Average Error: 29.4 → 14.0
Time: 54.0s
Precision: 64
Internal Precision: 1344
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.5397285693473385 \cdot 10^{+116}:\\ \;\;\;\;\frac{e^{\left(a + a\right) \cdot \left(x + x\right)} - 1}{\left(1 + e^{a \cdot x}\right) \cdot \left(e^{a \cdot \left(x + x\right)} + 1\right)}\\ \mathbf{elif}\;x \le 2.554810378889993 \cdot 10^{+102}:\\ \;\;\;\;\frac{\left(2 + \left(a \cdot x\right) \cdot \frac{4}{3}\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) + 2 \cdot \left(a \cdot x\right)}{1 + e^{a \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(a + a\right) \cdot \left(x + x\right)} - 1}{\left(1 + e^{a \cdot x}\right) \cdot \left(e^{a \cdot \left(x + x\right)} + 1\right)}\\ \end{array}\]

Error

Bits error versus a

Bits error versus x

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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 2 regimes

  2. if x < -4.5397285693473385e+116 or 2.554810378889993e+102 < x

    1. Initial program 15.8

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

    3. Applied flip--15.9

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

    5. Applied prod-exp15.8

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

    6. Simplified15.8

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

    8. Applied flip--15.9

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

    9. Applied associate-/l/15.9

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

    10. Simplified15.8

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

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

    1. Initial program 34.0

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

    3. Applied flip--34.0

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

    4. Taylor expanded around 0 20.2

      \[\leadsto \frac{\color{blue}{2 \cdot \left({a}^{2} \cdot {x}^{2}\right) + \left(2 \cdot \left(a \cdot x\right) + \frac{4}{3} \cdot \left({a}^{3} \cdot {x}^{3}\right)\right)}}{e^{a \cdot x} + 1}\]

    5. Simplified13.4

      \[\leadsto \frac{\color{blue}{\left(\left(x \cdot a\right) \cdot \left(x \cdot a\right)\right) \cdot \left(2 + \frac{4}{3} \cdot \left(x \cdot a\right)\right) + \left(x \cdot a\right) \cdot 2}}{e^{a \cdot x} + 1}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification14.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.5397285693473385 \cdot 10^{+116}:\\ \;\;\;\;\frac{e^{\left(a + a\right) \cdot \left(x + x\right)} - 1}{\left(1 + e^{a \cdot x}\right) \cdot \left(e^{a \cdot \left(x + x\right)} + 1\right)}\\ \mathbf{elif}\;x \le 2.554810378889993 \cdot 10^{+102}:\\ \;\;\;\;\frac{\left(2 + \left(a \cdot x\right) \cdot \frac{4}{3}\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) + 2 \cdot \left(a \cdot x\right)}{1 + e^{a \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(a + a\right) \cdot \left(x + x\right)} - 1}{\left(1 + e^{a \cdot x}\right) \cdot \left(e^{a \cdot \left(x + x\right)} + 1\right)}\\ \end{array}\]

Runtime

Time bar (total: 54.0s)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))