Average Error: 60.6 → 0.1
Time: 13.6s
Precision: binary64
Cost: 87304
\[-1 < \varepsilon \land \varepsilon < 1\]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
\[\begin{array}{l} t_0 := \sqrt[3]{\mathsf{expm1}\left(\varepsilon \cdot b\right)}\\ t_1 := \varepsilon \cdot \left(a + b\right)\\ t_2 := \frac{\varepsilon \cdot \left(e^{t_1} + -1\right)}{\left(e^{\varepsilon \cdot a} + -1\right) \cdot \left(e^{\varepsilon \cdot b} + -1\right)}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{\frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)} \cdot \mathsf{expm1}\left(t_1\right)}{{t_0}^{2}}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \left(\frac{1}{a} + \varepsilon \cdot 0.5\right)\\ \end{array} \]
(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 (cbrt (expm1 (* eps b))))
        (t_1 (* eps (+ a b)))
        (t_2
         (/
          (* eps (+ (exp t_1) -1.0))
          (* (+ (exp (* eps a)) -1.0) (+ (exp (* eps b)) -1.0)))))
   (if (<= t_2 (- INFINITY))
     (+ (/ 1.0 b) (/ 1.0 a))
     (if (<= t_2 2e-19)
       (/ (/ (* (/ eps (expm1 (* eps a))) (expm1 t_1)) (pow t_0 2.0)) t_0)
       (+ (/ 1.0 b) (+ (/ 1.0 a) (* eps 0.5)))))))
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 = cbrt(expm1((eps * b)));
	double t_1 = eps * (a + b);
	double t_2 = (eps * (exp(t_1) + -1.0)) / ((exp((eps * a)) + -1.0) * (exp((eps * b)) + -1.0));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (1.0 / b) + (1.0 / a);
	} else if (t_2 <= 2e-19) {
		tmp = (((eps / expm1((eps * a))) * expm1(t_1)) / pow(t_0, 2.0)) / t_0;
	} else {
		tmp = (1.0 / b) + ((1.0 / a) + (eps * 0.5));
	}
	return tmp;
}
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 = Math.cbrt(Math.expm1((eps * b)));
	double t_1 = eps * (a + b);
	double t_2 = (eps * (Math.exp(t_1) + -1.0)) / ((Math.exp((eps * a)) + -1.0) * (Math.exp((eps * b)) + -1.0));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = (1.0 / b) + (1.0 / a);
	} else if (t_2 <= 2e-19) {
		tmp = (((eps / Math.expm1((eps * a))) * Math.expm1(t_1)) / Math.pow(t_0, 2.0)) / t_0;
	} else {
		tmp = (1.0 / b) + ((1.0 / a) + (eps * 0.5));
	}
	return tmp;
}
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 = cbrt(expm1(Float64(eps * b)))
	t_1 = Float64(eps * Float64(a + b))
	t_2 = Float64(Float64(eps * Float64(exp(t_1) + -1.0)) / Float64(Float64(exp(Float64(eps * a)) + -1.0) * Float64(exp(Float64(eps * b)) + -1.0)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(1.0 / b) + Float64(1.0 / a));
	elseif (t_2 <= 2e-19)
		tmp = Float64(Float64(Float64(Float64(eps / expm1(Float64(eps * a))) * expm1(t_1)) / (t_0 ^ 2.0)) / t_0);
	else
		tmp = Float64(Float64(1.0 / b) + Float64(Float64(1.0 / a) + Float64(eps * 0.5)));
	end
	return tmp
end
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[Power[N[(Exp[N[(eps * b), $MachinePrecision]] - 1), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[(eps * N[(a + b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(eps * N[(N[Exp[t$95$1], $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$2, (-Infinity)], N[(N[(1.0 / b), $MachinePrecision] + N[(1.0 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e-19], N[(N[(N[(N[(eps / N[(Exp[N[(eps * a), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] * N[(Exp[t$95$1] - 1), $MachinePrecision]), $MachinePrecision] / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(1.0 / b), $MachinePrecision] + N[(N[(1.0 / a), $MachinePrecision] + N[(eps * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]
\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 := \sqrt[3]{\mathsf{expm1}\left(\varepsilon \cdot b\right)}\\
t_1 := \varepsilon \cdot \left(a + b\right)\\
t_2 := \frac{\varepsilon \cdot \left(e^{t_1} + -1\right)}{\left(e^{\varepsilon \cdot a} + -1\right) \cdot \left(e^{\varepsilon \cdot b} + -1\right)}\\
\mathbf{if}\;t_2 \leq -\infty:\\
\;\;\;\;\frac{1}{b} + \frac{1}{a}\\

\mathbf{elif}\;t_2 \leq 2 \cdot 10^{-19}:\\
\;\;\;\;\frac{\frac{\frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)} \cdot \mathsf{expm1}\left(t_1\right)}{{t_0}^{2}}}{t_0}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{b} + \left(\frac{1}{a} + \varepsilon \cdot 0.5\right)\\


\end{array}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original60.6
Target14.8
Herbie0.1
\[\frac{a + b}{a \cdot b} \]

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. Simplified17.5

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)}} \]
      Proof
      (*.f64 (expm1.f64 (*.f64 eps (+.f64 a b))) (/.f64 eps (*.f64 (expm1.f64 (*.f64 eps a)) (expm1.f64 (*.f64 eps b))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (expm1.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 a b) eps))) (/.f64 eps (*.f64 (expm1.f64 (*.f64 eps a)) (expm1.f64 (*.f64 eps b))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1)) (/.f64 eps (*.f64 (expm1.f64 (*.f64 eps a)) (expm1.f64 (*.f64 eps b))))): 52 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1) (/.f64 eps (*.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 (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1) (/.f64 eps (*.f64 (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (*.f64 a eps)) 1)) (expm1.f64 (*.f64 eps b))))): 73 points increase in error, 6 points decrease in error
      (*.f64 (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1) (/.f64 eps (*.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 (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1) (/.f64 eps (*.f64 (-.f64 (exp.f64 (*.f64 a eps)) 1) (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (*.f64 b eps)) 1))))): 28 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1) eps) (*.f64 (-.f64 (exp.f64 (*.f64 a eps)) 1) (-.f64 (exp.f64 (*.f64 b eps)) 1)))): 0 points increase in error, 1 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.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 6.8

      \[\leadsto \color{blue}{\frac{a + b}{a \cdot b}} \]
    4. Simplified15.2

      \[\leadsto \color{blue}{\frac{\frac{a + b}{b}}{a}} \]
      Proof
      (/.f64 (/.f64 (+.f64 a b) b) a): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r*_binary64 (/.f64 (+.f64 a b) (*.f64 b a))): 90 points increase in error, 40 points decrease in error
      (/.f64 (+.f64 a b) (Rewrite<= *-commutative_binary64 (*.f64 a b))): 0 points increase in error, 0 points decrease in error
    5. Taylor expanded in a around 0 0.0

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

    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))) < 2e-19

    1. Initial program 3.8

      \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
    2. Simplified0.1

      \[\leadsto \color{blue}{\frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)} \cdot \frac{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\mathsf{expm1}\left(\varepsilon \cdot b\right)}} \]
      Proof
      (*.f64 (/.f64 eps (expm1.f64 (*.f64 eps a))) (/.f64 (expm1.f64 (*.f64 eps (+.f64 a b))) (expm1.f64 (*.f64 eps b)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 eps (expm1.f64 (Rewrite<= *-commutative_binary64 (*.f64 a eps)))) (/.f64 (expm1.f64 (*.f64 eps (+.f64 a b))) (expm1.f64 (*.f64 eps b)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 eps (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (*.f64 a eps)) 1))) (/.f64 (expm1.f64 (*.f64 eps (+.f64 a b))) (expm1.f64 (*.f64 eps b)))): 118 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 eps (-.f64 (exp.f64 (*.f64 a eps)) 1)) (/.f64 (expm1.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 a b) eps))) (expm1.f64 (*.f64 eps b)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 eps (-.f64 (exp.f64 (*.f64 a eps)) 1)) (/.f64 (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1)) (expm1.f64 (*.f64 eps b)))): 1 points increase in error, 6 points decrease in error
      (*.f64 (/.f64 eps (-.f64 (exp.f64 (*.f64 a eps)) 1)) (/.f64 (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1) (expm1.f64 (Rewrite<= *-commutative_binary64 (*.f64 b eps))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 eps (-.f64 (exp.f64 (*.f64 a eps)) 1)) (/.f64 (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1) (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (*.f64 b eps)) 1)))): 27 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_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, 2 points decrease in error
    3. Applied egg-rr0.2

      \[\leadsto \color{blue}{\frac{\frac{\frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)} \cdot \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{{\left(\sqrt[3]{\mathsf{expm1}\left(\varepsilon \cdot b\right)}\right)}^{2}}}{\sqrt[3]{\mathsf{expm1}\left(\varepsilon \cdot b\right)}}} \]

    if 2e-19 < (/.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 63.9

      \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
    2. Simplified33.2

      \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\mathsf{expm1}\left(\varepsilon \cdot a\right)} \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot b\right)}} \]
      Proof
      (*.f64 (/.f64 (expm1.f64 (*.f64 eps (+.f64 a b))) (expm1.f64 (*.f64 eps a))) (/.f64 eps (expm1.f64 (*.f64 eps b)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (expm1.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 a b) eps))) (expm1.f64 (*.f64 eps a))) (/.f64 eps (expm1.f64 (*.f64 eps b)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1)) (expm1.f64 (*.f64 eps a))) (/.f64 eps (expm1.f64 (*.f64 eps b)))): 91 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1) (expm1.f64 (Rewrite<= *-commutative_binary64 (*.f64 a eps)))) (/.f64 eps (expm1.f64 (*.f64 eps b)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1) (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (*.f64 a eps)) 1))) (/.f64 eps (expm1.f64 (*.f64 eps b)))): 110 points increase in error, 6 points decrease in error
      (*.f64 (/.f64 (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1) (-.f64 (exp.f64 (*.f64 a eps)) 1)) (/.f64 eps (expm1.f64 (Rewrite<= *-commutative_binary64 (*.f64 b eps))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1) (-.f64 (exp.f64 (*.f64 a eps)) 1)) (/.f64 eps (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (*.f64 b eps)) 1)))): 28 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1) eps) (*.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
      (/.f64 (Rewrite<= *-commutative_binary64 (*.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 23.6

      \[\leadsto \frac{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\mathsf{expm1}\left(\varepsilon \cdot a\right)} \cdot \color{blue}{\frac{1}{b}} \]
    4. Taylor expanded in b around 0 58.9

      \[\leadsto \color{blue}{\frac{\varepsilon \cdot e^{\varepsilon \cdot a}}{e^{\varepsilon \cdot a} - 1} + \frac{1}{b}} \]
    5. Taylor expanded in eps around 0 0.1

      \[\leadsto \color{blue}{\left(0.5 \cdot \varepsilon + \frac{1}{a}\right)} + \frac{1}{b} \]
  3. Recombined 3 regimes into one program.
  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} + -1\right)}{\left(e^{\varepsilon \cdot a} + -1\right) \cdot \left(e^{\varepsilon \cdot b} + -1\right)} \leq -\infty:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{elif}\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} + -1\right)}{\left(e^{\varepsilon \cdot a} + -1\right) \cdot \left(e^{\varepsilon \cdot b} + -1\right)} \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{\frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)} \cdot \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{{\left(\sqrt[3]{\mathsf{expm1}\left(\varepsilon \cdot b\right)}\right)}^{2}}}{\sqrt[3]{\mathsf{expm1}\left(\varepsilon \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \left(\frac{1}{a} + \varepsilon \cdot 0.5\right)\\ \end{array} \]

Alternatives

Alternative 1
Error0.1
Cost74248
\[\begin{array}{l} t_0 := \varepsilon \cdot \left(a + b\right)\\ t_1 := \frac{\varepsilon \cdot \left(e^{t_0} + -1\right)}{\left(e^{\varepsilon \cdot a} + -1\right) \cdot \left(e^{\varepsilon \cdot b} + -1\right)}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(t_0\right)}{\mathsf{expm1}\left(\varepsilon \cdot a\right)} \cdot \frac{\varepsilon}{{\left(\sqrt[3]{\mathsf{expm1}\left(\varepsilon \cdot b\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \left(\frac{1}{a} + \varepsilon \cdot 0.5\right)\\ \end{array} \]
Alternative 2
Error0.1
Cost61384
\[\begin{array}{l} t_0 := \varepsilon \cdot \left(a + b\right)\\ t_1 := \frac{\varepsilon \cdot \left(e^{t_0} + -1\right)}{\left(e^{\varepsilon \cdot a} + -1\right) \cdot \left(e^{\varepsilon \cdot b} + -1\right)}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(t_0\right)}{\mathsf{expm1}\left(\varepsilon \cdot a\right)} \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \left(\frac{1}{a} + \varepsilon \cdot 0.5\right)\\ \end{array} \]
Alternative 3
Error27.5
Cost580
\[\begin{array}{l} \mathbf{if}\;b \leq 6.5 \cdot 10^{-168}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} + \varepsilon \cdot 0.5\\ \end{array} \]
Alternative 4
Error3.1
Cost448
\[\frac{1}{b} + \frac{1}{a} \]
Alternative 5
Error27.5
Cost324
\[\begin{array}{l} \mathbf{if}\;b \leq 2.4 \cdot 10^{-168}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a}\\ \end{array} \]
Alternative 6
Error33.6
Cost192
\[\frac{1}{a} \]

Error

Reproduce

herbie shell --seed 2022338 
(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))))