Average Error: 29.8 → 0.3
Time: 3.4s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -4.1023891212148034 \cdot 10^{-4}:\\ \;\;\;\;\left(\sqrt{e^{a \cdot x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{a \cdot x}} - \sqrt{1}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{6}, {\left(x \cdot a\right)}^{3}, \mathsf{fma}\left(\frac{1}{2}, {\left(x \cdot a\right)}^{2}, x \cdot a\right)\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -4.1023891212148034 \cdot 10^{-4}:\\
\;\;\;\;\left(\sqrt{e^{a \cdot x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{a \cdot x}} - \sqrt{1}\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{6}, {\left(x \cdot a\right)}^{3}, \mathsf{fma}\left(\frac{1}{2}, {\left(x \cdot a\right)}^{2}, x \cdot a\right)\right)\\

\end{array}
double f(double a, double x) {
        double r79440 = a;
        double r79441 = x;
        double r79442 = r79440 * r79441;
        double r79443 = exp(r79442);
        double r79444 = 1.0;
        double r79445 = r79443 - r79444;
        return r79445;
}
↓
double f(double a, double x) {
        double r79446 = a;
        double r79447 = x;
        double r79448 = r79446 * r79447;
        double r79449 = -0.00041023891212148034;
        bool r79450 = r79448 <= r79449;
        double r79451 = exp(r79448);
        double r79452 = sqrt(r79451);
        double r79453 = 1.0;
        double r79454 = sqrt(r79453);
        double r79455 = r79452 + r79454;
        double r79456 = r79452 - r79454;
        double r79457 = r79455 * r79456;
        double r79458 = 0.16666666666666666;
        double r79459 = r79447 * r79446;
        double r79460 = 3.0;
        double r79461 = pow(r79459, r79460);
        double r79462 = 0.5;
        double r79463 = 2.0;
        double r79464 = pow(r79459, r79463);
        double r79465 = fma(r79462, r79464, r79459);
        double r79466 = fma(r79458, r79461, r79465);
        double r79467 = r79450 ? r79457 : r79466;
        return r79467;
}

Error

Bits error versus a

Bits error versus x

Target

Original29.8
Target0.2
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.10000000000000001:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -0.00041023891212148034

    1. Initial program 0.0

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

    3. Applied add-sqr-sqrt0.0

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

    4. Applied add-sqr-sqrt0.0

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

    5. Applied difference-of-squares0.0

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

    if -0.00041023891212148034 < (* a x)

    1. Initial program 44.8

      \[e^{a \cdot x} - 1\]

    2. Taylor expanded around 0 14.6

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

    3. Simplified14.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {x}^{3}, a \cdot x\right)\right)}\]
    4. Using strategy rm

    5. Applied pow-prod-down8.6

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

    6. Simplified8.6

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

    7. Taylor expanded around inf 14.6

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

    8. Simplified0.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{6}, {\left(x \cdot a\right)}^{3}, \mathsf{fma}\left(\frac{1}{2}, {\left(x \cdot a\right)}^{2}, x \cdot a\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -4.1023891212148034 \cdot 10^{-4}:\\ \;\;\;\;\left(\sqrt{e^{a \cdot x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{a \cdot x}} - \sqrt{1}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{6}, {\left(x \cdot a\right)}^{3}, \mathsf{fma}\left(\frac{1}{2}, {\left(x \cdot a\right)}^{2}, x \cdot a\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 +o rules:numerics
(FPCore (a x)
  :name "expax (section 3.5)"
  :precision binary64
  :herbie-expected 14

  :herbie-target
  (if (< (fabs (* a x)) 0.1) (* (* a x) (+ 1 (+ (/ (* a x) 2) (/ (pow (* a x) 2) 6)))) (- (exp (* a x)) 1))

  (- (exp (* a x)) 1))