Octave 3.8, jcobi/3

Average Error: 3.6 → 3.9

Time: 7.3s

Precision: binary64

\[\alpha > -1 \land \beta > -1\]

Math

\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 3.681462720248197 \cdot 10^{+89}:\\ \;\;\;\;\frac{1}{\alpha + \left(\beta + 1 \cdot 2\right)} \cdot \frac{\frac{\alpha + \left(\beta + \left(1 + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 1 \cdot 2\right)}}{\alpha + \left(\beta + \left(1 + 1 \cdot 2\right)\right)}\\ \mathbf{elif}\;\alpha \leq 4.090496957446828 \cdot 10^{+129}:\\ \;\;\;\;\frac{\frac{1 + \left(\frac{2}{\alpha \cdot \alpha} - \frac{1}{\alpha}\right)}{1 \cdot 2 + \left(\alpha + \beta\right)}}{1 + \left(1 \cdot 2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}{1 \cdot 2 + \left(\alpha + \beta\right)}}{1 \cdot 2 + \left(\alpha + \beta\right)}}{\sqrt{\alpha + \left(\beta + \left(1 + 1 \cdot 2\right)\right)} \cdot \sqrt{\alpha + \left(\beta + \left(1 + 1 \cdot 2\right)\right)}}\\ \end{array}\]

C

double code(double alpha, double beta) {
	return (((((double) (((double) (((double) (alpha + beta)) + ((double) (beta * alpha)))) + 1.0)) / ((double) (((double) (alpha + beta)) + ((double) (2.0 * 1.0))))) / ((double) (((double) (alpha + beta)) + ((double) (2.0 * 1.0))))) / ((double) (((double) (((double) (alpha + beta)) + ((double) (2.0 * 1.0)))) + 1.0)));
}

↓

double code(double alpha, double beta) {
	double VAR;
	if ((alpha <= 3.681462720248197e+89)) {
		VAR = ((double) ((1.0 / ((double) (alpha + ((double) (beta + ((double) (1.0 * 2.0))))))) * ((((double) (alpha + ((double) (beta + ((double) (1.0 + ((double) (alpha * beta)))))))) / ((double) (alpha + ((double) (beta + ((double) (1.0 * 2.0))))))) / ((double) (alpha + ((double) (beta + ((double) (1.0 + ((double) (1.0 * 2.0)))))))))));
	} else {
		double VAR_1;
		if ((alpha <= 4.090496957446828e+129)) {
			VAR_1 = ((((double) (1.0 + ((double) ((2.0 / ((double) (alpha * alpha))) - (1.0 / alpha))))) / ((double) (((double) (1.0 * 2.0)) + ((double) (alpha + beta))))) / ((double) (1.0 + ((double) (((double) (1.0 * 2.0)) + ((double) (alpha + beta)))))));
		} else {
			VAR_1 = (((((double) (1.0 + ((double) (((double) (alpha * beta)) + ((double) (alpha + beta)))))) / ((double) (((double) (1.0 * 2.0)) + ((double) (alpha + beta))))) / ((double) (((double) (1.0 * 2.0)) + ((double) (alpha + beta))))) / ((double) (((double) sqrt(((double) (alpha + ((double) (beta + ((double) (1.0 + ((double) (1.0 * 2.0)))))))))) * ((double) sqrt(((double) (alpha + ((double) (beta + ((double) (1.0 + ((double) (1.0 * 2.0)))))))))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

1. Split input into 3 regimes

2. if alpha < 3.68146272024819717e89

   1. Initial program 0.4

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 0.4

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)}}\]

   4. Applied add-sqr-sqrt 1.1

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)}\]

   5. Applied add-sqr-sqrt 1.4

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\color{blue}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)}\]

   6. Applied *-un-lft-identity 1.4

      \[\leadsto \frac{\frac{\frac{\color{blue}{1 \cdot \left(\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1\right)}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)}\]

   7. Applied times-frac 1.4

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}} \cdot \frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)}\]

   8. Applied times-frac 1.2

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}} \cdot \frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)}\]

   9. Applied times-frac 1.2

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{1} \cdot \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}\]

   10. Simplified 0.6

       \[\leadsto \color{blue}{\frac{1}{\alpha + \left(\beta + 1 \cdot 2\right)}} \cdot \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]

   11. Simplified 0.5

       \[\leadsto \frac{1}{\alpha + \left(\beta + 1 \cdot 2\right)} \cdot \color{blue}{\frac{\frac{\alpha + \left(\beta + \left(\alpha \cdot \beta + 1\right)\right)}{\alpha + \left(\beta + 1 \cdot 2\right)}}{\alpha + \left(\beta + \left(1 + 1 \cdot 2\right)\right)}}\]

   if 3.68146272024819717e89 < alpha < 4.0904969574468277e129

   1. Initial program 8.7

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]

   2. Taylor expanded around inf 14.4

      \[\leadsto \frac{\frac{\color{blue}{\left(2 \cdot \frac{1}{{\alpha}^{2}} + 1\right) - 1 \cdot \frac{1}{\alpha}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]

   3. Simplified 14.4

      \[\leadsto \frac{\frac{\color{blue}{1 + \left(\frac{2}{\alpha \cdot \alpha} - \frac{1}{\alpha}\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]

   if 4.0904969574468277e129 < alpha

   1. Initial program 15.1

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 15.2

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \cdot \sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}}\]

   4. Simplified 15.2

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\sqrt{\alpha + \left(\beta + \left(1 + 1 \cdot 2\right)\right)}} \cdot \sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}\]

   5. Simplified 15.2

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\sqrt{\alpha + \left(\beta + \left(1 + 1 \cdot 2\right)\right)} \cdot \color{blue}{\sqrt{\alpha + \left(\beta + \left(1 + 1 \cdot 2\right)\right)}}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 3.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 3.681462720248197 \cdot 10^{+89}:\\ \;\;\;\;\frac{1}{\alpha + \left(\beta + 1 \cdot 2\right)} \cdot \frac{\frac{\alpha + \left(\beta + \left(1 + \alpha \cdot \beta\right)\right)}{\alpha + \left(\beta + 1 \cdot 2\right)}}{\alpha + \left(\beta + \left(1 + 1 \cdot 2\right)\right)}\\ \mathbf{elif}\;\alpha \leq 4.090496957446828 \cdot 10^{+129}:\\ \;\;\;\;\frac{\frac{1 + \left(\frac{2}{\alpha \cdot \alpha} - \frac{1}{\alpha}\right)}{1 \cdot 2 + \left(\alpha + \beta\right)}}{1 + \left(1 \cdot 2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1 + \left(\alpha \cdot \beta + \left(\alpha + \beta\right)\right)}{1 \cdot 2 + \left(\alpha + \beta\right)}}{1 \cdot 2 + \left(\alpha + \beta\right)}}{\sqrt{\alpha + \left(\beta + \left(1 + 1 \cdot 2\right)\right)} \cdot \sqrt{\alpha + \left(\beta + \left(1 + 1 \cdot 2\right)\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020199 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :precision binary64
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ (+ alpha beta) (* 2.0 1.0)) 1.0)))