expq3 (problem 3.4.2)

Average Accuracy: 5.7% → 85.0%

Time: 16.7s

Precision: binary64

Cost: 964

\[-1 < \varepsilon \land \varepsilon < 1\]

\[ \begin{array}{c}[a, b] = \mathsf{sort}([a, b])\\ \end{array} \]

Math

\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

↓

\[\begin{array}{l} \mathbf{if}\;b \leq 2.5 \cdot 10^{-141}:\\ \;\;\;\;\frac{1}{b} - \varepsilon \cdot \left(\varepsilon \cdot \left(b \cdot -0.08333333333333333\right) + 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a}{b \cdot a}\\ \end{array} \]

FPCore

(FPCore (a b eps)
 :precision binary64
 (/
  (* eps (- (exp (* (+ a b) eps)) 1.0))
  (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0))))

↓

(FPCore (a b eps)
 :precision binary64
 (if (<= b 2.5e-141)
   (- (/ 1.0 b) (* eps (+ (* eps (* b -0.08333333333333333)) 0.5)))
   (/ (+ b a) (* b a))))

C

double code(double a, double b, double eps) {
	return (eps * (exp(((a + b) * eps)) - 1.0)) / ((exp((a * eps)) - 1.0) * (exp((b * eps)) - 1.0));
}

↓

double code(double a, double b, double eps) {
	double tmp;
	if (b <= 2.5e-141) {
		tmp = (1.0 / b) - (eps * ((eps * (b * -0.08333333333333333)) + 0.5));
	} else {
		tmp = (b + a) / (b * a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, eps)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: eps
    code = (eps * (exp(((a + b) * eps)) - 1.0d0)) / ((exp((a * eps)) - 1.0d0) * (exp((b * eps)) - 1.0d0))
end function

↓

real(8) function code(a, b, eps)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (b <= 2.5d-141) then
        tmp = (1.0d0 / b) - (eps * ((eps * (b * (-0.08333333333333333d0))) + 0.5d0))
    else
        tmp = (b + a) / (b * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double eps) {
	return (eps * (Math.exp(((a + b) * eps)) - 1.0)) / ((Math.exp((a * eps)) - 1.0) * (Math.exp((b * eps)) - 1.0));
}

↓

public static double code(double a, double b, double eps) {
	double tmp;
	if (b <= 2.5e-141) {
		tmp = (1.0 / b) - (eps * ((eps * (b * -0.08333333333333333)) + 0.5));
	} else {
		tmp = (b + a) / (b * a);
	}
	return tmp;
}

Python

def code(a, b, eps):
	return (eps * (math.exp(((a + b) * eps)) - 1.0)) / ((math.exp((a * eps)) - 1.0) * (math.exp((b * eps)) - 1.0))

↓

def code(a, b, eps):
	tmp = 0
	if b <= 2.5e-141:
		tmp = (1.0 / b) - (eps * ((eps * (b * -0.08333333333333333)) + 0.5))
	else:
		tmp = (b + a) / (b * a)
	return tmp

Julia

function code(a, b, eps)
	return Float64(Float64(eps * Float64(exp(Float64(Float64(a + b) * eps)) - 1.0)) / Float64(Float64(exp(Float64(a * eps)) - 1.0) * Float64(exp(Float64(b * eps)) - 1.0)))
end

↓

function code(a, b, eps)
	tmp = 0.0
	if (b <= 2.5e-141)
		tmp = Float64(Float64(1.0 / b) - Float64(eps * Float64(Float64(eps * Float64(b * -0.08333333333333333)) + 0.5)));
	else
		tmp = Float64(Float64(b + a) / Float64(b * a));
	end
	return tmp
end

MATLAB

function tmp = code(a, b, eps)
	tmp = (eps * (exp(((a + b) * eps)) - 1.0)) / ((exp((a * eps)) - 1.0) * (exp((b * eps)) - 1.0));
end

↓

function tmp_2 = code(a, b, eps)
	tmp = 0.0;
	if (b <= 2.5e-141)
		tmp = (1.0 / b) - (eps * ((eps * (b * -0.08333333333333333)) + 0.5));
	else
		tmp = (b + a) / (b * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, eps_] := N[(N[(eps * N[(N[Exp[N[(N[(a + b), $MachinePrecision] * eps), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Exp[N[(a * eps), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision] * N[(N[Exp[N[(b * eps), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[a_, b_, eps_] := If[LessEqual[b, 2.5e-141], N[(N[(1.0 / b), $MachinePrecision] - N[(eps * N[(N[(eps * N[(b * -0.08333333333333333), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + a), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision]]

Target

Original	5.7%
Target	76.4%
Herbie	85.0%

\[\frac{a + b}{a \cdot b} \]

Derivation

1. Split input into 2 regimes

2. if b < 2.5e-141

   1. Initial program 3.1%

      \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

   2. Simplified 24.8%

      \[\leadsto \color{blue}{\frac{\varepsilon \cdot \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)}} \]
      Proof

      [Start] 3.1	\[ \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      expm1-def [=>] 3.1	\[ \frac{\varepsilon \cdot \color{blue}{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)}}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      *-commutative [=>] 3.1	\[ \frac{\varepsilon \cdot \mathsf{expm1}\left(\color{blue}{\varepsilon \cdot \left(a + b\right)}\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      expm1-def [=>] 3.1	\[ \frac{\varepsilon \cdot \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\color{blue}{\mathsf{expm1}\left(a \cdot \varepsilon\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      *-commutative [=>] 3.1	\[ \frac{\varepsilon \cdot \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\mathsf{expm1}\left(\color{blue}{\varepsilon \cdot a}\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      expm1-def [=>] 24.8	\[ \frac{\varepsilon \cdot \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \color{blue}{\mathsf{expm1}\left(b \cdot \varepsilon\right)}} \]
      *-commutative [=>] 24.8	\[ \frac{\varepsilon \cdot \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\color{blue}{\varepsilon \cdot b}\right)} \]

   3. Taylor expanded in b around 0 16.4%

      \[\leadsto \frac{\color{blue}{\left(e^{\varepsilon \cdot a} - 1\right) \cdot \varepsilon}}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)} \]

   4. Simplified 24.5%

      \[\leadsto \frac{\color{blue}{\varepsilon \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)}}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)} \]
      Proof

      [Start] 16.4	\[ \frac{\left(e^{\varepsilon \cdot a} - 1\right) \cdot \varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)} \]
      expm1-def [=>] 24.5	\[ \frac{\color{blue}{\mathsf{expm1}\left(\varepsilon \cdot a\right)} \cdot \varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)} \]
      *-commutative [=>] 24.5	\[ \frac{\color{blue}{\varepsilon \cdot \mathsf{expm1}\left(\varepsilon \cdot a\right)}}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)} \]
      *-commutative [=>] 24.5	\[ \frac{\varepsilon \cdot \mathsf{expm1}\left(\color{blue}{a \cdot \varepsilon}\right)}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)} \]

   5. Taylor expanded in eps around 0 85.2%

      \[\leadsto \color{blue}{-0.5 \cdot \varepsilon + \left(-1 \cdot \left({\varepsilon}^{2} \cdot \left(0.16666666666666666 \cdot b + -0.25 \cdot b\right)\right) + \frac{1}{b}\right)} \]

   6. Simplified 85.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \varepsilon, \frac{1}{b} - b \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot -0.08333333333333333\right)\right)\right)} \]
      Proof

      [Start] 85.2	\[ -0.5 \cdot \varepsilon + \left(-1 \cdot \left({\varepsilon}^{2} \cdot \left(0.16666666666666666 \cdot b + -0.25 \cdot b\right)\right) + \frac{1}{b}\right) \]
      fma-def [=>] 85.2	\[ \color{blue}{\mathsf{fma}\left(-0.5, \varepsilon, -1 \cdot \left({\varepsilon}^{2} \cdot \left(0.16666666666666666 \cdot b + -0.25 \cdot b\right)\right) + \frac{1}{b}\right)} \]
      +-commutative [=>] 85.2	\[ \mathsf{fma}\left(-0.5, \varepsilon, \color{blue}{\frac{1}{b} + -1 \cdot \left({\varepsilon}^{2} \cdot \left(0.16666666666666666 \cdot b + -0.25 \cdot b\right)\right)}\right) \]
      mul-1-neg [=>] 85.2	\[ \mathsf{fma}\left(-0.5, \varepsilon, \frac{1}{b} + \color{blue}{\left(-{\varepsilon}^{2} \cdot \left(0.16666666666666666 \cdot b + -0.25 \cdot b\right)\right)}\right) \]
      unsub-neg [=>] 85.2	\[ \mathsf{fma}\left(-0.5, \varepsilon, \color{blue}{\frac{1}{b} - {\varepsilon}^{2} \cdot \left(0.16666666666666666 \cdot b + -0.25 \cdot b\right)}\right) \]
      *-commutative [=>] 85.2	\[ \mathsf{fma}\left(-0.5, \varepsilon, \frac{1}{b} - \color{blue}{\left(0.16666666666666666 \cdot b + -0.25 \cdot b\right) \cdot {\varepsilon}^{2}}\right) \]
      distribute-rgt-out [=>] 85.2	\[ \mathsf{fma}\left(-0.5, \varepsilon, \frac{1}{b} - \color{blue}{\left(b \cdot \left(0.16666666666666666 + -0.25\right)\right)} \cdot {\varepsilon}^{2}\right) \]
      metadata-eval [=>] 85.2	\[ \mathsf{fma}\left(-0.5, \varepsilon, \frac{1}{b} - \left(b \cdot \color{blue}{-0.08333333333333333}\right) \cdot {\varepsilon}^{2}\right) \]
      associate-*l* [=>] 85.2	\[ \mathsf{fma}\left(-0.5, \varepsilon, \frac{1}{b} - \color{blue}{b \cdot \left(-0.08333333333333333 \cdot {\varepsilon}^{2}\right)}\right) \]
      *-commutative [<=] 85.2	\[ \mathsf{fma}\left(-0.5, \varepsilon, \frac{1}{b} - b \cdot \color{blue}{\left({\varepsilon}^{2} \cdot -0.08333333333333333\right)}\right) \]
      unpow2 [=>] 85.2	\[ \mathsf{fma}\left(-0.5, \varepsilon, \frac{1}{b} - b \cdot \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot -0.08333333333333333\right)\right) \]
      associate-*l* [=>] 85.2	\[ \mathsf{fma}\left(-0.5, \varepsilon, \frac{1}{b} - b \cdot \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot -0.08333333333333333\right)\right)}\right) \]

   7. Taylor expanded in eps around 0 85.2%

      \[\leadsto \color{blue}{-0.5 \cdot \varepsilon + \left(\frac{1}{b} + 0.08333333333333333 \cdot \left({\varepsilon}^{2} \cdot b\right)\right)} \]

   8. Simplified 85.2%

      \[\leadsto \color{blue}{\frac{1}{b} - \varepsilon \cdot \mathsf{fma}\left(\varepsilon, -0.08333333333333333 \cdot b, 0.5\right)} \]
      Proof

      [Start] 85.2	\[ -0.5 \cdot \varepsilon + \left(\frac{1}{b} + 0.08333333333333333 \cdot \left({\varepsilon}^{2} \cdot b\right)\right) \]
      *-commutative [=>] 85.2	\[ \color{blue}{\varepsilon \cdot -0.5} + \left(\frac{1}{b} + 0.08333333333333333 \cdot \left({\varepsilon}^{2} \cdot b\right)\right) \]
      metadata-eval [<=] 85.2	\[ \varepsilon \cdot -0.5 + \left(\frac{1}{b} + \color{blue}{\left(--0.08333333333333333\right)} \cdot \left({\varepsilon}^{2} \cdot b\right)\right) \]
      cancel-sign-sub-inv [<=] 85.2	\[ \varepsilon \cdot -0.5 + \color{blue}{\left(\frac{1}{b} - -0.08333333333333333 \cdot \left({\varepsilon}^{2} \cdot b\right)\right)} \]
      unpow2 [=>] 85.2	\[ \varepsilon \cdot -0.5 + \left(\frac{1}{b} - -0.08333333333333333 \cdot \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot b\right)\right) \]
      associate-*r* [=>] 85.2	\[ \varepsilon \cdot -0.5 + \left(\frac{1}{b} - \color{blue}{\left(-0.08333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot b}\right) \]
      *-commutative [=>] 85.2	\[ \varepsilon \cdot -0.5 + \left(\frac{1}{b} - \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.08333333333333333\right)} \cdot b\right) \]
      associate-*r* [<=] 85.2	\[ \varepsilon \cdot -0.5 + \left(\frac{1}{b} - \color{blue}{\left(\varepsilon \cdot \varepsilon\right) \cdot \left(-0.08333333333333333 \cdot b\right)}\right) \]
      +-commutative [=>] 85.2	\[ \color{blue}{\left(\frac{1}{b} - \left(\varepsilon \cdot \varepsilon\right) \cdot \left(-0.08333333333333333 \cdot b\right)\right) + \varepsilon \cdot -0.5} \]
      associate-+l- [=>] 85.2	\[ \color{blue}{\frac{1}{b} - \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(-0.08333333333333333 \cdot b\right) - \varepsilon \cdot -0.5\right)} \]
      associate-*l* [=>] 85.2	\[ \frac{1}{b} - \left(\color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(-0.08333333333333333 \cdot b\right)\right)} - \varepsilon \cdot -0.5\right) \]
      distribute-lft-out-- [=>] 85.2	\[ \frac{1}{b} - \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(-0.08333333333333333 \cdot b\right) - -0.5\right)} \]
      fma-neg [=>] 85.2	\[ \frac{1}{b} - \varepsilon \cdot \color{blue}{\mathsf{fma}\left(\varepsilon, -0.08333333333333333 \cdot b, --0.5\right)} \]
      metadata-eval [=>] 85.2	\[ \frac{1}{b} - \varepsilon \cdot \mathsf{fma}\left(\varepsilon, -0.08333333333333333 \cdot b, \color{blue}{0.5}\right) \]

   9. Applied egg-rr 85.2%

      \[\leadsto \frac{1}{b} - \varepsilon \cdot \color{blue}{\left(\left(-0.08333333333333333 \cdot b\right) \cdot \varepsilon + 0.5\right)} \]
      Proof

      [Start] 85.2	\[ \frac{1}{b} - \varepsilon \cdot \mathsf{fma}\left(\varepsilon, -0.08333333333333333 \cdot b, 0.5\right) \]
      fma-udef [=>] 85.2	\[ \frac{1}{b} - \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \left(-0.08333333333333333 \cdot b\right) + 0.5\right)} \]
      *-commutative [=>] 85.2	\[ \frac{1}{b} - \varepsilon \cdot \left(\color{blue}{\left(-0.08333333333333333 \cdot b\right) \cdot \varepsilon} + 0.5\right) \]

   if 2.5e-141 < b

   1. Initial program 7.6%

      \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

   2. Simplified 47.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)}} \]
      Proof

      [Start] 7.6	\[ \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      associate-*l/ [<=] 7.6	\[ \color{blue}{\frac{\varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)} \]
      *-commutative [=>] 7.6	\[ \color{blue}{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right) \cdot \frac{\varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}} \]
      expm1-def [=>] 7.6	\[ \color{blue}{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)} \cdot \frac{\varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      *-commutative [=>] 7.6	\[ \mathsf{expm1}\left(\color{blue}{\varepsilon \cdot \left(a + b\right)}\right) \cdot \frac{\varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      expm1-def [=>] 23.4	\[ \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\color{blue}{\mathsf{expm1}\left(a \cdot \varepsilon\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      *-commutative [=>] 23.4	\[ \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\color{blue}{\varepsilon \cdot a}\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      expm1-def [=>] 47.8	\[ \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \color{blue}{\mathsf{expm1}\left(b \cdot \varepsilon\right)}} \]
      *-commutative [=>] 47.8	\[ \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\color{blue}{\varepsilon \cdot b}\right)} \]

   3. Taylor expanded in eps around 0 84.9%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.5 \cdot 10^{-141}:\\ \;\;\;\;\frac{1}{b} - \varepsilon \cdot \left(\varepsilon \cdot \left(b \cdot -0.08333333333333333\right) + 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a}{b \cdot a}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	77.1%
Cost	845

\[\begin{array}{l} \mathbf{if}\;b \leq 1.75 \cdot 10^{-110} \lor \neg \left(b \leq 2.85 \cdot 10^{-83}\right) \land b \leq 6.2 \cdot 10^{-39}:\\ \;\;\;\;\frac{1}{b} + \varepsilon \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a}\\ \end{array} \]

Alternative 2
Accuracy	76.8%
Cost	589

\[\begin{array}{l} \mathbf{if}\;b \leq 2.55 \cdot 10^{-110} \lor \neg \left(b \leq 3.5 \cdot 10^{-83}\right) \land b \leq 1.4 \cdot 10^{-35}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a}\\ \end{array} \]

Alternative 3
Accuracy	85.1%
Cost	580

\[\begin{array}{l} \mathbf{if}\;b \leq 4.8 \cdot 10^{-141}:\\ \;\;\;\;\frac{1}{b} + \varepsilon \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a}{b \cdot a}\\ \end{array} \]

Alternative 4
Accuracy	3.1%
Cost	192

\[\varepsilon \cdot -0.5 \]

Alternative 5
Accuracy	47.8%
Cost	192

\[\frac{1}{a} \]

Reproduce

herbie shell --seed 2023136 
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :precision binary64
  :pre (and (< -1.0 eps) (< eps 1.0))

  :herbie-target
  (/ (+ a b) (* a b))

  (/ (* eps (- (exp (* (+ a b) eps)) 1.0)) (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0))))