Average Error: 61.9 → 0.0
Time: 19.4s
Precision: 64
Internal precision: 128
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
⬇
\[\frac{1}{a} + \frac{1}{b}\]

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original61.9
Comparison15.4
Herbie0.0
\[ \frac{a + b}{a \cdot b} \]

Derivation

  1. Initial program 61.9

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

  2. Applied taylor 0.0

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

  3. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{\frac{1}{a} + \frac{1}{b}}\]
  4. Removed slow pow expressions

Runtime

Time bar (total: 19.4s) Debug log

Please include this information when filing a bug report:

herbie --seed '#(1452020878 2550675912 2875856812 1052815057 1313707707 496417834)'
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :pre (and (< -1 eps) (< eps 1))

  :target
  (/ (+ a b) (* a b))

  (/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))))