expax (section 3.5)

Average Error: 29.2 → 0.3

Time: 20.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -1.933321263417943109298030579523697269906 \cdot 10^{-4}:\\ \;\;\;\;\left(\sqrt{1} + \sqrt{e^{a \cdot x}}\right) \cdot \frac{\log \left(e^{e^{a \cdot x} \cdot \sqrt{e^{a \cdot x}} - 1 \cdot \sqrt{1}}\right)}{\left(\sqrt{e^{a \cdot x}} \cdot \sqrt{1} + 1\right) + e^{a \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + \left(\left(\left(a \cdot x\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) \cdot \frac{1}{6} + \frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right)}{2}\right)\\ \end{array}\]

C

double f(double a, double x) {
        double r4937761 = a;
        double r4937762 = x;
        double r4937763 = r4937761 * r4937762;
        double r4937764 = exp(r4937763);
        double r4937765 = 1.0;
        double r4937766 = r4937764 - r4937765;
        return r4937766;
}

↓

double f(double a, double x) {
        double r4937767 = a;
        double r4937768 = x;
        double r4937769 = r4937767 * r4937768;
        double r4937770 = -0.0001933321263417943;
        bool r4937771 = r4937769 <= r4937770;
        double r4937772 = 1.0;
        double r4937773 = sqrt(r4937772);
        double r4937774 = exp(r4937769);
        double r4937775 = sqrt(r4937774);
        double r4937776 = r4937773 + r4937775;
        double r4937777 = r4937774 * r4937775;
        double r4937778 = r4937772 * r4937773;
        double r4937779 = r4937777 - r4937778;
        double r4937780 = exp(r4937779);
        double r4937781 = log(r4937780);
        double r4937782 = r4937775 * r4937773;
        double r4937783 = r4937782 + r4937772;
        double r4937784 = r4937783 + r4937774;
        double r4937785 = r4937781 / r4937784;
        double r4937786 = r4937776 * r4937785;
        double r4937787 = r4937769 * r4937769;
        double r4937788 = r4937769 * r4937787;
        double r4937789 = 0.16666666666666666;
        double r4937790 = r4937788 * r4937789;
        double r4937791 = 2.0;
        double r4937792 = r4937787 / r4937791;
        double r4937793 = r4937790 + r4937792;
        double r4937794 = r4937769 + r4937793;
        double r4937795 = r4937771 ? r4937786 : r4937794;
        return r4937795;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.2
Target	0.2
Herbie	0.3

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

Derivation

1. Split input into 2 regimes

2. if (* a x) < -0.0001933321263417943

   1. Initial program 0.0

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

   3. Applied add-sqr-sqrt 0.0

      \[\leadsto e^{a \cdot x} - \color{blue}{\sqrt{1} \cdot \sqrt{1}}\]

   4. Applied add-sqr-sqrt 0.0

      \[\leadsto \color{blue}{\sqrt{e^{a \cdot x}} \cdot \sqrt{e^{a \cdot x}}} - \sqrt{1} \cdot \sqrt{1}\]

   5. Applied difference-of-squares 0.0

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

   7. Applied flip3-- 0.0

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

   8. Simplified 0.0

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

   9. Simplified 0.0

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

   11. Applied add-log-exp 0.0

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

   if -0.0001933321263417943 < (* a x)

   1. Initial program 44.6

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

   2. Taylor expanded around 0 14.4

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

   3. Simplified 0.5

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -1.933321263417943109298030579523697269906 \cdot 10^{-4}:\\ \;\;\;\;\left(\sqrt{1} + \sqrt{e^{a \cdot x}}\right) \cdot \frac{\log \left(e^{e^{a \cdot x} \cdot \sqrt{e^{a \cdot x}} - 1 \cdot \sqrt{1}}\right)}{\left(\sqrt{e^{a \cdot x}} \cdot \sqrt{1} + 1\right) + e^{a \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + \left(\left(\left(a \cdot x\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) \cdot \frac{1}{6} + \frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right)}{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 
(FPCore (a x)
  :name "expax (section 3.5)"
  :herbie-expected 14

  :herbie-target
  (if (< (fabs (* a x)) 0.1) (* (* a x) (+ 1.0 (+ (/ (* a x) 2.0) (/ (pow (* a x) 2.0) 6.0)))) (- (exp (* a x)) 1.0))

  (- (exp (* a x)) 1.0))