Average Error: 29.4 → 0.4
Time: 3.5s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.00103720060762227677:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, {\left(x \cdot a\right)}^{2}, \mathsf{fma}\left(x, a, {\left(a \cdot x\right)}^{3} \cdot \frac{1}{6}\right)\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -0.00103720060762227677:\\
\;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\

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

\end{array}
double f(double a, double x) {
        double r98460 = a;
        double r98461 = x;
        double r98462 = r98460 * r98461;
        double r98463 = exp(r98462);
        double r98464 = 1.0;
        double r98465 = r98463 - r98464;
        return r98465;
}
↓
double f(double a, double x) {
        double r98466 = a;
        double r98467 = x;
        double r98468 = r98466 * r98467;
        double r98469 = -0.0010372006076222768;
        bool r98470 = r98468 <= r98469;
        double r98471 = exp(r98468);
        double r98472 = 1.0;
        double r98473 = r98471 - r98472;
        double r98474 = exp(r98473);
        double r98475 = log(r98474);
        double r98476 = 0.5;
        double r98477 = r98467 * r98466;
        double r98478 = 2.0;
        double r98479 = pow(r98477, r98478);
        double r98480 = 3.0;
        double r98481 = pow(r98468, r98480);
        double r98482 = 0.16666666666666666;
        double r98483 = r98481 * r98482;
        double r98484 = fma(r98467, r98466, r98483);
        double r98485 = fma(r98476, r98479, r98484);
        double r98486 = r98470 ? r98475 : r98485;
        return r98486;
}

Error

Bits error versus a

Bits error versus x

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) < -0.0010372006076222768

    1. Initial program 0.0

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

    3. Applied add-log-exp0.0

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

    4. Applied add-log-exp0.0

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

    5. Applied diff-log0.0

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

    6. Simplified0.0

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

    if -0.0010372006076222768 < (* 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

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

    5. Applied pow-prod-down8.3

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

    6. Taylor expanded around inf 14.7

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

    7. Simplified0.6

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

    8. Taylor expanded around inf 14.7

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

    9. Simplified0.5

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.00103720060762227677:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, {\left(x \cdot a\right)}^{2}, \mathsf{fma}\left(x, a, {\left(a \cdot x\right)}^{3} \cdot \frac{1}{6}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 +o rules:numerics
(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))