expq3 (problem 3.4.2)

Average Error: 60.5 → 53.3

Time: 13.0s

Precision: 64

• could not determine a ground truth for program body

  1. a = -1.5612152647874022e+129
  2. b = -1.623787643110698e+39
  3. eps = -5.263347100161414e-116

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -2942943340030.65479:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(\frac{1}{6} \cdot {\left(b \cdot \varepsilon\right)}^{3} + \left(\frac{1}{2} \cdot {\left(b \cdot \varepsilon\right)}^{2} + \varepsilon \cdot b\right)\right)}\\ \mathbf{elif}\;a \le 8.895710189523327 \cdot 10^{43}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(\frac{1}{6} \cdot \left({a}^{3} \cdot {\varepsilon}^{3}\right) + \left(\frac{1}{2} \cdot \left({a}^{2} \cdot {\varepsilon}^{2}\right) + a \cdot \varepsilon\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(\sqrt[3]{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot {b}^{3}\right) + \left(\frac{1}{2} \cdot \left({\varepsilon}^{2} \cdot {b}^{2}\right) + \varepsilon \cdot b\right)\right)} \cdot \sqrt[3]{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot {b}^{3}\right) + \left(\frac{1}{2} \cdot \left({\varepsilon}^{2} \cdot {b}^{2}\right) + \varepsilon \cdot b\right)\right)}\right) \cdot \sqrt[3]{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot {b}^{3}\right) + \left(\frac{1}{2} \cdot \left({\varepsilon}^{2} \cdot {b}^{2}\right) + \varepsilon \cdot b\right)\right)}}\\ \end{array}\]

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 VAR;
	if ((a <= -2942943340030.655)) {
		VAR = ((eps * (exp(((a + b) * eps)) - 1.0)) / ((exp((a * eps)) - 1.0) * ((0.16666666666666666 * pow((b * eps), 3.0)) + ((0.5 * pow((b * eps), 2.0)) + (eps * b)))));
	} else {
		double VAR_1;
		if ((a <= 8.895710189523327e+43)) {
			VAR_1 = ((eps * (exp(((a + b) * eps)) - 1.0)) / (((0.16666666666666666 * (pow(a, 3.0) * pow(eps, 3.0))) + ((0.5 * (pow(a, 2.0) * pow(eps, 2.0))) + (a * eps))) * (exp((b * eps)) - 1.0)));
		} else {
			VAR_1 = ((eps * (exp(((a + b) * eps)) - 1.0)) / ((cbrt(((exp((a * eps)) - 1.0) * ((0.16666666666666666 * (pow(eps, 3.0) * pow(b, 3.0))) + ((0.5 * (pow(eps, 2.0) * pow(b, 2.0))) + (eps * b))))) * cbrt(((exp((a * eps)) - 1.0) * ((0.16666666666666666 * (pow(eps, 3.0) * pow(b, 3.0))) + ((0.5 * (pow(eps, 2.0) * pow(b, 2.0))) + (eps * b)))))) * cbrt(((exp((a * eps)) - 1.0) * ((0.16666666666666666 * (pow(eps, 3.0) * pow(b, 3.0))) + ((0.5 * (pow(eps, 2.0) * pow(b, 2.0))) + (eps * b)))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	60.5
Target	15.0
Herbie	53.3

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

Derivation

1. Split input into 3 regimes

2. if a < -2942943340030.655

   1. Initial program 55.5

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

   2. Taylor expanded around 0 49.9

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

   4. Applied pow-prod-down 48.8

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

   5. Simplified 48.8

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

   7. Applied pow-prod-down 47.1

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

   8. Simplified 47.1

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

   if -2942943340030.655 < a < 8.895710189523327e+43

   1. Initial program 63.9

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

   2. Taylor expanded around 0 56.5

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

   if 8.895710189523327e+43 < a

   1. Initial program 55.1

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

   2. Taylor expanded around 0 49.3

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

   4. Applied add-cube-cbrt 49.5

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

4. Final simplification 53.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -2942943340030.65479:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(\frac{1}{6} \cdot {\left(b \cdot \varepsilon\right)}^{3} + \left(\frac{1}{2} \cdot {\left(b \cdot \varepsilon\right)}^{2} + \varepsilon \cdot b\right)\right)}\\ \mathbf{elif}\;a \le 8.895710189523327 \cdot 10^{43}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(\frac{1}{6} \cdot \left({a}^{3} \cdot {\varepsilon}^{3}\right) + \left(\frac{1}{2} \cdot \left({a}^{2} \cdot {\varepsilon}^{2}\right) + a \cdot \varepsilon\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(\sqrt[3]{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot {b}^{3}\right) + \left(\frac{1}{2} \cdot \left({\varepsilon}^{2} \cdot {b}^{2}\right) + \varepsilon \cdot b\right)\right)} \cdot \sqrt[3]{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot {b}^{3}\right) + \left(\frac{1}{2} \cdot \left({\varepsilon}^{2} \cdot {b}^{2}\right) + \varepsilon \cdot b\right)\right)}\right) \cdot \sqrt[3]{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot {b}^{3}\right) + \left(\frac{1}{2} \cdot \left({\varepsilon}^{2} \cdot {b}^{2}\right) + \varepsilon \cdot b\right)\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020079 
(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))))