expq3 (problem 3.4.2)

Average Error: 60.4 → 3.5

Time: 13.6s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -2.950754027435784 \cdot 10^{-43} \lor \neg \left(b \le 3.98465799141235432 \cdot 10^{-43}\right):\\ \;\;\;\;\frac{\frac{b + a}{b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} \cdot \frac{b + a}{a}\\ \end{array}\]

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 (((b <= -2.9507540274357845e-43) || !(b <= 3.984657991412354e-43))) {
		VAR = ((double) (((double) (((double) (b + a)) / b)) / a));
	} else {
		VAR = ((double) (((double) (1.0 / b)) * ((double) (((double) (b + a)) / a))));
	}
	return VAR;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	60.4
Target	15.2
Herbie	3.5

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

Derivation

1. Split input into 2 regimes

2. if b < -2.950754027435784e-43 or 3.98465799141235432e-43 < b

   1. Initial program 57.2

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

   2. Taylor expanded around 0 58.7

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

   3. Simplified 57.3

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

   4. Taylor expanded around 0 6.2

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

   6. Applied frac-add 10.7

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

   7. Simplified 10.7

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

   9. Applied associate-/r* 6.4

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

   if -2.950754027435784e-43 < b < 3.98465799141235432e-43

   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 57.4

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

   3. Simplified 57.4

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

   4. Taylor expanded around 0 0.0

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

   6. Applied frac-add 20.5

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

   7. Simplified 20.5

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

   9. Applied *-un-lft-identity 20.5

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

   10. Applied times-frac 0.1

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

4. Final simplification 3.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.950754027435784 \cdot 10^{-43} \lor \neg \left(b \le 3.98465799141235432 \cdot 10^{-43}\right):\\ \;\;\;\;\frac{\frac{b + a}{b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} \cdot \frac{b + a}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020171 
(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))))