expax (section 3.5)

Average Error: 29.7 → 3.6

Time: 11.4s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -6.39764424539762286 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{e^{a \cdot x}}, \sqrt{e^{a \cdot x}}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, a, \log \left({\left(e^{x \cdot x}\right)}^{\left(\mathsf{fma}\left(\frac{1}{2}, a \cdot a, \left(\frac{1}{6} \cdot {a}^{3}\right) \cdot x\right)\right)}\right)\right)\\ \end{array}\]

C

double f(double a, double x) {
        double r145882 = a;
        double r145883 = x;
        double r145884 = r145882 * r145883;
        double r145885 = exp(r145884);
        double r145886 = 1.0;
        double r145887 = r145885 - r145886;
        return r145887;
}

↓

double f(double a, double x) {
        double r145888 = a;
        double r145889 = x;
        double r145890 = r145888 * r145889;
        double r145891 = -6.397644245397623e-09;
        bool r145892 = r145890 <= r145891;
        double r145893 = exp(r145890);
        double r145894 = sqrt(r145893);
        double r145895 = 1.0;
        double r145896 = -r145895;
        double r145897 = fma(r145894, r145894, r145896);
        double r145898 = r145889 * r145889;
        double r145899 = exp(r145898);
        double r145900 = 0.5;
        double r145901 = r145888 * r145888;
        double r145902 = 0.16666666666666666;
        double r145903 = 3.0;
        double r145904 = pow(r145888, r145903);
        double r145905 = r145902 * r145904;
        double r145906 = r145905 * r145889;
        double r145907 = fma(r145900, r145901, r145906);
        double r145908 = pow(r145899, r145907);
        double r145909 = log(r145908);
        double r145910 = fma(r145889, r145888, r145909);
        double r145911 = r145892 ? r145897 : r145910;
        return r145911;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.7
Target	0.2
Herbie	3.6

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

Derivation

1. Split input into 2 regimes

2. if (* a x) < -6.397644245397623e-09

   1. Initial program 0.3

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

   3. Applied add-sqr-sqrt 0.3

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

   4. Applied fma-neg 0.3

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

   if -6.397644245397623e-09 < (* a x)

   1. Initial program 44.7

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

   2. Taylor expanded around 0 14.1

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

   3. Simplified 11.3

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

   5. Applied add-log-exp 11.4

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

   6. Simplified 5.2

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

4. Final simplification 3.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -6.39764424539762286 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{e^{a \cdot x}}, \sqrt{e^{a \cdot x}}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, a, \log \left({\left(e^{x \cdot x}\right)}^{\left(\mathsf{fma}\left(\frac{1}{2}, a \cdot a, \left(\frac{1}{6} \cdot {a}^{3}\right) \cdot x\right)\right)}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 +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))