Quotient of sum of exps

Average Error: 0.8 → 0.4

Time: 10.6s

Precision: binary64

Cost: 20040

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{1 + e^{b}}\\ \mathbf{if}\;e^{b} \leq 0:\\ \;\;\;\;t_0\\ \mathbf{elif}\;e^{b} \leq 2:\\ \;\;\;\;\frac{1}{1 + \frac{b + 1}{e^{a}}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

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 (<= (exp b) 0.0)
     t_0
     (if (<= (exp b) 2.0) (/ 1.0 (+ 1.0 (/ (+ b 1.0) (exp a)))) 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 (exp(b) <= 0.0) {
		tmp = t_0;
	} else if (exp(b) <= 2.0) {
		tmp = 1.0 / (1.0 + ((b + 1.0) / exp(a)));
	} 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 (exp(b) <= 0.0d0) then
        tmp = t_0
    else if (exp(b) <= 2.0d0) then
        tmp = 1.0d0 / (1.0d0 + ((b + 1.0d0) / exp(a)))
    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 (Math.exp(b) <= 0.0) {
		tmp = t_0;
	} else if (Math.exp(b) <= 2.0) {
		tmp = 1.0 / (1.0 + ((b + 1.0) / Math.exp(a)));
	} 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 math.exp(b) <= 0.0:
		tmp = t_0
	elif math.exp(b) <= 2.0:
		tmp = 1.0 / (1.0 + ((b + 1.0) / math.exp(a)))
	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 (exp(b) <= 0.0)
		tmp = t_0;
	elseif (exp(b) <= 2.0)
		tmp = Float64(1.0 / Float64(1.0 + Float64(Float64(b + 1.0) / exp(a))));
	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 (exp(b) <= 0.0)
		tmp = t_0;
	elseif (exp(b) <= 2.0)
		tmp = 1.0 / (1.0 + ((b + 1.0) / exp(a)));
	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[N[Exp[b], $MachinePrecision], 0.0], t$95$0, If[LessEqual[N[Exp[b], $MachinePrecision], 2.0], N[(1.0 / N[(1.0 + N[(N[(b + 1.0), $MachinePrecision] / N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Target

Original	0.8
Target	0.0
Herbie	0.4

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 b) < 0.0 or 2 < (exp.f64 b)

   1. Initial program 1.2

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

   2. Taylor expanded in a around 0 0.3

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

   if 0.0 < (exp.f64 b) < 2

   1. Initial program 0.5

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

   2. Applied egg-rr 0.5

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

   3. Taylor expanded in b around 0 10.5

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

   4. Simplified 0.5

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

Alternatives

Alternative 1
Error	0.8
Cost	19648

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

Alternative 2
Error	0.8
Cost	19520

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

Alternative 3
Error	0.7
Cost	7048

\[\begin{array}{l} t_0 := \frac{1}{1 + e^{b}}\\ \mathbf{if}\;b \leq -460000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 0.085:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	1.1
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a \leq -780000:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]

Alternative 5
Error	22.3
Cost	6592

\[\frac{e^{a}}{2} \]

Alternative 6
Error	27.7
Cost	1860

\[\begin{array}{l} \mathbf{if}\;b \leq -7500:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \left(1 + \left(\left(-a \cdot \left(1 + b\right)\right) + \left(b + \left(-\left(1 + b\right) \cdot \left(-1 + 0.5\right)\right) \cdot \left(a \cdot a\right)\right)\right)\right)}\\ \end{array} \]

Alternative 7
Error	34.6
Cost	900

\[\begin{array}{l} \mathbf{if}\;b \leq -10.5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(-a \cdot \left(1 + b\right)\right) + b\right) + 2}\\ \end{array} \]

Alternative 8
Error	38.4
Cost	320

\[0.5 + 0.25 \cdot a \]

Alternative 9
Error	38.5
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023010 
(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))))