expax (section 3.5)

Average Error: 29.3 → 9.0

Time: 4.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -1.9961401647367767 \cdot 10^{-21}:\\ \;\;\;\;\sqrt[3]{{\left(\sqrt[3]{\log \left(e^{e^{a \cdot x} - 1}\right) \cdot \left(e^{a \cdot x} - 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\sqrt{e^{a \cdot x}}, \sqrt{e^{a \cdot x}}, -1\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {x}^{3}, a \cdot x\right)\right)\\ \end{array}\]

C

double f(double a, double x) {
        double r99562 = a;
        double r99563 = x;
        double r99564 = r99562 * r99563;
        double r99565 = exp(r99564);
        double r99566 = 1.0;
        double r99567 = r99565 - r99566;
        return r99567;
}

↓

double f(double a, double x) {
        double r99568 = a;
        double r99569 = x;
        double r99570 = r99568 * r99569;
        double r99571 = -1.9961401647367767e-21;
        bool r99572 = r99570 <= r99571;
        double r99573 = exp(r99570);
        double r99574 = 1.0;
        double r99575 = r99573 - r99574;
        double r99576 = exp(r99575);
        double r99577 = log(r99576);
        double r99578 = r99577 * r99575;
        double r99579 = cbrt(r99578);
        double r99580 = sqrt(r99573);
        double r99581 = -r99574;
        double r99582 = fma(r99580, r99580, r99581);
        double r99583 = cbrt(r99582);
        double r99584 = r99579 * r99583;
        double r99585 = 3.0;
        double r99586 = pow(r99584, r99585);
        double r99587 = cbrt(r99586);
        double r99588 = 0.5;
        double r99589 = 2.0;
        double r99590 = pow(r99568, r99589);
        double r99591 = pow(r99569, r99589);
        double r99592 = r99590 * r99591;
        double r99593 = 0.16666666666666666;
        double r99594 = pow(r99568, r99585);
        double r99595 = pow(r99569, r99585);
        double r99596 = r99594 * r99595;
        double r99597 = fma(r99593, r99596, r99570);
        double r99598 = fma(r99588, r99592, r99597);
        double r99599 = r99572 ? r99587 : r99598;
        return r99599;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.3
Target	0.2
Herbie	9.0

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

Derivation

1. Split input into 2 regimes

2. if (* a x) < -1.9961401647367767e-21

   1. Initial program 2.0

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

   3. Applied add-cbrt-cube 2.0

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

   4. Simplified 2.0

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

   6. Applied add-cube-cbrt 2.0

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

   7. Simplified 2.0

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

   9. Applied add-sqr-sqrt 2.0

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

   10. Applied fma-neg 2.0

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

   12. Applied add-log-exp 2.0

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

   13. Applied add-log-exp 2.0

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

   14. Applied diff-log 2.0

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

   15. Simplified 2.0

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

   if -1.9961401647367767e-21 < (* a x)

   1. Initial program 44.5

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

   2. Taylor expanded around 0 13.0

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

   3. Simplified 13.0

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

4. Final simplification 9.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -1.9961401647367767 \cdot 10^{-21}:\\ \;\;\;\;\sqrt[3]{{\left(\sqrt[3]{\log \left(e^{e^{a \cdot x} - 1}\right) \cdot \left(e^{a \cdot x} - 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\sqrt{e^{a \cdot x}}, \sqrt{e^{a \cdot x}}, -1\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {x}^{3}, a \cdot x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 +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))