Average Error: 40.0 → 0.2
Time: 28.2s
Precision: 64
Internal Precision: 1408
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt{\frac{1}{6} \cdot {x}^{2} + \left(1 + \frac{1}{2} \cdot x\right)} \cdot \log \left(e^{\sqrt{\frac{1}{6} \cdot {x}^{2} + \left(1 + \frac{1}{2} \cdot x\right)}}\right) \le 1.0000562928638548:\\ \;\;\;\;\frac{\frac{1}{2} \cdot {x}^{2} + \left(\frac{1}{6} \cdot {x}^{3} + x\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{e^{x + x} - 1}{e^{x} + 1}}{x}\\ \end{array}\]

Error

Bits error versus x

Target

Original40.0
Target39.2
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 (* (sqrt (+ (* 1/6 (pow x 2)) (+ 1 (* 1/2 x)))) (log (exp (sqrt (+ (* 1/6 (pow x 2)) (+ 1 (* 1/2 x))))))) < 1.0000562928638548

    1. Initial program 60.3

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

    2. Taylor expanded around 0 0.3

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot {x}^{2} + \left(\frac{1}{6} \cdot {x}^{3} + x\right)}}{x}\]

    if 1.0000562928638548 < (* (sqrt (+ (* 1/6 (pow x 2)) (+ 1 (* 1/2 x)))) (log (exp (sqrt (+ (* 1/6 (pow x 2)) (+ 1 (* 1/2 x)))))))

    1. Initial program 0.1

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

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

Runtime

Time bar (total: 28.2s)Debug logProfile

herbie shell --seed '#(1070578969 3140398606 632207097 462683394 1189254563 964980650)' 
(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))