expq3 (problem 3.4.2)

Average Error: 60.3 → 54.4

Time: 12.3s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \leq -1.8812294181881513 \cdot 10^{+209} \lor \neg \left(a \leq 1.0020287718930736 \cdot 10^{+105}\right):\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot {b}^{3}\right) + b \cdot \left(\varepsilon + b \cdot \left(0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \sqrt[3]{{\left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}^{3}}}{\left(\varepsilon \cdot \left(a + \varepsilon \cdot \left(0.5 \cdot \left(a \cdot a\right)\right)\right)\right) \cdot \sqrt[3]{{\left(-1 + e^{\varepsilon \cdot b}\right)}^{3}}}\\ \end{array}\]

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 (or (<= a -1.8812294181881513e+209) (not (<= a 1.0020287718930736e+105)))
   (/
    (* eps (- (exp (* eps (+ a b))) 1.0))
    (*
     (- (exp (* a eps)) 1.0)
     (+
      (* 0.16666666666666666 (* (pow eps 3.0) (pow b 3.0)))
      (* b (+ eps (* b (* 0.5 (* eps eps))))))))
   (/
    (* eps (cbrt (pow (- (exp (* eps (+ a b))) 1.0) 3.0)))
    (*
     (* eps (+ a (* eps (* 0.5 (* a a)))))
     (cbrt (pow (+ -1.0 (exp (* eps b))) 3.0))))))

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 <= -1.8812294181881513e+209) || !(a <= 1.0020287718930736e+105)) {
		tmp = (eps * (exp(eps * (a + b)) - 1.0)) / ((exp(a * eps) - 1.0) * ((0.16666666666666666 * (pow(eps, 3.0) * pow(b, 3.0))) + (b * (eps + (b * (0.5 * (eps * eps)))))));
	} else {
		tmp = (eps * cbrt(pow((exp(eps * (a + b)) - 1.0), 3.0))) / ((eps * (a + (eps * (0.5 * (a * a))))) * cbrt(pow((-1.0 + exp(eps * b)), 3.0)));
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	60.3
Target	15.3
Herbie	54.4

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

Derivation

1. Split input into 2 regimes

2. if a < -1.88122e209 or 1.00202e105 < a

   1. Initial program 52.6

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

   2. Taylor expanded around 0 44.4

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

   3. Simplified 44.4

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

   if -1.88122e209 < a < 1.00202e105

   1. Initial program 62.2

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

   2. Taylor expanded around 0 57.1

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

   3. Simplified 57.1

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

   4. Taylor expanded around 0 56.9

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

   5. Simplified 56.8

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

   7. Applied add-cbrt-cube_binary64_546 56.9

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

   8. Simplified 56.9

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

   10. Applied add-cbrt-cube_binary64_546 56.8

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

   11. Simplified 56.8

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

4. Final simplification 54.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.8812294181881513 \cdot 10^{+209} \lor \neg \left(a \leq 1.0020287718930736 \cdot 10^{+105}\right):\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot {b}^{3}\right) + b \cdot \left(\varepsilon + b \cdot \left(0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \sqrt[3]{{\left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}^{3}}}{\left(\varepsilon \cdot \left(a + \varepsilon \cdot \left(0.5 \cdot \left(a \cdot a\right)\right)\right)\right) \cdot \sqrt[3]{{\left(-1 + e^{\varepsilon \cdot b}\right)}^{3}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020231 
(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))))