Average Error: 61.4 → 0.0
Time: 51.1s
Precision: 64
Internal precision: 2432
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
⬇
\[\begin{array}{l} \mathbf{if}\;b \cdot \varepsilon \le 5.909251882114744 \cdot 10^{+35}:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\sqrt[3]{{\left(e^{a \cdot \varepsilon} - 1\right)}^3} \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original61.4
Comparison14.3
Herbie0.0
\[ \frac{a + b}{a \cdot b} \]

Derivation

  1. Split input into 2 regimes.

  2. if (* b eps) < 5.909251882114744e+35

    1. Initial program 62.0

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

    2. Applied taylor 0.0

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

    3. Taylor expanded around 0 0.0

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

    if 5.909251882114744e+35 < (* b eps)

    1. Initial program 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. Using strategy rm

    3. Applied add-cbrt-cube 0

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\color{blue}{\sqrt[3]{{\left(e^{a \cdot \varepsilon} - 1\right)}^3}} \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
  3. Recombined 2 regimes into one program.
  4. Removed slow pow expressions

Runtime

Time bar (total: 51.1s) Debug log

Please include this information when filing a bug report:

herbie shell --seed '#(2277612311 2645429965 1090895633 2857793080 2144184008 3989768357)'
(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))))