sqrtexp (problem 3.4.4)

Average Error: 4.7 → 0.8

Time: 6.1s

Precision: 64

• could not determine a ground truth for program body

  1. x = 1.611838088146848e+293

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -4.46518960337222232 \cdot 10^{-14}:\\ \;\;\;\;\sqrt{{\left(e^{2 \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}\right)}^{\left(\frac{\sqrt[3]{x}}{2}\right)} + \sqrt{1}} \cdot \sqrt{\frac{{\left(e^{2}\right)}^{\left(\frac{1}{2} \cdot x\right)} - \sqrt{1}}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\\ \end{array}\]

C

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

↓

double code(double x) {
	double VAR;
	if ((x <= -4.4651896033722223e-14)) {
		VAR = ((double) (((double) sqrt(((double) (((double) pow(((double) exp(((double) (2.0 * ((double) (((double) cbrt(x)) * ((double) cbrt(x)))))))), ((double) (((double) cbrt(x)) / 2.0)))) + ((double) sqrt(1.0)))))) * ((double) sqrt(((double) (((double) (((double) pow(((double) exp(2.0)), ((double) (0.5 * x)))) - ((double) sqrt(1.0)))) / ((double) (((double) exp(x)) - 1.0))))))));
	} else {
		VAR = ((double) (((double) (0.5 * ((double) (x / ((double) sqrt(2.0)))))) + ((double) (((double) sqrt(2.0)) + ((double) (((double) (((double) pow(x, 2.0)) / ((double) sqrt(2.0)))) * ((double) (0.25 - ((double) (0.125 / 2.0))))))))));
	}
	return VAR;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -4.4651896033722223e-14

   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. Applied sqrt-prod 0.2

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

   9. Simplified 0.2

      \[\leadsto \color{blue}{\sqrt{\sqrt{e^{2 \cdot x}} + \sqrt{1}}} \cdot \sqrt{\frac{\sqrt{e^{2 \cdot x}} - \sqrt{1}}{e^{x} - 1}}\]
   10. Using strategy rm

   11. Applied add-log-exp 0.2

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

   12. Applied exp-to-pow 0.2

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

   13. Applied sqrt-pow1 0.0

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

   14. Simplified 0.0

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

   16. Applied add-cube-cbrt 0.0

       \[\leadsto \sqrt{\sqrt{e^{2 \cdot \color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)}}} + \sqrt{1}} \cdot \sqrt{\frac{{\left(e^{2}\right)}^{\left(\frac{1}{2} \cdot x\right)} - \sqrt{1}}{e^{x} - 1}}\]

   17. Applied associate-*r* 0.0

       \[\leadsto \sqrt{\sqrt{e^{\color{blue}{\left(2 \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right) \cdot \sqrt[3]{x}}}} + \sqrt{1}} \cdot \sqrt{\frac{{\left(e^{2}\right)}^{\left(\frac{1}{2} \cdot x\right)} - \sqrt{1}}{e^{x} - 1}}\]

   18. Applied exp-prod 0.0

       \[\leadsto \sqrt{\sqrt{\color{blue}{{\left(e^{2 \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}\right)}^{\left(\sqrt[3]{x}\right)}}} + \sqrt{1}} \cdot \sqrt{\frac{{\left(e^{2}\right)}^{\left(\frac{1}{2} \cdot x\right)} - \sqrt{1}}{e^{x} - 1}}\]

   19. Applied sqrt-pow1 0.0

       \[\leadsto \sqrt{\color{blue}{{\left(e^{2 \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}\right)}^{\left(\frac{\sqrt[3]{x}}{2}\right)}} + \sqrt{1}} \cdot \sqrt{\frac{{\left(e^{2}\right)}^{\left(\frac{1}{2} \cdot x\right)} - \sqrt{1}}{e^{x} - 1}}\]

   if -4.4651896033722223e-14 < x

   1. Initial program 37.0

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

   2. Taylor expanded around 0 7.5

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

   3. Simplified 7.5

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.46518960337222232 \cdot 10^{-14}:\\ \;\;\;\;\sqrt{{\left(e^{2 \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}\right)}^{\left(\frac{\sqrt[3]{x}}{2}\right)} + \sqrt{1}} \cdot \sqrt{\frac{{\left(e^{2}\right)}^{\left(\frac{1}{2} \cdot x\right)} - \sqrt{1}}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020113 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))