Average Error: 40.1 → 0.2
Time: 42.7s
Precision: 64
Internal Precision: 1344
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt{\frac{1}{6} \cdot {x}^{2} + \left(1 + \frac{1}{2} \cdot x\right)} \cdot \left(\log \left(e^{\frac{5}{96} \cdot {x}^{2}}\right) + \left(1 + \frac{1}{4} \cdot x\right)\right) \le 1.0262104794077462:\\ \;\;\;\;\sqrt{\frac{1}{6} \cdot {x}^{2} + \left(1 + \frac{1}{2} \cdot x\right)} \cdot \left(\log \left(e^{\frac{5}{96} \cdot {x}^{2}}\right) + \left(1 + \frac{1}{4} \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x}}{x} - \frac{1}{x}\\ \end{array}\]

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

Derivation

  1. Split input into 2 regimes

  2. if (* (sqrt (+ (* 1/6 (pow x 2)) (+ 1 (* 1/2 x)))) (+ (log (exp (* 5/96 (pow x 2)))) (+ 1 (* 1/4 x)))) < 1.0262104794077462

    1. Initial program 60.0

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

    2. Taylor expanded around 0 0.4

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

    4. Applied add-sqr-sqrt0.4

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

    5. Taylor expanded around 0 0.4

      \[\leadsto \sqrt{\frac{1}{6} \cdot {x}^{2} + \left(1 + \frac{1}{2} \cdot x\right)} \cdot \color{blue}{\left(\frac{5}{96} \cdot {x}^{2} + \left(1 + \frac{1}{4} \cdot x\right)\right)}\]
    6. Using strategy rm

    7. Applied add-log-exp0.4

      \[\leadsto \sqrt{\frac{1}{6} \cdot {x}^{2} + \left(1 + \frac{1}{2} \cdot x\right)} \cdot \left(\color{blue}{\log \left(e^{\frac{5}{96} \cdot {x}^{2}}\right)} + \left(1 + \frac{1}{4} \cdot x\right)\right)\]

    if 1.0262104794077462 < (* (sqrt (+ (* 1/6 (pow x 2)) (+ 1 (* 1/2 x)))) (+ (log (exp (* 5/96 (pow x 2)))) (+ 1 (* 1/4 x))))

    1. Initial program 0.0

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

    3. Applied div-sub0.0

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

Runtime

Time bar (total: 42.7s)Debug logProfile

herbie shell --seed 2018206 
(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))