Average Error: 58.8 → 2.6
Time: 44.3s
Precision: 64
Internal Precision: 2432
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{b} \cdot \frac{a + b}{a} \le -1.9821640126467075 \cdot 10^{-187}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{if}\;\frac{1}{b} \cdot \frac{a + b}{a} \le 1.4979562851105258 \cdot 10^{-85}:\\ \;\;\;\;\frac{(\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)}\right) + \left(-\varepsilon\right))_*}{(e^{\varepsilon \cdot b} - 1)^* \cdot (e^{a \cdot \varepsilon} - 1)^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original58.8
Target13.9
Herbie2.6
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Split input into 2 regimes

  2. if (* (/ 1 b) (/ (+ a b) a)) < -1.9821640126467075e-187 or 1.4979562851105258e-85 < (* (/ 1 b) (/ (+ a b) a))

    1. Initial program 60.6

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

    2. Applied simplify60.0

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

    3. Taylor expanded around 0 1.5

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

    if -1.9821640126467075e-187 < (* (/ 1 b) (/ (+ a b) a)) < 1.4979562851105258e-85

    1. Initial program 31.8

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

    2. Applied simplify45.0

      \[\leadsto \color{blue}{\frac{(\varepsilon \cdot \left({\left(e^{\varepsilon}\right)}^{\left(a + b\right)}\right) + \left(-\varepsilon\right))_*}{(e^{\varepsilon \cdot b} - 1)^* \cdot (e^{a \cdot \varepsilon} - 1)^*}}\]
    3. Using strategy rm

    4. Applied pow-exp19.4

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

Runtime

Time bar (total: 44.3s)Debug logProfile

herbie shell --seed '#(1070386091 2509006183 1430610344 1025408621 36622005 1425925650)' +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))))