Average Error: 29.7 → 9.8
Time: 4.0s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -6.883670130696454110254072018681462839622 \cdot 10^{-19}:\\ \;\;\;\;\sqrt[3]{{\left(\mathsf{fma}\left(\sqrt{e^{a \cdot x}}, \sqrt{\left(\sqrt[3]{e^{a \cdot x}} \cdot \sqrt[3]{e^{a \cdot x}}\right) \cdot \sqrt[3]{e^{a \cdot x}}}, -1\right)\right)}^{3}}\\ \mathbf{elif}\;a \cdot x \le 5.145272381710981521082629097103889956475 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {x}^{3}, a \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\mathsf{fma}\left(\sqrt{e^{a \cdot x}}, \sqrt{e^{a \cdot x}}, -1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\sqrt{e^{a \cdot x}}, \sqrt{e^{a \cdot x}}, -1\right)}\right) \cdot \left(\sqrt[3]{1} \cdot {\left(e^{a \cdot x} - 1\right)}^{\frac{1}{3}}\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -6.883670130696454110254072018681462839622 \cdot 10^{-19}:\\
\;\;\;\;\sqrt[3]{{\left(\mathsf{fma}\left(\sqrt{e^{a \cdot x}}, \sqrt{\left(\sqrt[3]{e^{a \cdot x}} \cdot \sqrt[3]{e^{a \cdot x}}\right) \cdot \sqrt[3]{e^{a \cdot x}}}, -1\right)\right)}^{3}}\\

\mathbf{elif}\;a \cdot x \le 5.145272381710981521082629097103889956475 \cdot 10^{-33}:\\
\;\;\;\;\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)\\

\mathbf{else}:\\
\;\;\;\;\left(\sqrt[3]{\mathsf{fma}\left(\sqrt{e^{a \cdot x}}, \sqrt{e^{a \cdot x}}, -1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\sqrt{e^{a \cdot x}}, \sqrt{e^{a \cdot x}}, -1\right)}\right) \cdot \left(\sqrt[3]{1} \cdot {\left(e^{a \cdot x} - 1\right)}^{\frac{1}{3}}\right)\\

\end{array}
double f(double a, double x) {
        double r128388 = a;
        double r128389 = x;
        double r128390 = r128388 * r128389;
        double r128391 = exp(r128390);
        double r128392 = 1.0;
        double r128393 = r128391 - r128392;
        return r128393;
}
↓
double f(double a, double x) {
        double r128394 = a;
        double r128395 = x;
        double r128396 = r128394 * r128395;
        double r128397 = -6.883670130696454e-19;
        bool r128398 = r128396 <= r128397;
        double r128399 = exp(r128396);
        double r128400 = sqrt(r128399);
        double r128401 = cbrt(r128399);
        double r128402 = r128401 * r128401;
        double r128403 = r128402 * r128401;
        double r128404 = sqrt(r128403);
        double r128405 = 1.0;
        double r128406 = -r128405;
        double r128407 = fma(r128400, r128404, r128406);
        double r128408 = 3.0;
        double r128409 = pow(r128407, r128408);
        double r128410 = cbrt(r128409);
        double r128411 = 5.1452723817109815e-33;
        bool r128412 = r128396 <= r128411;
        double r128413 = 0.5;
        double r128414 = 2.0;
        double r128415 = pow(r128394, r128414);
        double r128416 = pow(r128395, r128414);
        double r128417 = r128415 * r128416;
        double r128418 = 0.16666666666666666;
        double r128419 = pow(r128394, r128408);
        double r128420 = pow(r128395, r128408);
        double r128421 = r128419 * r128420;
        double r128422 = fma(r128418, r128421, r128396);
        double r128423 = fma(r128413, r128417, r128422);
        double r128424 = fma(r128400, r128400, r128406);
        double r128425 = cbrt(r128424);
        double r128426 = r128425 * r128425;
        double r128427 = 1.0;
        double r128428 = cbrt(r128427);
        double r128429 = r128399 - r128405;
        double r128430 = 0.3333333333333333;
        double r128431 = pow(r128429, r128430);
        double r128432 = r128428 * r128431;
        double r128433 = r128426 * r128432;
        double r128434 = r128412 ? r128423 : r128433;
        double r128435 = r128398 ? r128410 : r128434;
        return r128435;
}

Error

Bits error versus a

Bits error versus x

Target

Original29.7
Target0.2
Herbie9.8
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.1000000000000000055511151231257827021182:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* a x) < -6.883670130696454e-19

    1. Initial program 1.4

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

    3. Applied add-sqr-sqrt1.5

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

    4. Applied fma-neg1.5

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

    6. Applied add-cube-cbrt1.5

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

    8. Applied add-cbrt-cube1.5

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

    9. Simplified1.5

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

    if -6.883670130696454e-19 < (* a x) < 5.1452723817109815e-33

    1. Initial program 45.3

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

    2. Taylor expanded around 0 12.6

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

    3. Simplified12.6

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

    if 5.1452723817109815e-33 < (* a x)

    1. Initial program 38.5

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

    3. Applied add-sqr-sqrt39.1

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

    4. Applied fma-neg39.1

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

    6. Applied add-cube-cbrt39.1

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

    8. Applied *-un-lft-identity39.1

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

    9. Applied cbrt-prod39.1

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

    10. Simplified38.8

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

  4. Final simplification9.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -6.883670130696454110254072018681462839622 \cdot 10^{-19}:\\ \;\;\;\;\sqrt[3]{{\left(\mathsf{fma}\left(\sqrt{e^{a \cdot x}}, \sqrt{\left(\sqrt[3]{e^{a \cdot x}} \cdot \sqrt[3]{e^{a \cdot x}}\right) \cdot \sqrt[3]{e^{a \cdot x}}}, -1\right)\right)}^{3}}\\ \mathbf{elif}\;a \cdot x \le 5.145272381710981521082629097103889956475 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {x}^{3}, a \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\mathsf{fma}\left(\sqrt{e^{a \cdot x}}, \sqrt{e^{a \cdot x}}, -1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\sqrt{e^{a \cdot x}}, \sqrt{e^{a \cdot x}}, -1\right)}\right) \cdot \left(\sqrt[3]{1} \cdot {\left(e^{a \cdot x} - 1\right)}^{\frac{1}{3}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +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))