Average Error: 58.6 → 0.1
Time: 34.7s
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 r10733077 = x;
        double r10733078 = exp(r10733077);
        double r10733079 = 1.0;
        double r10733080 = r10733078 - r10733079;
        return r10733080;
}
↓
double f(double x) {
        double r10733081 = x;
        double r10733082 = 0.00023583480866303395;
        bool r10733083 = r10733081 <= r10733082;
        double r10733084 = r10733081 * r10733081;
        double r10733085 = 0.16666666666666666;
        double r10733086 = r10733085 * r10733081;
        double r10733087 = 0.5;
        double r10733088 = r10733086 + r10733087;
        double r10733089 = r10733084 * r10733088;
        double r10733090 = r10733081 + r10733089;
        double r10733091 = exp(r10733081);
        double r10733092 = sqrt(r10733091);
        double r10733093 = 1.0;
        double r10733094 = r10733092 - r10733093;
        double r10733095 = r10733093 + r10733092;
        double r10733096 = r10733094 * r10733095;
        double r10733097 = sqrt(r10733096);
        double r10733098 = r10733097 * r10733097;
        double r10733099 = r10733083 ? r10733090 : r10733098;
        return r10733099;
}

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

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

    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))