expax (section 3.5)

Average Error: 29.2 → 0.3

Time: 20.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -1.819921510314330350029099481190542064724 \cdot 10^{-4}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\left(\sqrt{1} + \sqrt{e^{a \cdot x}}\right) \cdot \frac{e^{a \cdot x} \cdot \sqrt{e^{a \cdot x}} - 1 \cdot \sqrt{1}}{\mathsf{fma}\left(\sqrt{e^{a \cdot x}}, \sqrt{1}, e^{a \cdot x} + 1\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \left(a \cdot x\right) \cdot \left(a \cdot x\right), \mathsf{fma}\left(\left(a \cdot x\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right), \frac{1}{6}, a \cdot x\right)\right)\\ \end{array}\]

C

double f(double a, double x) {
        double r4712157 = a;
        double r4712158 = x;
        double r4712159 = r4712157 * r4712158;
        double r4712160 = exp(r4712159);
        double r4712161 = 1.0;
        double r4712162 = r4712160 - r4712161;
        return r4712162;
}

↓

double f(double a, double x) {
        double r4712163 = a;
        double r4712164 = x;
        double r4712165 = r4712163 * r4712164;
        double r4712166 = -0.00018199215103143304;
        bool r4712167 = r4712165 <= r4712166;
        double r4712168 = 1.0;
        double r4712169 = sqrt(r4712168);
        double r4712170 = exp(r4712165);
        double r4712171 = sqrt(r4712170);
        double r4712172 = r4712169 + r4712171;
        double r4712173 = r4712170 * r4712171;
        double r4712174 = r4712168 * r4712169;
        double r4712175 = r4712173 - r4712174;
        double r4712176 = r4712170 + r4712168;
        double r4712177 = fma(r4712171, r4712169, r4712176);
        double r4712178 = r4712175 / r4712177;
        double r4712179 = r4712172 * r4712178;
        double r4712180 = expm1(r4712179);
        double r4712181 = log1p(r4712180);
        double r4712182 = 0.5;
        double r4712183 = r4712165 * r4712165;
        double r4712184 = r4712165 * r4712183;
        double r4712185 = 0.16666666666666666;
        double r4712186 = fma(r4712184, r4712185, r4712165);
        double r4712187 = fma(r4712182, r4712183, r4712186);
        double r4712188 = r4712167 ? r4712181 : r4712187;
        return r4712188;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.2
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) < -0.00018199215103143304

   1. Initial program 0.0

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

   3. Applied log1p-expm1-u 0.0

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(e^{a \cdot x} - 1\right)\right)}\]
   4. Using strategy rm

   5. Applied add-sqr-sqrt 0.0

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(e^{a \cdot x} - \color{blue}{\sqrt{1} \cdot \sqrt{1}}\right)\right)\]

   6. Applied add-sqr-sqrt 0.1

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\sqrt{e^{a \cdot x}} \cdot \sqrt{e^{a \cdot x}}} - \sqrt{1} \cdot \sqrt{1}\right)\right)\]

   7. Applied difference-of-squares 0.0

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left(\sqrt{e^{a \cdot x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{a \cdot x}} - \sqrt{1}\right)}\right)\right)\]
   8. Using strategy rm

   9. Applied flip3-- 0.1

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\left(\sqrt{e^{a \cdot x}} + \sqrt{1}\right) \cdot \color{blue}{\frac{{\left(\sqrt{e^{a \cdot x}}\right)}^{3} - {\left(\sqrt{1}\right)}^{3}}{\sqrt{e^{a \cdot x}} \cdot \sqrt{e^{a \cdot x}} + \left(\sqrt{1} \cdot \sqrt{1} + \sqrt{e^{a \cdot x}} \cdot \sqrt{1}\right)}}\right)\right)\]

   10. Simplified 0.0

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\left(\sqrt{e^{a \cdot x}} + \sqrt{1}\right) \cdot \frac{\color{blue}{e^{a \cdot x} \cdot \sqrt{e^{a \cdot x}} - \sqrt{1} \cdot 1}}{\sqrt{e^{a \cdot x}} \cdot \sqrt{e^{a \cdot x}} + \left(\sqrt{1} \cdot \sqrt{1} + \sqrt{e^{a \cdot x}} \cdot \sqrt{1}\right)}\right)\right)\]

   11. Simplified 0.0

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\left(\sqrt{e^{a \cdot x}} + \sqrt{1}\right) \cdot \frac{e^{a \cdot x} \cdot \sqrt{e^{a \cdot x}} - \sqrt{1} \cdot 1}{\color{blue}{\mathsf{fma}\left(\sqrt{e^{a \cdot x}}, \sqrt{1}, e^{a \cdot x} + 1\right)}}\right)\right)\]

   if -0.00018199215103143304 < (* a x)

   1. Initial program 44.6

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

   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. Simplified 0.5

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -1.819921510314330350029099481190542064724 \cdot 10^{-4}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\left(\sqrt{1} + \sqrt{e^{a \cdot x}}\right) \cdot \frac{e^{a \cdot x} \cdot \sqrt{e^{a \cdot x}} - 1 \cdot \sqrt{1}}{\mathsf{fma}\left(\sqrt{e^{a \cdot x}}, \sqrt{1}, e^{a \cdot x} + 1\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \left(a \cdot x\right) \cdot \left(a \cdot x\right), \mathsf{fma}\left(\left(a \cdot x\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right), \frac{1}{6}, a \cdot x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 +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))