) (/ (exp a) (+ (exp a) 1.0)) (exp (- (log t_0)))))))

C

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

↓

double code(double a, double b) {
	double t_0 = exp(b) + 1.0;
	double tmp;
	if (exp(b) <= 0.98) {
		tmp = 1.0 / t_0;
	} else if (exp(b) <= 1.0) {
		tmp = exp(a) / (exp(a) + 1.0);
	} else {
		tmp = exp(-log(t_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) :: t_0
    real(8) :: tmp
    t_0 = exp(b) + 1.0d0
    if (exp(b) <= 0.98d0) then
        tmp = 1.0d0 / t_0
    else if (exp(b) <= 1.0d0) then
        tmp = exp(a) / (exp(a) + 1.0d0)
    else
        tmp = exp(-log(t_0))
    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 t_0 = Math.exp(b) + 1.0;
	double tmp;
	if (Math.exp(b) <= 0.98) {
		tmp = 1.0 / t_0;
	} else if (Math.exp(b) <= 1.0) {
		tmp = Math.exp(a) / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.exp(-Math.log(t_0));
	}
	return tmp;
}

Python

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

↓

def code(a, b):
	t_0 = math.exp(b) + 1.0
	tmp = 0
	if math.exp(b) <= 0.98:
		tmp = 1.0 / t_0
	elif math.exp(b) <= 1.0:
		tmp = math.exp(a) / (math.exp(a) + 1.0)
	else:
		tmp = math.exp(-math.log(t_0))
	return tmp

Julia

function code(a, b)
	return Float64(exp(a) / Float64(exp(a) + exp(b)))
end

↓

function code(a, b)
	t_0 = Float64(exp(b) + 1.0)
	tmp = 0.0
	if (exp(b) <= 0.98)
		tmp = Float64(1.0 / t_0);
	elseif (exp(b) <= 1.0)
		tmp = Float64(exp(a) / Float64(exp(a) + 1.0));
	else
		tmp = exp(Float64(-log(t_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)
	t_0 = exp(b) + 1.0;
	tmp = 0.0;
	if (exp(b) <= 0.98)
		tmp = 1.0 / t_0;
	elseif (exp(b) <= 1.0)
		tmp = exp(a) / (exp(a) + 1.0);
	else
		tmp = exp(-log(t_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_] := Block[{t$95$0 = N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[N[Exp[b], $MachinePrecision], 0.98], N[(1.0 / t$95$0), $MachinePrecision], If[LessEqual[N[Exp[b], $MachinePrecision], 1.0], N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Exp[(-N[Log[t$95$0], $MachinePrecision])], $MachinePrecision]]]]

Target

Original	0.7
Target	0.0
Herbie	1.1

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

Derivation

1. Split input into 3 regimes

2. if (exp.f64 b) < 0.97999999999999998

   1. Initial program 1.5

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

   2. Taylor expanded in a around 0 1.1

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

   if 0.97999999999999998 < (exp.f64 b) < 1

   1. Initial program 0.5

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

   2. Taylor expanded in b around 0 0.9

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

   if 1 < (exp.f64 b)

   1. Initial program 0.5

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

   2. Applied egg-rr 0.0

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

   3. Taylor expanded in a around 0 1.2

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

4. Final simplification 1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{b} \leq 0.98:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \mathbf{elif}\;e^{b} \leq 1:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;e^{-\log \left(e^{b} + 1\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.1
Cost	26184

\[\begin{array}{l} t_0 := \frac{1}{e^{b} + 1}\\ \mathbf{if}\;e^{b} \leq 0.98:\\ \;\;\;\;t_0\\ \mathbf{elif}\;e^{b} \leq 1:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.7
Cost	19520

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

Alternative 3
Error	1.0
Cost	13508

\[\begin{array}{l} \mathbf{if}\;e^{a} \leq 0.0001:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{1}{e^{b} + 1}\right) + -1\\ \end{array} \]

Alternative 4
Error	1.0
Cost	13252

\[\begin{array}{l} \mathbf{if}\;e^{a} \leq 0.0001:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]

Alternative 5
Error	15.3
Cost	7124

\[\begin{array}{l} \mathbf{if}\;b \leq -1474655.4947532765:\\ \;\;\;\;e^{b} + 1\\ \mathbf{elif}\;b \leq -9.390390219898622 \cdot 10^{-30}:\\ \;\;\;\;\left(1.5 + b \cdot -0.25\right) + -1\\ \mathbf{elif}\;b \leq -8.073838721941194 \cdot 10^{-73}:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;b \leq -3.458280205294541 \cdot 10^{-94}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq -2.38 \cdot 10^{-243}:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{1}{b + 2}\right) + -1\\ \end{array} \]

Alternative 6
Error	15.9
Cost	6860

\[\begin{array}{l} \mathbf{if}\;b \leq -8.073838721941194 \cdot 10^{-73}:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;b \leq -3.458280205294541 \cdot 10^{-94}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq -2.38 \cdot 10^{-243}:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{1}{b + 2}\right) + -1\\ \end{array} \]

Alternative 7
Error	23.3
Cost	716

\[\begin{array}{l} \mathbf{if}\;b \leq -3.458280205294541 \cdot 10^{-94}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq -4.727176996354683 \cdot 10^{-191}:\\ \;\;\;\;0\\ \mathbf{elif}\;b \leq 0.0336326306206187:\\ \;\;\;\;b \cdot -0.25 + 0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 8
Error	23.3
Cost	716

\[\begin{array}{l} \mathbf{if}\;b \leq -3.458280205294541 \cdot 10^{-94}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{elif}\;b \leq -4.727176996354683 \cdot 10^{-191}:\\ \;\;\;\;0\\ \mathbf{elif}\;b \leq 0.0336326306206187:\\ \;\;\;\;b \cdot -0.25 + 0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 9
Error	13.8
Cost	708

\[\begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{+30}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{1}{b + 2}\right) + -1\\ \end{array} \]

Alternative 10
Error	23.4
Cost	460

\[\begin{array}{l} \mathbf{if}\;b \leq -3.458280205294541 \cdot 10^{-94}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq -4.727176996354683 \cdot 10^{-191}:\\ \;\;\;\;0\\ \mathbf{elif}\;b \leq 0.0336326306206187:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 11
Error	38.8
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2022306 
(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))))