Average Error: 58.6 → 0.1
Time: 8.7s
Precision: 64
\[-0.00017 \lt x\]
\[e^{x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 0.00023583480866303395:\\ \;\;\;\;x + \left(x \cdot x\right) \cdot \left(\frac{1}{6} \cdot x + \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\sqrt{e^{x}} - 1\right) \cdot \left(1 + \sqrt{e^{x}}\right)} \cdot \sqrt{\left(\sqrt{e^{x}} - 1\right) \cdot \left(1 + \sqrt{e^{x}}\right)}\\ \end{array}\]
e^{x} - 1
↓
\begin{array}{l}
\mathbf{if}\;x \le 0.00023583480866303395:\\
\;\;\;\;x + \left(x \cdot x\right) \cdot \left(\frac{1}{6} \cdot x + \frac{1}{2}\right)\\

\mathbf{else}:\\
\;\;\;\;\sqrt{\left(\sqrt{e^{x}} - 1\right) \cdot \left(1 + \sqrt{e^{x}}\right)} \cdot \sqrt{\left(\sqrt{e^{x}} - 1\right) \cdot \left(1 + \sqrt{e^{x}}\right)}\\

\end{array}
double f(double x) {
        double r1209416 = x;
        double r1209417 = exp(r1209416);
        double r1209418 = 1.0;
        double r1209419 = r1209417 - r1209418;
        return r1209419;
}
↓
double f(double x) {
        double r1209420 = x;
        double r1209421 = 0.00023583480866303395;
        bool r1209422 = r1209420 <= r1209421;
        double r1209423 = r1209420 * r1209420;
        double r1209424 = 0.16666666666666666;
        double r1209425 = r1209424 * r1209420;
        double r1209426 = 0.5;
        double r1209427 = r1209425 + r1209426;
        double r1209428 = r1209423 * r1209427;
        double r1209429 = r1209420 + r1209428;
        double r1209430 = exp(r1209420);
        double r1209431 = sqrt(r1209430);
        double r1209432 = 1.0;
        double r1209433 = r1209431 - r1209432;
        double r1209434 = r1209432 + r1209431;
        double r1209435 = r1209433 * r1209434;
        double r1209436 = sqrt(r1209435);
        double r1209437 = r1209436 * r1209436;
        double r1209438 = r1209422 ? r1209429 : r1209437;
        return r1209438;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.6
Target0.5
Herbie0.1
\[x \cdot \left(\left(1 + \frac{x}{2}\right) + \frac{x \cdot x}{6}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if x < 0.00023583480866303395

    1. Initial program 59.3

      \[e^{x} - 1\]

    2. Taylor expanded around 0 0.0

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

    3. Simplified0.0

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

    if 0.00023583480866303395 < x

    1. Initial program 2.0

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

    3. Applied *-un-lft-identity2.0

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

    4. Applied add-sqr-sqrt2.7

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

    5. Applied difference-of-squares2.8

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

    7. Applied add-sqr-sqrt2.8

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 0.00023583480866303395:\\ \;\;\;\;x + \left(x \cdot x\right) \cdot \left(\frac{1}{6} \cdot x + \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\sqrt{e^{x}} - 1\right) \cdot \left(1 + \sqrt{e^{x}}\right)} \cdot \sqrt{\left(\sqrt{e^{x}} - 1\right) \cdot \left(1 + \sqrt{e^{x}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 
(FPCore (x)
  :name "expm1 (example 3.7)"
  :pre (< -0.00017 x)

  :herbie-target
  (* x (+ (+ 1 (/ x 2)) (/ (* x x) 6)))

  (- (exp x) 1))