expax (section 3.5)

Average Error: 29.4 → 0.4

Time: 3.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -5.8466223410137292:\\ \;\;\;\;\left(\sqrt[3]{e^{a \cdot x}} \cdot \sqrt[3]{e^{a \cdot x}}\right) \cdot \sqrt[3]{e^{a \cdot x}} - 1\\ \mathbf{elif}\;a \cdot x \le 8.9719548727940208 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \left(\left|x\right| \cdot a\right) \cdot \left(\left|x\right| \cdot a\right), x \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{e^{a \cdot x} - 1} \cdot \sqrt{e^{a \cdot x} - 1}\\ \end{array}\]

C

double f(double a, double x) {
        double r83362 = a;
        double r83363 = x;
        double r83364 = r83362 * r83363;
        double r83365 = exp(r83364);
        double r83366 = 1.0;
        double r83367 = r83365 - r83366;
        return r83367;
}

↓

double f(double a, double x) {
        double r83368 = a;
        double r83369 = x;
        double r83370 = r83368 * r83369;
        double r83371 = -5.846622341013729;
        bool r83372 = r83370 <= r83371;
        double r83373 = exp(r83370);
        double r83374 = cbrt(r83373);
        double r83375 = r83374 * r83374;
        double r83376 = r83375 * r83374;
        double r83377 = 1.0;
        double r83378 = r83376 - r83377;
        double r83379 = 8.971954872794021e-08;
        bool r83380 = r83370 <= r83379;
        double r83381 = 0.5;
        double r83382 = fabs(r83369);
        double r83383 = r83382 * r83368;
        double r83384 = r83383 * r83383;
        double r83385 = r83369 * r83368;
        double r83386 = fma(r83381, r83384, r83385);
        double r83387 = r83373 - r83377;
        double r83388 = sqrt(r83387);
        double r83389 = r83388 * r83388;
        double r83390 = r83380 ? r83386 : r83389;
        double r83391 = r83372 ? r83378 : r83390;
        return r83391;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.4
Target	0.2
Herbie	0.4

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

Derivation

1. Split input into 3 regimes

2. if (* a x) < -5.846622341013729

   1. Initial program 0

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

   3. Applied add-cube-cbrt 0

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

   if -5.846622341013729 < (* a x) < 8.971954872794021e-08

   1. Initial program 44.5

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

   2. Taylor expanded around 0 14.0

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

   3. Simplified 14.0

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

   4. Taylor expanded around 0 7.9

      \[\leadsto \mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {x}^{2}, \color{blue}{a \cdot x}\right)\]

   5. Simplified 7.9

      \[\leadsto \mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {x}^{2}, \color{blue}{x \cdot a}\right)\]
   6. Using strategy rm

   7. Applied add-sqr-sqrt 7.9

      \[\leadsto \mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot \color{blue}{\left(\sqrt{{x}^{2}} \cdot \sqrt{{x}^{2}}\right)}, x \cdot a\right)\]

   8. Applied add-sqr-sqrt 35.4

      \[\leadsto \mathsf{fma}\left(\frac{1}{2}, {\color{blue}{\left(\sqrt{a} \cdot \sqrt{a}\right)}}^{2} \cdot \left(\sqrt{{x}^{2}} \cdot \sqrt{{x}^{2}}\right), x \cdot a\right)\]

   9. Applied unpow-prod-down 35.4

      \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left({\left(\sqrt{a}\right)}^{2} \cdot {\left(\sqrt{a}\right)}^{2}\right)} \cdot \left(\sqrt{{x}^{2}} \cdot \sqrt{{x}^{2}}\right), x \cdot a\right)\]

   10. Applied unswap-sqr 33.5

       \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left({\left(\sqrt{a}\right)}^{2} \cdot \sqrt{{x}^{2}}\right) \cdot \left({\left(\sqrt{a}\right)}^{2} \cdot \sqrt{{x}^{2}}\right)}, x \cdot a\right)\]

   11. Simplified 33.5

       \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left(\left|x\right| \cdot a\right)} \cdot \left({\left(\sqrt{a}\right)}^{2} \cdot \sqrt{{x}^{2}}\right), x \cdot a\right)\]

   12. Simplified 0.5

       \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \left(\left|x\right| \cdot a\right) \cdot \color{blue}{\left(\left|x\right| \cdot a\right)}, x \cdot a\right)\]

   if 8.971954872794021e-08 < (* a x)

   1. Initial program 7.6

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

   3. Applied add-sqr-sqrt 7.6

      \[\leadsto \color{blue}{\sqrt{e^{a \cdot x} - 1} \cdot \sqrt{e^{a \cdot x} - 1}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -5.8466223410137292:\\ \;\;\;\;\left(\sqrt[3]{e^{a \cdot x}} \cdot \sqrt[3]{e^{a \cdot x}}\right) \cdot \sqrt[3]{e^{a \cdot x}} - 1\\ \mathbf{elif}\;a \cdot x \le 8.9719548727940208 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \left(\left|x\right| \cdot a\right) \cdot \left(\left|x\right| \cdot a\right), x \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{e^{a \cdot x} - 1} \cdot \sqrt{e^{a \cdot x} - 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 +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))