expax (section 3.5)

Average Error: 28.8 → 9.3

Time: 4.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.05343806011964837:\\ \;\;\;\;\left(\left(\sqrt[3]{\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}} \cdot \sqrt[3]{\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{e^{a \cdot x} - 1}} \cdot \sqrt[3]{\sqrt[3]{e^{a \cdot x} - 1}}\right)\right) \cdot \sqrt[3]{\log \left(e^{e^{a \cdot x} - 1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {x}^{3}, a \cdot x\right)\right)\\ \end{array}\]

C

double f(double a, double x) {
        double r117153 = a;
        double r117154 = x;
        double r117155 = r117153 * r117154;
        double r117156 = exp(r117155);
        double r117157 = 1.0;
        double r117158 = r117156 - r117157;
        return r117158;
}

↓

double f(double a, double x) {
        double r117159 = a;
        double r117160 = x;
        double r117161 = r117159 * r117160;
        double r117162 = -0.05343806011964837;
        bool r117163 = r117161 <= r117162;
        double r117164 = exp(r117161);
        double r117165 = 1.0;
        double r117166 = r117164 - r117165;
        double r117167 = cbrt(r117166);
        double r117168 = r117167 * r117167;
        double r117169 = cbrt(r117168);
        double r117170 = r117169 * r117169;
        double r117171 = cbrt(r117167);
        double r117172 = r117171 * r117171;
        double r117173 = r117170 * r117172;
        double r117174 = exp(r117166);
        double r117175 = log(r117174);
        double r117176 = cbrt(r117175);
        double r117177 = r117173 * r117176;
        double r117178 = 0.5;
        double r117179 = 2.0;
        double r117180 = pow(r117159, r117179);
        double r117181 = pow(r117160, r117179);
        double r117182 = r117180 * r117181;
        double r117183 = 0.16666666666666666;
        double r117184 = 3.0;
        double r117185 = pow(r117159, r117184);
        double r117186 = pow(r117160, r117184);
        double r117187 = r117185 * r117186;
        double r117188 = fma(r117183, r117187, r117161);
        double r117189 = fma(r117178, r117182, r117188);
        double r117190 = r117163 ? r117177 : r117189;
        return r117190;
}

Error

Bits error versus a

Bits error versus x

Target

Original	28.8
Target	0.2
Herbie	9.3

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

Derivation

1. Split input into 2 regimes

2. if (* a x) < -0.05343806011964837

   1. Initial program 0.0

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

   3. Applied add-cube-cbrt 0.0

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

   5. Applied add-cube-cbrt 0.0

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

   6. Applied cbrt-prod 0.0

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

   7. Applied add-cube-cbrt 0.0

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

   8. Applied cbrt-prod 0.0

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

   9. Applied swap-sqr 0.0

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

   11. Applied add-log-exp 0.0

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

   12. Applied add-log-exp 0.0

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

   13. Applied diff-log 0.0

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

   14. Simplified 0.0

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

   if -0.05343806011964837 < (* a x)

   1. Initial program 44.0

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

   2. Taylor expanded around 0 14.2

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

   3. Simplified 14.2

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

4. Final simplification 9.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.05343806011964837:\\ \;\;\;\;\left(\left(\sqrt[3]{\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}} \cdot \sqrt[3]{\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{e^{a \cdot x} - 1}} \cdot \sqrt[3]{\sqrt[3]{e^{a \cdot x} - 1}}\right)\right) \cdot \sqrt[3]{\log \left(e^{e^{a \cdot x} - 1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {x}^{3}, a \cdot x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 +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))