Quotient of sum of exps

Average Error: 0.6 → 0.9

Time: 4.7s

Precision: binary64

Cost: 26184

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{e^{b} + 1}\\ \mathbf{if}\;e^{b} \leq 5 \cdot 10^{-55}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;e^{b} \leq 2:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ (exp b) 1.0))))
   (if (<= (exp b) 5e-55)
     t_0
     (if (<= (exp b) 2.0) (/ (exp a) (+ (exp a) 1.0)) 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 = 1.0 / (exp(b) + 1.0);
	double tmp;
	if (exp(b) <= 5e-55) {
		tmp = t_0;
	} else if (exp(b) <= 2.0) {
		tmp = exp(a) / (exp(a) + 1.0);
	} else {
		tmp = 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 = 1.0d0 / (exp(b) + 1.0d0)
    if (exp(b) <= 5d-55) then
        tmp = t_0
    else if (exp(b) <= 2.0d0) then
        tmp = exp(a) / (exp(a) + 1.0d0)
    else
        tmp = 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 = 1.0 / (Math.exp(b) + 1.0);
	double tmp;
	if (Math.exp(b) <= 5e-55) {
		tmp = t_0;
	} else if (Math.exp(b) <= 2.0) {
		tmp = Math.exp(a) / (Math.exp(a) + 1.0);
	} else {
		tmp = 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 = 1.0 / (math.exp(b) + 1.0)
	tmp = 0
	if math.exp(b) <= 5e-55:
		tmp = t_0
	elif math.exp(b) <= 2.0:
		tmp = math.exp(a) / (math.exp(a) + 1.0)
	else:
		tmp = 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(1.0 / Float64(exp(b) + 1.0))
	tmp = 0.0
	if (exp(b) <= 5e-55)
		tmp = t_0;
	elseif (exp(b) <= 2.0)
		tmp = Float64(exp(a) / Float64(exp(a) + 1.0));
	else
		tmp = 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 = 1.0 / (exp(b) + 1.0);
	tmp = 0.0;
	if (exp(b) <= 5e-55)
		tmp = t_0;
	elseif (exp(b) <= 2.0)
		tmp = exp(a) / (exp(a) + 1.0);
	else
		tmp = 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[(1.0 / N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Exp[b], $MachinePrecision], 5e-55], t$95$0, If[LessEqual[N[Exp[b], $MachinePrecision], 2.0], N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Target

Original	0.6
Target	0.0
Herbie	0.9

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 b) < 5.0000000000000002e-55 or 2 < (exp.f64 b)

   1. Initial program 0.8

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

   2. Taylor expanded in a around 0 0.4

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

   if 5.0000000000000002e-55 < (exp.f64 b) < 2

   1. Initial program 0.4

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

   2. Taylor expanded in b around 0 1.4

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

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{b} \leq 5 \cdot 10^{-55}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \mathbf{elif}\;e^{b} \leq 2:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.5
Cost	25920

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

Alternative 2
Error	1.1
Cost	19912

\[\begin{array}{l} t_0 := \frac{1}{e^{b} + 1}\\ \mathbf{if}\;e^{b} \leq 5 \cdot 10^{-55}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;e^{b} \leq 1.00001:\\ \;\;\;\;e^{a} \cdot \left(0.5 + a \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.6
Cost	19520

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

Alternative 4
Error	13.3
Cost	6860

\[\begin{array}{l} t_0 := 0.5 + a \cdot 0.25\\ \mathbf{if}\;b \leq -4 \cdot 10^{-10}:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-193}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{-166}:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;b \leq 1.8:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{2}{b \cdot b}\right) + -1\\ \end{array} \]

Alternative 5
Error	0.9
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a \leq -25000000:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]

Alternative 6
Error	23.0
Cost	708

\[\begin{array}{l} \mathbf{if}\;b \leq 5.9 \cdot 10^{-65}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{1}{b + 2}\right) + -1\\ \end{array} \]

Alternative 7
Error	22.6
Cost	708

\[\begin{array}{l} \mathbf{if}\;b \leq 1.9:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{2}{b \cdot b}\right) + -1\\ \end{array} \]

Alternative 8
Error	23.1
Cost	580

\[\begin{array}{l} \mathbf{if}\;b \leq 1.5:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{1}{b}\right) + -1\\ \end{array} \]

Alternative 9
Error	30.6
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 1.75:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \]

Alternative 10
Error	39.2
Cost	320

\[0.5 + a \cdot 0.25 \]

Alternative 11
Error	39.3
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2022338 
(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))))