expax (section 3.5)

Average Error: 30.0 → 0.5

Time: 3.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.001626719355739147552727952295015256822808:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, {\left(x \cdot a\right)}^{2}, x \cdot a\right)\\ \end{array}\]

C

double f(double a, double x) {
        double r87706 = a;
        double r87707 = x;
        double r87708 = r87706 * r87707;
        double r87709 = exp(r87708);
        double r87710 = 1.0;
        double r87711 = r87709 - r87710;
        return r87711;
}

↓

double f(double a, double x) {
        double r87712 = a;
        double r87713 = x;
        double r87714 = r87712 * r87713;
        double r87715 = -0.0016267193557391476;
        bool r87716 = r87714 <= r87715;
        double r87717 = exp(r87714);
        double r87718 = 1.0;
        double r87719 = r87717 - r87718;
        double r87720 = exp(r87719);
        double r87721 = log(r87720);
        double r87722 = 0.5;
        double r87723 = r87713 * r87712;
        double r87724 = 2.0;
        double r87725 = pow(r87723, r87724);
        double r87726 = fma(r87722, r87725, r87723);
        double r87727 = r87716 ? r87721 : r87726;
        return r87727;
}

Error

Bits error versus a

Bits error versus x

Target

Original	30.0
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) < -0.0016267193557391476

   1. Initial program 0.0

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

   3. Applied add-log-exp 0.0

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

   4. Applied add-log-exp 0.0

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

   5. Applied diff-log 0.0

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

   6. Simplified 0.0

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

   if -0.0016267193557391476 < (* a x)

   1. Initial program 44.7

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

   2. Taylor expanded around 0 14.7

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

   3. Simplified 14.7

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

   4. Taylor expanded around 0 8.4

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

   5. Simplified 8.4

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

   7. Applied pow-prod-down 0.7

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

   8. Simplified 0.7

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.001626719355739147552727952295015256822808:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, {\left(x \cdot a\right)}^{2}, x \cdot a\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +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))