Average Error: 60.3 → 0.9
Time: 13.0s
Precision: binary64
\[-1 < \varepsilon \land \varepsilon < 1\]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(e^{\varepsilon \cdot a} - 1\right) \cdot \left(e^{\varepsilon \cdot b} - 1\right)} \leq -\infty \lor \neg \left(\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(e^{\varepsilon \cdot a} - 1\right) \cdot \left(e^{\varepsilon \cdot b} - 1\right)} \leq 7.680910706340266 \cdot 10^{-134}\right):\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(e^{\varepsilon \cdot b} - 1\right) \cdot \left(\left(1 + \sqrt{e^{\varepsilon \cdot a}}\right) \cdot \left(\sqrt{e^{\varepsilon \cdot a}} - 1\right)\right)}\\ \end{array}\]
\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 \lor \neg \left(\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 7.680910706340266 \cdot 10^{-134}\right):\\
\;\;\;\;\frac{1}{b} + \frac{1}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(e^{\varepsilon \cdot b} - 1\right) \cdot \left(\left(1 + \sqrt{e^{\varepsilon \cdot a}}\right) \cdot \left(\sqrt{e^{\varepsilon \cdot a}} - 1\right)\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
 (if (or (<=
          (/
           (* eps (- (exp (* eps (+ a b))) 1.0))
           (* (- (exp (* eps a)) 1.0) (- (exp (* eps b)) 1.0)))
          (- INFINITY))
         (not
          (<=
           (/
            (* eps (- (exp (* eps (+ a b))) 1.0))
            (* (- (exp (* eps a)) 1.0) (- (exp (* eps b)) 1.0)))
           7.680910706340266e-134)))
   (+ (/ 1.0 b) (/ 1.0 a))
   (/
    (* eps (- (exp (* eps (+ a b))) 1.0))
    (*
     (- (exp (* eps b)) 1.0)
     (* (+ 1.0 (sqrt (exp (* eps a)))) (- (sqrt (exp (* eps a))) 1.0))))))
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)) || !(((eps * (exp(eps * (a + b)) - 1.0)) / ((exp(eps * a) - 1.0) * (exp(eps * b) - 1.0))) <= 7.680910706340266e-134)) {
		tmp = (1.0 / b) + (1.0 / a);
	} else {
		tmp = (eps * (exp(eps * (a + b)) - 1.0)) / ((exp(eps * b) - 1.0) * ((1.0 + sqrt(exp(eps * a))) * (sqrt(exp(eps * a)) - 1.0)));
	}
	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

Original60.3
Target14.8
Herbie0.9
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Split input into 2 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 or 7.6809107063402663e-134 < (/.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 63.1

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

    2. Taylor expanded around 0 57.6

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

    3. Simplified57.4

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

    4. Taylor expanded around 0 13.9

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

    5. Simplified13.9

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

    6. Taylor expanded around 0 0.8

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

    7. Simplified0.8

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

    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))) < 7.6809107063402663e-134

    1. Initial program 3.4

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

    3. Applied add-sqr-sqrt_binary64_11233.4

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

    4. Applied difference-of-sqr-1_binary64_10713.5

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\color{blue}{\left(\left(\sqrt{e^{a \cdot \varepsilon}} + 1\right) \cdot \left(\sqrt{e^{a \cdot \varepsilon}} - 1\right)\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(e^{\varepsilon \cdot a} - 1\right) \cdot \left(e^{\varepsilon \cdot b} - 1\right)} \leq -\infty \lor \neg \left(\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(e^{\varepsilon \cdot a} - 1\right) \cdot \left(e^{\varepsilon \cdot b} - 1\right)} \leq 7.680910706340266 \cdot 10^{-134}\right):\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(e^{\varepsilon \cdot b} - 1\right) \cdot \left(\left(1 + \sqrt{e^{\varepsilon \cdot a}}\right) \cdot \left(\sqrt{e^{\varepsilon \cdot a}} - 1\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2021045 
(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))))