expax (section 3.5)

Average Error: 29.6 → 9.1

Time: 15.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -1.030443421023876352958167923468124627107 \cdot 10^{-6} \lor \neg \left(a \cdot x \le 2.106168723861143240310319851371754358155 \cdot 10^{-28}\right):\\ \;\;\;\;\frac{{\left(e^{a \cdot x}\right)}^{3} - {1}^{3}}{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \end{array}\]

C

double f(double a, double x) {
        double r87974 = a;
        double r87975 = x;
        double r87976 = r87974 * r87975;
        double r87977 = exp(r87976);
        double r87978 = 1.0;
        double r87979 = r87977 - r87978;
        return r87979;
}

↓

double f(double a, double x) {
        double r87980 = a;
        double r87981 = x;
        double r87982 = r87980 * r87981;
        double r87983 = -1.0304434210238764e-06;
        bool r87984 = r87982 <= r87983;
        double r87985 = 2.1061687238611432e-28;
        bool r87986 = r87982 <= r87985;
        double r87987 = !r87986;
        bool r87988 = r87984 || r87987;
        double r87989 = exp(r87982);
        double r87990 = 3.0;
        double r87991 = pow(r87989, r87990);
        double r87992 = 1.0;
        double r87993 = pow(r87992, r87990);
        double r87994 = r87991 - r87993;
        double r87995 = r87989 + r87992;
        double r87996 = r87989 * r87995;
        double r87997 = r87992 * r87992;
        double r87998 = r87996 + r87997;
        double r87999 = r87994 / r87998;
        double r88000 = 0.5;
        double r88001 = 2.0;
        double r88002 = pow(r87980, r88001);
        double r88003 = r88000 * r88002;
        double r88004 = r88003 * r87981;
        double r88005 = r87980 + r88004;
        double r88006 = r87981 * r88005;
        double r88007 = 0.16666666666666666;
        double r88008 = pow(r87980, r87990);
        double r88009 = pow(r87981, r87990);
        double r88010 = r88008 * r88009;
        double r88011 = r88007 * r88010;
        double r88012 = r88006 + r88011;
        double r88013 = r87988 ? r87999 : r88012;
        return r88013;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.6
Target	0.2
Herbie	9.1

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

Derivation

1. Split input into 3 regimes

2. if (* a x) < -1.0304434210238764e-06

   1. Initial program 0.2

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

   3. Applied flip3-- 0.2

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

   4. Simplified 0.2

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

   6. Applied add-cube-cbrt 0.2

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

   8. Applied add-log-exp 0.2

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

   9. Applied add-log-exp 0.2

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

   10. Applied diff-log 0.2

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

   11. Simplified 0.2

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

   if -1.0304434210238764e-06 < (* a x) < 2.1061687238611432e-28

   1. Initial program 45.1

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

   2. Taylor expanded around 0 12.6

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

   3. Simplified 12.6

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

   if 2.1061687238611432e-28 < (* a x)

   1. Initial program 36.0

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

   3. Applied flip3-- 37.6

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

   4. Simplified 37.6

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

   6. Applied add-cbrt-cube 38.2

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

   7. Simplified 38.2

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

4. Final simplification 9.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -1.030443421023876352958167923468124627107 \cdot 10^{-6} \lor \neg \left(a \cdot x \le 2.106168723861143240310319851371754358155 \cdot 10^{-28}\right):\\ \;\;\;\;\frac{{\left(e^{a \cdot x}\right)}^{3} - {1}^{3}}{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019291 
(FPCore (a x)
  :name "expax (section 3.5)"
  :precision binary64
  :herbie-expected 14

  :herbie-target
  (if (< (fabs (* a x)) 0.10000000000000001) (* (* a x) (+ 1 (+ (/ (* a x) 2) (/ (pow (* a x) 2) 6)))) (- (exp (* a x)) 1))

  (- (exp (* a x)) 1))