expq3 (problem 3.4.2)

Average Error: 60.3 → 52.2

Time: 15.7s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -3.3763543922335703 \cdot 10^{65}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(\left(\sqrt[3]{e^{\left(a + b\right) \cdot \varepsilon} - 1} \cdot \sqrt[3]{e^{\left(a + b\right) \cdot \varepsilon} - 1}\right) \cdot \sqrt[3]{e^{\left(a + b\right) \cdot \varepsilon} - 1}\right)}{\left(\frac{1}{6} \cdot {\left(\varepsilon \cdot a\right)}^{3} + \left(\frac{1}{2} \cdot {\left(\varepsilon \cdot a\right)}^{2} + a \cdot \varepsilon\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \mathbf{elif}\;b \le 4.13103216890765473 \cdot 10^{66}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot {b}^{3}\right) + \left(\frac{1}{2} \cdot \left({\varepsilon}^{2} \cdot {b}^{2}\right) + \varepsilon \cdot b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(\frac{1}{6} \cdot {\left(\varepsilon \cdot a\right)}^{3} + \left(\frac{1}{2} \cdot {\left(\varepsilon \cdot a\right)}^{2} + a \cdot \varepsilon\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \end{array}\]

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 <= -3.3763543922335703e+65)) {
		VAR = ((double) (((double) (eps * ((double) (((double) (((double) cbrt(((double) (((double) exp(((double) (((double) (a + b)) * eps)))) - 1.0)))) * ((double) cbrt(((double) (((double) exp(((double) (((double) (a + b)) * eps)))) - 1.0)))))) * ((double) cbrt(((double) (((double) exp(((double) (((double) (a + b)) * eps)))) - 1.0)))))))) / ((double) (((double) (((double) (0.16666666666666666 * ((double) pow(((double) (eps * a)), 3.0)))) + ((double) (((double) (0.5 * ((double) pow(((double) (eps * a)), 2.0)))) + ((double) (a * eps)))))) * ((double) (((double) exp(((double) (b * eps)))) - 1.0))))));
	} else {
		double VAR_1;
		if ((b <= 4.131032168907655e+66)) {
			VAR_1 = ((double) (((double) (eps * ((double) (((double) exp(((double) (((double) (a + b)) * eps)))) - 1.0)))) / ((double) (((double) (((double) exp(((double) (a * eps)))) - 1.0)) * ((double) (((double) (0.16666666666666666 * ((double) (((double) pow(eps, 3.0)) * ((double) pow(b, 3.0)))))) + ((double) (((double) (0.5 * ((double) (((double) pow(eps, 2.0)) * ((double) pow(b, 2.0)))))) + ((double) (eps * b))))))))));
		} else {
			VAR_1 = ((double) (((double) (eps * ((double) (((double) exp(((double) (((double) (a + b)) * eps)))) - 1.0)))) / ((double) (((double) (((double) (0.16666666666666666 * ((double) pow(((double) (eps * a)), 3.0)))) + ((double) (((double) (0.5 * ((double) pow(((double) (eps * a)), 2.0)))) + ((double) (a * eps)))))) * ((double) (((double) exp(((double) (b * eps)))) - 1.0))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	60.3
Target	14.9
Herbie	52.2

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

Derivation

1. Split input into 3 regimes

2. if b < -3.3763543922335703e65

   1. Initial program 53.4

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

   2. Taylor expanded around 0 48.3

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

   4. Applied pow-prod-down 47.2

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

   5. Simplified 47.2

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

   7. Applied pow-prod-down 45.4

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

   8. Simplified 45.4

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

   10. Applied add-cube-cbrt 45.4

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

   if -3.3763543922335703e65 < b < 4.13103216890765473e66

   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

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

   if 4.13103216890765473e66 < b

   1. Initial program 54.0

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

   2. Taylor expanded around 0 46.2

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

   4. Applied pow-prod-down 44.9

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

   5. Simplified 44.9

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

   7. Applied pow-prod-down 43.2

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

   8. Simplified 43.2

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

4. Final simplification 52.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.3763543922335703 \cdot 10^{65}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(\left(\sqrt[3]{e^{\left(a + b\right) \cdot \varepsilon} - 1} \cdot \sqrt[3]{e^{\left(a + b\right) \cdot \varepsilon} - 1}\right) \cdot \sqrt[3]{e^{\left(a + b\right) \cdot \varepsilon} - 1}\right)}{\left(\frac{1}{6} \cdot {\left(\varepsilon \cdot a\right)}^{3} + \left(\frac{1}{2} \cdot {\left(\varepsilon \cdot a\right)}^{2} + a \cdot \varepsilon\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \mathbf{elif}\;b \le 4.13103216890765473 \cdot 10^{66}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot {b}^{3}\right) + \left(\frac{1}{2} \cdot \left({\varepsilon}^{2} \cdot {b}^{2}\right) + \varepsilon \cdot b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(\frac{1}{6} \cdot {\left(\varepsilon \cdot a\right)}^{3} + \left(\frac{1}{2} \cdot {\left(\varepsilon \cdot a\right)}^{2} + a \cdot \varepsilon\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020177 
(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))))