Average Error: 40.3 → 0.2
Time: 42.9s
Precision: 64
Internal Precision: 1344
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x + \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(x \cdot x\right)}{x} \le 1.0187894117188008:\\ \;\;\;\;\frac{x + \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(x \cdot x\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\left(e^{x} \cdot e^{x}\right) \cdot \left(e^{x} \cdot e^{x}\right) - 1}{\sqrt[3]{1 + e^{x} \cdot e^{x}} \cdot \sqrt[3]{1 + e^{x} \cdot e^{x}}}}{\sqrt[3]{1 + e^{x} \cdot e^{x}}}}{x \cdot \left(e^{x} + 1\right)}\\ \end{array}\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.3
Target39.6
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 (/ (+ x (* (* x x) (+ (* x 1/6) 1/2))) x) < 1.0187894117188008

    1. Initial program 60.3

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

    2. Taylor expanded around 0 0.3

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

    3. Simplified0.3

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

    if 1.0187894117188008 < (/ (+ x (* (* x x) (+ (* x 1/6) 1/2))) x)

    1. Initial program 0.0

      \[\frac{e^{x} - 1}{x}\]
    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 associate-/l/0.1

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

    5. Simplified0.1

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

    7. Applied flip--0.1

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

    9. Applied add-cube-cbrt0.1

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

    10. Applied associate-/r*0.1

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(x \cdot x\right)}{x} \le 1.0187894117188008:\\ \;\;\;\;\frac{x + \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(x \cdot x\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\left(e^{x} \cdot e^{x}\right) \cdot \left(e^{x} \cdot e^{x}\right) - 1}{\sqrt[3]{1 + e^{x} \cdot e^{x}} \cdot \sqrt[3]{1 + e^{x} \cdot e^{x}}}}{\sqrt[3]{1 + e^{x} \cdot e^{x}}}}{x \cdot \left(e^{x} + 1\right)}\\ \end{array}\]

Runtime

Time bar (total: 42.9s)Debug logProfile

herbie shell --seed 2018216 
(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))