expax (section 3.5)

Average Error: 28.9 → 7.2

Time: 20.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -6.875657620123969931839940451833229073664 \cdot 10^{-9}:\\ \;\;\;\;\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \sqrt[3]{\frac{\frac{e^{\left(a \cdot x\right) \cdot 4} - {1}^{4}}{\mathsf{fma}\left(1, 1, {\left(e^{2}\right)}^{\left(a \cdot x\right)}\right)}}{1 + e^{a \cdot x}}}\\ \mathbf{elif}\;a \cdot x \le 8.25458109042831203908792331233248718025 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left({x}^{2}, \mathsf{fma}\left(\frac{1}{6} \cdot {a}^{3}, x, \frac{1}{2} \cdot {a}^{2}\right), a \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \frac{\sqrt[3]{e^{2 \cdot \left(a \cdot x\right)} - 1 \cdot 1}}{{\left(e^{a \cdot x} + 1\right)}^{\frac{1}{3}}}\right) \cdot \sqrt[3]{\frac{e^{2 \cdot \left(a \cdot x\right)} - 1 \cdot 1}{1 + e^{a \cdot x}}}\\ \end{array}\]

C

double f(double a, double x) {
        double r80169 = a;
        double r80170 = x;
        double r80171 = r80169 * r80170;
        double r80172 = exp(r80171);
        double r80173 = 1.0;
        double r80174 = r80172 - r80173;
        return r80174;
}

↓

double f(double a, double x) {
        double r80175 = a;
        double r80176 = x;
        double r80177 = r80175 * r80176;
        double r80178 = -6.87565762012397e-09;
        bool r80179 = r80177 <= r80178;
        double r80180 = exp(r80177);
        double r80181 = 1.0;
        double r80182 = r80180 - r80181;
        double r80183 = cbrt(r80182);
        double r80184 = r80183 * r80183;
        double r80185 = 4.0;
        double r80186 = r80177 * r80185;
        double r80187 = exp(r80186);
        double r80188 = pow(r80181, r80185);
        double r80189 = r80187 - r80188;
        double r80190 = 2.0;
        double r80191 = exp(r80190);
        double r80192 = pow(r80191, r80177);
        double r80193 = fma(r80181, r80181, r80192);
        double r80194 = r80189 / r80193;
        double r80195 = r80181 + r80180;
        double r80196 = r80194 / r80195;
        double r80197 = cbrt(r80196);
        double r80198 = r80184 * r80197;
        double r80199 = 8.254581090428312e-23;
        bool r80200 = r80177 <= r80199;
        double r80201 = pow(r80176, r80190);
        double r80202 = 0.16666666666666666;
        double r80203 = 3.0;
        double r80204 = pow(r80175, r80203);
        double r80205 = r80202 * r80204;
        double r80206 = 0.5;
        double r80207 = pow(r80175, r80190);
        double r80208 = r80206 * r80207;
        double r80209 = fma(r80205, r80176, r80208);
        double r80210 = fma(r80201, r80209, r80177);
        double r80211 = r80190 * r80177;
        double r80212 = exp(r80211);
        double r80213 = r80181 * r80181;
        double r80214 = r80212 - r80213;
        double r80215 = cbrt(r80214);
        double r80216 = r80180 + r80181;
        double r80217 = 0.3333333333333333;
        double r80218 = pow(r80216, r80217);
        double r80219 = r80215 / r80218;
        double r80220 = r80183 * r80219;
        double r80221 = r80214 / r80195;
        double r80222 = cbrt(r80221);
        double r80223 = r80220 * r80222;
        double r80224 = r80200 ? r80210 : r80223;
        double r80225 = r80179 ? r80198 : r80224;
        return r80225;
}

Error

Bits error versus a

Bits error versus x

Target

Original	28.9
Target	0.2
Herbie	7.2

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

Derivation

1. Split input into 3 regimes

2. if (* a x) < -6.87565762012397e-09

   1. Initial program 0.3

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

   3. Applied add-cube-cbrt 0.3

      \[\leadsto \color{blue}{\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \sqrt[3]{e^{a \cdot x} - 1}}\]
   4. Using strategy rm

   5. Applied flip-- 0.3

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

   6. Simplified 0.3

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

   7. Simplified 0.3

      \[\leadsto \left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \sqrt[3]{\frac{e^{2 \cdot \left(a \cdot x\right)} - 1 \cdot 1}{\color{blue}{1 + e^{a \cdot x}}}}\]
   8. Using strategy rm

   9. Applied flip-- 0.3

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

   10. Simplified 0.3

       \[\leadsto \left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \sqrt[3]{\frac{\frac{\color{blue}{e^{\left(a \cdot x\right) \cdot 4} - {1}^{4}}}{e^{2 \cdot \left(a \cdot x\right)} + 1 \cdot 1}}{1 + e^{a \cdot x}}}\]

   11. Simplified 0.3

       \[\leadsto \left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \sqrt[3]{\frac{\frac{e^{\left(a \cdot x\right) \cdot 4} - {1}^{4}}{\color{blue}{\mathsf{fma}\left(1, 1, {\left(e^{2}\right)}^{\left(a \cdot x\right)}\right)}}}{1 + e^{a \cdot x}}}\]

   if -6.87565762012397e-09 < (* a x) < 8.254581090428312e-23

   1. Initial program 44.5

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

   2. Taylor expanded around 0 13.0

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

   3. Simplified 9.9

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

   if 8.254581090428312e-23 < (* a x)

   1. Initial program 31.6

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

   3. Applied add-cube-cbrt 31.6

      \[\leadsto \color{blue}{\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \sqrt[3]{e^{a \cdot x} - 1}}\]
   4. Using strategy rm

   5. Applied flip-- 32.5

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

   6. Simplified 32.2

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

   7. Simplified 32.2

      \[\leadsto \left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \sqrt[3]{\frac{e^{2 \cdot \left(a \cdot x\right)} - 1 \cdot 1}{\color{blue}{1 + e^{a \cdot x}}}}\]
   8. Using strategy rm

   9. Applied flip-- 32.3

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

   10. Applied cbrt-div 32.3

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

   11. Simplified 31.7

       \[\leadsto \left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \frac{\color{blue}{\sqrt[3]{e^{2 \cdot \left(a \cdot x\right)} - 1 \cdot 1}}}{\sqrt[3]{e^{a \cdot x} + 1}}\right) \cdot \sqrt[3]{\frac{e^{2 \cdot \left(a \cdot x\right)} - 1 \cdot 1}{1 + e^{a \cdot x}}}\]
   12. Using strategy rm

   13. Applied pow1/3 31.7

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

4. Final simplification 7.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -6.875657620123969931839940451833229073664 \cdot 10^{-9}:\\ \;\;\;\;\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \sqrt[3]{\frac{\frac{e^{\left(a \cdot x\right) \cdot 4} - {1}^{4}}{\mathsf{fma}\left(1, 1, {\left(e^{2}\right)}^{\left(a \cdot x\right)}\right)}}{1 + e^{a \cdot x}}}\\ \mathbf{elif}\;a \cdot x \le 8.25458109042831203908792331233248718025 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left({x}^{2}, \mathsf{fma}\left(\frac{1}{6} \cdot {a}^{3}, x, \frac{1}{2} \cdot {a}^{2}\right), a \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \frac{\sqrt[3]{e^{2 \cdot \left(a \cdot x\right)} - 1 \cdot 1}}{{\left(e^{a \cdot x} + 1\right)}^{\frac{1}{3}}}\right) \cdot \sqrt[3]{\frac{e^{2 \cdot \left(a \cdot x\right)} - 1 \cdot 1}{1 + e^{a \cdot x}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +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))