Average Error: 29.4 → 0.4
Time: 6.8s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -1.4477752003782106 \cdot 10^{-4}:\\ \;\;\;\;\frac{{\left(e^{a \cdot x}\right)}^{3} - {1}^{3}}{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot x + \frac{1}{2} \cdot {\left(a \cdot x\right)}^{2}\right) + \frac{1}{6} \cdot {\left(a \cdot x\right)}^{3}\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -1.4477752003782106 \cdot 10^{-4}:\\
\;\;\;\;\frac{{\left(e^{a \cdot x}\right)}^{3} - {1}^{3}}{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}\\

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

\end{array}
double f(double a, double x) {
        double r454 = a;
        double r455 = x;
        double r456 = r454 * r455;
        double r457 = exp(r456);
        double r458 = 1.0;
        double r459 = r457 - r458;
        return r459;
}
↓
double f(double a, double x) {
        double r460 = a;
        double r461 = x;
        double r462 = r460 * r461;
        double r463 = -0.00014477752003782106;
        bool r464 = r462 <= r463;
        double r465 = exp(r462);
        double r466 = 3.0;
        double r467 = pow(r465, r466);
        double r468 = 1.0;
        double r469 = pow(r468, r466);
        double r470 = r467 - r469;
        double r471 = r465 + r468;
        double r472 = r465 * r471;
        double r473 = r468 * r468;
        double r474 = r472 + r473;
        double r475 = r470 / r474;
        double r476 = 0.5;
        double r477 = 2.0;
        double r478 = pow(r462, r477);
        double r479 = r476 * r478;
        double r480 = r462 + r479;
        double r481 = 0.16666666666666666;
        double r482 = pow(r462, r466);
        double r483 = r481 * r482;
        double r484 = r480 + r483;
        double r485 = r464 ? r475 : r484;
        return r485;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.4
Target0.2
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.10000000000000001:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -0.00014477752003782106

    1. Initial program 0.1

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

    3. Applied flip3--0.1

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

    4. Simplified0.1

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

    if -0.00014477752003782106 < (* a x)

    1. Initial program 44.4

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

    2. Taylor expanded around 0 14.7

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

    3. Simplified14.7

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

    5. Applied pow-prod-down5.0

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

    7. Applied distribute-lft-in5.0

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

    8. Simplified5.0

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

    9. Simplified0.5

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -1.4477752003782106 \cdot 10^{-4}:\\ \;\;\;\;\frac{{\left(e^{a \cdot x}\right)}^{3} - {1}^{3}}{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot x + \frac{1}{2} \cdot {\left(a \cdot x\right)}^{2}\right) + \frac{1}{6} \cdot {\left(a \cdot x\right)}^{3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 
(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))