expq3 (problem 3.4.2)

Average Error: 60.5 → 52.3

Time: 11.0s

Precision: 64

• could not determine a ground truth for program body

  1. a = -1.5170616589416367e-08
  2. b = 2.940293384203835e+300
  3. eps = 2.3524677729413477e-261

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -1.32961010093359485 \cdot 10^{37} \lor \neg \left(a \le 1.5041808259877258 \cdot 10^{32}\right):\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(b \cdot \left(\left(\left(\frac{1}{2} \cdot {\varepsilon}^{2}\right) \cdot b + \left(\frac{1}{6} \cdot \left(\varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot b\right)\right)\right) \cdot {b}^{\left(\frac{2}{2}\right)}\right) + \varepsilon\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(\varepsilon \cdot \left(\left(\left(\frac{1}{2} \cdot {a}^{2}\right) \cdot \varepsilon + \left(\frac{1}{6} \cdot {a}^{3}\right) \cdot {\varepsilon}^{2}\right) + a\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \end{array}\]

C

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

↓

double code(double a, double b, double eps) {
	double temp;
	if (((a <= -1.3296101009335949e+37) || !(a <= 1.5041808259877258e+32))) {
		temp = ((eps * (exp(((a + b) * eps)) - 1.0)) / ((exp((a * eps)) - 1.0) * (b * ((((0.5 * pow(eps, 2.0)) * b) + ((0.16666666666666666 * (eps * ((eps * eps) * b))) * pow(b, (2.0 / 2.0)))) + eps))));
	} else {
		temp = ((eps * (exp(((a + b) * eps)) - 1.0)) / ((eps * ((((0.5 * pow(a, 2.0)) * eps) + ((0.16666666666666666 * pow(a, 3.0)) * pow(eps, 2.0))) + a)) * (exp((b * eps)) - 1.0)));
	}
	return temp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	60.5
Target	14.7
Herbie	52.3

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

Derivation

1. Split input into 2 regimes

2. if a < -1.3296101009335949e+37 or 1.5041808259877258e+32 < a

   1. Initial program 55.0

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

   2. Taylor expanded around 0 49.5

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

   3. Simplified 47.5

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

   5. Applied sqr-pow 47.5

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

   6. Applied associate-*r* 45.6

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

   7. Simplified 45.6

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

   9. Applied cube-mult 45.6

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

   10. Applied associate-*l* 45.5

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

   if -1.3296101009335949e+37 < a < 1.5041808259877258e+32

   1. Initial program 63.8

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

   2. Taylor expanded around 0 56.4

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

   3. Simplified 56.4

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

4. Final simplification 52.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -1.32961010093359485 \cdot 10^{37} \lor \neg \left(a \le 1.5041808259877258 \cdot 10^{32}\right):\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(b \cdot \left(\left(\left(\frac{1}{2} \cdot {\varepsilon}^{2}\right) \cdot b + \left(\frac{1}{6} \cdot \left(\varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot b\right)\right)\right) \cdot {b}^{\left(\frac{2}{2}\right)}\right) + \varepsilon\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(\varepsilon \cdot \left(\left(\left(\frac{1}{2} \cdot {a}^{2}\right) \cdot \varepsilon + \left(\frac{1}{6} \cdot {a}^{3}\right) \cdot {\varepsilon}^{2}\right) + a\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020053 
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :precision binary64
  :pre (and (< -1 eps) (< eps 1))

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

  (/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))))