Average Error: 58.4 → 2.8
Time: 1.7m
Precision: 64
Internal Precision: 2432
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \cdot \varepsilon \le 16130229.558168741:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\frac{{\left(e^{a}\right)}^{\left(\varepsilon + \varepsilon\right)} - 1}{e^{a \cdot \varepsilon} + 1} \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original58.4
Target14.5
Herbie2.8
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Split input into 2 regimes

  2. if (* b eps) < 16130229.558168741

    1. Initial program 59.2

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

    2. Taylor expanded around 0 2.7

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

    if 16130229.558168741 < (* b eps)

    1. Initial program 6.5

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

    3. Applied flip--6.5

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

    4. Applied simplify6.5

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

Runtime

Time bar (total: 1.7m)Debug logProfile

herbie shell --seed '#(1063282112 2455465480 4141627379 3773598652 1647277307 776739644)' 
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :pre (and (< -1 eps) (< eps 1))

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

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