Average Error: 58.7 → 0.1
Time: 2.9s
Precision: binary64
\[-1.7 \cdot 10^{-4} \lt x\]
\[e^{x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 1.00000003539203353:\\ \;\;\;\;{x}^{2} \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) + x\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{e^{x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{x}} - \sqrt{1}\right)\\ \end{array}\]
e^{x} - 1
↓
\begin{array}{l}
\mathbf{if}\;e^{x} \le 1.00000003539203353:\\
\;\;\;\;{x}^{2} \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) + x\\

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

\end{array}
double code(double x) {
	return ((double) (((double) exp(x)) - 1.0));
}
↓
double code(double x) {
	double VAR;
	if ((((double) exp(x)) <= 1.0000000353920335)) {
		VAR = ((double) (((double) (((double) pow(x, 2.0)) * ((double) (((double) (x * 0.16666666666666666)) + 0.5)))) + x));
	} else {
		VAR = ((double) (((double) (((double) sqrt(((double) exp(x)))) + ((double) sqrt(1.0)))) * ((double) (((double) sqrt(((double) exp(x)))) - ((double) sqrt(1.0))))));
	}
	return VAR;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.7
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 (exp x) < 1.00000003539203353

    1. Initial program 59.5

      \[e^{x} - 1\]

    2. Taylor expanded around 0 0.0

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

    3. Simplified0.0

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

    if 1.00000003539203353 < (exp x)

    1. Initial program 7.3

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

    3. Applied add-sqr-sqrt7.3

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

    4. Applied add-sqr-sqrt8.1

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

    5. Applied difference-of-squares8.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 1.00000003539203353:\\ \;\;\;\;{x}^{2} \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) + x\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{e^{x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{x}} - \sqrt{1}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020173 
(FPCore (x)
  :name "expm1 (example 3.7)"
  :precision binary64
  :pre (< -0.00017 x)

  :herbie-target
  (* x (+ (+ 1.0 (/ x 2.0)) (/ (* x x) 6.0)))

  (- (exp x) 1.0))