Average Error: 39.9 → 0.3
Time: 2.8m
Precision: 64
Internal Precision: 1344
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\log \left(e^{\left(1 + \frac{1}{2} \cdot x\right) + \left(x \cdot x\right) \cdot \frac{1}{6}}\right) \le 1.0076251944592607:\\ \;\;\;\;\log \left(e^{\left(1 + \frac{1}{2} \cdot x\right) + \left(x \cdot x\right) \cdot \frac{1}{6}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot 3} - {1}^{3}}{x + \left(e^{x} \cdot x\right) \cdot \left(1 + e^{x}\right)}\\ \end{array}\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.9
Target39.0
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;x \lt 1 \land x \gt -1:\\ \;\;\;\;\frac{e^{x} - 1}{\log \left(e^{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - 1}{x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (log (exp (+ (+ 1 (* 1/2 x)) (* (* x x) 1/6)))) < 1.0076251944592607

    1. Initial program 60.1

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

    2. Taylor expanded around 0 0.3

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

    4. Applied add-log-exp0.3

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

    5. Applied add-log-exp0.3

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

    6. Applied sum-log0.3

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

    7. Applied simplify0.3

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

    if 1.0076251944592607 < (log (exp (+ (+ 1 (* 1/2 x)) (* (* x x) 1/6))))

    1. Initial program 0.0

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

    3. Applied flip3--0.2

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

    4. Applied associate-/l/0.2

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

    5. Applied simplify0.2

      \[\leadsto \frac{{\left(e^{x}\right)}^{3} - {1}^{3}}{\color{blue}{x + \left(e^{x} \cdot x\right) \cdot \left(1 + e^{x}\right)}}\]
    6. Using strategy rm

    7. Applied pow-exp0.2

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

Runtime

Time bar (total: 2.8m)Debug logProfile

herbie shell --seed 2019053 
(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))