Average Error: 40.1 → 0.1
Time: 12.8s
Precision: 64
Internal Precision: 1344
\[\frac{e^{x} - 1}{x}\]
↓
\[\log_* (1 + (e^{\frac{(e^{x} - 1)^*}{x}} - 1)^*)\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.1
Target39.2
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;x \lt 1 \land x \gt -1:\\ \;\;\;\;\frac{e^{x} - 1}{\log \left(e^{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - 1}{x}\\ \end{array}\]

Derivation

  1. Initial program 40.1

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

  2. Applied simplify0.0

    \[\leadsto \color{blue}{\frac{(e^{x} - 1)^*}{x}}\]
  3. Using strategy rm

  4. Applied log1p-expm1-u0.1

    \[\leadsto \color{blue}{\log_* (1 + (e^{\frac{(e^{x} - 1)^*}{x}} - 1)^*)}\]

Runtime

Time bar (total: 12.8s)Debug logProfile

herbie shell --seed 2018206 +o rules:numerics
(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))