Average Error: 29.7 → 0.4
Time: 35.5s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -2.550482612080378535049374200127658696147 \cdot 10^{-4}:\\ \;\;\;\;\frac{e^{\mathsf{fma}\left(2, a, a\right) \cdot x} - 1 \cdot \left(1 \cdot 1\right)}{\mathsf{fma}\left(\left(\sqrt[3]{e^{a \cdot x}} \cdot \sqrt[3]{e^{a \cdot x}}\right) \cdot \sqrt[3]{e^{a \cdot x}}, e^{a \cdot x} + 1, 1 \cdot 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right), \frac{1}{2}, \mathsf{fma}\left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot x\right), \frac{1}{6}, a \cdot x\right)\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -2.550482612080378535049374200127658696147 \cdot 10^{-4}:\\
\;\;\;\;\frac{e^{\mathsf{fma}\left(2, a, a\right) \cdot x} - 1 \cdot \left(1 \cdot 1\right)}{\mathsf{fma}\left(\left(\sqrt[3]{e^{a \cdot x}} \cdot \sqrt[3]{e^{a \cdot x}}\right) \cdot \sqrt[3]{e^{a \cdot x}}, e^{a \cdot x} + 1, 1 \cdot 1\right)}\\

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

\end{array}
double f(double a, double x) {
        double r4391363 = a;
        double r4391364 = x;
        double r4391365 = r4391363 * r4391364;
        double r4391366 = exp(r4391365);
        double r4391367 = 1.0;
        double r4391368 = r4391366 - r4391367;
        return r4391368;
}
↓
double f(double a, double x) {
        double r4391369 = a;
        double r4391370 = x;
        double r4391371 = r4391369 * r4391370;
        double r4391372 = -0.00025504826120803785;
        bool r4391373 = r4391371 <= r4391372;
        double r4391374 = 2.0;
        double r4391375 = fma(r4391374, r4391369, r4391369);
        double r4391376 = r4391375 * r4391370;
        double r4391377 = exp(r4391376);
        double r4391378 = 1.0;
        double r4391379 = r4391378 * r4391378;
        double r4391380 = r4391378 * r4391379;
        double r4391381 = r4391377 - r4391380;
        double r4391382 = exp(r4391371);
        double r4391383 = cbrt(r4391382);
        double r4391384 = r4391383 * r4391383;
        double r4391385 = r4391384 * r4391383;
        double r4391386 = r4391382 + r4391378;
        double r4391387 = fma(r4391385, r4391386, r4391379);
        double r4391388 = r4391381 / r4391387;
        double r4391389 = r4391371 * r4391371;
        double r4391390 = 0.5;
        double r4391391 = r4391389 * r4391371;
        double r4391392 = 0.16666666666666666;
        double r4391393 = fma(r4391391, r4391392, r4391371);
        double r4391394 = fma(r4391389, r4391390, r4391393);
        double r4391395 = r4391373 ? r4391388 : r4391394;
        return r4391395;
}

Error

Bits error versus a

Bits error versus x

Target

Original29.7
Target0.2
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.1000000000000000055511151231257827021182:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -0.00025504826120803785

    1. Initial program 0.1

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

    3. Applied flip3--0.1

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

    4. Simplified0.0

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

    5. Simplified0.0

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

    7. Applied add-cube-cbrt0.0

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

    if -0.00025504826120803785 < (* a x)

    1. Initial program 44.4

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

    2. Taylor expanded around 0 14.6

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

    3. Simplified0.6

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -2.550482612080378535049374200127658696147 \cdot 10^{-4}:\\ \;\;\;\;\frac{e^{\mathsf{fma}\left(2, a, a\right) \cdot x} - 1 \cdot \left(1 \cdot 1\right)}{\mathsf{fma}\left(\left(\sqrt[3]{e^{a \cdot x}} \cdot \sqrt[3]{e^{a \cdot x}}\right) \cdot \sqrt[3]{e^{a \cdot x}}, e^{a \cdot x} + 1, 1 \cdot 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right), \frac{1}{2}, \mathsf{fma}\left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot x\right), \frac{1}{6}, a \cdot x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (a x)
  :name "expax (section 3.5)"
  :herbie-expected 14

  :herbie-target
  (if (< (fabs (* a x)) 0.1) (* (* a x) (+ 1.0 (+ (/ (* a x) 2.0) (/ (pow (* a x) 2.0) 6.0)))) (- (exp (* a x)) 1.0))

  (- (exp (* a x)) 1.0))