Average Error: 60.2 → 3.5
Time: 11.7s
Precision: 64
\[-1 \lt \varepsilon \land \varepsilon \lt 1\]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[1 \cdot \left(\frac{1}{b} + \frac{1}{a}\right)\]
\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)}
↓
1 \cdot \left(\frac{1}{b} + \frac{1}{a}\right)
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) (1.0 * ((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

Original60.2
Target15.4
Herbie3.5
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 60.2

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

  2. Taylor expanded around 0 58.4

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

  3. Simplified57.5

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

  5. Applied unpow257.5

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

  6. Applied associate-*r*57.0

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

  7. Simplified57.0

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

  9. Applied flip--57.9

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

  10. Simplified58.3

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

  11. Taylor expanded around 0 3.5

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

  12. Simplified3.5

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

  13. Final simplification3.5

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

Reproduce

herbie shell --seed 2020124 
(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))))