expq3 (problem 3.4.2)

Average Error: 60.4 → 52.3

Time: 11.1s

Precision: binary64

\[-1 < \varepsilon \land \varepsilon < 1\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \leq -4497903.061841475:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(\varepsilon \cdot b + b \cdot \left(b \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(b \cdot 0.16666666666666666\right) + 0.5\right)\right)\right)\right)}\\ \mathbf{elif}\;a \leq 8.249043684066252 \cdot 10^{+42}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(a \cdot \varepsilon + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(0.5 \cdot \left(a \cdot a\right) + \varepsilon \cdot \left(0.16666666666666666 \cdot {a}^{3}\right)\right)\right) \cdot \left(e^{\varepsilon \cdot b} - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(\varepsilon \cdot b + b \cdot \left(b \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(b \cdot 0.16666666666666666\right) + 0.5\right)\right)\right)\right)}\\ \end{array}\]

FPCore

(FPCore (a b eps)
 :precision binary64
 (/
  (* eps (- (exp (* (+ a b) eps)) 1.0))
  (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0))))

↓

(FPCore (a b eps)
 :precision binary64
 (if (<= a -4497903.061841475)
   (/
    (* eps (- (exp (* eps (+ a b))) 1.0))
    (*
     (- (exp (* a eps)) 1.0)
     (+
      (* eps b)
      (* b (* b (* (* eps eps) (+ (* eps (* b 0.16666666666666666)) 0.5)))))))
   (if (<= a 8.249043684066252e+42)
     (/
      (* eps (- (exp (* eps (+ a b))) 1.0))
      (*
       (+
        (* a eps)
        (*
         (* eps eps)
         (+ (* 0.5 (* a a)) (* eps (* 0.16666666666666666 (pow a 3.0))))))
       (- (exp (* eps b)) 1.0)))
     (/
      (* eps (- (exp (* eps (+ a b))) 1.0))
      (*
       (- (exp (* a eps)) 1.0)
       (+
        (* eps b)
        (*
         b
         (* b (* (* eps eps) (+ (* eps (* b 0.16666666666666666)) 0.5))))))))))

C

double code(double a, double b, double eps) {
	return (eps * (exp((a + b) * eps) - 1.0)) / ((exp(a * eps) - 1.0) * (exp(b * eps) - 1.0));
}

↓

double code(double a, double b, double eps) {
	double tmp;
	if (a <= -4497903.061841475) {
		tmp = (eps * (exp(eps * (a + b)) - 1.0)) / ((exp(a * eps) - 1.0) * ((eps * b) + (b * (b * ((eps * eps) * ((eps * (b * 0.16666666666666666)) + 0.5))))));
	} else if (a <= 8.249043684066252e+42) {
		tmp = (eps * (exp(eps * (a + b)) - 1.0)) / (((a * eps) + ((eps * eps) * ((0.5 * (a * a)) + (eps * (0.16666666666666666 * pow(a, 3.0)))))) * (exp(eps * b) - 1.0));
	} else {
		tmp = (eps * (exp(eps * (a + b)) - 1.0)) / ((exp(a * eps) - 1.0) * ((eps * b) + (b * (b * ((eps * eps) * ((eps * (b * 0.16666666666666666)) + 0.5))))));
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	60.4
Target	15.5
Herbie	52.3

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

Derivation

1. Split input into 2 regimes

2. if a < -4497903.0618414748 or 8.2490436840662522e42 < a

   1. Initial program 55.0

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

   2. Taylor expanded around 0 49.5

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \color{blue}{\left(\varepsilon \cdot b + \left(0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot {b}^{3}\right) + 0.5 \cdot \left({\varepsilon}^{2} \cdot {b}^{2}\right)\right)\right)}}\]

   3. Simplified 47.1

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \color{blue}{\left(b \cdot \varepsilon + \left(b \cdot b\right) \cdot \left(b \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3}\right) + 0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)}}\]
   4. Using strategy rm

   5. Applied associate-*l*_binary64 46.2

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(b \cdot \varepsilon + \color{blue}{b \cdot \left(b \cdot \left(b \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3}\right) + 0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)}\right)}\]

   6. Simplified 46.1

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(b \cdot \varepsilon + b \cdot \color{blue}{\left(b \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(b \cdot 0.16666666666666666\right) \cdot \varepsilon + 0.5\right)\right)\right)}\right)}\]

   if -4497903.0618414748 < a < 8.2490436840662522e42

   1. Initial program 63.9

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

   2. Taylor expanded around 0 56.5

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\color{blue}{\left(a \cdot \varepsilon + \left(0.5 \cdot \left({a}^{2} \cdot {\varepsilon}^{2}\right) + 0.16666666666666666 \cdot \left({a}^{3} \cdot {\varepsilon}^{3}\right)\right)\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]

   3. Simplified 56.5

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\color{blue}{\left(a \cdot \varepsilon + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(0.5 \cdot \left(a \cdot a\right) + \varepsilon \cdot \left(0.16666666666666666 \cdot {a}^{3}\right)\right)\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 52.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4497903.061841475:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(\varepsilon \cdot b + b \cdot \left(b \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(b \cdot 0.16666666666666666\right) + 0.5\right)\right)\right)\right)}\\ \mathbf{elif}\;a \leq 8.249043684066252 \cdot 10^{+42}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(a \cdot \varepsilon + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(0.5 \cdot \left(a \cdot a\right) + \varepsilon \cdot \left(0.16666666666666666 \cdot {a}^{3}\right)\right)\right) \cdot \left(e^{\varepsilon \cdot b} - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(\varepsilon \cdot b + b \cdot \left(b \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(b \cdot 0.16666666666666666\right) + 0.5\right)\right)\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020273 
(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))))