Average Error: 29.8 → 9.8
Time: 4.3s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -3.6128960162996795 \cdot 10^{-4}:\\ \;\;\;\;\sqrt[3]{\left(e^{a \cdot x} - 1\right) \cdot \left(e^{a \cdot x} - 1\right)} \cdot \sqrt[3]{e^{a \cdot x} - 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {x}^{3}, a \cdot x\right)\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -3.6128960162996795 \cdot 10^{-4}:\\
\;\;\;\;\sqrt[3]{\left(e^{a \cdot x} - 1\right) \cdot \left(e^{a \cdot x} - 1\right)} \cdot \sqrt[3]{e^{a \cdot x} - 1}\\

\mathbf{else}:\\
\;\;\;\;\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)\\

\end{array}
double f(double a, double x) {
        double r101420 = a;
        double r101421 = x;
        double r101422 = r101420 * r101421;
        double r101423 = exp(r101422);
        double r101424 = 1.0;
        double r101425 = r101423 - r101424;
        return r101425;
}
↓
double f(double a, double x) {
        double r101426 = a;
        double r101427 = x;
        double r101428 = r101426 * r101427;
        double r101429 = -0.00036128960162996795;
        bool r101430 = r101428 <= r101429;
        double r101431 = exp(r101428);
        double r101432 = 1.0;
        double r101433 = r101431 - r101432;
        double r101434 = r101433 * r101433;
        double r101435 = cbrt(r101434);
        double r101436 = cbrt(r101433);
        double r101437 = r101435 * r101436;
        double r101438 = 0.5;
        double r101439 = 2.0;
        double r101440 = pow(r101426, r101439);
        double r101441 = pow(r101427, r101439);
        double r101442 = r101440 * r101441;
        double r101443 = 0.16666666666666666;
        double r101444 = 3.0;
        double r101445 = pow(r101426, r101444);
        double r101446 = pow(r101427, r101444);
        double r101447 = r101445 * r101446;
        double r101448 = fma(r101443, r101447, r101428);
        double r101449 = fma(r101438, r101442, r101448);
        double r101450 = r101430 ? r101437 : r101449;
        return r101450;
}

Error

Bits error versus a

Bits error versus x

Target

Original29.8
Target0.2
Herbie9.8
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.10000000000000001:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -0.00036128960162996795

    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

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

    6. Applied add-cbrt-cube0.0

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

    7. Simplified0.0

      \[\leadsto \sqrt[3]{\color{blue}{{\left(e^{a \cdot x} - 1\right)}^{3}}}\]
    8. Using strategy rm

    9. Applied add-cube-cbrt0.0

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

    10. Applied unpow-prod-down0.0

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

    11. Applied cbrt-prod0.0

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

    12. Simplified0.0

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

    13. Simplified0.0

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

    if -0.00036128960162996795 < (* a x)

    1. Initial program 44.5

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

    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

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

  4. Final simplification9.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -3.6128960162996795 \cdot 10^{-4}:\\ \;\;\;\;\sqrt[3]{\left(e^{a \cdot x} - 1\right) \cdot \left(e^{a \cdot x} - 1\right)} \cdot \sqrt[3]{e^{a \cdot x} - 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {x}^{3}, a \cdot x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 +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))