expq3 (problem 3.4.2)

Average Error: 60.4 → 31.7

Time: 13.4s

Precision: binary64

\[-1 \lt \varepsilon \land \varepsilon \lt 1\]

Math

\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;b \le -8.9086780986168568 \cdot 10^{104}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(b + a\right)} - 1\right)}{\left(\varepsilon \cdot a + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(a \cdot \left(a \cdot \frac{1}{2}\right) + \varepsilon \cdot \left(\frac{1}{6} \cdot {a}^{3}\right)\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b}\\ \end{array}\]

C

double code(double a, double b, double eps) {
	return ((double) (((double) (eps * ((double) (((double) exp(((double) (((double) (a + b)) * eps)))) - 1.0)))) / ((double) (((double) (((double) exp(((double) (a * eps)))) - 1.0)) * ((double) (((double) exp(((double) (b * eps)))) - 1.0))))));
}

↓

double code(double a, double b, double eps) {
	double VAR;
	if ((b <= -8.908678098616857e+104)) {
		VAR = ((double) (((double) (eps * ((double) (((double) exp(((double) (eps * ((double) (b + a)))))) - 1.0)))) / ((double) (((double) (((double) (eps * a)) + ((double) (((double) (eps * eps)) * ((double) (((double) (a * ((double) (a * 0.5)))) + ((double) (eps * ((double) (0.16666666666666666 * ((double) pow(a, 3.0)))))))))))) * ((double) (((double) exp(((double) (b * eps)))) - 1.0))))));
	} else {
		VAR = ((double) (1.0 / b));
	}
	return VAR;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	60.4
Target	14.9
Herbie	31.7

\[\frac{a + b}{a \cdot b}\]

Derivation

1. Split input into 2 regimes

2. if b < -8.9086780986168568e104

   1. Initial program 53.0

      \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]

   2. Taylor expanded around 0 45.3

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

   3. Simplified 45.0

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

   if -8.9086780986168568e104 < b

   1. Initial program 61.7

      \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]

   2. Taylor expanded around 0 56.9

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

   3. Simplified 56.4

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \color{blue}{\left(b \cdot \varepsilon + \left(b \cdot b\right) \cdot \left(b \cdot \left(\frac{1}{6} \cdot {\varepsilon}^{3}\right) + \varepsilon \cdot \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right)}}\]
   4. Using strategy rm

   5. Applied associate-*l* 56.2

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

   6. Simplified 56.2

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(b \cdot \varepsilon + b \cdot \color{blue}{\left(b \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(b \cdot \frac{1}{6}\right) \cdot \varepsilon + \frac{1}{2}\right)\right)\right)}\right)}\]

   7. Taylor expanded around inf 56.2

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\color{blue}{\left(e^{a \cdot \varepsilon} - 1\right)} \cdot \left(b \cdot \varepsilon + b \cdot \left(b \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(b \cdot \frac{1}{6}\right) \cdot \varepsilon + \frac{1}{2}\right)\right)\right)\right)}\]

   8. Simplified 56.9

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\color{blue}{\left({\left(e^{a}\right)}^{\varepsilon} - 1\right)} \cdot \left(b \cdot \varepsilon + b \cdot \left(b \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(b \cdot \frac{1}{6}\right) \cdot \varepsilon + \frac{1}{2}\right)\right)\right)\right)}\]

   9. Taylor expanded around 0 29.4

      \[\leadsto \color{blue}{\frac{1}{b}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 31.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -8.9086780986168568 \cdot 10^{104}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(b + a\right)} - 1\right)}{\left(\varepsilon \cdot a + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(a \cdot \left(a \cdot \frac{1}{2}\right) + \varepsilon \cdot \left(\frac{1}{6} \cdot {a}^{3}\right)\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020179 
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :precision binary64
  :pre (and (< -1.0 eps) (< eps 1.0))

  :herbie-target
  (/ (+ a b) (* a b))

  (/ (* eps (- (exp (* (+ a b) eps)) 1.0)) (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0))))