Average Error: 58.7 → 0.5
Time: 47.6s
Precision: 64
Internal Precision: 2432
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{a + b}{b}}{a} \le -1918333522.815699:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{if}\;\frac{\frac{a + b}{b}}{a} \le 5.227390913475924 \cdot 10^{-90}:\\ \;\;\;\;\frac{\frac{(e^{\left(a + b\right) \cdot \varepsilon} - 1)^*}{(e^{\varepsilon \cdot b} - 1)^*}}{\frac{(e^{a \cdot \varepsilon} - 1)^*}{\varepsilon}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original58.7
Target14.1
Herbie0.5
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (/ (+ a b) b) a) < -1918333522.815699 or 5.227390913475924e-90 < (/ (/ (+ a b) b) a)

    1. Initial program 61.8

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

    2. Applied simplify32.3

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

    3. Taylor expanded around 0 0.4

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

    if -1918333522.815699 < (/ (/ (+ a b) b) a) < 5.227390913475924e-90

    1. Initial program 42.5

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

    2. Applied simplify1.1

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

Runtime

Time bar (total: 47.6s)Debug logProfile

herbie shell --seed '#(1064300848 3212030778 2049303162 3567222883 2277747821 1384278011)' +o rules:numerics
(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))))