Average Error: 40.2 → 0.2
Time: 9.3s
Precision: 64
Internal Precision: 1408
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{6} \cdot {x}^{2} + \left(1 + \frac{1}{2} \cdot x\right) \le 2.681588727493412:\\ \;\;\;\;\frac{1}{6} \cdot {x}^{2} + \left(1 + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x}}{x} - \frac{1}{x}\\ \end{array}\]

Error

Bits error versus x

Target

Original40.2
Target39.4
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))) < 2.681588727493412

    1. Initial program 60.2

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

    2. Taylor expanded around 0 0.4

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

    if 2.681588727493412 < (+ (* 1/6 (pow x 2)) (+ 1 (* 1/2 x)))

    1. Initial program 0.0

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

    3. Applied div-sub0.0

      \[\leadsto \color{blue}{\frac{e^{x}}{x} - \frac{1}{x}}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 9.3s)Debug logProfile

herbie shell --seed '#(1064300848 3212030778 2049303162 3567222883 2277747821 1384278011)' 
(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))