expq3 (problem 3.4.2)

Average Error: 60.5 → 3.6

Time: 47.4s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -3.258139005630207 \cdot 10^{-81} \lor \neg \left(a \le 5.49877905837764948 \cdot 10^{-13}\right):\\ \;\;\;\;\frac{1}{b} \cdot \frac{b + a}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a}{b} \cdot \frac{1}{a}\\ \end{array}\]

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 <= -3.2581390056302074e-81) || !(a <= 5.498779058377649e-13))) {
		VAR = ((double) (((double) (1.0 / b)) * ((double) (((double) (b + a)) / a))));
	} else {
		VAR = ((double) (((double) (((double) (b + a)) / b)) * ((double) (1.0 / a))));
	}
	return VAR;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	60.5
Target	14.6
Herbie	3.6

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

Derivation

1. Split input into 2 regimes

2. if a < -3.258139005630207e-81 or 5.49877905837764948e-13 < a

   1. Initial program 57.3

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

   2. Taylor expanded around 0 6.2

      \[\leadsto \color{blue}{\frac{1}{b} + \frac{1}{a}}\]
   3. Using strategy rm

   4. Applied frac-add 10.3

      \[\leadsto \color{blue}{\frac{1 \cdot a + b \cdot 1}{b \cdot a}}\]

   5. Simplified 10.3

      \[\leadsto \frac{\color{blue}{b + a}}{b \cdot a}\]
   6. Using strategy rm

   7. Applied *-un-lft-identity 10.3

      \[\leadsto \frac{\color{blue}{1 \cdot \left(b + a\right)}}{b \cdot a}\]

   8. Applied times-frac 6.7

      \[\leadsto \color{blue}{\frac{1}{b} \cdot \frac{b + a}{a}}\]

   if -3.258139005630207e-81 < a < 5.49877905837764948e-13

   1. Initial program 64.0

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

   2. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\frac{1}{b} + \frac{1}{a}}\]
   3. Using strategy rm

   4. Applied frac-add 19.4

      \[\leadsto \color{blue}{\frac{1 \cdot a + b \cdot 1}{b \cdot a}}\]

   5. Simplified 19.4

      \[\leadsto \frac{\color{blue}{b + a}}{b \cdot a}\]
   6. Using strategy rm

   7. Applied *-un-lft-identity 19.4

      \[\leadsto \frac{\color{blue}{1 \cdot \left(b + a\right)}}{b \cdot a}\]

   8. Applied times-frac 14.5

      \[\leadsto \color{blue}{\frac{1}{b} \cdot \frac{b + a}{a}}\]
   9. Using strategy rm

   10. Applied div-inv 14.5

       \[\leadsto \frac{1}{b} \cdot \color{blue}{\left(\left(b + a\right) \cdot \frac{1}{a}\right)}\]

   11. Applied associate-*r* 0.2

       \[\leadsto \color{blue}{\left(\frac{1}{b} \cdot \left(b + a\right)\right) \cdot \frac{1}{a}}\]

   12. Simplified 0.1

       \[\leadsto \color{blue}{\frac{b + a}{b}} \cdot \frac{1}{a}\]
3. Recombined 2 regimes into one program.

4. Final simplification 3.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -3.258139005630207 \cdot 10^{-81} \lor \neg \left(a \le 5.49877905837764948 \cdot 10^{-13}\right):\\ \;\;\;\;\frac{1}{b} \cdot \frac{b + a}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a}{b} \cdot \frac{1}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020163 
(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))))