Average Error: 29.4 → 14.0
Time: 31.1s
Precision: 64
Internal Precision: 1344
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.5397285693473385 \cdot 10^{+116}:\\ \;\;\;\;\frac{\frac{e^{\left(x + x\right) \cdot \left(a + a\right)} \cdot e^{\left(x + x\right) \cdot \left(a + a\right)} - 1}{1 + e^{\left(x + x\right) \cdot \left(a + a\right)}}}{\left(1 + e^{a \cdot x}\right) \cdot \left(e^{a \cdot x} \cdot e^{a \cdot x} + 1\right)}\\ \mathbf{elif}\;x \le 2.554810378889993 \cdot 10^{+102}:\\ \;\;\;\;\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot \left(x \cdot \frac{1}{6}\right) + \frac{1}{2}\right) + a \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\frac{e^{\left(x + x\right) \cdot \left(a + a\right)} - 1}{\left(1 + e^{a \cdot x}\right) \cdot \left(e^{a \cdot x} \cdot e^{a \cdot x} + 1\right)}} \cdot \sqrt[3]{\frac{e^{\left(x + x\right) \cdot \left(a + a\right)} - 1}{\left(1 + e^{a \cdot x}\right) \cdot \left(e^{a \cdot x} \cdot e^{a \cdot x} + 1\right)}}\right) \cdot \sqrt[3]{\frac{e^{\left(x + x\right) \cdot \left(a + a\right)} - 1}{\left(1 + e^{a \cdot x}\right) \cdot \left(e^{a \cdot x} \cdot e^{a \cdot x} + 1\right)}}\\ \end{array}\]

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.4
Target0.2
Herbie14.0
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt \frac{1}{10}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -4.5397285693473385e+116

    1. Initial program 14.9

      \[e^{a \cdot x} - 1\]

    2. Initial simplification14.9

      \[\leadsto e^{a \cdot x} - 1\]
    3. Using strategy rm

    4. Applied flip--14.9

      \[\leadsto \color{blue}{\frac{e^{a \cdot x} \cdot e^{a \cdot x} - 1 \cdot 1}{e^{a \cdot x} + 1}}\]
    5. Using strategy rm

    6. Applied flip--15.0

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

    7. Applied associate-/l/15.0

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

    8. Simplified14.9

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

    10. Applied flip--14.9

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

    if -4.5397285693473385e+116 < x < 2.554810378889993e+102

    1. Initial program 34.0

      \[e^{a \cdot x} - 1\]

    2. Initial simplification34.0

      \[\leadsto e^{a \cdot x} - 1\]

    3. Taylor expanded around 0 20.2

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({a}^{2} \cdot {x}^{2}\right) + \left(a \cdot x + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\right)}\]

    4. Simplified13.4

      \[\leadsto \color{blue}{\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(\frac{1}{2} + a \cdot \left(x \cdot \frac{1}{6}\right)\right) + a \cdot x}\]

    if 2.554810378889993e+102 < x

    1. Initial program 16.7

      \[e^{a \cdot x} - 1\]

    2. Initial simplification16.7

      \[\leadsto e^{a \cdot x} - 1\]
    3. Using strategy rm

    4. Applied flip--16.7

      \[\leadsto \color{blue}{\frac{e^{a \cdot x} \cdot e^{a \cdot x} - 1 \cdot 1}{e^{a \cdot x} + 1}}\]
    5. Using strategy rm

    6. Applied flip--16.7

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

    7. Applied associate-/l/16.7

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

    8. Simplified16.6

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

    10. Applied add-cube-cbrt16.6

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

  4. Final simplification14.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.5397285693473385 \cdot 10^{+116}:\\ \;\;\;\;\frac{\frac{e^{\left(x + x\right) \cdot \left(a + a\right)} \cdot e^{\left(x + x\right) \cdot \left(a + a\right)} - 1}{1 + e^{\left(x + x\right) \cdot \left(a + a\right)}}}{\left(1 + e^{a \cdot x}\right) \cdot \left(e^{a \cdot x} \cdot e^{a \cdot x} + 1\right)}\\ \mathbf{elif}\;x \le 2.554810378889993 \cdot 10^{+102}:\\ \;\;\;\;\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot \left(x \cdot \frac{1}{6}\right) + \frac{1}{2}\right) + a \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\frac{e^{\left(x + x\right) \cdot \left(a + a\right)} - 1}{\left(1 + e^{a \cdot x}\right) \cdot \left(e^{a \cdot x} \cdot e^{a \cdot x} + 1\right)}} \cdot \sqrt[3]{\frac{e^{\left(x + x\right) \cdot \left(a + a\right)} - 1}{\left(1 + e^{a \cdot x}\right) \cdot \left(e^{a \cdot x} \cdot e^{a \cdot x} + 1\right)}}\right) \cdot \sqrt[3]{\frac{e^{\left(x + x\right) \cdot \left(a + a\right)} - 1}{\left(1 + e^{a \cdot x}\right) \cdot \left(e^{a \cdot x} \cdot e^{a \cdot x} + 1\right)}}\\ \end{array}\]

Runtime

Time bar (total: 31.1s)Debug logProfile

herbie shell --seed 2018234 
(FPCore (a x)
  :name "expax (section 3.5)"

  :herbie-target
  (if (< (fabs (* a x)) 1/10) (* (* a x) (+ 1 (+ (/ (* a x) 2) (/ (pow (* a x) 2) 6)))) (- (exp (* a x)) 1))

  (- (exp (* a x)) 1))