Error

Target

Derivation

1. Split input into 2 regimes

2. if (+ (/ 1 b) (/ 1 a)) < -2.557574743704027e-52 or 3.091720763301976e-19 < (+ (/ 1 b) (/ 1 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 simplify33.1

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

   3. Taylor expanded around 0 0.2

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

   if -2.557574743704027e-52 < (+ (/ 1 b) (/ 1 a)) < 3.091720763301976e-19

   1. Initial program 40.8

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

   2. Applied simplify1.0

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

   4. Applied div-inv1.0

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

   5. Applied *-un-lft-identity1.0

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

   6. Applied *-un-lft-identity1.0

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

   7. Applied times-frac1.0

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

   8. Applied times-frac1.1

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

   9. Applied simplify1.1

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

   10. Applied simplify1.0

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

Runtime

Time bar (total: 40.4s)Debug logProfile

herbie shell --seed '#(1064269945 2896236262 301053905 1701069080 1701464310 1614783279)' +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))))