expax (section 3.5)

Average Error: 29.8 → 8.9

Time: 3.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -7.7613608644350317 \cdot 10^{-23}:\\ \;\;\;\;\frac{\left({\left(e^{a \cdot x}\right)}^{\frac{3}{2}} + \sqrt{{1}^{3}}\right) \cdot \log \left(e^{{\left(e^{a \cdot x}\right)}^{\frac{3}{2}} - \sqrt{{1}^{3}}}\right)}{e^{a \cdot x} \cdot \left(\sqrt[3]{{\left(e^{a \cdot x}\right)}^{3}} + 1\right) + 1 \cdot 1}\\ \mathbf{elif}\;a \cdot x \le 4.00137271357986445 \cdot 10^{-17}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\sqrt[3]{{\left(e^{a \cdot x}\right)}^{3}}\right)}^{3} - {1}^{3}}{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}\\ \end{array}\]

C

double f(double a, double x) {
        double r95643 = a;
        double r95644 = x;
        double r95645 = r95643 * r95644;
        double r95646 = exp(r95645);
        double r95647 = 1.0;
        double r95648 = r95646 - r95647;
        return r95648;
}

↓

double f(double a, double x) {
        double r95649 = a;
        double r95650 = x;
        double r95651 = r95649 * r95650;
        double r95652 = -7.761360864435032e-23;
        bool r95653 = r95651 <= r95652;
        double r95654 = exp(r95651);
        double r95655 = 1.5;
        double r95656 = pow(r95654, r95655);
        double r95657 = 1.0;
        double r95658 = 3.0;
        double r95659 = pow(r95657, r95658);
        double r95660 = sqrt(r95659);
        double r95661 = r95656 + r95660;
        double r95662 = r95656 - r95660;
        double r95663 = exp(r95662);
        double r95664 = log(r95663);
        double r95665 = r95661 * r95664;
        double r95666 = pow(r95654, r95658);
        double r95667 = cbrt(r95666);
        double r95668 = r95667 + r95657;
        double r95669 = r95654 * r95668;
        double r95670 = r95657 * r95657;
        double r95671 = r95669 + r95670;
        double r95672 = r95665 / r95671;
        double r95673 = 4.0013727135798645e-17;
        bool r95674 = r95651 <= r95673;
        double r95675 = 0.5;
        double r95676 = 2.0;
        double r95677 = pow(r95649, r95676);
        double r95678 = r95675 * r95677;
        double r95679 = r95678 * r95650;
        double r95680 = r95649 + r95679;
        double r95681 = r95650 * r95680;
        double r95682 = 0.16666666666666666;
        double r95683 = pow(r95649, r95658);
        double r95684 = pow(r95650, r95658);
        double r95685 = r95683 * r95684;
        double r95686 = r95682 * r95685;
        double r95687 = r95681 + r95686;
        double r95688 = pow(r95667, r95658);
        double r95689 = r95688 - r95659;
        double r95690 = r95654 + r95657;
        double r95691 = r95654 * r95690;
        double r95692 = r95691 + r95670;
        double r95693 = r95689 / r95692;
        double r95694 = r95674 ? r95687 : r95693;
        double r95695 = r95653 ? r95672 : r95694;
        return r95695;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.8
Target	0.2
Herbie	8.9

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

Derivation

1. Split input into 3 regimes

2. if (* a x) < -7.761360864435032e-23

   1. Initial program 2.1

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

   3. Applied flip3-- 2.1

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

   4. Simplified 2.1

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

   6. Applied add-cbrt-cube 2.1

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

   7. Simplified 2.1

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

   9. Applied add-sqr-sqrt 2.1

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

   10. Applied sqr-pow 2.1

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

   11. Applied difference-of-squares 2.1

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

   12. Simplified 2.1

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

   13. Simplified 2.1

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

   15. Applied add-log-exp 2.1

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

   16. Applied add-log-exp 2.1

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

   17. Applied diff-log 2.1

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

   18. Simplified 2.1

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

   if -7.761360864435032e-23 < (* a x) < 4.0013727135798645e-17

   1. Initial program 45.6

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

   2. Taylor expanded around 0 12.2

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

   3. Simplified 12.2

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

   if 4.0013727135798645e-17 < (* a x)

   1. Initial program 22.5

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

   3. Applied flip3-- 25.2

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

   4. Simplified 25.2

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

   6. Applied add-cbrt-cube 26.0

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

   7. Simplified 26.0

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

4. Final simplification 8.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -7.7613608644350317 \cdot 10^{-23}:\\ \;\;\;\;\frac{\left({\left(e^{a \cdot x}\right)}^{\frac{3}{2}} + \sqrt{{1}^{3}}\right) \cdot \log \left(e^{{\left(e^{a \cdot x}\right)}^{\frac{3}{2}} - \sqrt{{1}^{3}}}\right)}{e^{a \cdot x} \cdot \left(\sqrt[3]{{\left(e^{a \cdot x}\right)}^{3}} + 1\right) + 1 \cdot 1}\\ \mathbf{elif}\;a \cdot x \le 4.00137271357986445 \cdot 10^{-17}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\sqrt[3]{{\left(e^{a \cdot x}\right)}^{3}}\right)}^{3} - {1}^{3}}{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 
(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))