Average Error: 29.3 → 9.1
Time: 3.3s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -2.9814260104564775 \cdot 10^{-15}:\\ \;\;\;\;\frac{e^{\left(a \cdot x\right) \cdot 3} - {1}^{3}}{\frac{{\left(e^{a \cdot x}\right)}^{3} + e^{a \cdot x} \cdot \left(-1 \cdot 1\right)}{e^{a \cdot x} - 1} + 1 \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -2.9814260104564775 \cdot 10^{-15}:\\
\;\;\;\;\frac{e^{\left(a \cdot x\right) \cdot 3} - {1}^{3}}{\frac{{\left(e^{a \cdot x}\right)}^{3} + e^{a \cdot x} \cdot \left(-1 \cdot 1\right)}{e^{a \cdot x} - 1} + 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 r111485 = a;
        double r111486 = x;
        double r111487 = r111485 * r111486;
        double r111488 = exp(r111487);
        double r111489 = 1.0;
        double r111490 = r111488 - r111489;
        return r111490;
}
↓
double f(double a, double x) {
        double r111491 = a;
        double r111492 = x;
        double r111493 = r111491 * r111492;
        double r111494 = -2.9814260104564775e-15;
        bool r111495 = r111493 <= r111494;
        double r111496 = 3.0;
        double r111497 = r111493 * r111496;
        double r111498 = exp(r111497);
        double r111499 = 1.0;
        double r111500 = pow(r111499, r111496);
        double r111501 = r111498 - r111500;
        double r111502 = exp(r111493);
        double r111503 = pow(r111502, r111496);
        double r111504 = r111499 * r111499;
        double r111505 = -r111504;
        double r111506 = r111502 * r111505;
        double r111507 = r111503 + r111506;
        double r111508 = r111502 - r111499;
        double r111509 = r111507 / r111508;
        double r111510 = r111509 + r111504;
        double r111511 = r111501 / r111510;
        double r111512 = 0.5;
        double r111513 = 2.0;
        double r111514 = pow(r111491, r111513);
        double r111515 = r111512 * r111514;
        double r111516 = r111515 * r111492;
        double r111517 = r111491 + r111516;
        double r111518 = r111492 * r111517;
        double r111519 = 0.16666666666666666;
        double r111520 = pow(r111491, r111496);
        double r111521 = pow(r111492, r111496);
        double r111522 = r111520 * r111521;
        double r111523 = r111519 * r111522;
        double r111524 = r111518 + r111523;
        double r111525 = r111495 ? r111511 : r111524;
        return r111525;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.3
Target0.2
Herbie9.1
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.10000000000000001:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -2.9814260104564775e-15

    1. Initial program 0.8

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

    3. Applied flip3--0.8

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

    4. Simplified0.8

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

    6. Applied pow-exp0.7

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

    8. Applied flip-+0.7

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

    9. Applied associate-*r/0.7

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

    10. Simplified0.7

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

    if -2.9814260104564775e-15 < (* a x)

    1. Initial program 44.6

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

    2. Taylor expanded around 0 13.5

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

    3. Simplified13.5

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

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -2.9814260104564775 \cdot 10^{-15}:\\ \;\;\;\;\frac{e^{\left(a \cdot x\right) \cdot 3} - {1}^{3}}{\frac{{\left(e^{a \cdot x}\right)}^{3} + e^{a \cdot x} \cdot \left(-1 \cdot 1\right)}{e^{a \cdot x} - 1} + 1 \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 
(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))