Average Error: 29.7 → 15.1
Time: 34.2s
Precision: 64
Internal Precision: 128
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -2.8689996172778207 \cdot 10^{-08}:\\ \;\;\;\;\frac{\log \left(e^{e^{\left(x + x\right) \cdot \left(a + a\right)} + -1}\right)}{\left(e^{x \cdot a} \cdot e^{x \cdot a} + 1\right) \cdot \left(e^{x \cdot a} + 1\right)}\\ \mathbf{elif}\;a \le 1.1626810969887635 \cdot 10^{-23}:\\ \;\;\;\;\left(\left(x \cdot a\right) \cdot \left(x \cdot a\right)\right) \cdot \left(\frac{1}{2} + a \cdot \left(\frac{1}{6} \cdot x\right)\right) + x \cdot a\\ \mathbf{elif}\;a \le 1.1586552917250053 \cdot 10^{+22}:\\ \;\;\;\;\left(\sqrt[3]{\frac{e^{\left(x + x\right) \cdot \left(a + a\right)} + -1}{\left(e^{x \cdot a} \cdot e^{x \cdot a} + 1\right) \cdot \left(e^{x \cdot a} + 1\right)}} \cdot \sqrt[3]{\frac{e^{\left(x + x\right) \cdot \left(a + a\right)} + -1}{\left(e^{x \cdot a} \cdot e^{x \cdot a} + 1\right) \cdot \left(e^{x \cdot a} + 1\right)}}\right) \cdot \sqrt[3]{\frac{e^{\left(x + x\right) \cdot \left(a + a\right)} + -1}{\left(e^{x \cdot a} \cdot e^{x \cdot a} + 1\right) \cdot \left(e^{x \cdot a} + 1\right)}}\\ \mathbf{elif}\;a \le 1.2934075903718333 \cdot 10^{+60}:\\ \;\;\;\;\left(\left(x \cdot a\right) \cdot \left(x \cdot a\right)\right) \cdot \left(\frac{1}{2} + a \cdot \left(\frac{1}{6} \cdot x\right)\right) + x \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{e^{\left(x + x\right) \cdot \left(a + a\right)} + -1}\right)}{\left(e^{x \cdot a} \cdot e^{x \cdot a} + 1\right) \cdot \left(e^{x \cdot a} + 1\right)}\\ \end{array}\]

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.7
Target0.2
Herbie15.1
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt \frac{1}{10}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if a < -2.8689996172778207e-08 or 1.2934075903718333e+60 < a

    1. Initial program 21.4

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

    3. Applied flip--21.5

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

    5. Applied flip--21.5

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

    6. Applied associate-/l/21.5

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

    7. Simplified21.3

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

    9. Applied add-log-exp21.6

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

    if -2.8689996172778207e-08 < a < 1.1626810969887635e-23 or 1.1586552917250053e+22 < a < 1.2934075903718333e+60

    1. Initial program 34.3

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

    2. Taylor expanded around 0 17.5

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

    3. Simplified9.5

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

    if 1.1626810969887635e-23 < a < 1.1586552917250053e+22

    1. Initial program 40.2

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

    3. Applied flip--40.2

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

    5. Applied flip--40.2

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

    6. Applied associate-/l/40.2

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

    7. Simplified40.1

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

    9. Applied add-cube-cbrt40.1

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

  4. Final simplification15.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -2.8689996172778207 \cdot 10^{-08}:\\ \;\;\;\;\frac{\log \left(e^{e^{\left(x + x\right) \cdot \left(a + a\right)} + -1}\right)}{\left(e^{x \cdot a} \cdot e^{x \cdot a} + 1\right) \cdot \left(e^{x \cdot a} + 1\right)}\\ \mathbf{elif}\;a \le 1.1626810969887635 \cdot 10^{-23}:\\ \;\;\;\;\left(\left(x \cdot a\right) \cdot \left(x \cdot a\right)\right) \cdot \left(\frac{1}{2} + a \cdot \left(\frac{1}{6} \cdot x\right)\right) + x \cdot a\\ \mathbf{elif}\;a \le 1.1586552917250053 \cdot 10^{+22}:\\ \;\;\;\;\left(\sqrt[3]{\frac{e^{\left(x + x\right) \cdot \left(a + a\right)} + -1}{\left(e^{x \cdot a} \cdot e^{x \cdot a} + 1\right) \cdot \left(e^{x \cdot a} + 1\right)}} \cdot \sqrt[3]{\frac{e^{\left(x + x\right) \cdot \left(a + a\right)} + -1}{\left(e^{x \cdot a} \cdot e^{x \cdot a} + 1\right) \cdot \left(e^{x \cdot a} + 1\right)}}\right) \cdot \sqrt[3]{\frac{e^{\left(x + x\right) \cdot \left(a + a\right)} + -1}{\left(e^{x \cdot a} \cdot e^{x \cdot a} + 1\right) \cdot \left(e^{x \cdot a} + 1\right)}}\\ \mathbf{elif}\;a \le 1.2934075903718333 \cdot 10^{+60}:\\ \;\;\;\;\left(\left(x \cdot a\right) \cdot \left(x \cdot a\right)\right) \cdot \left(\frac{1}{2} + a \cdot \left(\frac{1}{6} \cdot x\right)\right) + x \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{e^{\left(x + x\right) \cdot \left(a + a\right)} + -1}\right)}{\left(e^{x \cdot a} \cdot e^{x \cdot a} + 1\right) \cdot \left(e^{x \cdot a} + 1\right)}\\ \end{array}\]

Runtime

Time bar (total: 34.2s)Debug logProfile

herbie shell --seed 2018346 
(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))