Average Error: 28.9 → 0.3
Time: 17.3s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.01601638977115915629667597386287525296211:\\ \;\;\;\;\frac{\frac{e^{\left(a \cdot x\right) \cdot 4} - {1}^{4}}{\mathsf{fma}\left(1, 1, e^{2 \cdot \left(a \cdot x\right)}\right)}}{e^{a \cdot x} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 1 \cdot a, {\left(a \cdot x\right)}^{2} \cdot \left(0.1666666666666666296592325124947819858789 \cdot \left(x \cdot a\right) + 0.5\right)\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -0.01601638977115915629667597386287525296211:\\
\;\;\;\;\frac{\frac{e^{\left(a \cdot x\right) \cdot 4} - {1}^{4}}{\mathsf{fma}\left(1, 1, e^{2 \cdot \left(a \cdot x\right)}\right)}}{e^{a \cdot x} + 1}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(x, 1 \cdot a, {\left(a \cdot x\right)}^{2} \cdot \left(0.1666666666666666296592325124947819858789 \cdot \left(x \cdot a\right) + 0.5\right)\right)\\

\end{array}
double f(double a, double x) {
        double r92472 = a;
        double r92473 = x;
        double r92474 = r92472 * r92473;
        double r92475 = exp(r92474);
        double r92476 = 1.0;
        double r92477 = r92475 - r92476;
        return r92477;
}
↓
double f(double a, double x) {
        double r92478 = a;
        double r92479 = x;
        double r92480 = r92478 * r92479;
        double r92481 = -0.016016389771159156;
        bool r92482 = r92480 <= r92481;
        double r92483 = 4.0;
        double r92484 = r92480 * r92483;
        double r92485 = exp(r92484);
        double r92486 = 1.0;
        double r92487 = pow(r92486, r92483);
        double r92488 = r92485 - r92487;
        double r92489 = 2.0;
        double r92490 = r92489 * r92480;
        double r92491 = exp(r92490);
        double r92492 = fma(r92486, r92486, r92491);
        double r92493 = r92488 / r92492;
        double r92494 = exp(r92480);
        double r92495 = r92494 + r92486;
        double r92496 = r92493 / r92495;
        double r92497 = r92486 * r92478;
        double r92498 = pow(r92480, r92489);
        double r92499 = 0.16666666666666663;
        double r92500 = r92479 * r92478;
        double r92501 = r92499 * r92500;
        double r92502 = 0.5;
        double r92503 = r92501 + r92502;
        double r92504 = r92498 * r92503;
        double r92505 = fma(r92479, r92497, r92504);
        double r92506 = r92482 ? r92496 : r92505;
        return r92506;
}

Error

Bits error versus a

Bits error versus x

Target

Original28.9
Target0.2
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.1000000000000000055511151231257827021182:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -0.016016389771159156

    1. Initial program 0.0

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

    3. Applied flip--0.0

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

    4. Simplified0.0

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

    6. Applied flip--0.0

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

    7. Simplified0.0

      \[\leadsto \frac{\frac{\color{blue}{e^{\left(a \cdot x\right) \cdot 4} - {1}^{4}}}{e^{2 \cdot \left(a \cdot x\right)} + 1 \cdot 1}}{e^{a \cdot x} + 1}\]

    8. Simplified0.0

      \[\leadsto \frac{\frac{e^{\left(a \cdot x\right) \cdot 4} - {1}^{4}}{\color{blue}{\mathsf{fma}\left(1, 1, e^{2 \cdot \left(a \cdot x\right)}\right)}}}{e^{a \cdot x} + 1}\]

    if -0.016016389771159156 < (* a x)

    1. Initial program 43.8

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

    3. Applied flip--43.8

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

    4. Simplified43.7

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

    5. Taylor expanded around 0 14.6

      \[\leadsto \color{blue}{0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right) + \left(0.1666666666666666296592325124947819858789 \cdot \left({a}^{3} \cdot {x}^{3}\right) + 1 \cdot \left(a \cdot x\right)\right)}\]

    6. Simplified0.5

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(a \cdot x\right)}^{3}, 0.1666666666666666296592325124947819858789, \left(a \cdot x\right) \cdot \left(1 + \left(a \cdot x\right) \cdot 0.5\right)\right)}\]
    7. Using strategy rm

    8. Applied distribute-lft-in0.5

      \[\leadsto \mathsf{fma}\left({\left(a \cdot x\right)}^{3}, 0.1666666666666666296592325124947819858789, \color{blue}{\left(a \cdot x\right) \cdot 1 + \left(a \cdot x\right) \cdot \left(\left(a \cdot x\right) \cdot 0.5\right)}\right)\]

    9. Simplified0.5

      \[\leadsto \mathsf{fma}\left({\left(a \cdot x\right)}^{3}, 0.1666666666666666296592325124947819858789, \color{blue}{1 \cdot \left(a \cdot x\right)} + \left(a \cdot x\right) \cdot \left(\left(a \cdot x\right) \cdot 0.5\right)\right)\]

    10. Simplified0.5

      \[\leadsto \mathsf{fma}\left({\left(a \cdot x\right)}^{3}, 0.1666666666666666296592325124947819858789, 1 \cdot \left(a \cdot x\right) + \color{blue}{0.5 \cdot {\left(x \cdot a\right)}^{2}}\right)\]

    11. Taylor expanded around inf 14.6

      \[\leadsto \color{blue}{0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right) + \left(1 \cdot \left(a \cdot x\right) + 0.1666666666666666296592325124947819858789 \cdot \left({a}^{3} \cdot {x}^{3}\right)\right)}\]

    12. Simplified0.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1 \cdot a, {\left(a \cdot x\right)}^{2} \cdot \left(0.1666666666666666296592325124947819858789 \cdot \left(x \cdot a\right) + 0.5\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.01601638977115915629667597386287525296211:\\ \;\;\;\;\frac{\frac{e^{\left(a \cdot x\right) \cdot 4} - {1}^{4}}{\mathsf{fma}\left(1, 1, e^{2 \cdot \left(a \cdot x\right)}\right)}}{e^{a \cdot x} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 1 \cdot a, {\left(a \cdot x\right)}^{2} \cdot \left(0.1666666666666666296592325124947819858789 \cdot \left(x \cdot a\right) + 0.5\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 +o rules:numerics
(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))