Average Error: 29.5 → 5.1
Time: 17.5s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -3.667914775029957871162604216182245853162 \cdot 10^{-8}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(\frac{1}{2} \cdot {a}^{2} + \left(\frac{1}{6} \cdot {a}^{3}\right) \cdot x\right)\right) + a \cdot x\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -3.667914775029957871162604216182245853162 \cdot 10^{-8}:\\
\;\;\;\;e^{a \cdot x} - 1\\

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

\end{array}
double f(double a, double x) {
        double r85404 = a;
        double r85405 = x;
        double r85406 = r85404 * r85405;
        double r85407 = exp(r85406);
        double r85408 = 1.0;
        double r85409 = r85407 - r85408;
        return r85409;
}
↓
double f(double a, double x) {
        double r85410 = a;
        double r85411 = x;
        double r85412 = r85410 * r85411;
        double r85413 = -3.667914775029958e-08;
        bool r85414 = r85412 <= r85413;
        double r85415 = exp(r85412);
        double r85416 = 1.0;
        double r85417 = r85415 - r85416;
        double r85418 = 0.5;
        double r85419 = 2.0;
        double r85420 = pow(r85410, r85419);
        double r85421 = r85418 * r85420;
        double r85422 = 0.16666666666666666;
        double r85423 = 3.0;
        double r85424 = pow(r85410, r85423);
        double r85425 = r85422 * r85424;
        double r85426 = r85425 * r85411;
        double r85427 = r85421 + r85426;
        double r85428 = r85411 * r85427;
        double r85429 = r85411 * r85428;
        double r85430 = r85429 + r85412;
        double r85431 = r85414 ? r85417 : r85430;
        return r85431;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.5
Target0.2
Herbie5.1
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.1000000000000000055511151231257827021182:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -3.667914775029958e-08

    1. Initial program 0.2

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

    if -3.667914775029958e-08 < (* a x)

    1. Initial program 45.0

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

    2. Taylor expanded around 0 14.4

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

    3. Simplified11.4

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

    5. Applied sqr-pow11.4

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

    6. Applied associate-*l*7.6

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

    7. Simplified7.6

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

  4. Final simplification5.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -3.667914775029957871162604216182245853162 \cdot 10^{-8}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(\frac{1}{2} \cdot {a}^{2} + \left(\frac{1}{6} \cdot {a}^{3}\right) \cdot x\right)\right) + a \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2019305 
(FPCore (a x)
  :name "expax (section 3.5)"
  :precision binary64
  :herbie-expected 14

  :herbie-target
  (if (< (fabs (* a x)) 0.10000000000000001) (* (* a x) (+ 1 (+ (/ (* a x) 2) (/ (pow (* a x) 2) 6)))) (- (exp (* a x)) 1))

  (- (exp (* a x)) 1))