expax (section 3.5)

Average Error: 29.5 → 0.9

Time: 21.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -3.257644604903133386550957879823631202498 \cdot 10^{-7}:\\ \;\;\;\;\frac{{\left(e^{a \cdot x}\right)}^{3} - {1}^{3}}{\mathsf{fma}\left(1, e^{a \cdot x} + 1, e^{2 \cdot \left(a \cdot x\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{9}{2}, a \cdot 0, 3 \cdot \left(a \cdot x\right)\right)}{\mathsf{fma}\left(1, e^{a \cdot x} + 1, e^{2 \cdot \left(a \cdot x\right)}\right)}\\ \end{array}\]

C

double f(double a, double x) {
        double r90455 = a;
        double r90456 = x;
        double r90457 = r90455 * r90456;
        double r90458 = exp(r90457);
        double r90459 = 1.0;
        double r90460 = r90458 - r90459;
        return r90460;
}

↓

double f(double a, double x) {
        double r90461 = a;
        double r90462 = x;
        double r90463 = r90461 * r90462;
        double r90464 = -3.2576446049031334e-07;
        bool r90465 = r90463 <= r90464;
        double r90466 = exp(r90463);
        double r90467 = 3.0;
        double r90468 = pow(r90466, r90467);
        double r90469 = 1.0;
        double r90470 = pow(r90469, r90467);
        double r90471 = r90468 - r90470;
        double r90472 = r90466 + r90469;
        double r90473 = 2.0;
        double r90474 = r90473 * r90463;
        double r90475 = exp(r90474);
        double r90476 = fma(r90469, r90472, r90475);
        double r90477 = r90471 / r90476;
        double r90478 = 4.5;
        double r90479 = 0.0;
        double r90480 = r90461 * r90479;
        double r90481 = r90467 * r90463;
        double r90482 = fma(r90478, r90480, r90481);
        double r90483 = r90482 / r90476;
        double r90484 = r90465 ? r90477 : r90483;
        return r90484;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.5
Target	0.2
Herbie	0.9

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

Derivation

1. Split input into 2 regimes

2. if (* a x) < -3.2576446049031334e-07

   1. Initial program 0.2

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

   3. Applied flip3-- 0.2

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

   4. Simplified 0.2

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

   if -3.2576446049031334e-07 < (* a x)

   1. Initial program 44.5

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

   3. Applied flip3-- 44.6

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

   4. Simplified 44.6

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

   5. Taylor expanded around 0 14.1

      \[\leadsto \frac{\color{blue}{\frac{9}{2} \cdot \left({a}^{2} \cdot {x}^{2}\right) + \left(\frac{9}{2} \cdot \left({a}^{3} \cdot {x}^{3}\right) + 3 \cdot \left(a \cdot x\right)\right)}}{\mathsf{fma}\left(1, e^{a \cdot x} + 1, e^{2 \cdot \left(a \cdot x\right)}\right)}\]

   6. Simplified 11.1

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{9}{2}, {a}^{2} \cdot \left({x}^{2} + {x}^{3} \cdot a\right), 3 \cdot \left(a \cdot x\right)\right)}}{\mathsf{fma}\left(1, e^{a \cdot x} + 1, e^{2 \cdot \left(a \cdot x\right)}\right)}\]
   7. Using strategy rm

   8. Applied sqr-pow 11.1

      \[\leadsto \frac{\mathsf{fma}\left(\frac{9}{2}, \color{blue}{\left({a}^{\left(\frac{2}{2}\right)} \cdot {a}^{\left(\frac{2}{2}\right)}\right)} \cdot \left({x}^{2} + {x}^{3} \cdot a\right), 3 \cdot \left(a \cdot x\right)\right)}{\mathsf{fma}\left(1, e^{a \cdot x} + 1, e^{2 \cdot \left(a \cdot x\right)}\right)}\]

   9. Applied associate-*l* 7.4

      \[\leadsto \frac{\mathsf{fma}\left(\frac{9}{2}, \color{blue}{{a}^{\left(\frac{2}{2}\right)} \cdot \left({a}^{\left(\frac{2}{2}\right)} \cdot \left({x}^{2} + {x}^{3} \cdot a\right)\right)}, 3 \cdot \left(a \cdot x\right)\right)}{\mathsf{fma}\left(1, e^{a \cdot x} + 1, e^{2 \cdot \left(a \cdot x\right)}\right)}\]

   10. Simplified 7.4

       \[\leadsto \frac{\mathsf{fma}\left(\frac{9}{2}, {a}^{\left(\frac{2}{2}\right)} \cdot \color{blue}{\left(\mathsf{fma}\left(x, x, {x}^{3} \cdot a\right) \cdot a\right)}, 3 \cdot \left(a \cdot x\right)\right)}{\mathsf{fma}\left(1, e^{a \cdot x} + 1, e^{2 \cdot \left(a \cdot x\right)}\right)}\]

   11. Taylor expanded around 0 1.2

       \[\leadsto \frac{\mathsf{fma}\left(\frac{9}{2}, {a}^{\left(\frac{2}{2}\right)} \cdot \color{blue}{0}, 3 \cdot \left(a \cdot x\right)\right)}{\mathsf{fma}\left(1, e^{a \cdot x} + 1, e^{2 \cdot \left(a \cdot x\right)}\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -3.257644604903133386550957879823631202498 \cdot 10^{-7}:\\ \;\;\;\;\frac{{\left(e^{a \cdot x}\right)}^{3} - {1}^{3}}{\mathsf{fma}\left(1, e^{a \cdot x} + 1, e^{2 \cdot \left(a \cdot x\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{9}{2}, a \cdot 0, 3 \cdot \left(a \cdot x\right)\right)}{\mathsf{fma}\left(1, e^{a \cdot x} + 1, e^{2 \cdot \left(a \cdot x\right)}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +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))