Average Error: 39.9 → 0.5
Time: 24.7s
Precision: 64
Internal Precision: 1344
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{e^{x} - 1}{x} \le 0.0:\\ \;\;\;\;\frac{1}{6} \cdot {x}^{2} + \left(1 + \frac{1}{2} \cdot x\right)\\ \mathbf{if}\;\frac{e^{x} - 1}{x} \le 0.049221242662298555:\\ \;\;\;\;\frac{\frac{e^{x + x} - 1}{e^{x} + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{6} \cdot {x}^{2} + \left(1 + \frac{1}{2} \cdot x\right)\\ \end{array}\]

Error

Bits error versus x

Target

Original39.9
Target39.0
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;x \lt 1 \land x \gt -1:\\ \;\;\;\;\frac{e^{x} - 1}{\log \left(e^{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - 1}{x}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (- (exp x) 1) x) < 0.0

    1. Initial program 62.0

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

    2. Taylor expanded around 0 0

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

    if 0.0 < (/ (- (exp x) 1) x) < 0.049221242662298555

    1. Initial program 0.0

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

    3. Applied flip--0

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

    4. Applied simplify0

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

    if 0.049221242662298555 < (/ (- (exp x) 1) x)

    1. Initial program 22.3

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

    2. Taylor expanded around 0 11.1

      \[\leadsto \color{blue}{\frac{1}{6} \cdot {x}^{2} + \left(1 + \frac{1}{2} \cdot x\right)}\]
  3. Recombined 3 regimes into one program.

Runtime

Time bar (total: 24.7s)Debug logProfile

herbie shell --seed '#(1071246582 2318319007 2683472949 3810440501 3233274817 2724848749)' 
(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))