Average Error: 40.1 → 0.5
Time: 30.8s
Precision: 64
Internal Precision: 1408
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{e^{x} - 1}{x} \le -0.0:\\ \;\;\;\;\frac{1}{6} \cdot {x}^{2} + \left(1 + \frac{1}{2} \cdot x\right)\\ \mathbf{if}\;\frac{e^{x} - 1}{x} \le 0.985336851063437:\\ \;\;\;\;\frac{\log \left(e^{e^{x} - 1}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{6} \cdot {x}^{2} + \left(1 + \frac{1}{2} \cdot x\right)\\ \end{array}\]

Error

Bits error versus x

Target

Original40.1
Target39.2
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 2 regimes

  2. if (/ (- (exp x) 1) x) < -0.0 or 0.985336851063437 < (/ (- (exp x) 1) x)

    1. Initial program 59.8

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

    2. Taylor expanded around 0 0.6

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

    if -0.0 < (/ (- (exp x) 1) x) < 0.985336851063437

    1. Initial program 0.2

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

    3. Applied add-log-exp0.2

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

Runtime

Time bar (total: 30.8s)Debug logProfile

herbie shell --seed '#(1070258749 1877548225 2229079127 1588002776 3179087814 1886870650)' 
(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))