expax (section 3.5)

Average Error: 29.1 → 0.4

Time: 3.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.26310033146711964:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, {\left(x \cdot a\right)}^{2}, \mathsf{fma}\left(\frac{1}{6}, {\left(x \cdot a\right)}^{3}, a \cdot x\right)\right)\\ \end{array}\]

C

double code(double a, double x) {
	return ((double) (((double) exp(((double) (a * x)))) - 1.0));
}

↓

double code(double a, double x) {
	double VAR;
	if ((((double) (a * x)) <= -0.26310033146711964)) {
		VAR = ((double) log(((double) exp(((double) (((double) exp(((double) (a * x)))) - 1.0))))));
	} else {
		VAR = ((double) fma(0.5, ((double) pow(((double) (x * a)), 2.0)), ((double) fma(0.16666666666666666, ((double) pow(((double) (x * a)), 3.0)), ((double) (a * x))))));
	}
	return VAR;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.1
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 2 regimes

2. if (* a x) < -0.26310033146711964

   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 -0.26310033146711964 < (* a x)

   1. Initial program 43.6

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

   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. Using strategy rm

   5. Applied pow-prod-down 8.4

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

   6. Simplified 8.4

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

   8. Applied pow-prod-down 0.7

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

   9. Simplified 0.7

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.26310033146711964:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, {\left(x \cdot a\right)}^{2}, \mathsf{fma}\left(\frac{1}{6}, {\left(x \cdot a\right)}^{3}, a \cdot x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020121 +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))