Average Error: 40.4 → 0.2
Time: 1.5m
Precision: 64
Internal Precision: 1408
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{6} \cdot {x}^{2} + \left(1 + \frac{1}{2} \cdot x\right) \le 1.0085994019832611:\\ \;\;\;\;\frac{1}{6} \cdot {x}^{2} + \left(1 + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{e^{x + x}}{e^{x} + 1}}{x} - \frac{\frac{1}{e^{x} + 1}}{x}\\ \end{array}\]

Error

Bits error versus x

Target

Original40.4
Target39.6
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))) < 1.0085994019832611

    1. Initial program 60.2

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

    2. Taylor expanded around 0 0.3

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

    if 1.0085994019832611 < (+ (* 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 flip--0.1

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

    4. Applied simplify0.1

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

    6. Applied div-sub0.1

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

    7. Applied div-sub0.1

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

Runtime

Time bar (total: 1.5m)Debug log

herbie shell --seed '#(991339738 1419949195 2842012120 4157638069 1320221275 2092628673)' 
(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))