expax (section 3.5)

Average Error: 29.6 → 0.5

Time: 3.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -8.695234510422765288051305532235346618108 \cdot 10^{-4}:\\ \;\;\;\;\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \sqrt[3]{e^{a \cdot x} - 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{1}{2} \cdot a, x, 1\right) \cdot a\right) \cdot x\\ \end{array}\]

C

double f(double a, double x) {
        double r102401 = a;
        double r102402 = x;
        double r102403 = r102401 * r102402;
        double r102404 = exp(r102403);
        double r102405 = 1.0;
        double r102406 = r102404 - r102405;
        return r102406;
}

↓

double f(double a, double x) {
        double r102407 = a;
        double r102408 = x;
        double r102409 = r102407 * r102408;
        double r102410 = -0.0008695234510422765;
        bool r102411 = r102409 <= r102410;
        double r102412 = exp(r102409);
        double r102413 = 1.0;
        double r102414 = r102412 - r102413;
        double r102415 = cbrt(r102414);
        double r102416 = r102415 * r102415;
        double r102417 = r102416 * r102415;
        double r102418 = 0.5;
        double r102419 = r102418 * r102407;
        double r102420 = 1.0;
        double r102421 = fma(r102419, r102408, r102420);
        double r102422 = r102421 * r102407;
        double r102423 = r102422 * r102408;
        double r102424 = r102411 ? r102417 : r102423;
        return r102424;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.6
Target	0.2
Herbie	0.5

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

Derivation

1. Split input into 2 regimes

2. if (* a x) < -0.0008695234510422765

   1. Initial program 0.0

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

   3. Applied add-cube-cbrt 0.0

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

   if -0.0008695234510422765 < (* a x)

   1. Initial program 44.4

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

   2. Taylor expanded around 0 14.5

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

   3. Simplified 14.5

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

   4. Taylor expanded around 0 8.0

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

   5. Simplified 8.0

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

   7. Applied add-sqr-sqrt 36.4

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

   8. Applied unpow-prod-down 36.4

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

   9. Applied add-sqr-sqrt 50.1

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

   10. Applied unpow-prod-down 50.1

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

   11. Applied unswap-sqr 48.4

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

   12. Simplified 48.4

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

   13. Simplified 0.8

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

   14. Taylor expanded around inf 8.0

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

   15. Simplified 0.8

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -8.695234510422765288051305532235346618108 \cdot 10^{-4}:\\ \;\;\;\;\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \sqrt[3]{e^{a \cdot x} - 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{1}{2} \cdot a, x, 1\right) \cdot a\right) \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2019356 +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))