Average Error: 29.4 → 0.3
Time: 7.9s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -9.4276480012748349 \cdot 10^{-4}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{e^{a \cdot x}}, \sqrt{e^{a \cdot x}}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({\left(x \cdot a\right)}^{3}, \frac{1}{6}, \frac{1}{2} \cdot {\left(a \cdot x\right)}^{2} + a \cdot x\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -9.4276480012748349 \cdot 10^{-4}:\\
\;\;\;\;\mathsf{fma}\left(\sqrt{e^{a \cdot x}}, \sqrt{e^{a \cdot x}}, -1\right)\\

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

\end{array}
double f(double a, double x) {
        double r104769 = a;
        double r104770 = x;
        double r104771 = r104769 * r104770;
        double r104772 = exp(r104771);
        double r104773 = 1.0;
        double r104774 = r104772 - r104773;
        return r104774;
}
↓
double f(double a, double x) {
        double r104775 = a;
        double r104776 = x;
        double r104777 = r104775 * r104776;
        double r104778 = -0.0009427648001274835;
        bool r104779 = r104777 <= r104778;
        double r104780 = exp(r104777);
        double r104781 = sqrt(r104780);
        double r104782 = 1.0;
        double r104783 = -r104782;
        double r104784 = fma(r104781, r104781, r104783);
        double r104785 = r104776 * r104775;
        double r104786 = 3.0;
        double r104787 = pow(r104785, r104786);
        double r104788 = 0.16666666666666666;
        double r104789 = 0.5;
        double r104790 = 2.0;
        double r104791 = pow(r104777, r104790);
        double r104792 = r104789 * r104791;
        double r104793 = r104792 + r104777;
        double r104794 = fma(r104787, r104788, r104793);
        double r104795 = r104779 ? r104784 : r104794;
        return r104795;
}

Error

Bits error versus a

Bits error versus x

Target

Original29.4
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) < -0.0009427648001274835

    1. Initial program 0.0

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

    3. Applied add-sqr-sqrt0.0

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

    4. Applied fma-neg0.0

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

    if -0.0009427648001274835 < (* a x)

    1. Initial program 44.3

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

    2. Taylor expanded around 0 14.2

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

    3. Simplified14.2

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

    5. Applied pow-prod-down4.3

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

    6. Simplified4.3

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

    8. Applied distribute-lft-in4.3

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

    9. Simplified0.4

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

    10. Simplified0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -9.4276480012748349 \cdot 10^{-4}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{e^{a \cdot x}}, \sqrt{e^{a \cdot x}}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({\left(x \cdot a\right)}^{3}, \frac{1}{6}, \frac{1}{2} \cdot {\left(a \cdot x\right)}^{2} + a \cdot x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 +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))