expax (section 3.5)

Average Error: 29.1 → 0.4

Time: 4.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -3.02916961651562611:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{e^{a \cdot x}}, \sqrt{e^{a \cdot x}}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{6}, {\left(x \cdot a\right)}^{3}, \mathsf{fma}\left(x, a, {\left(x \cdot a\right)}^{2} \cdot \frac{1}{2}\right)\right)\\ \end{array}\]

C

double f(double a, double x) {
        double r80628 = a;
        double r80629 = x;
        double r80630 = r80628 * r80629;
        double r80631 = exp(r80630);
        double r80632 = 1.0;
        double r80633 = r80631 - r80632;
        return r80633;
}

↓

double f(double a, double x) {
        double r80634 = a;
        double r80635 = x;
        double r80636 = r80634 * r80635;
        double r80637 = -3.029169616515626;
        bool r80638 = r80636 <= r80637;
        double r80639 = exp(r80636);
        double r80640 = sqrt(r80639);
        double r80641 = 1.0;
        double r80642 = -r80641;
        double r80643 = fma(r80640, r80640, r80642);
        double r80644 = 0.16666666666666666;
        double r80645 = r80635 * r80634;
        double r80646 = 3.0;
        double r80647 = pow(r80645, r80646);
        double r80648 = 2.0;
        double r80649 = pow(r80645, r80648);
        double r80650 = 0.5;
        double r80651 = r80649 * r80650;
        double r80652 = fma(r80635, r80634, r80651);
        double r80653 = fma(r80644, r80647, r80652);
        double r80654 = r80638 ? r80643 : r80653;
        return r80654;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.1
Target	0.2
Herbie	0.4

\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.10000000000000001:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (* a x) < -3.029169616515626

   1. Initial program 0

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

   3. Applied add-sqr-sqrt 0

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

   4. Applied fma-neg 0

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

   if -3.029169616515626 < (* a x)

   1. Initial program 43.7

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

   2. Taylor expanded around 0 14.6

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

   3. Simplified 14.6

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

   5. Applied pow-prod-down 8.5

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

   6. Simplified 8.5

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

   8. Applied add-sqr-sqrt 36.2

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

   9. Applied unpow-prod-down 36.2

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

   10. Applied add-sqr-sqrt 49.9

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

   11. Applied unpow-prod-down 49.9

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

   12. Applied unswap-sqr 47.7

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

   13. Simplified 47.7

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

   14. Simplified 0.6

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

   15. Taylor expanded around inf 14.6

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

   16. Simplified 0.6

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -3.02916961651562611:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{e^{a \cdot x}}, \sqrt{e^{a \cdot x}}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{6}, {\left(x \cdot a\right)}^{3}, \mathsf{fma}\left(x, a, {\left(x \cdot a\right)}^{2} \cdot \frac{1}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 +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))