expq3 (problem 3.4.2)

Average Error: 60.2 → 25.3

Time: 20.2s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \leq -1.1449809168537672 \cdot 10^{+112} \lor \neg \left(a \leq 1.6259980530357452 \cdot 10^{+21}\right):\\ \;\;\;\;\frac{1}{\frac{{\left(e^{a}\right)}^{\varepsilon} - 1}{\frac{{\left(e^{a + b}\right)}^{\varepsilon} - 1}{b + \left(\log 1 + \varepsilon \cdot \left(0.5 \cdot {\left(\log 1\right)}^{2}\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a + \left(\log 1 + \left(a \cdot \varepsilon\right) \cdot \left(\log 1 \cdot 0.49999999999999994\right)\right)}\\ \end{array}\]

C

double code(double a, double b, double eps) {
	return (((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 <= -1.1449809168537672e+112) || !(a <= 1.6259980530357452e+21))) {
		VAR = (1.0 / (((double) (((double) pow(((double) exp(a)), eps)) - 1.0)) / (((double) (((double) pow(((double) exp(((double) (a + b)))), eps)) - 1.0)) / ((double) (b + ((double) (((double) log(1.0)) + ((double) (eps * ((double) (0.5 * ((double) pow(((double) log(1.0)), 2.0)))))))))))));
	} else {
		VAR = (1.0 / ((double) (a + ((double) (((double) log(1.0)) + ((double) (((double) (a * eps)) * ((double) (((double) log(1.0)) * 0.49999999999999994)))))))));
	}
	return VAR;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	60.2
Target	15.3
Herbie	25.3

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

Derivation

1. Split input into 2 regimes

2. if a < -1.1449809168537672e112 or 1625998053035745210000 < a

   1. Initial program 53.9

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

   2. Simplified 54.0

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

   3. Taylor expanded around 0 40.4

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

   4. Simplified 40.4

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

   6. Applied associate-*r/ 39.7

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

   8. Applied clear-num 39.6

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

   9. Simplified 29.3

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

   if -1.1449809168537672e112 < a < 1625998053035745210000

   1. Initial program 63.3

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

   2. Simplified 63.4

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

   3. Taylor expanded around 0 61.5

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

   4. Simplified 61.5

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

   6. Applied associate-*r/ 61.4

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

   8. Applied clear-num 61.3

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

   9. Simplified 59.7

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

   10. Taylor expanded around 0 23.3

       \[\leadsto \frac{1}{\color{blue}{\log 1 + \left(a + 0.49999999999999994 \cdot \left(\log 1 \cdot \left(a \cdot \varepsilon\right)\right)\right)}}\]

   11. Simplified 23.3

       \[\leadsto \frac{1}{\color{blue}{a + \left(\left(a \cdot \varepsilon\right) \cdot \left(\log 1 \cdot 0.49999999999999994\right) + \log 1\right)}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 25.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.1449809168537672 \cdot 10^{+112} \lor \neg \left(a \leq 1.6259980530357452 \cdot 10^{+21}\right):\\ \;\;\;\;\frac{1}{\frac{{\left(e^{a}\right)}^{\varepsilon} - 1}{\frac{{\left(e^{a + b}\right)}^{\varepsilon} - 1}{b + \left(\log 1 + \varepsilon \cdot \left(0.5 \cdot {\left(\log 1\right)}^{2}\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a + \left(\log 1 + \left(a \cdot \varepsilon\right) \cdot \left(\log 1 \cdot 0.49999999999999994\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020196 
(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))))