Quotient of sum of exps

Average Error: 0.7 → 1.1

Time: 12.2s

Precision: binary64

Cost: 13512

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{1 + e^{b}}\\ \mathbf{if}\;b \leq -22000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 4.3 \cdot 10^{-23}:\\ \;\;\;\;\frac{e^{a}}{e^{a} + \left(b + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 (exp b)))))
   (if (<= b -22000000.0)
     t_0
     (if (<= b 4.3e-23) (/ (exp a) (+ (exp a) (+ b 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 / (1.0 + exp(b));
	double tmp;
	if (b <= -22000000.0) {
		tmp = t_0;
	} else if (b <= 4.3e-23) {
		tmp = exp(a) / (exp(a) + (b + 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 / (1.0d0 + exp(b))
    if (b <= (-22000000.0d0)) then
        tmp = t_0
    else if (b <= 4.3d-23) then
        tmp = exp(a) / (exp(a) + (b + 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 / (1.0 + Math.exp(b));
	double tmp;
	if (b <= -22000000.0) {
		tmp = t_0;
	} else if (b <= 4.3e-23) {
		tmp = Math.exp(a) / (Math.exp(a) + (b + 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 / (1.0 + math.exp(b))
	tmp = 0
	if b <= -22000000.0:
		tmp = t_0
	elif b <= 4.3e-23:
		tmp = math.exp(a) / (math.exp(a) + (b + 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(1.0 + exp(b)))
	tmp = 0.0
	if (b <= -22000000.0)
		tmp = t_0;
	elseif (b <= 4.3e-23)
		tmp = Float64(exp(a) / Float64(exp(a) + Float64(b + 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 / (1.0 + exp(b));
	tmp = 0.0;
	if (b <= -22000000.0)
		tmp = t_0;
	elseif (b <= 4.3e-23)
		tmp = exp(a) / (exp(a) + (b + 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[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -22000000.0], t$95$0, If[LessEqual[b, 4.3e-23], N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[(b + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Target

Original	0.7
Target	0.0
Herbie	1.1

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

Derivation

1. Split input into 2 regimes

2. if b < -2.2e7 or 4.30000000000000002e-23 < b

   1. Initial program 0.9

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

   2. Taylor expanded in a around 0 1.1

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

   if -2.2e7 < b < 4.30000000000000002e-23

   1. Initial program 0.5

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

   2. Taylor expanded in b around 0 1.0

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

   3. Simplified 1.0

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

      [Start] 1.0	\[ \frac{e^{a}}{1 + \left(e^{a} + b\right)} \]
      rational.json-simplify-41 [=>] 1.0	\[ \frac{e^{a}}{\color{blue}{e^{a} + \left(b + 1\right)}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -22000000:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \mathbf{elif}\;b \leq 4.3 \cdot 10^{-23}:\\ \;\;\;\;\frac{e^{a}}{e^{a} + \left(b + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.7
Cost	19520

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

Alternative 2
Error	1.1
Cost	13384

\[\begin{array}{l} t_0 := \frac{1}{1 + e^{b}}\\ \mathbf{if}\;b \leq -22000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 4.3 \cdot 10^{-23}:\\ \;\;\;\;\frac{e^{a}}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.9
Cost	13252

\[\begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]

Alternative 4
Error	14.3
Cost	6724

\[\begin{array}{l} \mathbf{if}\;b \leq 55000000000:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{a}{b} - \left(-1 - \frac{1}{b}\right)\right) - 1\\ \end{array} \]

Alternative 5
Error	25.3
Cost	836

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

Alternative 6
Error	33.0
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 310:\\ \;\;\;\;0.5 + 0.25 \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{b}\\ \end{array} \]

Alternative 7
Error	33.1
Cost	324

\[\begin{array}{l} \mathbf{if}\;b \leq 360:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{b}\\ \end{array} \]

Alternative 8
Error	38.5
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023077 
(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))))