expax (section 3.5)

Average Error: 29.6 → 0.5

Time: 4.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \leq -0.05266943121646845:\\ \;\;\;\;\left(1 + \sqrt{e^{a \cdot x}}\right) \cdot \left(\sqrt{e^{a \cdot x}} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x + \left(x \cdot \left(a \cdot x\right)\right) \cdot 0.5\right)\\ \end{array}\]

FPCore

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

↓

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -0.05266943121646845)
   (* (+ 1.0 (sqrt (exp (* a x)))) (+ (sqrt (exp (* a x))) -1.0))
   (* a (+ x (* (* x (* a 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) <= -0.05266943121646845) {
		tmp = (1.0 + sqrt(exp(a * x))) * (sqrt(exp(a * x)) + -1.0);
	} else {
		tmp = a * (x + ((x * (a * x)) * 0.5));
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.6
Target	0.2
Herbie	0.5

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -0.05266943121646845

   1. Initial program 0.0

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

   3. Applied add-sqr-sqrt_binary64_1464 0.0

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

   4. Applied difference-of-sqr-1_binary64_1412 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 -0.05266943121646845 < (*.f64 a x)

   1. Initial program 44.5

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

   3. Applied flip3--_binary64_1446 44.6

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

   4. Simplified 44.6

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

   5. Simplified 44.6

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

   6. Taylor expanded around 0 8.6

      \[\leadsto \frac{\color{blue}{3 \cdot \left(a \cdot x\right) + 4.5 \cdot \left({a}^{2} \cdot {x}^{2}\right)}}{{\left(e^{a \cdot x}\right)}^{2} + \left(e^{a \cdot x} + 1\right)}\]

   7. Simplified 0.9

      \[\leadsto \frac{\color{blue}{\left(a \cdot x\right) \cdot \left(3 + 4.5 \cdot \left(a \cdot x\right)\right)}}{{\left(e^{a \cdot x}\right)}^{2} + \left(e^{a \cdot x} + 1\right)}\]

   8. Taylor expanded around 0 8.5

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

   9. Simplified 0.7

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -0.05266943121646845:\\ \;\;\;\;\left(1 + \sqrt{e^{a \cdot x}}\right) \cdot \left(\sqrt{e^{a \cdot x}} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x + \left(x \cdot \left(a \cdot x\right)\right) \cdot 0.5\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021032 
(FPCore (a x)
  :name "expax (section 3.5)"
  :precision binary64

  :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))