sqrtexp (problem 3.4.4)

Average Error: 5.3 → 0.9

Time: 8.2s

Precision: 64

• could not determine a ground truth for program body

  1. x = 8.598898606995893e+135

Math

\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.62109741641090893 \cdot 10^{-13}:\\ \;\;\;\;\sqrt{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \frac{\sqrt{e^{2 \cdot x}} - \sqrt{1}}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left(\sqrt{2} \cdot \sqrt{2} - \left(\frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right) \cdot \left(\frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\right) + 0.5 \cdot \left(x \cdot \left(\sqrt{2} - \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\right)}{\sqrt{2} \cdot \left(\sqrt{2} - \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)}\\ \end{array}\]

C

double code(double x) {
	return sqrt(((exp((2.0 * x)) - 1.0) / (exp(x) - 1.0)));
}

↓

double code(double x) {
	double temp;
	if ((x <= -1.621097416410909e-13)) {
		temp = sqrt(((sqrt(exp((2.0 * x))) + sqrt(1.0)) * ((sqrt(exp((2.0 * x))) - sqrt(1.0)) / (exp(x) - 1.0))));
	} else {
		temp = (((sqrt(2.0) * ((sqrt(2.0) * sqrt(2.0)) - (((pow(x, 2.0) / sqrt(2.0)) * (0.25 - (0.125 / 2.0))) * ((pow(x, 2.0) / sqrt(2.0)) * (0.25 - (0.125 / 2.0)))))) + (0.5 * (x * (sqrt(2.0) - ((pow(x, 2.0) / sqrt(2.0)) * (0.25 - (0.125 / 2.0))))))) / (sqrt(2.0) * (sqrt(2.0) - ((pow(x, 2.0) / sqrt(2.0)) * (0.25 - (0.125 / 2.0))))));
	}
	return temp;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -1.621097416410909e-13

   1. Initial program 0.6

      \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 0.6

      \[\leadsto \sqrt{\frac{e^{2 \cdot x} - 1}{\color{blue}{1 \cdot \left(e^{x} - 1\right)}}}\]

   4. Applied add-sqr-sqrt 0.6

      \[\leadsto \sqrt{\frac{e^{2 \cdot x} - \color{blue}{\sqrt{1} \cdot \sqrt{1}}}{1 \cdot \left(e^{x} - 1\right)}}\]

   5. Applied add-sqr-sqrt 0.6

      \[\leadsto \sqrt{\frac{\color{blue}{\sqrt{e^{2 \cdot x}} \cdot \sqrt{e^{2 \cdot x}}} - \sqrt{1} \cdot \sqrt{1}}{1 \cdot \left(e^{x} - 1\right)}}\]

   6. Applied difference-of-squares 0.2

      \[\leadsto \sqrt{\frac{\color{blue}{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{2 \cdot x}} - \sqrt{1}\right)}}{1 \cdot \left(e^{x} - 1\right)}}\]

   7. Applied times-frac 0.2

      \[\leadsto \sqrt{\color{blue}{\frac{\sqrt{e^{2 \cdot x}} + \sqrt{1}}{1} \cdot \frac{\sqrt{e^{2 \cdot x}} - \sqrt{1}}{e^{x} - 1}}}\]

   8. Simplified 0.2

      \[\leadsto \sqrt{\color{blue}{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right)} \cdot \frac{\sqrt{e^{2 \cdot x}} - \sqrt{1}}{e^{x} - 1}}\]

   if -1.621097416410909e-13 < x

   1. Initial program 39.5

      \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]

   2. Taylor expanded around 0 6.2

      \[\leadsto \color{blue}{\left(0.25 \cdot \frac{{x}^{2}}{\sqrt{2}} + \left(\sqrt{2} + 0.5 \cdot \frac{x}{\sqrt{2}}\right)\right) - 0.125 \cdot \frac{{x}^{2}}{{\left(\sqrt{2}\right)}^{3}}}\]

   3. Simplified 6.2

      \[\leadsto \color{blue}{0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)}\]
   4. Using strategy rm

   5. Applied flip-+ 6.9

      \[\leadsto 0.5 \cdot \frac{x}{\sqrt{2}} + \color{blue}{\frac{\sqrt{2} \cdot \sqrt{2} - \left(\frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right) \cdot \left(\frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)}{\sqrt{2} - \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)}}\]

   6. Applied associate-*r/ 6.9

      \[\leadsto \color{blue}{\frac{0.5 \cdot x}{\sqrt{2}}} + \frac{\sqrt{2} \cdot \sqrt{2} - \left(\frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right) \cdot \left(\frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)}{\sqrt{2} - \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)}\]

   7. Applied frac-add 6.2

      \[\leadsto \color{blue}{\frac{\left(0.5 \cdot x\right) \cdot \left(\sqrt{2} - \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right) + \sqrt{2} \cdot \left(\sqrt{2} \cdot \sqrt{2} - \left(\frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right) \cdot \left(\frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\right)}{\sqrt{2} \cdot \left(\sqrt{2} - \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)}}\]

   8. Simplified 6.2

      \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \left(\sqrt{2} \cdot \sqrt{2} - \left(\frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right) \cdot \left(\frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\right) + 0.5 \cdot \left(x \cdot \left(\sqrt{2} - \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\right)}}{\sqrt{2} \cdot \left(\sqrt{2} - \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.62109741641090893 \cdot 10^{-13}:\\ \;\;\;\;\sqrt{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \frac{\sqrt{e^{2 \cdot x}} - \sqrt{1}}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left(\sqrt{2} \cdot \sqrt{2} - \left(\frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right) \cdot \left(\frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\right) + 0.5 \cdot \left(x \cdot \left(\sqrt{2} - \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\right)}{\sqrt{2} \cdot \left(\sqrt{2} - \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))