expq3 (problem 3.4.2)

Average Error: 60.3 → 52.2

Time: 12.3s

Precision: 64

• could not determine a ground truth for program body

  1. a = -1.0886249986002586e+19
  2. b = 1.3590711598542108e+288
  3. eps = 1.2264523170516892e-147

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -1.53739497967038146 \cdot 10^{77} \lor \neg \left(a \le 4.30903653173779036 \cdot 10^{51}\right):\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\sqrt[3]{{\left(e^{a \cdot \varepsilon} - 1\right)}^{3}} \cdot \mathsf{fma}\left(\frac{1}{6}, \left(\left({\varepsilon}^{3} \cdot b\right) \cdot b\right) \cdot b, \mathsf{fma}\left(\frac{1}{2}, \left(\left|\varepsilon\right| \cdot b\right) \cdot \left(\left|\varepsilon\right| \cdot b\right), \varepsilon \cdot b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {\varepsilon}^{3}, \mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {\varepsilon}^{2}, a \cdot \varepsilon\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \end{array}\]

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 (((a <= -1.5373949796703815e+77) || !(a <= 4.3090365317377904e+51))) {
		VAR = ((double) (((double) (eps * ((double) (((double) exp(((double) (((double) (a + b)) * eps)))) - 1.0)))) / ((double) (((double) cbrt(((double) pow(((double) (((double) exp(((double) (a * eps)))) - 1.0)), 3.0)))) * ((double) fma(0.16666666666666666, ((double) (((double) (((double) (((double) pow(eps, 3.0)) * b)) * b)) * b)), ((double) fma(0.5, ((double) (((double) (((double) fabs(eps)) * b)) * ((double) (((double) fabs(eps)) * b)))), ((double) (eps * b))))))))));
	} else {
		VAR = ((double) (((double) (eps * ((double) (((double) exp(((double) (((double) (a + b)) * eps)))) - 1.0)))) / ((double) (((double) fma(0.16666666666666666, ((double) (((double) pow(a, 3.0)) * ((double) pow(eps, 3.0)))), ((double) fma(0.5, ((double) (((double) pow(a, 2.0)) * ((double) pow(eps, 2.0)))), ((double) (a * eps)))))) * ((double) (((double) exp(((double) (b * eps)))) - 1.0))))));
	}
	return VAR;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	60.3
Target	14.6
Herbie	52.2

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

Derivation

1. Split input into 2 regimes

2. if a < -1.5373949796703815e+77 or 4.3090365317377904e+51 < a

   1. Initial program 53.8

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

   2. Taylor expanded around 0 48.5

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

   3. Simplified 48.5

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

   5. Applied unpow3 48.5

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

   6. Applied associate-*r* 47.5

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

   7. Simplified 46.7

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

   9. Applied add-sqr-sqrt 55.0

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

   10. Applied unpow-prod-down 55.0

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

   11. Applied add-sqr-sqrt 55.0

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

   12. Applied unswap-sqr 54.4

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

   13. Simplified 54.4

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

   14. Simplified 45.1

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

   16. Applied add-cbrt-cube 45.1

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

   17. Simplified 45.1

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

   if -1.5373949796703815e+77 < a < 4.3090365317377904e+51

   1. Initial program 63.6

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

   2. Taylor expanded around 0 55.8

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

   3. Simplified 55.8

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

4. Final simplification 52.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -1.53739497967038146 \cdot 10^{77} \lor \neg \left(a \le 4.30903653173779036 \cdot 10^{51}\right):\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\sqrt[3]{{\left(e^{a \cdot \varepsilon} - 1\right)}^{3}} \cdot \mathsf{fma}\left(\frac{1}{6}, \left(\left({\varepsilon}^{3} \cdot b\right) \cdot b\right) \cdot b, \mathsf{fma}\left(\frac{1}{2}, \left(\left|\varepsilon\right| \cdot b\right) \cdot \left(\left|\varepsilon\right| \cdot b\right), \varepsilon \cdot b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {\varepsilon}^{3}, \mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {\varepsilon}^{2}, a \cdot \varepsilon\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020114 +o rules:numerics
(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))))