sqrtexp (problem 3.4.4)

Average Error: 4.7 → 0.2

Time: 6.1s

Precision: 64

• could not determine a ground truth for program body

  1. x = 3048118974.682059

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -8.3512322087182569 \cdot 10^{-12} \lor \neg \left(x \le 6.561572677279099 \cdot 10^{-15}\right):\\ \;\;\;\;\sqrt{\frac{\left(e^{x} + 1\right) \cdot \left(e^{2 \cdot x} - 1\right)}{e^{2 \cdot x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \frac{{x}^{2}}{{\left(\sqrt{2}\right)}^{3}}, \mathsf{fma}\left(\frac{{x}^{2}}{\sqrt{2}}, 0.25, \mathsf{fma}\left(0.5, \frac{x}{\sqrt{2}}, \sqrt{2}\right)\right)\right)\\ \end{array}\]

C

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

↓

double code(double x) {
	double VAR;
	if (((x <= -8.351232208718257e-12) || !(x <= 6.561572677279099e-15))) {
		VAR = sqrt((((exp(x) + 1.0) * (exp((2.0 * x)) - 1.0)) / (exp((2.0 * x)) - 1.0)));
	} else {
		VAR = fma(-0.125, (pow(x, 2.0) / pow(sqrt(2.0), 3.0)), fma((pow(x, 2.0) / sqrt(2.0)), 0.25, fma(0.5, (x / sqrt(2.0)), sqrt(2.0))));
	}
	return VAR;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -8.351232208718257e-12 or 6.561572677279099e-15 < x

   1. Initial program 1.2

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

   3. Applied flip-- 0.8

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

   4. Applied associate-/r/ 0.8

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

   5. Simplified 0.1

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

   6. Taylor expanded around inf 0.2

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

   if -8.351232208718257e-12 < x < 6.561572677279099e-15

   1. Initial program 54.9

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

   3. Applied flip-- 54.4

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

   4. Applied associate-/r/ 54.4

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

   5. Simplified 46.3

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

   6. Taylor expanded around 0 0.1

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

   7. Simplified 0.1

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -8.3512322087182569 \cdot 10^{-12} \lor \neg \left(x \le 6.561572677279099 \cdot 10^{-15}\right):\\ \;\;\;\;\sqrt{\frac{\left(e^{x} + 1\right) \cdot \left(e^{2 \cdot x} - 1\right)}{e^{2 \cdot x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \frac{{x}^{2}}{{\left(\sqrt{2}\right)}^{3}}, \mathsf{fma}\left(\frac{{x}^{2}}{\sqrt{2}}, 0.25, \mathsf{fma}\left(0.5, \frac{x}{\sqrt{2}}, \sqrt{2}\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020091 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))