Average Error: 29.7 → 9.7
Time: 3.6s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -6.883670130696454110254072018681462839622 \cdot 10^{-19} \lor \neg \left(a \cdot x \le 5.145272381710981521082629097103889956475 \cdot 10^{-33}\right):\\ \;\;\;\;\frac{\left({\left(e^{a \cdot x}\right)}^{\frac{3}{2}} + \sqrt{{1}^{3}}\right) \cdot \left(e^{\frac{3}{2} \cdot \left(a \cdot x\right)} - \sqrt{1}\right)}{e^{a \cdot x} \cdot \left(\sqrt{e^{a \cdot x}} \cdot \sqrt{e^{a \cdot x}} + 1\right) + 1 \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -6.883670130696454110254072018681462839622 \cdot 10^{-19} \lor \neg \left(a \cdot x \le 5.145272381710981521082629097103889956475 \cdot 10^{-33}\right):\\
\;\;\;\;\frac{\left({\left(e^{a \cdot x}\right)}^{\frac{3}{2}} + \sqrt{{1}^{3}}\right) \cdot \left(e^{\frac{3}{2} \cdot \left(a \cdot x\right)} - \sqrt{1}\right)}{e^{a \cdot x} \cdot \left(\sqrt{e^{a \cdot x}} \cdot \sqrt{e^{a \cdot x}} + 1\right) + 1 \cdot 1}\\

\mathbf{else}:\\
\;\;\;\;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)\\

\end{array}
double f(double a, double x) {
        double r85558 = a;
        double r85559 = x;
        double r85560 = r85558 * r85559;
        double r85561 = exp(r85560);
        double r85562 = 1.0;
        double r85563 = r85561 - r85562;
        return r85563;
}
↓
double f(double a, double x) {
        double r85564 = a;
        double r85565 = x;
        double r85566 = r85564 * r85565;
        double r85567 = -6.883670130696454e-19;
        bool r85568 = r85566 <= r85567;
        double r85569 = 5.1452723817109815e-33;
        bool r85570 = r85566 <= r85569;
        double r85571 = !r85570;
        bool r85572 = r85568 || r85571;
        double r85573 = exp(r85566);
        double r85574 = 1.5;
        double r85575 = pow(r85573, r85574);
        double r85576 = 1.0;
        double r85577 = 3.0;
        double r85578 = pow(r85576, r85577);
        double r85579 = sqrt(r85578);
        double r85580 = r85575 + r85579;
        double r85581 = r85574 * r85566;
        double r85582 = exp(r85581);
        double r85583 = sqrt(r85576);
        double r85584 = r85582 - r85583;
        double r85585 = r85580 * r85584;
        double r85586 = sqrt(r85573);
        double r85587 = r85586 * r85586;
        double r85588 = r85587 + r85576;
        double r85589 = r85573 * r85588;
        double r85590 = r85576 * r85576;
        double r85591 = r85589 + r85590;
        double r85592 = r85585 / r85591;
        double r85593 = 0.5;
        double r85594 = 2.0;
        double r85595 = pow(r85564, r85594);
        double r85596 = r85593 * r85595;
        double r85597 = r85596 * r85565;
        double r85598 = r85564 + r85597;
        double r85599 = r85565 * r85598;
        double r85600 = 0.16666666666666666;
        double r85601 = pow(r85564, r85577);
        double r85602 = pow(r85565, r85577);
        double r85603 = r85601 * r85602;
        double r85604 = r85600 * r85603;
        double r85605 = r85599 + r85604;
        double r85606 = r85572 ? r85592 : r85605;
        return r85606;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Split input into 2 regimes

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

    1. Initial program 5.2

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

    3. Applied flip3--5.3

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

    4. Simplified5.3

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

    6. Applied add-sqr-sqrt5.3

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

    7. Applied sqr-pow5.3

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

    8. Applied difference-of-squares5.3

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

    9. Simplified5.3

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

    10. Simplified5.3

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

    11. Taylor expanded around inf 5.2

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

    13. Applied add-sqr-sqrt5.2

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

    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

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

  4. Final simplification9.7

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

Reproduce

herbie shell --seed 2019353 
(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))