Average Error: 58.8 → 0.0
Time: 31.9s
Precision: 64
\[-1.700000000000000122124532708767219446599 \cdot 10^{-4} \lt x\]
\[e^{x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 1.000116207589260675092646124539896845818:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), x \cdot x, x\right)\\ \mathbf{else}:\\ \;\;\;\;e^{x} + \left(-1\right)\\ \end{array}\]
e^{x} - 1
↓
\begin{array}{l}
\mathbf{if}\;e^{x} \le 1.000116207589260675092646124539896845818:\\
\;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), x \cdot x, x\right)\\

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

\end{array}
double f(double x) {
        double r4449295 = x;
        double r4449296 = exp(r4449295);
        double r4449297 = 1.0;
        double r4449298 = r4449296 - r4449297;
        return r4449298;
}
↓
double f(double x) {
        double r4449299 = x;
        double r4449300 = exp(r4449299);
        double r4449301 = 1.0001162075892607;
        bool r4449302 = r4449300 <= r4449301;
        double r4449303 = 0.16666666666666666;
        double r4449304 = 0.5;
        double r4449305 = fma(r4449303, r4449299, r4449304);
        double r4449306 = r4449299 * r4449299;
        double r4449307 = fma(r4449305, r4449306, r4449299);
        double r4449308 = 1.0;
        double r4449309 = -r4449308;
        double r4449310 = r4449300 + r4449309;
        double r4449311 = r4449302 ? r4449307 : r4449310;
        return r4449311;
}

Error

Bits error versus x

Target

Original58.8
Target0.4
Herbie0.0
\[x \cdot \left(\left(1 + \frac{x}{2}\right) + \frac{x \cdot x}{6}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if (exp x) < 1.0001162075892607

    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

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

    if 1.0001162075892607 < (exp x)

    1. Initial program 2.7

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

    3. Applied sub-neg2.7

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 1.000116207589260675092646124539896845818:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), x \cdot x, x\right)\\ \mathbf{else}:\\ \;\;\;\;e^{x} + \left(-1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x)
  :name "expm1 (example 3.7)"
  :pre (< -0.00017 x)

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

  (- (exp x) 1.0))