Average Error: 60.4 → 0.4
Time: 9.8s
Precision: binary64
\[-1 < \varepsilon \land \varepsilon < 1\]
\[[a, b]=\mathsf{sort}([a, b])\]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(e^{\varepsilon \cdot a} - 1\right) \cdot \left(e^{\varepsilon \cdot b} - 1\right)} \leq -\infty:\\ \;\;\;\;\frac{1}{a} + \left(\frac{1}{b} + 0.08333333333333333 \cdot \left(a \cdot {\varepsilon}^{2}\right)\right)\\ \mathbf{elif}\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(e^{\varepsilon \cdot a} - 1\right) \cdot \left(e^{\varepsilon \cdot b} - 1\right)} \leq 1.0095335246879834 \cdot 10^{-14}:\\ \;\;\;\;\frac{\varepsilon}{e^{\varepsilon \cdot a} - 1} \cdot \frac{e^{\varepsilon \cdot \left(a + b\right)} - 1}{e^{\varepsilon \cdot b} - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \end{array}\]
\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}\;\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}{a} + \left(\frac{1}{b} + 0.08333333333333333 \cdot \left(a \cdot {\varepsilon}^{2}\right)\right)\\

\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 1.0095335246879834 \cdot 10^{-14}:\\
\;\;\;\;\frac{\varepsilon}{e^{\varepsilon \cdot a} - 1} \cdot \frac{e^{\varepsilon \cdot \left(a + b\right)} - 1}{e^{\varepsilon \cdot b} - 1}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{a} + \frac{1}{b}\\

\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
 (if (<=
      (/
       (* eps (- (exp (* eps (+ a b))) 1.0))
       (* (- (exp (* eps a)) 1.0) (- (exp (* eps b)) 1.0)))
      (- INFINITY))
   (+ (/ 1.0 a) (+ (/ 1.0 b) (* 0.08333333333333333 (* a (pow eps 2.0)))))
   (if (<=
        (/
         (* eps (- (exp (* eps (+ a b))) 1.0))
         (* (- (exp (* eps a)) 1.0) (- (exp (* eps b)) 1.0)))
        1.0095335246879834e-14)
     (*
      (/ eps (- (exp (* eps a)) 1.0))
      (/ (- (exp (* eps (+ a b))) 1.0) (- (exp (* eps b)) 1.0)))
     (+ (/ 1.0 a) (/ 1.0 b)))))
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 (((eps * (exp(eps * (a + b)) - 1.0)) / ((exp(eps * a) - 1.0) * (exp(eps * b) - 1.0))) <= -((double) INFINITY)) {
		tmp = (1.0 / a) + ((1.0 / b) + (0.08333333333333333 * (a * pow(eps, 2.0))));
	} else if (((eps * (exp(eps * (a + b)) - 1.0)) / ((exp(eps * a) - 1.0) * (exp(eps * b) - 1.0))) <= 1.0095335246879834e-14) {
		tmp = (eps / (exp(eps * a) - 1.0)) * ((exp(eps * (a + b)) - 1.0) / (exp(eps * b) - 1.0));
	} else {
		tmp = (1.0 / a) + (1.0 / b);
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original60.4
Target14.8
Herbie0.4
\[\frac{a + b}{a \cdot b}\]

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. Taylor expanded around 0 10.0

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

    3. Simplified10.9

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

    4. Taylor expanded around 0 0

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

    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))) < 1.0095335246879834e-14

    1. Initial program 4.2

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

    3. Applied times-frac_binary644.2

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

    if 1.0095335246879834e-14 < (/.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. Taylor expanded around 0 1.7

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

    3. Simplified1.8

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

    4. Taylor expanded around 0 0.2

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

    5. Simplified0.2

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(e^{\varepsilon \cdot a} - 1\right) \cdot \left(e^{\varepsilon \cdot b} - 1\right)} \leq -\infty:\\ \;\;\;\;\frac{1}{a} + \left(\frac{1}{b} + 0.08333333333333333 \cdot \left(a \cdot {\varepsilon}^{2}\right)\right)\\ \mathbf{elif}\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(e^{\varepsilon \cdot a} - 1\right) \cdot \left(e^{\varepsilon \cdot b} - 1\right)} \leq 1.0095335246879834 \cdot 10^{-14}:\\ \;\;\;\;\frac{\varepsilon}{e^{\varepsilon \cdot a} - 1} \cdot \frac{e^{\varepsilon \cdot \left(a + b\right)} - 1}{e^{\varepsilon \cdot b} - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2021176 
(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))))