Quotient of sum of exps

Average Error: 0.7 → 0.7

Time: 25.2s

Precision: binary64

Cost: 26560

Math

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

↓

\[\frac{e^{a}}{\frac{e^{a} \cdot 3}{4} - \left(\frac{-e^{a}}{4} + \left(-e^{b}\right)\right)} \]

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (/ (exp a) (- (/ (* (exp a) 3.0) 4.0) (+ (/ (- (exp a)) 4.0) (- (exp b))))))

C

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

↓

double code(double a, double b) {
	return exp(a) / (((exp(a) * 3.0) / 4.0) - ((-exp(a) / 4.0) + -exp(b)));
}

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
    code = exp(a) / (((exp(a) * 3.0d0) / 4.0d0) - ((-exp(a) / 4.0d0) + -exp(b)))
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) {
	return Math.exp(a) / (((Math.exp(a) * 3.0) / 4.0) - ((-Math.exp(a) / 4.0) + -Math.exp(b)));
}

Python

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

↓

def code(a, b):
	return math.exp(a) / (((math.exp(a) * 3.0) / 4.0) - ((-math.exp(a) / 4.0) + -math.exp(b)))

Julia

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

↓

function code(a, b)
	return Float64(exp(a) / Float64(Float64(Float64(exp(a) * 3.0) / 4.0) - Float64(Float64(Float64(-exp(a)) / 4.0) + Float64(-exp(b)))))
end

MATLAB

function tmp = code(a, b)
	tmp = exp(a) / (exp(a) + exp(b));
end

↓

function tmp = code(a, b)
	tmp = exp(a) / (((exp(a) * 3.0) / 4.0) - ((-exp(a) / 4.0) + -exp(b)));
end

Wolfram

code[a_, b_] := N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[a_, b_] := N[(N[Exp[a], $MachinePrecision] / N[(N[(N[(N[Exp[a], $MachinePrecision] * 3.0), $MachinePrecision] / 4.0), $MachinePrecision] - N[(N[((-N[Exp[a], $MachinePrecision]) / 4.0), $MachinePrecision] + (-N[Exp[b], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	0.7
Target	0.0
Herbie	0.7

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

Derivation

1. Initial program 0.7

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

2. Applied egg-rr 0.7

   \[\leadsto \frac{e^{a}}{\color{blue}{\frac{e^{a} \cdot 3}{4} - \left(\frac{-e^{a}}{4} + \left(-e^{b}\right)\right)}} \]

3. Final simplification 0.7

   \[\leadsto \frac{e^{a}}{\frac{e^{a} \cdot 3}{4} - \left(\frac{-e^{a}}{4} + \left(-e^{b}\right)\right)} \]

Alternatives

Alternative 1
Error	12.3
Cost	26184

\[\begin{array}{l} t_0 := e^{b} \leq 1\\ t_1 := \frac{1}{1 + e^{b}}\\ \mathbf{if}\;t_0:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0:\\ \;\;\;\;\frac{e^{a}}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	12.3
Cost	19784

\[\begin{array}{l} t_0 := e^{b} \leq 1\\ t_1 := \frac{1}{1 + e^{b}}\\ \mathbf{if}\;t_0:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	0.7
Cost	19520

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

Alternative 4
Error	21.7
Cost	7052

\[\begin{array}{l} t_0 := \frac{e^{a}}{2}\\ t_1 := -0.020833333333333332 \cdot {a}^{3}\\ \mathbf{if}\;b \leq 7.8 \cdot 10^{+43}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{+151}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 1.65 \cdot 10^{+252}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	22.2
Cost	6592

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

Alternative 6
Error	38.9
Cost	320

\[0.5 + 0.25 \cdot a \]

Alternative 7
Error	39.0
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023099 
(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))))