1. Split input into 2 regimes

2. if (/.f64 (*.f64 eps (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1)) (*.f64 (-.f64 (exp.f64 (*.f64 a eps)) 1) (-.f64 (exp.f64 (*.f64 b eps)) 1))) < -inf.0 or 0.0400000000000000008 < (/.f64 (*.f64 eps (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1)) (*.f64 (-.f64 (exp.f64 (*.f64 a eps)) 1) (-.f64 (exp.f64 (*.f64 b eps)) 1)))

   1. Initial program 0.5%

      \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l/0.5%

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

      2. *-commutative0.5%

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

      3. expm1-def2.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)} \cdot \frac{\varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

      4. *-commutative2.4%

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\varepsilon \cdot \left(a + b\right)}\right) \cdot \frac{\varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

      5. expm1-def9.1%

         \[\leadsto \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\color{blue}{\mathsf{expm1}\left(a \cdot \varepsilon\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

      6. *-commutative9.1%

         \[\leadsto \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\color{blue}{\varepsilon \cdot a}\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

      7. expm1-def32.6%

         \[\leadsto \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \color{blue}{\mathsf{expm1}\left(b \cdot \varepsilon\right)}} \]

      8. *-commutative32.6%

         \[\leadsto \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\color{blue}{\varepsilon \cdot b}\right)} \]

   3. Simplified32.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)}} \]

   4. Taylor expanded in eps around 0 77.8%

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

   5. Taylor expanded in a around 0 100.0%

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

   if -inf.0 < (/.f64 (*.f64 eps (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1)) (*.f64 (-.f64 (exp.f64 (*.f64 a eps)) 1) (-.f64 (exp.f64 (*.f64 b eps)) 1))) < 0.0400000000000000008

   1. Initial program 91.1%

      \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l/91.1%

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

      2. *-commutative91.1%

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

      3. expm1-def91.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)} \cdot \frac{\varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

      4. *-commutative91.1%

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\varepsilon \cdot \left(a + b\right)}\right) \cdot \frac{\varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

      5. expm1-def96.6%

         \[\leadsto \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\color{blue}{\mathsf{expm1}\left(a \cdot \varepsilon\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

      6. *-commutative96.6%

         \[\leadsto \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\color{blue}{\varepsilon \cdot a}\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

      7. expm1-def99.8%

         \[\leadsto \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \color{blue}{\mathsf{expm1}\left(b \cdot \varepsilon\right)}} \]

      8. *-commutative99.8%

         \[\leadsto \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\color{blue}{\varepsilon \cdot b}\right)} \]

   3. Simplified99.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} + -1\right)}{\left(e^{\varepsilon \cdot a} + -1\right) \cdot \left(e^{\varepsilon \cdot b} + -1\right)} \leq -\infty \lor \neg \left(\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} + -1\right)}{\left(e^{\varepsilon \cdot a} + -1\right) \cdot \left(e^{\varepsilon \cdot b} + -1\right)} \leq 0.04\right):\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)}\\ \end{array} \]

1. Split input into 2 regimes

2. if eps < -5.2000000000000002e-86

   1. Initial program 28.1%

      \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
   2. Step-by-step derivation

      1. times-frac28.1%

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

      2. expm1-def41.6%

         \[\leadsto \frac{\varepsilon}{\color{blue}{\mathsf{expm1}\left(a \cdot \varepsilon\right)}} \cdot \frac{e^{\left(a + b\right) \cdot \varepsilon} - 1}{e^{b \cdot \varepsilon} - 1} \]

      3. *-commutative41.6%

         \[\leadsto \frac{\varepsilon}{\mathsf{expm1}\left(\color{blue}{\varepsilon \cdot a}\right)} \cdot \frac{e^{\left(a + b\right) \cdot \varepsilon} - 1}{e^{b \cdot \varepsilon} - 1} \]

      4. expm1-def43.2%

         \[\leadsto \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)} \cdot \frac{\color{blue}{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)}}{e^{b \cdot \varepsilon} - 1} \]

      5. *-commutative43.2%

         \[\leadsto \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)} \cdot \frac{\mathsf{expm1}\left(\color{blue}{\varepsilon \cdot \left(a + b\right)}\right)}{e^{b \cdot \varepsilon} - 1} \]

      6. expm1-def92.7%

         \[\leadsto \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)} \cdot \frac{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\color{blue}{\mathsf{expm1}\left(b \cdot \varepsilon\right)}} \]

      7. *-commutative92.7%

         \[\leadsto \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)} \cdot \frac{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\mathsf{expm1}\left(\color{blue}{\varepsilon \cdot b}\right)} \]

   3. Simplified92.7%

      \[\leadsto \color{blue}{\frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)} \cdot \frac{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\mathsf{expm1}\left(\varepsilon \cdot b\right)}} \]

   4. Taylor expanded in eps around 0 69.3%

      \[\leadsto \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)} \cdot \color{blue}{\frac{a + b}{b}} \]

   if -5.2000000000000002e-86 < eps

   1. Initial program 4.1%

      \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l/4.1%

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

      2. *-commutative4.1%

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

      3. expm1-def5.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)} \cdot \frac{\varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

      4. *-commutative5.9%

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\varepsilon \cdot \left(a + b\right)}\right) \cdot \frac{\varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

      5. expm1-def11.6%

         \[\leadsto \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\color{blue}{\mathsf{expm1}\left(a \cdot \varepsilon\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

      6. *-commutative11.6%

         \[\leadsto \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\color{blue}{\varepsilon \cdot a}\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

      7. expm1-def33.5%

         \[\leadsto \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \color{blue}{\mathsf{expm1}\left(b \cdot \varepsilon\right)}} \]

      8. *-commutative33.5%

         \[\leadsto \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\color{blue}{\varepsilon \cdot b}\right)} \]

   3. Simplified33.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)}} \]

   4. Taylor expanded in eps around 0 76.5%

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

   5. Taylor expanded in a around 0 97.0%

      \[\leadsto \color{blue}{\frac{1}{b} + \frac{1}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.2 \cdot 10^{-86}:\\ \;\;\;\;\frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)} \cdot \frac{a + b}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array} \]

1. Split input into 2 regimes

2. if a < 5.8000000000000003e116

   1. Initial program 4.2%

      \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l/4.2%

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

      2. *-commutative4.2%

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

      3. expm1-def6.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)} \cdot \frac{\varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

      4. *-commutative6.1%

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\varepsilon \cdot \left(a + b\right)}\right) \cdot \frac{\varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

      5. expm1-def13.3%

         \[\leadsto \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\color{blue}{\mathsf{expm1}\left(a \cdot \varepsilon\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

      6. *-commutative13.3%

         \[\leadsto \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\color{blue}{\varepsilon \cdot a}\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

      7. expm1-def33.4%

         \[\leadsto \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \color{blue}{\mathsf{expm1}\left(b \cdot \varepsilon\right)}} \]

      8. *-commutative33.4%

         \[\leadsto \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\color{blue}{\varepsilon \cdot b}\right)} \]

   3. Simplified33.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)}} \]

   4. Taylor expanded in eps around 0 75.0%

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

   5. Taylor expanded in a around 0 96.9%

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

   if 5.8000000000000003e116 < a

   1. Initial program 25.0%

      \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
   2. Step-by-step derivation

      1. times-frac25.0%

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

      2. expm1-def28.0%

         \[\leadsto \frac{\varepsilon}{\color{blue}{\mathsf{expm1}\left(a \cdot \varepsilon\right)}} \cdot \frac{e^{\left(a + b\right) \cdot \varepsilon} - 1}{e^{b \cdot \varepsilon} - 1} \]

      3. *-commutative28.0%

         \[\leadsto \frac{\varepsilon}{\mathsf{expm1}\left(\color{blue}{\varepsilon \cdot a}\right)} \cdot \frac{e^{\left(a + b\right) \cdot \varepsilon} - 1}{e^{b \cdot \varepsilon} - 1} \]

      4. expm1-def28.5%

         \[\leadsto \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)} \cdot \frac{\color{blue}{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)}}{e^{b \cdot \varepsilon} - 1} \]

      5. *-commutative28.5%

         \[\leadsto \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)} \cdot \frac{\mathsf{expm1}\left(\color{blue}{\varepsilon \cdot \left(a + b\right)}\right)}{e^{b \cdot \varepsilon} - 1} \]

      6. expm1-def67.9%

         \[\leadsto \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)} \cdot \frac{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\color{blue}{\mathsf{expm1}\left(b \cdot \varepsilon\right)}} \]

      7. *-commutative67.9%

         \[\leadsto \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)} \cdot \frac{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\mathsf{expm1}\left(\color{blue}{\varepsilon \cdot b}\right)} \]

   3. Simplified67.9%

      \[\leadsto \color{blue}{\frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)} \cdot \frac{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\mathsf{expm1}\left(\varepsilon \cdot b\right)}} \]

   4. Taylor expanded in a around 0 26.7%

      \[\leadsto \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)} \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5.8 \cdot 10^{+116}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)}\\ \end{array} \]

1. Split input into 2 regimes

2. if a < -3.40000000000000005e-130

   1. Initial program 10.0%

      \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
   2. Step-by-step derivation

      1. expm1-def10.7%

         \[\leadsto \frac{\varepsilon \cdot \color{blue}{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)}}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

      2. *-commutative10.7%

         \[\leadsto \frac{\varepsilon \cdot \mathsf{expm1}\left(\color{blue}{\varepsilon \cdot \left(a + b\right)}\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

      3. expm1-def17.6%

         \[\leadsto \frac{\varepsilon \cdot \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\color{blue}{\mathsf{expm1}\left(a \cdot \varepsilon\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

      4. *-commutative17.6%

         \[\leadsto \frac{\varepsilon \cdot \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\mathsf{expm1}\left(\color{blue}{\varepsilon \cdot a}\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

      5. expm1-def47.4%

         \[\leadsto \frac{\varepsilon \cdot \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \color{blue}{\mathsf{expm1}\left(b \cdot \varepsilon\right)}} \]

      6. *-commutative47.4%

         \[\leadsto \frac{\varepsilon \cdot \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\color{blue}{\varepsilon \cdot b}\right)} \]

   3. Simplified47.4%

      \[\leadsto \color{blue}{\frac{\varepsilon \cdot \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)}} \]

   4. Taylor expanded in b around 0 21.5%

      \[\leadsto \frac{\varepsilon \cdot \color{blue}{\left(e^{\varepsilon \cdot a} - 1\right)}}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)} \]
   5. Step-by-step derivation

      1. expm1-def29.7%

         \[\leadsto \frac{\varepsilon \cdot \color{blue}{\mathsf{expm1}\left(\varepsilon \cdot a\right)}}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)} \]

      2. *-commutative29.7%

         \[\leadsto \frac{\varepsilon \cdot \mathsf{expm1}\left(\color{blue}{a \cdot \varepsilon}\right)}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)} \]

   6. Simplified29.7%

      \[\leadsto \frac{\varepsilon \cdot \color{blue}{\mathsf{expm1}\left(a \cdot \varepsilon\right)}}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)} \]

   7. Taylor expanded in eps around 0 62.6%

      \[\leadsto \color{blue}{-0.5 \cdot \varepsilon + \frac{1}{b}} \]

   if -3.40000000000000005e-130 < a

   1. Initial program 5.2%

      \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l/5.2%

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

      2. *-commutative5.2%

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

      3. expm1-def7.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)} \cdot \frac{\varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

      4. *-commutative7.0%

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\varepsilon \cdot \left(a + b\right)}\right) \cdot \frac{\varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

      5. expm1-def13.6%

         \[\leadsto \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\color{blue}{\mathsf{expm1}\left(a \cdot \varepsilon\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

      6. *-commutative13.6%

         \[\leadsto \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\color{blue}{\varepsilon \cdot a}\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

      7. expm1-def30.5%

         \[\leadsto \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \color{blue}{\mathsf{expm1}\left(b \cdot \varepsilon\right)}} \]

      8. *-commutative30.5%

         \[\leadsto \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\color{blue}{\varepsilon \cdot b}\right)} \]

   3. Simplified30.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)}} \]

   4. Taylor expanded in a around 0 60.1%

      \[\leadsto \color{blue}{\frac{1}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.4 \cdot 10^{-130}:\\ \;\;\;\;\frac{1}{b} + \varepsilon \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a}\\ \end{array} \]

1. Initial program 6.8%

   \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
2. Step-by-step derivation

   1. associate-*l/6.8%

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

   2. *-commutative6.8%

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

   3. expm1-def8.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)} \cdot \frac{\varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

   4. *-commutative8.6%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\varepsilon \cdot \left(a + b\right)}\right) \cdot \frac{\varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

   5. expm1-def15.2%

      \[\leadsto \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\color{blue}{\mathsf{expm1}\left(a \cdot \varepsilon\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

   6. *-commutative15.2%

      \[\leadsto \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\color{blue}{\varepsilon \cdot a}\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

   7. expm1-def37.3%

      \[\leadsto \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \color{blue}{\mathsf{expm1}\left(b \cdot \varepsilon\right)}} \]

   8. *-commutative37.3%

      \[\leadsto \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\color{blue}{\varepsilon \cdot b}\right)} \]

3. Simplified37.3%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)}} \]

4. Taylor expanded in eps around 0 75.1%

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

5. Taylor expanded in a around 0 94.4%

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

6. Final simplification94.4%

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

1. Split input into 2 regimes

2. if a < -3.40000000000000005e-130

   1. Initial program 10.0%

      \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l/10.0%

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

      2. *-commutative10.0%

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

      3. expm1-def11.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)} \cdot \frac{\varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

      4. *-commutative11.6%

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\varepsilon \cdot \left(a + b\right)}\right) \cdot \frac{\varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

      5. expm1-def18.3%

         \[\leadsto \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\color{blue}{\mathsf{expm1}\left(a \cdot \varepsilon\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

      6. *-commutative18.3%

         \[\leadsto \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\color{blue}{\varepsilon \cdot a}\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

      7. expm1-def50.0%

         \[\leadsto \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \color{blue}{\mathsf{expm1}\left(b \cdot \varepsilon\right)}} \]

      8. *-commutative50.0%

         \[\leadsto \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\color{blue}{\varepsilon \cdot b}\right)} \]

   3. Simplified50.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)}} \]

   4. Taylor expanded in b around 0 62.1%

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

   if -3.40000000000000005e-130 < a

   1. Initial program 5.2%

      \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l/5.2%

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

      2. *-commutative5.2%

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

      3. expm1-def7.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)} \cdot \frac{\varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

      4. *-commutative7.0%

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\varepsilon \cdot \left(a + b\right)}\right) \cdot \frac{\varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

      5. expm1-def13.6%

         \[\leadsto \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\color{blue}{\mathsf{expm1}\left(a \cdot \varepsilon\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

      6. *-commutative13.6%

         \[\leadsto \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\color{blue}{\varepsilon \cdot a}\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

      7. expm1-def30.5%

         \[\leadsto \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \color{blue}{\mathsf{expm1}\left(b \cdot \varepsilon\right)}} \]

      8. *-commutative30.5%

         \[\leadsto \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\color{blue}{\varepsilon \cdot b}\right)} \]

   3. Simplified30.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)}} \]

   4. Taylor expanded in a around 0 60.1%

      \[\leadsto \color{blue}{\frac{1}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.4 \cdot 10^{-130}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a}\\ \end{array} \]

1. Initial program 6.8%

   \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
2. Step-by-step derivation

   1. expm1-def7.7%

      \[\leadsto \frac{\varepsilon \cdot \color{blue}{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)}}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

   2. *-commutative7.7%

      \[\leadsto \frac{\varepsilon \cdot \mathsf{expm1}\left(\color{blue}{\varepsilon \cdot \left(a + b\right)}\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

   3. expm1-def14.4%

      \[\leadsto \frac{\varepsilon \cdot \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\color{blue}{\mathsf{expm1}\left(a \cdot \varepsilon\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

   4. *-commutative14.4%

      \[\leadsto \frac{\varepsilon \cdot \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\mathsf{expm1}\left(\color{blue}{\varepsilon \cdot a}\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

   5. expm1-def34.3%

      \[\leadsto \frac{\varepsilon \cdot \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \color{blue}{\mathsf{expm1}\left(b \cdot \varepsilon\right)}} \]

   6. *-commutative34.3%

      \[\leadsto \frac{\varepsilon \cdot \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\color{blue}{\varepsilon \cdot b}\right)} \]

3. Simplified34.3%

   \[\leadsto \color{blue}{\frac{\varepsilon \cdot \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)}} \]

4. Taylor expanded in b around 0 13.9%

   \[\leadsto \frac{\varepsilon \cdot \color{blue}{\left(e^{\varepsilon \cdot a} - 1\right)}}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)} \]
5. Step-by-step derivation

   1. expm1-def19.7%

      \[\leadsto \frac{\varepsilon \cdot \color{blue}{\mathsf{expm1}\left(\varepsilon \cdot a\right)}}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)} \]

   2. *-commutative19.7%

      \[\leadsto \frac{\varepsilon \cdot \mathsf{expm1}\left(\color{blue}{a \cdot \varepsilon}\right)}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)} \]

6. Simplified19.7%

   \[\leadsto \frac{\varepsilon \cdot \color{blue}{\mathsf{expm1}\left(a \cdot \varepsilon\right)}}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)} \]

7. Taylor expanded in eps around 0 45.7%

   \[\leadsto \color{blue}{-0.5 \cdot \varepsilon + \frac{1}{b}} \]

8. Taylor expanded in eps around inf 3.2%

   \[\leadsto \color{blue}{-0.5 \cdot \varepsilon} \]

9. Final simplification3.2%

   \[\leadsto \varepsilon \cdot -0.5 \]

1. Initial program 6.8%

   \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
2. Step-by-step derivation

   1. associate-*l/6.8%

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

   2. *-commutative6.8%

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

   3. expm1-def8.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)} \cdot \frac{\varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

   4. *-commutative8.6%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\varepsilon \cdot \left(a + b\right)}\right) \cdot \frac{\varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

   5. expm1-def15.2%

      \[\leadsto \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\color{blue}{\mathsf{expm1}\left(a \cdot \varepsilon\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

   6. *-commutative15.2%

      \[\leadsto \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\color{blue}{\varepsilon \cdot a}\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

   7. expm1-def37.3%

      \[\leadsto \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \color{blue}{\mathsf{expm1}\left(b \cdot \varepsilon\right)}} \]

   8. *-commutative37.3%

      \[\leadsto \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\color{blue}{\varepsilon \cdot b}\right)} \]

3. Simplified37.3%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)}} \]

4. Taylor expanded in a around 0 49.8%

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

5. Final simplification49.8%

   \[\leadsto \frac{1}{a} \]

herbie shell --seed 2023166 
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :precision binary64
  :pre (and (< -1.0 eps) (< eps 1.0))

  :herbie-target
  (/ (+ a b) (* a b))

  (/ (* eps (- (exp (* (+ a b) eps)) 1.0)) (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0))))