expax (section 3.5)

Average Error: 28.8 → 9.3

Time: 3.5s

Precision: 64

Math

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

↓

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

C

double f(double a, double x) {
        double r96656 = a;
        double r96657 = x;
        double r96658 = r96656 * r96657;
        double r96659 = exp(r96658);
        double r96660 = 1.0;
        double r96661 = r96659 - r96660;
        return r96661;
}

↓

double f(double a, double x) {
        double r96662 = a;
        double r96663 = x;
        double r96664 = r96662 * r96663;
        double r96665 = -0.05343806011964837;
        bool r96666 = r96664 <= r96665;
        double r96667 = exp(r96664);
        double r96668 = 1.0;
        double r96669 = r96667 - r96668;
        double r96670 = cbrt(r96669);
        double r96671 = r96670 * r96670;
        double r96672 = cbrt(r96671);
        double r96673 = r96672 * r96672;
        double r96674 = cbrt(r96670);
        double r96675 = r96674 * r96674;
        double r96676 = r96673 * r96675;
        double r96677 = exp(r96669);
        double r96678 = log(r96677);
        double r96679 = cbrt(r96678);
        double r96680 = r96676 * r96679;
        double r96681 = 0.5;
        double r96682 = 2.0;
        double r96683 = pow(r96662, r96682);
        double r96684 = r96681 * r96683;
        double r96685 = r96684 * r96663;
        double r96686 = r96662 + r96685;
        double r96687 = r96663 * r96686;
        double r96688 = 0.16666666666666666;
        double r96689 = 3.0;
        double r96690 = pow(r96662, r96689);
        double r96691 = pow(r96663, r96689);
        double r96692 = r96690 * r96691;
        double r96693 = r96688 * r96692;
        double r96694 = r96687 + r96693;
        double r96695 = r96666 ? r96680 : r96694;
        return r96695;
}

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

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

4. Final simplification 9.3

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

Reproduce

herbie shell --seed 2020018 
(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))