expax (section 3.5)

Average Error: 29.3 → 0.3

Time: 20.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.07174886411196652880040147692852769978344:\\ \;\;\;\;\frac{\log \left(e^{e^{\left(3 \cdot a\right) \cdot x} - 1 \cdot \left(1 \cdot 1\right)}\right)}{\mathsf{fma}\left(e^{a \cdot x}, e^{a \cdot x}, \left(1 + e^{a \cdot x}\right) \cdot 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{6}, \left(a \cdot x\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right), \mathsf{fma}\left(\frac{1}{2}, \left(a \cdot x\right) \cdot \left(a \cdot x\right), a \cdot x\right)\right)\\ \end{array}\]

C

double f(double a, double x) {
        double r5499503 = a;
        double r5499504 = x;
        double r5499505 = r5499503 * r5499504;
        double r5499506 = exp(r5499505);
        double r5499507 = 1.0;
        double r5499508 = r5499506 - r5499507;
        return r5499508;
}

↓

double f(double a, double x) {
        double r5499509 = a;
        double r5499510 = x;
        double r5499511 = r5499509 * r5499510;
        double r5499512 = -0.07174886411196653;
        bool r5499513 = r5499511 <= r5499512;
        double r5499514 = 3.0;
        double r5499515 = r5499514 * r5499509;
        double r5499516 = r5499515 * r5499510;
        double r5499517 = exp(r5499516);
        double r5499518 = 1.0;
        double r5499519 = r5499518 * r5499518;
        double r5499520 = r5499518 * r5499519;
        double r5499521 = r5499517 - r5499520;
        double r5499522 = exp(r5499521);
        double r5499523 = log(r5499522);
        double r5499524 = exp(r5499511);
        double r5499525 = r5499518 + r5499524;
        double r5499526 = r5499525 * r5499518;
        double r5499527 = fma(r5499524, r5499524, r5499526);
        double r5499528 = r5499523 / r5499527;
        double r5499529 = 0.16666666666666666;
        double r5499530 = r5499511 * r5499511;
        double r5499531 = r5499511 * r5499530;
        double r5499532 = 0.5;
        double r5499533 = fma(r5499532, r5499530, r5499511);
        double r5499534 = fma(r5499529, r5499531, r5499533);
        double r5499535 = r5499513 ? r5499528 : r5499534;
        return r5499535;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.3
Target	0.2
Herbie	0.3

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

Derivation

1. Split input into 2 regimes

2. if (* a x) < -0.07174886411196653

   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. Simplified 0.0

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

   5. Simplified 0.0

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

   7. Applied add-log-exp 0.0

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

   8. Applied add-log-exp 0.0

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

   9. Applied diff-log 0.0

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

   10. Simplified 0.0

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

   if -0.07174886411196653 < (* a x)

   1. Initial program 44.2

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

   2. Taylor expanded around 0 13.8

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

   3. Simplified 0.5

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.07174886411196652880040147692852769978344:\\ \;\;\;\;\frac{\log \left(e^{e^{\left(3 \cdot a\right) \cdot x} - 1 \cdot \left(1 \cdot 1\right)}\right)}{\mathsf{fma}\left(e^{a \cdot x}, e^{a \cdot x}, \left(1 + e^{a \cdot x}\right) \cdot 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{6}, \left(a \cdot x\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right), \mathsf{fma}\left(\frac{1}{2}, \left(a \cdot x\right) \cdot \left(a \cdot x\right), a \cdot x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +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))