Average Error: 61.1 → 1.5
Time: 14.1s
Precision: 64
Internal precision: 2432
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
⬇
\[\begin{array}{l} \mathbf{if}\;a \cdot \varepsilon \le -2.9125533655836386 \cdot 10^{+137}:\\ \;\;\;\;\frac{\varepsilon \cdot {\left(\sqrt[3]{e^{\left(a + b\right) \cdot \varepsilon} - 1}\right)}^3}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original61.1
Comparison14.3
Herbie1.5
\[ \frac{a + b}{a \cdot b} \]

Derivation

  1. Split input into 2 regimes.

  2. if (* a eps) < -2.9125533655836386e+137

    1. Initial program 38.3

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

    3. Applied add-cube-cbrt 38.3

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

    if -2.9125533655836386e+137 < (* a eps)

    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

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

    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: 14.1s) Debug logProfile

Please include this information when filing a bug report:

herbie shell --seed '#(1065003094 2074156664 2352254222 753858891 3745550101 3374585842)'
(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))))