Average Error: 29.8 → 7.5
Time: 29.5s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -2.63503256833740627 \cdot 10^{-11}:\\ \;\;\;\;\frac{\left(\sqrt[3]{e^{2 \cdot \left(a \cdot x\right)} - 1 \cdot 1} \cdot \sqrt[3]{e^{2 \cdot \left(a \cdot x\right)} - 1 \cdot 1}\right) \cdot \sqrt[3]{e^{2 \cdot \left(a \cdot x\right)} - 1 \cdot 1}}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{a \cdot x} + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({x}^{2}, \mathsf{fma}\left(x, \frac{1}{6} \cdot {a}^{3}, \frac{1}{2} \cdot {a}^{2}\right), a \cdot x\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -2.63503256833740627 \cdot 10^{-11}:\\
\;\;\;\;\frac{\left(\sqrt[3]{e^{2 \cdot \left(a \cdot x\right)} - 1 \cdot 1} \cdot \sqrt[3]{e^{2 \cdot \left(a \cdot x\right)} - 1 \cdot 1}\right) \cdot \sqrt[3]{e^{2 \cdot \left(a \cdot x\right)} - 1 \cdot 1}}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{a \cdot x} + 1\right)\right)}\\

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

\end{array}
double f(double a, double x) {
        double r73554 = a;
        double r73555 = x;
        double r73556 = r73554 * r73555;
        double r73557 = exp(r73556);
        double r73558 = 1.0;
        double r73559 = r73557 - r73558;
        return r73559;
}
↓
double f(double a, double x) {
        double r73560 = a;
        double r73561 = x;
        double r73562 = r73560 * r73561;
        double r73563 = -2.6350325683374063e-11;
        bool r73564 = r73562 <= r73563;
        double r73565 = 2.0;
        double r73566 = r73565 * r73562;
        double r73567 = exp(r73566);
        double r73568 = 1.0;
        double r73569 = r73568 * r73568;
        double r73570 = r73567 - r73569;
        double r73571 = cbrt(r73570);
        double r73572 = r73571 * r73571;
        double r73573 = r73572 * r73571;
        double r73574 = exp(r73562);
        double r73575 = r73574 + r73568;
        double r73576 = log1p(r73575);
        double r73577 = expm1(r73576);
        double r73578 = r73573 / r73577;
        double r73579 = pow(r73561, r73565);
        double r73580 = 0.16666666666666666;
        double r73581 = 3.0;
        double r73582 = pow(r73560, r73581);
        double r73583 = r73580 * r73582;
        double r73584 = 0.5;
        double r73585 = pow(r73560, r73565);
        double r73586 = r73584 * r73585;
        double r73587 = fma(r73561, r73583, r73586);
        double r73588 = fma(r73579, r73587, r73562);
        double r73589 = r73564 ? r73578 : r73588;
        return r73589;
}

Error

Bits error versus a

Bits error versus x

Target

Original29.8
Target0.2
Herbie7.5
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.10000000000000001:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -2.6350325683374063e-11

    1. Initial program 0.5

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

    3. Applied flip--0.5

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

    4. Simplified0.4

      \[\leadsto \frac{\color{blue}{e^{2 \cdot \left(a \cdot x\right)} - 1 \cdot 1}}{e^{a \cdot x} + 1}\]
    5. Using strategy rm

    6. Applied expm1-log1p-u0.4

      \[\leadsto \frac{e^{2 \cdot \left(a \cdot x\right)} - 1 \cdot 1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{a \cdot x} + 1\right)\right)}}\]
    7. Using strategy rm

    8. Applied add-cube-cbrt0.4

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{e^{2 \cdot \left(a \cdot x\right)} - 1 \cdot 1} \cdot \sqrt[3]{e^{2 \cdot \left(a \cdot x\right)} - 1 \cdot 1}\right) \cdot \sqrt[3]{e^{2 \cdot \left(a \cdot x\right)} - 1 \cdot 1}}}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{a \cdot x} + 1\right)\right)}\]

    if -2.6350325683374063e-11 < (* a x)

    1. Initial program 45.0

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

    2. Taylor expanded around 0 14.5

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

    3. Simplified11.2

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

  4. Final simplification7.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -2.63503256833740627 \cdot 10^{-11}:\\ \;\;\;\;\frac{\left(\sqrt[3]{e^{2 \cdot \left(a \cdot x\right)} - 1 \cdot 1} \cdot \sqrt[3]{e^{2 \cdot \left(a \cdot x\right)} - 1 \cdot 1}\right) \cdot \sqrt[3]{e^{2 \cdot \left(a \cdot x\right)} - 1 \cdot 1}}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{a \cdot x} + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({x}^{2}, \mathsf{fma}\left(x, \frac{1}{6} \cdot {a}^{3}, \frac{1}{2} \cdot {a}^{2}\right), a \cdot x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 +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))