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Target

Derivation

1. Split input into 2 regimes

2. if (cbrt (* (* (log (exp (+ (+ (* 1/2 x) 1) (* (* x 1/6) x)))) (log (exp (+ (+ (* 1/2 x) 1) (* (* x 1/6) x))))) (log (exp (+ (+ (* 1/2 x) 1) (* (* x 1/6) x)))))) < 1.0470384144565212

   1. Initial program 60.0

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

   2. Taylor expanded around 0 0.4

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

   if 1.0470384144565212 < (cbrt (* (* (log (exp (+ (+ (* 1/2 x) 1) (* (* x 1/6) x)))) (log (exp (+ (+ (* 1/2 x) 1) (* (* x 1/6) x))))) (log (exp (+ (+ (* 1/2 x) 1) (* (* x 1/6) x))))))

   1. Initial program 0.0

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

   3. Applied flip3--0.2

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

4. Final simplification0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{\left(\log \left(e^{\left(\frac{1}{6} \cdot x\right) \cdot x + \left(1 + x \cdot \frac{1}{2}\right)}\right) \cdot \log \left(e^{\left(\frac{1}{6} \cdot x\right) \cdot x + \left(1 + x \cdot \frac{1}{2}\right)}\right)\right) \cdot \log \left(e^{\left(\frac{1}{6} \cdot x\right) \cdot x + \left(1 + x \cdot \frac{1}{2}\right)}\right)} \le 1.0470384144565212:\\ \;\;\;\;\frac{1}{6} \cdot {x}^{2} + \left(1 + x \cdot \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(e^{x}\right)}^{3} - {1}^{3}}{\left(e^{x} + 1\right) + e^{x} \cdot e^{x}}}{x}\\ \end{array}\]

Runtime

Time bar (total: 9.3s)Debug logProfile

herbie shell --seed 2018215 
(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))