expax (section 3.5)

Average Error: 29.5 → 0.4

Time: 3.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \leq -0.6342384098193586:\\ \;\;\;\;\left(1 + \sqrt{e^{a \cdot x}}\right) \cdot \left(\sqrt{e^{a \cdot x}} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + \left(a \cdot \left(\left(a \cdot \left(0.5 + a \cdot \left(x \cdot 0.16666666666666666\right)\right)\right) \cdot \left(x \cdot 0.5\right)\right) + a \cdot \left(\left(a \cdot \left(0.5 + a \cdot \left(x \cdot 0.16666666666666666\right)\right)\right) \cdot \left(x \cdot 0.5\right)\right)\right)\right)\\ \end{array}\]

FPCore

(FPCore (a x) :precision binary64 (- (exp (* a x)) 1.0))

↓

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -0.6342384098193586)
   (* (+ 1.0 (sqrt (exp (* a x)))) (+ (sqrt (exp (* a x))) -1.0))
   (*
    x
    (+
     a
     (+
      (* a (* (* a (+ 0.5 (* a (* x 0.16666666666666666)))) (* x 0.5)))
      (* a (* (* a (+ 0.5 (* a (* x 0.16666666666666666)))) (* x 0.5))))))))

C

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

↓

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -0.6342384098193586) {
		tmp = (1.0 + sqrt(exp(a * x))) * (sqrt(exp(a * x)) + -1.0);
	} else {
		tmp = x * (a + ((a * ((a * (0.5 + (a * (x * 0.16666666666666666)))) * (x * 0.5))) + (a * ((a * (0.5 + (a * (x * 0.16666666666666666)))) * (x * 0.5)))));
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.5
Target	0.2
Herbie	0.4

\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| < 0.1:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -0.634238409819358639

   1. Initial program 0.0

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

   3. Applied add-sqr-sqrt_binary64_1123 0.0

      \[\leadsto \color{blue}{\sqrt{e^{a \cdot x}} \cdot \sqrt{e^{a \cdot x}}} - 1\]

   4. Applied difference-of-sqr-1_binary64_1071 0.0

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

   5. Simplified 0.0

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

   6. Simplified 0.0

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

   if -0.634238409819358639 < (*.f64 a x)

   1. Initial program 44.2

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

   2. Taylor expanded around 0 14.2

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \left({a}^{3} \cdot {x}^{3}\right) + \left(a \cdot x + 0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right)\right)}\]

   3. Simplified 7.7

      \[\leadsto \color{blue}{x \cdot \left(a + x \cdot \left(0.5 \cdot \left(a \cdot a\right) + x \cdot \left(0.16666666666666666 \cdot {a}^{3}\right)\right)\right)}\]
   4. Using strategy rm

   5. Applied add-log-exp_binary64_1140 9.5

      \[\leadsto x \cdot \left(a + \color{blue}{\log \left(e^{x \cdot \left(0.5 \cdot \left(a \cdot a\right) + x \cdot \left(0.16666666666666666 \cdot {a}^{3}\right)\right)}\right)}\right)\]

   6. Simplified 5.8

      \[\leadsto x \cdot \left(a + \log \color{blue}{\left({\left(e^{x}\right)}^{\left(\left(a \cdot a\right) \cdot \left(0.5 + \left(x \cdot 0.16666666666666666\right) \cdot a\right)\right)}\right)}\right)\]
   7. Using strategy rm

   8. Applied add-sqr-sqrt_binary64_1123 5.8

      \[\leadsto x \cdot \left(a + \log \left({\color{blue}{\left(\sqrt{e^{x}} \cdot \sqrt{e^{x}}\right)}}^{\left(\left(a \cdot a\right) \cdot \left(0.5 + \left(x \cdot 0.16666666666666666\right) \cdot a\right)\right)}\right)\right)\]

   9. Applied unpow-prod-down_binary64_1180 5.8

      \[\leadsto x \cdot \left(a + \log \color{blue}{\left({\left(\sqrt{e^{x}}\right)}^{\left(\left(a \cdot a\right) \cdot \left(0.5 + \left(x \cdot 0.16666666666666666\right) \cdot a\right)\right)} \cdot {\left(\sqrt{e^{x}}\right)}^{\left(\left(a \cdot a\right) \cdot \left(0.5 + \left(x \cdot 0.16666666666666666\right) \cdot a\right)\right)}\right)}\right)\]

   10. Applied log-prod_binary64_1187 5.8

       \[\leadsto x \cdot \left(a + \color{blue}{\left(\log \left({\left(\sqrt{e^{x}}\right)}^{\left(\left(a \cdot a\right) \cdot \left(0.5 + \left(x \cdot 0.16666666666666666\right) \cdot a\right)\right)}\right) + \log \left({\left(\sqrt{e^{x}}\right)}^{\left(\left(a \cdot a\right) \cdot \left(0.5 + \left(x \cdot 0.16666666666666666\right) \cdot a\right)\right)}\right)\right)}\right)\]

   11. Simplified 5.8

       \[\leadsto x \cdot \left(a + \left(\color{blue}{a \cdot \left(\left(a \cdot \left(0.5 + a \cdot \left(x \cdot 0.16666666666666666\right)\right)\right) \cdot \left(x \cdot 0.5\right)\right)} + \log \left({\left(\sqrt{e^{x}}\right)}^{\left(\left(a \cdot a\right) \cdot \left(0.5 + \left(x \cdot 0.16666666666666666\right) \cdot a\right)\right)}\right)\right)\right)\]

   12. Simplified 0.6

       \[\leadsto x \cdot \left(a + \left(a \cdot \left(\left(a \cdot \left(0.5 + a \cdot \left(x \cdot 0.16666666666666666\right)\right)\right) \cdot \left(x \cdot 0.5\right)\right) + \color{blue}{a \cdot \left(\left(a \cdot \left(0.5 + a \cdot \left(x \cdot 0.16666666666666666\right)\right)\right) \cdot \left(x \cdot 0.5\right)\right)}\right)\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -0.6342384098193586:\\ \;\;\;\;\left(1 + \sqrt{e^{a \cdot x}}\right) \cdot \left(\sqrt{e^{a \cdot x}} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + \left(a \cdot \left(\left(a \cdot \left(0.5 + a \cdot \left(x \cdot 0.16666666666666666\right)\right)\right) \cdot \left(x \cdot 0.5\right)\right) + a \cdot \left(\left(a \cdot \left(0.5 + a \cdot \left(x \cdot 0.16666666666666666\right)\right)\right) \cdot \left(x \cdot 0.5\right)\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020355 
(FPCore (a x)
  :name "expax (section 3.5)"
  :precision binary64
  :herbie-expected 14

  :herbie-target
  (if (< (fabs (* a x)) 0.1) (* (* a x) (+ 1.0 (+ (/ (* a x) 2.0) (/ (pow (* a x) 2.0) 6.0)))) (- (exp (* a x)) 1.0))

  (- (exp (* a x)) 1.0))