expq3 (problem 3.4.2)

Average Error: 60.5 → 1.2

Time: 22.6s

Precision: binary64

Cost: 61384

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

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

Math

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

↓

\[\begin{array}{l} t_0 := \varepsilon \cdot \left(a + b\right)\\ t_1 := \frac{\varepsilon \cdot \left(e^{t_0} + -1\right)}{\left(e^{\varepsilon \cdot a} + -1\right) \cdot \left(e^{\varepsilon \cdot b} + -1\right)}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{\frac{a + b}{b}}{a}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{-29}:\\ \;\;\;\;\varepsilon \cdot \frac{\mathsf{expm1}\left(t_0\right)}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\left(a + b\right) \cdot \frac{\varepsilon}{a}\right) + \left(\frac{1}{a} + \frac{1}{b}\right)\\ \end{array} \]

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
 (let* ((t_0 (* eps (+ a b)))
        (t_1
         (/
          (* eps (+ (exp t_0) -1.0))
          (* (+ (exp (* eps a)) -1.0) (+ (exp (* eps b)) -1.0)))))
   (if (<= t_1 (- INFINITY))
     (/ (/ (+ a b) b) a)
     (if (<= t_1 2e-29)
       (* eps (/ (expm1 t_0) (* (expm1 (* eps a)) (expm1 (* eps b)))))
       (+ (* -0.5 (* (+ a b) (/ eps a))) (+ (/ 1.0 a) (/ 1.0 b)))))))

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 t_0 = eps * (a + b);
	double t_1 = (eps * (exp(t_0) + -1.0)) / ((exp((eps * a)) + -1.0) * (exp((eps * b)) + -1.0));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = ((a + b) / b) / a;
	} else if (t_1 <= 2e-29) {
		tmp = eps * (expm1(t_0) / (expm1((eps * a)) * expm1((eps * b))));
	} else {
		tmp = (-0.5 * ((a + b) * (eps / a))) + ((1.0 / a) + (1.0 / b));
	}
	return tmp;
}

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 t_0 = eps * (a + b);
	double t_1 = (eps * (Math.exp(t_0) + -1.0)) / ((Math.exp((eps * a)) + -1.0) * (Math.exp((eps * b)) + -1.0));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = ((a + b) / b) / a;
	} else if (t_1 <= 2e-29) {
		tmp = eps * (Math.expm1(t_0) / (Math.expm1((eps * a)) * Math.expm1((eps * b))));
	} else {
		tmp = (-0.5 * ((a + b) * (eps / a))) + ((1.0 / a) + (1.0 / b));
	}
	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):
	t_0 = eps * (a + b)
	t_1 = (eps * (math.exp(t_0) + -1.0)) / ((math.exp((eps * a)) + -1.0) * (math.exp((eps * b)) + -1.0))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = ((a + b) / b) / a
	elif t_1 <= 2e-29:
		tmp = eps * (math.expm1(t_0) / (math.expm1((eps * a)) * math.expm1((eps * b))))
	else:
		tmp = (-0.5 * ((a + b) * (eps / a))) + ((1.0 / a) + (1.0 / b))
	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)
	t_0 = Float64(eps * Float64(a + b))
	t_1 = Float64(Float64(eps * Float64(exp(t_0) + -1.0)) / Float64(Float64(exp(Float64(eps * a)) + -1.0) * Float64(exp(Float64(eps * b)) + -1.0)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(a + b) / b) / a);
	elseif (t_1 <= 2e-29)
		tmp = Float64(eps * Float64(expm1(t_0) / Float64(expm1(Float64(eps * a)) * expm1(Float64(eps * b)))));
	else
		tmp = Float64(Float64(-0.5 * Float64(Float64(a + b) * Float64(eps / a))) + Float64(Float64(1.0 / a) + Float64(1.0 / b)));
	end
	return 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_] := Block[{t$95$0 = N[(eps * N[(a + b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(eps * N[(N[Exp[t$95$0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Exp[N[(eps * a), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision] * N[(N[Exp[N[(eps * b), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(a + b), $MachinePrecision] / b), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[t$95$1, 2e-29], N[(eps * N[(N[(Exp[t$95$0] - 1), $MachinePrecision] / N[(N[(Exp[N[(eps * a), $MachinePrecision]] - 1), $MachinePrecision] * N[(Exp[N[(eps * b), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(N[(a + b), $MachinePrecision] * N[(eps / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / a), $MachinePrecision] + N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	60.5
Target	14.4
Herbie	1.2

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 eps (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1)) (*.f64 (-.f64 (exp.f64 (*.f64 a eps)) 1) (-.f64 (exp.f64 (*.f64 b eps)) 1))) < -inf.0

   1. Initial program 64.0

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

   2. Simplified 18.7

      \[\leadsto \color{blue}{\varepsilon \cdot \frac{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)}} \]
      Proof
      (*.f64 eps (/.f64 (expm1.f64 (*.f64 eps (+.f64 a b))) (*.f64 (expm1.f64 (*.f64 eps a)) (expm1.f64 (*.f64 eps b))))): 0 points increase in error, 0 points decrease in error
      (*.f64 eps (/.f64 (expm1.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 a b) eps))) (*.f64 (expm1.f64 (*.f64 eps a)) (expm1.f64 (*.f64 eps b))))): 0 points increase in error, 0 points decrease in error
      (*.f64 eps (/.f64 (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1)) (*.f64 (expm1.f64 (*.f64 eps a)) (expm1.f64 (*.f64 eps b))))): 39 points increase in error, 0 points decrease in error
      (*.f64 eps (/.f64 (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1) (*.f64 (expm1.f64 (Rewrite<= *-commutative_binary64 (*.f64 a eps))) (expm1.f64 (*.f64 eps b))))): 0 points increase in error, 0 points decrease in error
      (*.f64 eps (/.f64 (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1) (*.f64 (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (*.f64 a eps)) 1)) (expm1.f64 (*.f64 eps b))))): 62 points increase in error, 3 points decrease in error
      (*.f64 eps (/.f64 (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1) (*.f64 (-.f64 (exp.f64 (*.f64 a eps)) 1) (expm1.f64 (Rewrite<= *-commutative_binary64 (*.f64 b eps)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 eps (/.f64 (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1) (*.f64 (-.f64 (exp.f64 (*.f64 a eps)) 1) (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (*.f64 b eps)) 1))))): 24 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 eps (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1)) (*.f64 (-.f64 (exp.f64 (*.f64 a eps)) 1) (-.f64 (exp.f64 (*.f64 b eps)) 1)))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in eps around 0 7.4

      \[\leadsto \color{blue}{\frac{a + b}{a \cdot b}} \]

   4. Simplified 29.5

      \[\leadsto \color{blue}{\frac{\frac{a + b}{a}}{b}} \]
      Proof
      (/.f64 (/.f64 (+.f64 a b) a) b): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r*_binary64 (/.f64 (+.f64 a b) (*.f64 a b))): 93 points increase in error, 42 points decrease in error

   5. Applied egg-rr 29.5

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

   6. Applied egg-rr 2.2

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

   if -inf.0 < (/.f64 (*.f64 eps (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1)) (*.f64 (-.f64 (exp.f64 (*.f64 a eps)) 1) (-.f64 (exp.f64 (*.f64 b eps)) 1))) < 1.99999999999999989e-29

   1. Initial program 3.2

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

   2. Simplified 0.0

      \[\leadsto \color{blue}{\varepsilon \cdot \frac{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)}} \]
      Proof
      (*.f64 eps (/.f64 (expm1.f64 (*.f64 eps (+.f64 a b))) (*.f64 (expm1.f64 (*.f64 eps a)) (expm1.f64 (*.f64 eps b))))): 0 points increase in error, 0 points decrease in error
      (*.f64 eps (/.f64 (expm1.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 a b) eps))) (*.f64 (expm1.f64 (*.f64 eps a)) (expm1.f64 (*.f64 eps b))))): 0 points increase in error, 0 points decrease in error
      (*.f64 eps (/.f64 (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1)) (*.f64 (expm1.f64 (*.f64 eps a)) (expm1.f64 (*.f64 eps b))))): 39 points increase in error, 0 points decrease in error
      (*.f64 eps (/.f64 (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1) (*.f64 (expm1.f64 (Rewrite<= *-commutative_binary64 (*.f64 a eps))) (expm1.f64 (*.f64 eps b))))): 0 points increase in error, 0 points decrease in error
      (*.f64 eps (/.f64 (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1) (*.f64 (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (*.f64 a eps)) 1)) (expm1.f64 (*.f64 eps b))))): 62 points increase in error, 3 points decrease in error
      (*.f64 eps (/.f64 (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1) (*.f64 (-.f64 (exp.f64 (*.f64 a eps)) 1) (expm1.f64 (Rewrite<= *-commutative_binary64 (*.f64 b eps)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 eps (/.f64 (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1) (*.f64 (-.f64 (exp.f64 (*.f64 a eps)) 1) (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (*.f64 b eps)) 1))))): 24 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 eps (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1)) (*.f64 (-.f64 (exp.f64 (*.f64 a eps)) 1) (-.f64 (exp.f64 (*.f64 b eps)) 1)))): 0 points increase in error, 0 points decrease in error

   if 1.99999999999999989e-29 < (/.f64 (*.f64 eps (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1)) (*.f64 (-.f64 (exp.f64 (*.f64 a eps)) 1) (-.f64 (exp.f64 (*.f64 b eps)) 1)))

   1. Initial program 63.7

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

   2. Simplified 46.1

      \[\leadsto \color{blue}{\varepsilon \cdot \frac{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)}} \]
      Proof
      (*.f64 eps (/.f64 (expm1.f64 (*.f64 eps (+.f64 a b))) (*.f64 (expm1.f64 (*.f64 eps a)) (expm1.f64 (*.f64 eps b))))): 0 points increase in error, 0 points decrease in error
      (*.f64 eps (/.f64 (expm1.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 a b) eps))) (*.f64 (expm1.f64 (*.f64 eps a)) (expm1.f64 (*.f64 eps b))))): 0 points increase in error, 0 points decrease in error
      (*.f64 eps (/.f64 (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1)) (*.f64 (expm1.f64 (*.f64 eps a)) (expm1.f64 (*.f64 eps b))))): 39 points increase in error, 0 points decrease in error
      (*.f64 eps (/.f64 (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1) (*.f64 (expm1.f64 (Rewrite<= *-commutative_binary64 (*.f64 a eps))) (expm1.f64 (*.f64 eps b))))): 0 points increase in error, 0 points decrease in error
      (*.f64 eps (/.f64 (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1) (*.f64 (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (*.f64 a eps)) 1)) (expm1.f64 (*.f64 eps b))))): 62 points increase in error, 3 points decrease in error
      (*.f64 eps (/.f64 (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1) (*.f64 (-.f64 (exp.f64 (*.f64 a eps)) 1) (expm1.f64 (Rewrite<= *-commutative_binary64 (*.f64 b eps)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 eps (/.f64 (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1) (*.f64 (-.f64 (exp.f64 (*.f64 a eps)) 1) (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (*.f64 b eps)) 1))))): 24 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 eps (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1)) (*.f64 (-.f64 (exp.f64 (*.f64 a eps)) 1) (-.f64 (exp.f64 (*.f64 b eps)) 1)))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in eps around 0 52.3

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

   4. Simplified 52.3

      \[\leadsto \varepsilon \cdot \frac{\color{blue}{\left(a + b\right) \cdot \varepsilon}}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)} \]
      Proof
      (*.f64 (+.f64 a b) eps): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 eps (+.f64 a b))): 0 points increase in error, 0 points decrease in error

   5. Taylor expanded in a around 0 63.6

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

   6. Simplified 51.9

      \[\leadsto \varepsilon \cdot \frac{\left(a + b\right) \cdot \varepsilon}{\color{blue}{a \cdot \left(\varepsilon \cdot \mathsf{expm1}\left(b \cdot \varepsilon\right)\right)}} \]
      Proof
      (*.f64 a (*.f64 eps (expm1.f64 (*.f64 b eps)))): 0 points increase in error, 0 points decrease in error
      (*.f64 a (*.f64 eps (expm1.f64 (Rewrite<= *-commutative_binary64 (*.f64 eps b))))): 0 points increase in error, 0 points decrease in error
      (*.f64 a (*.f64 eps (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (*.f64 eps b)) 1)))): 14 points increase in error, 29 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 a eps) (-.f64 (exp.f64 (*.f64 eps b)) 1))): 2 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 eps a)) (-.f64 (exp.f64 (*.f64 eps b)) 1)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r*_binary64 (*.f64 eps (*.f64 a (-.f64 (exp.f64 (*.f64 eps b)) 1)))): 0 points increase in error, 4 points decrease in error

   7. Taylor expanded in eps around 0 1.2

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

   8. Applied egg-rr 1.1

      \[\leadsto -0.5 \cdot \color{blue}{\left(\frac{\varepsilon}{a} \cdot \left(a + b\right)\right)} + \left(\frac{1}{a} + \frac{1}{b}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} + -1\right)}{\left(e^{\varepsilon \cdot a} + -1\right) \cdot \left(e^{\varepsilon \cdot b} + -1\right)} \leq -\infty:\\ \;\;\;\;\frac{\frac{a + b}{b}}{a}\\ \mathbf{elif}\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} + -1\right)}{\left(e^{\varepsilon \cdot a} + -1\right) \cdot \left(e^{\varepsilon \cdot b} + -1\right)} \leq 2 \cdot 10^{-29}:\\ \;\;\;\;\varepsilon \cdot \frac{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\left(a + b\right) \cdot \frac{\varepsilon}{a}\right) + \left(\frac{1}{a} + \frac{1}{b}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	3.2
Cost	1220

\[\begin{array}{l} \mathbf{if}\;b \leq 9.520435249553234 \cdot 10^{-49}:\\ \;\;\;\;-0.5 \cdot \left(\left(a + b\right) \cdot \frac{\varepsilon}{a}\right) + \left(\frac{1}{a} + \frac{1}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} + \left(\frac{\varepsilon}{\varepsilon \cdot b} + \varepsilon \cdot -0.5\right)\\ \end{array} \]

Alternative 2
Error	3.0
Cost	1220

\[\begin{array}{l} \mathbf{if}\;b \leq 451691722538873:\\ \;\;\;\;\left(\frac{1}{a} + \frac{1}{b}\right) + -0.5 \cdot \frac{\varepsilon \cdot \left(a + b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} + \left(\frac{\varepsilon}{\varepsilon \cdot b} + \varepsilon \cdot -0.5\right)\\ \end{array} \]

Alternative 3
Error	3.3
Cost	964

\[\begin{array}{l} \mathbf{if}\;b \leq 9.520435249553234 \cdot 10^{-49}:\\ \;\;\;\;\frac{1}{b} \cdot \frac{a + b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} + \left(\frac{\varepsilon}{\varepsilon \cdot b} + \varepsilon \cdot -0.5\right)\\ \end{array} \]

Alternative 4
Error	13.2
Cost	844

\[\begin{array}{l} \mathbf{if}\;b \leq 4 \cdot 10^{-130}:\\ \;\;\;\;\frac{1}{b} + \varepsilon \cdot -0.5\\ \mathbf{elif}\;b \leq 1.130119473184844 \cdot 10^{-62}:\\ \;\;\;\;\frac{1}{a}\\ \mathbf{elif}\;b \leq 1.605140191044092 \cdot 10^{-56}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} + \varepsilon \cdot -0.5\\ \end{array} \]

Alternative 5
Error	3.3
Cost	708

\[\begin{array}{l} \mathbf{if}\;b \leq 7.986916818762323 \cdot 10^{-57}:\\ \;\;\;\;\frac{1}{b} \cdot \frac{a + b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a + b}{b}}{a}\\ \end{array} \]

Alternative 6
Error	13.5
Cost	588

\[\begin{array}{l} \mathbf{if}\;b \leq 4 \cdot 10^{-130}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{elif}\;b \leq 1.130119473184844 \cdot 10^{-62}:\\ \;\;\;\;\frac{1}{a}\\ \mathbf{elif}\;b \leq 1.605140191044092 \cdot 10^{-56}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a}\\ \end{array} \]

Alternative 7
Error	13.3
Cost	588

\[\begin{array}{l} \mathbf{if}\;b \leq 4 \cdot 10^{-130}:\\ \;\;\;\;\frac{1}{b} + \varepsilon \cdot -0.5\\ \mathbf{elif}\;b \leq 1.130119473184844 \cdot 10^{-62}:\\ \;\;\;\;\frac{1}{a}\\ \mathbf{elif}\;b \leq 1.605140191044092 \cdot 10^{-56}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a}\\ \end{array} \]

Alternative 8
Error	5.1
Cost	580

\[\begin{array}{l} \mathbf{if}\;a \leq -1.2877227664165489 \cdot 10^{+33}:\\ \;\;\;\;\frac{1}{b} + \varepsilon \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a + b}{b}}{a}\\ \end{array} \]

Alternative 9
Error	3.3
Cost	580

\[\begin{array}{l} \mathbf{if}\;b \leq 7.986916818762323 \cdot 10^{-57}:\\ \;\;\;\;\frac{\frac{a + b}{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a + b}{b}}{a}\\ \end{array} \]

Alternative 10
Error	62.0
Cost	192

\[\varepsilon \cdot -0.5 \]

Alternative 11
Error	32.7
Cost	192

\[\frac{1}{b} \]

Reproduce

herbie shell --seed 2022318 
(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))))