Average Error: 60.0 → 3.6
Time: 8.9s
Precision: binary64
\[-1 \lt \varepsilon \land \varepsilon \lt 1\]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\frac{1}{b} + \frac{1}{a}\]
\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

Original60.0
Target15.0
Herbie3.6
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 60.0

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

  2. Taylor expanded around 0 57.6

    \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\color{blue}{\left(0.16666666666666666 \cdot \left({a}^{3} \cdot {\varepsilon}^{3}\right) + \left(0.5 \cdot \left({a}^{2} \cdot {\varepsilon}^{2}\right) + a \cdot \varepsilon\right)\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]

  3. Simplified57.5

    \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\color{blue}{\left(a \cdot \varepsilon + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(a \cdot \left(a \cdot 0.5\right) + \varepsilon \cdot \left(0.16666666666666666 \cdot {a}^{3}\right)\right)\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
  4. Using strategy rm

  5. Applied associate-*l*57.4

    \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(a \cdot \varepsilon + \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(a \cdot \left(a \cdot 0.5\right) + \varepsilon \cdot \left(0.16666666666666666 \cdot {a}^{3}\right)\right)\right)}\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]

  6. Simplified56.2

    \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(a \cdot \varepsilon + \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \left(\left(a \cdot a\right) \cdot \left(0.5 + \left(\varepsilon \cdot 0.16666666666666666\right) \cdot a\right)\right)\right)}\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
  7. Using strategy rm

  8. Applied associate-*r*56.3

    \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(a \cdot \varepsilon + \varepsilon \cdot \color{blue}{\left(\left(\varepsilon \cdot \left(a \cdot a\right)\right) \cdot \left(0.5 + \left(\varepsilon \cdot 0.16666666666666666\right) \cdot a\right)\right)}\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]

  9. Simplified56.0

    \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(a \cdot \varepsilon + \varepsilon \cdot \left(\color{blue}{\left(a \cdot \left(a \cdot \varepsilon\right)\right)} \cdot \left(0.5 + \left(\varepsilon \cdot 0.16666666666666666\right) \cdot a\right)\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]

  10. Taylor expanded around 0 3.6

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

  11. Final simplification3.6

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

Reproduce

herbie shell --seed 2020184 
(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))))