expq3 (problem 3.4.2)

Average Error: 60.1 → 52.4

Time: 14.4s

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -8.6891423089943577 \cdot 10^{66}:\\ \;\;\;\;\frac{\varepsilon \cdot \sqrt[3]{{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}^{3}}}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(\left(b \cdot b\right) \cdot \left(\left(\frac{1}{6} \cdot {\varepsilon}^{3}\right) \cdot b + \frac{1}{2} \cdot {\varepsilon}^{2}\right) + b \cdot \varepsilon\right)}\\ \mathbf{elif}\;a \le 0.119145202101303846:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(a \cdot \varepsilon + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{2} \cdot {a}^{2} + \left(\frac{1}{6} \cdot {a}^{3}\right) \cdot \varepsilon\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(\sqrt[3]{b \cdot \left(b \cdot \left(\left(\frac{1}{6} \cdot {\varepsilon}^{3}\right) \cdot b + \frac{1}{2} \cdot {\varepsilon}^{2}\right) + \varepsilon\right)} \cdot \sqrt[3]{b \cdot \left(b \cdot \left(\left(\frac{1}{6} \cdot {\varepsilon}^{3}\right) \cdot b + \frac{1}{2} \cdot {\varepsilon}^{2}\right) + \varepsilon\right)}\right)\right) \cdot \sqrt[3]{\left(b \cdot b\right) \cdot \left(\left(\frac{1}{6} \cdot {\varepsilon}^{3}\right) \cdot b + \frac{1}{2} \cdot {\varepsilon}^{2}\right) + b \cdot \varepsilon}}\\ \end{array}\]

C

double code(double a, double b, double eps) {
	return (((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 ((a <= -8.689142308994358e+66)) {
		VAR = (((double) (eps * ((double) cbrt(((double) pow(((double) (((double) exp(((double) (((double) (a + b)) * eps)))) - 1.0)), 3.0)))))) / ((double) (((double) (((double) exp(((double) (a * eps)))) - 1.0)) * ((double) (((double) (((double) (b * b)) * ((double) (((double) (((double) (0.16666666666666666 * ((double) pow(eps, 3.0)))) * b)) + ((double) (0.5 * ((double) pow(eps, 2.0)))))))) + ((double) (b * eps)))))));
	} else {
		double VAR_1;
		if ((a <= 0.11914520210130385)) {
			VAR_1 = (((double) (eps * ((double) (((double) exp(((double) (((double) (a + b)) * eps)))) - 1.0)))) / ((double) (((double) (((double) (a * eps)) + ((double) (((double) (eps * eps)) * ((double) (((double) (0.5 * ((double) pow(a, 2.0)))) + ((double) (((double) (0.16666666666666666 * ((double) pow(a, 3.0)))) * eps)))))))) * ((double) (((double) exp(((double) (b * eps)))) - 1.0)))));
		} else {
			VAR_1 = (((double) (eps * ((double) (((double) exp(((double) (((double) (a + b)) * eps)))) - 1.0)))) / ((double) (((double) (((double) (((double) exp(((double) (a * eps)))) - 1.0)) * ((double) (((double) cbrt(((double) (b * ((double) (((double) (b * ((double) (((double) (((double) (0.16666666666666666 * ((double) pow(eps, 3.0)))) * b)) + ((double) (0.5 * ((double) pow(eps, 2.0)))))))) + eps)))))) * ((double) cbrt(((double) (b * ((double) (((double) (b * ((double) (((double) (((double) (0.16666666666666666 * ((double) pow(eps, 3.0)))) * b)) + ((double) (0.5 * ((double) pow(eps, 2.0)))))))) + eps)))))))))) * ((double) cbrt(((double) (((double) (((double) (b * b)) * ((double) (((double) (((double) (0.16666666666666666 * ((double) pow(eps, 3.0)))) * b)) + ((double) (0.5 * ((double) pow(eps, 2.0)))))))) + ((double) (b * eps)))))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	60.1
Target	14.8
Herbie	52.4

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

Derivation

1. Split input into 3 regimes

2. if a < -8.6891423089943577e66

   1. Initial program 53.5

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

   2. Taylor expanded around 0 49.4

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

   3. Simplified 45.7

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

   5. Applied add-cbrt-cube 45.7

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

   6. Simplified 45.7

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

   if -8.6891423089943577e66 < a < 0.119145202101303846

   1. Initial program 63.7

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

   2. Taylor expanded around 0 56.2

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

   3. Simplified 56.2

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

   if 0.119145202101303846 < a

   1. Initial program 55.9

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

   2. Taylor expanded around 0 49.6

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

   3. Simplified 47.3

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

   5. Applied add-cube-cbrt 47.6

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

   6. Applied associate-*r* 47.6

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

   7. Simplified 47.6

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

4. Final simplification 52.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -8.6891423089943577 \cdot 10^{66}:\\ \;\;\;\;\frac{\varepsilon \cdot \sqrt[3]{{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}^{3}}}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(\left(b \cdot b\right) \cdot \left(\left(\frac{1}{6} \cdot {\varepsilon}^{3}\right) \cdot b + \frac{1}{2} \cdot {\varepsilon}^{2}\right) + b \cdot \varepsilon\right)}\\ \mathbf{elif}\;a \le 0.119145202101303846:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(a \cdot \varepsilon + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{2} \cdot {a}^{2} + \left(\frac{1}{6} \cdot {a}^{3}\right) \cdot \varepsilon\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(\sqrt[3]{b \cdot \left(b \cdot \left(\left(\frac{1}{6} \cdot {\varepsilon}^{3}\right) \cdot b + \frac{1}{2} \cdot {\varepsilon}^{2}\right) + \varepsilon\right)} \cdot \sqrt[3]{b \cdot \left(b \cdot \left(\left(\frac{1}{6} \cdot {\varepsilon}^{3}\right) \cdot b + \frac{1}{2} \cdot {\varepsilon}^{2}\right) + \varepsilon\right)}\right)\right) \cdot \sqrt[3]{\left(b \cdot b\right) \cdot \left(\left(\frac{1}{6} \cdot {\varepsilon}^{3}\right) \cdot b + \frac{1}{2} \cdot {\varepsilon}^{2}\right) + b \cdot \varepsilon}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(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))))