Average Error: 29.5 → 0.4
Time: 19.7s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.03513580389188739050432275234925327822566:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{e^{\left(a \cdot x\right) \cdot 3}}, \sqrt{e^{\left(a \cdot x\right) \cdot 3}}, -1 \cdot \left(1 \cdot 1\right)\right)}{\mathsf{fma}\left(e^{a \cdot x}, e^{a \cdot x} + 1, 1 \cdot 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \left(a \cdot x\right) \cdot \left(a \cdot x\right), x \cdot \left(\left(\frac{1}{6} \cdot a\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) + a\right)\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -0.03513580389188739050432275234925327822566:\\
\;\;\;\;\frac{\mathsf{fma}\left(\sqrt{e^{\left(a \cdot x\right) \cdot 3}}, \sqrt{e^{\left(a \cdot x\right) \cdot 3}}, -1 \cdot \left(1 \cdot 1\right)\right)}{\mathsf{fma}\left(e^{a \cdot x}, e^{a \cdot x} + 1, 1 \cdot 1\right)}\\

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

\end{array}
double f(double a, double x) {
        double r4237462 = a;
        double r4237463 = x;
        double r4237464 = r4237462 * r4237463;
        double r4237465 = exp(r4237464);
        double r4237466 = 1.0;
        double r4237467 = r4237465 - r4237466;
        return r4237467;
}
↓
double f(double a, double x) {
        double r4237468 = a;
        double r4237469 = x;
        double r4237470 = r4237468 * r4237469;
        double r4237471 = -0.03513580389188739;
        bool r4237472 = r4237470 <= r4237471;
        double r4237473 = 3.0;
        double r4237474 = r4237470 * r4237473;
        double r4237475 = exp(r4237474);
        double r4237476 = sqrt(r4237475);
        double r4237477 = 1.0;
        double r4237478 = r4237477 * r4237477;
        double r4237479 = r4237477 * r4237478;
        double r4237480 = -r4237479;
        double r4237481 = fma(r4237476, r4237476, r4237480);
        double r4237482 = exp(r4237470);
        double r4237483 = r4237482 + r4237477;
        double r4237484 = fma(r4237482, r4237483, r4237478);
        double r4237485 = r4237481 / r4237484;
        double r4237486 = 0.5;
        double r4237487 = r4237470 * r4237470;
        double r4237488 = 0.16666666666666666;
        double r4237489 = r4237488 * r4237468;
        double r4237490 = r4237489 * r4237487;
        double r4237491 = r4237490 + r4237468;
        double r4237492 = r4237469 * r4237491;
        double r4237493 = fma(r4237486, r4237487, r4237492);
        double r4237494 = r4237472 ? r4237485 : r4237493;
        return r4237494;
}

Error

Bits error versus a

Bits error versus x

Target

Original29.5
Target0.2
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.1000000000000000055511151231257827021182:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -0.03513580389188739

    1. Initial program 0.0

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

    3. Applied flip3--0.0

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

    4. Simplified0.0

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

    5. Simplified0.0

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

    7. Applied add-sqr-sqrt0.0

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

    8. Applied fma-neg0.0

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

    if -0.03513580389188739 < (* a x)

    1. Initial program 44.4

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

    2. Taylor expanded around 0 14.4

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

    3. Simplified0.6

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.03513580389188739050432275234925327822566:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{e^{\left(a \cdot x\right) \cdot 3}}, \sqrt{e^{\left(a \cdot x\right) \cdot 3}}, -1 \cdot \left(1 \cdot 1\right)\right)}{\mathsf{fma}\left(e^{a \cdot x}, e^{a \cdot x} + 1, 1 \cdot 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \left(a \cdot x\right) \cdot \left(a \cdot x\right), x \cdot \left(\left(\frac{1}{6} \cdot a\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) + a\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (a x)
  :name "expax (section 3.5)"
  :herbie-expected 14

  :herbie-target
  (if (< (fabs (* a x)) 0.1) (* (* a x) (+ 1.0 (+ (/ (* a x) 2.0) (/ (pow (* a x) 2.0) 6.0)))) (- (exp (* a x)) 1.0))

  (- (exp (* a x)) 1.0))