Average Error: 60.3 → 3.4
Time: 11.1s
Precision: 64
\[-1 \lt \varepsilon \land \varepsilon \lt 1\]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\frac{1}{b} + \frac{1}{a}\]
\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)}
↓
\frac{1}{b} + \frac{1}{a}
double code(double a, double b, double eps) {
	return ((double) (((double) (eps * ((double) (((double) exp(((double) (((double) (a + b)) * eps)))) - 1.0)))) / ((double) (((double) (((double) exp(((double) (a * eps)))) - 1.0)) * ((double) (((double) exp(((double) (b * eps)))) - 1.0))))));
}
↓
double code(double a, double b, double eps) {
	return ((double) (((double) (1.0 / b)) + ((double) (1.0 / a))));
}

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
Target15.3
Herbie3.4
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 60.3

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

  2. Taylor expanded around 0 57.9

    \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \color{blue}{\left(\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot {b}^{3}\right) + \left(\frac{1}{2} \cdot \left({\varepsilon}^{2} \cdot {b}^{2}\right) + \varepsilon \cdot b\right)\right)}}\]
  3. Using strategy rm

  4. Applied pow-prod-down57.4

    \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(\frac{1}{6} \cdot \color{blue}{{\left(\varepsilon \cdot b\right)}^{3}} + \left(\frac{1}{2} \cdot \left({\varepsilon}^{2} \cdot {b}^{2}\right) + \varepsilon \cdot b\right)\right)}\]

  5. Simplified57.4

    \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(\frac{1}{6} \cdot {\color{blue}{\left(b \cdot \varepsilon\right)}}^{3} + \left(\frac{1}{2} \cdot \left({\varepsilon}^{2} \cdot {b}^{2}\right) + \varepsilon \cdot b\right)\right)}\]
  6. Using strategy rm

  7. Applied pow-prod-down56.7

    \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(\frac{1}{6} \cdot {\left(b \cdot \varepsilon\right)}^{3} + \left(\frac{1}{2} \cdot \color{blue}{{\left(\varepsilon \cdot b\right)}^{2}} + \varepsilon \cdot b\right)\right)}\]

  8. Simplified56.7

    \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(\frac{1}{6} \cdot {\left(b \cdot \varepsilon\right)}^{3} + \left(\frac{1}{2} \cdot {\color{blue}{\left(b \cdot \varepsilon\right)}}^{2} + \varepsilon \cdot b\right)\right)}\]

  9. Taylor expanded around 0 3.4

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

  10. Final simplification3.4

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

Reproduce

herbie shell --seed 2020140 
(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))))