Average Error: 39.4 → 0.2
Time: 3.8m
Precision: 64
Internal Precision: 1408
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{6} \cdot {x}^{2} + \left(1 + \frac{1}{2} \cdot x\right) \le 1.000005925336213:\\ \;\;\;\;\frac{1}{6} \cdot {x}^{2} + \left(1 + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({\left(e^{x}\right)}^{3} \cdot {\left(e^{x}\right)}^{3}\right)}^{3} - {\left({1}^{3} \cdot {1}^{3}\right)}^{3}}{\left({\left({\left(e^{x}\right)}^{3}\right)}^{\left(3 + 1\right)} + \left({\left(e^{x}\right)}^{3} \cdot {\left(e^{x}\right)}^{3} + 1\right)\right) \cdot \left(\left(\left(\left(1 + e^{x}\right) + e^{x} \cdot e^{x}\right) \cdot x\right) \cdot \left({\left(e^{x}\right)}^{3} + 1\right)\right)}\\ \end{array}\]

Error

Bits error versus x

Target

Original39.4
Target38.7
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;x \lt 1 \land x \gt -1:\\ \;\;\;\;\frac{e^{x} - 1}{\log \left(e^{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - 1}{x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (+ (* 1/6 (pow x 2)) (+ 1 (* 1/2 x))) < 1.000005925336213

    1. Initial program 60.5

      \[\frac{e^{x} - 1}{x}\]

    2. Taylor expanded around 0 0.2

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

    if 1.000005925336213 < (+ (* 1/6 (pow x 2)) (+ 1 (* 1/2 x)))

    1. Initial program 0.1

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

    3. Applied flip3--0.1

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

    4. Applied associate-/l/0.1

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

    5. Applied simplify0.1

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

    7. Applied flip--0.2

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

    8. Applied associate-/l/0.2

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

    10. Applied flip3--0.3

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

    11. Applied associate-/l/0.3

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

    12. Applied simplify0.3

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

Runtime

Time bar (total: 3.8m)Debug logProfile

herbie shell --seed '#(1070609872 3456127585 2380521889 2328837196 1765472538 734540918)' 
(FPCore (x)
  :name "Kahan's exp quotient"

  :herbie-target
  (if (and (< x 1) (> x -1)) (/ (- (exp x) 1) (log (exp x))) (/ (- (exp x) 1) x))

  (/ (- (exp x) 1) x))