Average Error: 52.4 → 2.7
Time: 24.9s
Precision: 64
Internal precision: 2432
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
⬇
\[\begin{array}{l} \mathbf{if}\;a \cdot \varepsilon \le -2.2151465087780744 \cdot 10^{+261}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \mathbf{if}\;a \cdot \varepsilon \le -5.3318909641188086 \cdot 10^{+141}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{if}\;a \cdot \varepsilon \le -3.8030378975762234 \cdot 10^{+73}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \mathbf{if}\;a \cdot \varepsilon \le 1059700.6866205125:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Derivation

  1. Split input into 2 regimes.

  2. if (* a eps) < -2.2151465087780744e+261 or -5.3318909641188086e+141 < (* a eps) < -3.8030378975762234e+73 or 1059700.6866205125 < (* a eps)

    1. Initial program 13.9

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

    if -2.2151465087780744e+261 < (* a eps) < -5.3318909641188086e+141 or -3.8030378975762234e+73 < (* a eps) < 1059700.6866205125

    1. Initial program 61.6

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

    2. Applied taylor 0.0

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

    3. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\frac{1}{b} + \frac{1}{a}}\]
  3. Recombined 2 regimes into one program.
  4. Removed slow pow expressions

Runtime

Time bar (total: 24.9s) Debug log

Please include this information when filing a bug report:

herbie shell --seed '#(3052192724 3812927732 3686175817 630908657 2373248591 511094450)'
(FPCore (a b eps)
  :name "NMSE problem 3.4.2"
  :pre (!= eps 0)
  (/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))))