sqrtexp (problem 3.4.4)

Average Error: 4.4 → 0.8

Time: 5.3s

Precision: 64

• could not determine a ground truth for program body

  1. x = 6.317567976138939e+21

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.11558615421983944 \cdot 10^{-16}:\\ \;\;\;\;\sqrt{\left({\left(\sqrt{e^{2}}\right)}^{\left(\frac{1}{2} \cdot x\right)} \cdot {\left(\sqrt{e^{2}}\right)}^{\left(\frac{1}{2} \cdot x\right)} + \sqrt{1}\right) \cdot \frac{{\left(e^{2}\right)}^{\left(\frac{1}{2} \cdot x\right)} - \sqrt{1}}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\\ \end{array}\]

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 <= -1.1155861542198394e-16)) {
		temp = sqrt((((pow(sqrt(exp(2.0)), (0.5 * x)) * pow(sqrt(exp(2.0)), (0.5 * x))) + sqrt(1.0)) * ((pow(exp(2.0), (0.5 * x)) - sqrt(1.0)) / (exp(x) - 1.0))));
	} else {
		temp = ((0.5 * (x / 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 < -1.1155861542198394e-16

   1. Initial program 0.8

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

   3. Applied *-un-lft-identity 0.8

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

   4. Applied add-sqr-sqrt 0.8

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

   5. Applied add-sqr-sqrt 0.7

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

   6. Applied difference-of-squares 0.3

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

   7. Applied times-frac 0.3

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

   8. Simplified 0.3

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

   10. Applied add-log-exp 0.3

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

   11. Applied exp-to-pow 0.3

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

   12. Applied sqrt-pow1 0.0

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

   13. Simplified 0.0

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

   15. Applied add-log-exp 0.0

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

   16. Applied exp-to-pow 0.0

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

   17. Applied sqrt-pow1 0.0

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

   18. Simplified 0.0

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

   20. Applied add-sqr-sqrt 0.0

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

   21. Applied unpow-prod-down 0.0

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

   if -1.1155861542198394e-16 < x

   1. Initial program 36.7

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

   2. Taylor expanded around 0 8.4

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

   3. Simplified 8.4

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.11558615421983944 \cdot 10^{-16}:\\ \;\;\;\;\sqrt{\left({\left(\sqrt{e^{2}}\right)}^{\left(\frac{1}{2} \cdot x\right)} \cdot {\left(\sqrt{e^{2}}\right)}^{\left(\frac{1}{2} \cdot x\right)} + \sqrt{1}\right) \cdot \frac{{\left(e^{2}\right)}^{\left(\frac{1}{2} \cdot x\right)} - \sqrt{1}}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))