Average Error: 40.1 → 0.1
Time: 35.7s
Precision: 64
Internal Precision: 1344
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{\left(\frac{1}{6} \cdot {x}^{3} + x\right) - \frac{1}{2} \cdot {x}^{2}} \le -147.5620355352164:\\ \;\;\;\;\frac{e^{x}}{\frac{1}{2} \cdot {x}^{2} + \left(\frac{1}{6} \cdot {x}^{3} + x\right)}\\ \mathbf{if}\;\frac{1}{\left(\frac{1}{6} \cdot {x}^{3} + x\right) - \frac{1}{2} \cdot {x}^{2}} \le 582800.4550585768:\\ \;\;\;\;\frac{1}{1 - e^{-x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{1}{6} \cdot {x}^{3} + x\right) - \frac{1}{2} \cdot {x}^{2}}\\ \end{array}\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.1
Target39.7
Herbie0.1
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ 1 (- (+ (* 1/6 (pow x 3)) x) (* 1/2 (pow x 2)))) < -147.5620355352164

    1. Initial program 61.8

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

    2. Taylor expanded around 0 0.1

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

    if -147.5620355352164 < (/ 1 (- (+ (* 1/6 (pow x 3)) x) (* 1/2 (pow x 2)))) < 582800.4550585768

    1. Initial program 1.3

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

    3. Applied clear-num1.3

      \[\leadsto \color{blue}{\frac{1}{\frac{e^{x} - 1}{e^{x}}}}\]

    4. Applied simplify0.2

      \[\leadsto \frac{1}{\color{blue}{1 - e^{-x}}}\]

    if 582800.4550585768 < (/ 1 (- (+ (* 1/6 (pow x 3)) x) (* 1/2 (pow x 2))))

    1. Initial program 59.5

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

    3. Applied clear-num59.5

      \[\leadsto \color{blue}{\frac{1}{\frac{e^{x} - 1}{e^{x}}}}\]

    4. Applied simplify59.4

      \[\leadsto \frac{1}{\color{blue}{1 - e^{-x}}}\]

    5. Taylor expanded around 0 0.0

      \[\leadsto \frac{1}{\color{blue}{\left(\frac{1}{6} \cdot {x}^{3} + x\right) - \frac{1}{2} \cdot {x}^{2}}}\]
  3. Recombined 3 regimes into one program.

Runtime

Time bar (total: 35.7s)Debug logProfile

herbie shell --seed 2018193 
(FPCore (x)
  :name "expq2 (section 3.11)"

  :herbie-target
  (/ 1 (- 1 (exp (- x))))

  (/ (exp x) (- (exp x) 1)))