Error

Target

Derivation

1. Split input into 2 regimes

2. if (/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))) < -inf.0 or 8.18217761357226e-13 < (/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1)))

   1. Initial program 62.0

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

   2. Applied simplify29.0

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

   3. Taylor expanded around 0 0.2

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

   if -inf.0 < (/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))) < 8.18217761357226e-13

   1. Initial program 4.4

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

   2. Applied simplify0.2

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

   4. Applied expm1-udef0.4

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

   5. Applied div-sub0.4

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

   6. Applied div-sub0.4

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

   7. Applied simplify0.3

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

Runtime

Time bar (total: 42.1s)Debug logProfile

herbie shell --seed '#(1070258749 1877548225 2229079127 1588002776 3179087814 1886870650)' +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))))