Average Error: 40.3 → 0.0
Time: 4.8s
Precision: 64
Internal Precision: 1344
\[\frac{e^{x} - 1}{x}\]
↓
\[(e^{\log_* (1 + \frac{(e^{x} - 1)^*}{x})} - 1)^*\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.3
Target39.6
Herbie0.0
\[\begin{array}{l} \mathbf{if}\;x \lt 1 \land x \gt -1:\\ \;\;\;\;\frac{e^{x} - 1}{\log \left(e^{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - 1}{x}\\ \end{array}\]

Derivation

  1. Initial program 40.3

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

  2. Initial simplification0.0

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

  4. Applied expm1-log1p-u0.0

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

  5. Final simplification0.0

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

Runtime

Time bar (total: 4.8s)Debug logProfile

herbie shell --seed 2018216 +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))