Average Error: 29.0 → 9.3
Time: 4.0s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -1.97221861134718174 \cdot 10^{-5}:\\ \;\;\;\;\sqrt[3]{{\left(\log \left(e^{e^{a \cdot x} - 1}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {x}^{3}, a \cdot x\right)\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -1.97221861134718174 \cdot 10^{-5}:\\
\;\;\;\;\sqrt[3]{{\left(\log \left(e^{e^{a \cdot x} - 1}\right)\right)}^{3}}\\

\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 r99400 = a;
        double r99401 = x;
        double r99402 = r99400 * r99401;
        double r99403 = exp(r99402);
        double r99404 = 1.0;
        double r99405 = r99403 - r99404;
        return r99405;
}
↓
double f(double a, double x) {
        double r99406 = a;
        double r99407 = x;
        double r99408 = r99406 * r99407;
        double r99409 = -1.9722186113471817e-05;
        bool r99410 = r99408 <= r99409;
        double r99411 = exp(r99408);
        double r99412 = 1.0;
        double r99413 = r99411 - r99412;
        double r99414 = exp(r99413);
        double r99415 = log(r99414);
        double r99416 = 3.0;
        double r99417 = pow(r99415, r99416);
        double r99418 = cbrt(r99417);
        double r99419 = 0.5;
        double r99420 = 2.0;
        double r99421 = pow(r99406, r99420);
        double r99422 = pow(r99407, r99420);
        double r99423 = r99421 * r99422;
        double r99424 = 0.16666666666666666;
        double r99425 = pow(r99406, r99416);
        double r99426 = pow(r99407, r99416);
        double r99427 = r99425 * r99426;
        double r99428 = fma(r99424, r99427, r99408);
        double r99429 = fma(r99419, r99423, r99428);
        double r99430 = r99410 ? r99418 : r99429;
        return r99430;
}

Error

Bits error versus a

Bits error versus x

Target

Original29.0
Target0.2
Herbie9.3
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.10000000000000001:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -1.9722186113471817e-05

    1. Initial program 0.1

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

    3. Applied add-cbrt-cube0.1

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

    4. Simplified0.1

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

    6. Applied add-log-exp0.1

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

    7. Applied add-log-exp0.1

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

    8. Applied diff-log0.1

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

    9. Simplified0.1

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

    if -1.9722186113471817e-05 < (* a x)

    1. Initial program 44.0

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

    2. Taylor expanded around 0 14.1

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

    3. Simplified14.1

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

  4. Final simplification9.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -1.97221861134718174 \cdot 10^{-5}:\\ \;\;\;\;\sqrt[3]{{\left(\log \left(e^{e^{a \cdot x} - 1}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {x}^{3}, a \cdot x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +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))