Error

Target

Derivation

1. Split input into 2 regimes

2. if (* a eps) < 6.507018077965857e-08

   1. Initial program 59.3

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

   2. Taylor expanded around 0 2.7

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

   if 6.507018077965857e-08 < (* a eps)

   1. Initial program 26.8

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

   3. Applied add-log-exp26.8

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

5. Applied simplify3.3

   \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;a \cdot \varepsilon \le 6.507018077965857 \cdot 10^{-08}:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(e^{\varepsilon}\right)}^{\left(b + a\right)} - 1}{e^{b \cdot \varepsilon} - 1}}{\frac{e^{a \cdot \varepsilon} - 1}{\varepsilon}}\\ \end{array}}\]

Runtime

Time bar (total: 42.8s)Debug log

herbie shell --seed '#(633950927 2092594946 1442981 2827247922 2812758452 390991499)' 
(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))))