Average Error: 29.4 → 5.2
Time: 16.4s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -1.335189100226837470869581266346825393354 \cdot 10^{-9}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, x, \mathsf{fma}\left(\frac{1}{2}, a \cdot a, \frac{1}{6} \cdot \left(\left(a \cdot \left(a \cdot x\right)\right) \cdot a\right)\right) \cdot \left(x \cdot x\right)\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -1.335189100226837470869581266346825393354 \cdot 10^{-9}:\\
\;\;\;\;e^{a \cdot x} - 1\\

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

\end{array}
double f(double a, double x) {
        double r56193 = a;
        double r56194 = x;
        double r56195 = r56193 * r56194;
        double r56196 = exp(r56195);
        double r56197 = 1.0;
        double r56198 = r56196 - r56197;
        return r56198;
}
↓
double f(double a, double x) {
        double r56199 = a;
        double r56200 = x;
        double r56201 = r56199 * r56200;
        double r56202 = -1.3351891002268375e-09;
        bool r56203 = r56201 <= r56202;
        double r56204 = exp(r56201);
        double r56205 = 1.0;
        double r56206 = r56204 - r56205;
        double r56207 = 0.5;
        double r56208 = r56199 * r56199;
        double r56209 = 0.16666666666666666;
        double r56210 = r56199 * r56201;
        double r56211 = r56210 * r56199;
        double r56212 = r56209 * r56211;
        double r56213 = fma(r56207, r56208, r56212);
        double r56214 = r56200 * r56200;
        double r56215 = r56213 * r56214;
        double r56216 = fma(r56199, r56200, r56215);
        double r56217 = r56203 ? r56206 : r56216;
        return r56217;
}

Error

Bits error versus a

Bits error versus x

Target

Original29.4
Target0.2
Herbie5.2
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.1000000000000000055511151231257827021182:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -1.3351891002268375e-09

    1. Initial program 0.3

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

    if -1.3351891002268375e-09 < (* a x)

    1. Initial program 44.4

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

    2. Taylor expanded around 0 14.1

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

    3. Simplified11.0

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

    5. Applied cube-mult11.0

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

    6. Applied associate-*l*7.8

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

    7. Simplified7.8

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

  4. Final simplification5.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -1.335189100226837470869581266346825393354 \cdot 10^{-9}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, x, \mathsf{fma}\left(\frac{1}{2}, a \cdot a, \frac{1}{6} \cdot \left(\left(a \cdot \left(a \cdot x\right)\right) \cdot a\right)\right) \cdot \left(x \cdot x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 +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))