Octave 3.8, jcobi/2

Average Error: 24.5 → 11.8

Time: 9.1s

Precision: 64

\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \le 3.24781343515455499 \cdot 10^{83}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\mathsf{fma}\left(\alpha + \beta, \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)\right)}^{3}}}{2}\\ \mathbf{elif}\;\alpha \le 2.2059526387587152 \cdot 10^{119}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}{2}\\ \mathbf{elif}\;\alpha \le 6.1120094823181713 \cdot 10^{200}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\frac{\alpha + \beta}{1}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}, \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}, 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}{2}\\ \end{array}\]

C

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

↓

double code(double alpha, double beta, double i) {
	double temp;
	if ((alpha <= 3.247813435154555e+83)) {
		temp = (cbrt(pow(fma((alpha + beta), (((beta - alpha) / ((alpha + beta) + (2.0 * i))) / (((alpha + beta) + (2.0 * i)) + 2.0)), 1.0), 3.0)) / 2.0);
	} else {
		double temp_1;
		if ((alpha <= 2.205952638758715e+119)) {
			temp_1 = (fma(2.0, (1.0 / alpha), ((8.0 * (1.0 / pow(alpha, 3.0))) - (4.0 * (1.0 / pow(alpha, 2.0))))) / 2.0);
		} else {
			double temp_2;
			if ((alpha <= 6.112009482318171e+200)) {
				temp_2 = (fma((((alpha + beta) / 1.0) / (cbrt((((alpha + beta) + (2.0 * i)) + 2.0)) * cbrt((((alpha + beta) + (2.0 * i)) + 2.0)))), (((beta - alpha) / ((alpha + beta) + (2.0 * i))) / cbrt((((alpha + beta) + (2.0 * i)) + 2.0))), 1.0) / 2.0);
			} else {
				temp_2 = (fma(2.0, (1.0 / alpha), ((8.0 * (1.0 / pow(alpha, 3.0))) - (4.0 * (1.0 / pow(alpha, 2.0))))) / 2.0);
			}
			temp_1 = temp_2;
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 3 regimes

2. if alpha < 3.247813435154555e+83

   1. Initial program 14.2

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

   3. Applied *-un-lft-identity 14.2

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

   4. Applied *-un-lft-identity 14.2

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

   5. Applied times-frac 2.7

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

   6. Applied times-frac 2.7

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

   7. Applied fma-def 2.6

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{\frac{\alpha + \beta}{1}}{1}, \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}}{2}\]
   8. Using strategy rm

   9. Applied add-cbrt-cube 2.7

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\mathsf{fma}\left(\frac{\frac{\alpha + \beta}{1}}{1}, \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right) \cdot \mathsf{fma}\left(\frac{\frac{\alpha + \beta}{1}}{1}, \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)\right) \cdot \mathsf{fma}\left(\frac{\frac{\alpha + \beta}{1}}{1}, \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}}}{2}\]

   10. Simplified 2.7

       \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(\mathsf{fma}\left(\alpha + \beta, \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)\right)}^{3}}}}{2}\]

   if 3.247813435154555e+83 < alpha < 2.205952638758715e+119 or 6.112009482318171e+200 < alpha

   1. Initial program 58.2

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

   2. Taylor expanded around inf 41.8

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}}{2}\]

   3. Simplified 41.8

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}}{2}\]

   if 2.205952638758715e+119 < alpha < 6.112009482318171e+200

   1. Initial program 55.5

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

   3. Applied add-cube-cbrt 55.4

      \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\color{blue}{\left(\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}\right) \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}} + 1}{2}\]

   4. Applied *-un-lft-identity 55.4

      \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\color{blue}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{\left(\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}\right) \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} + 1}{2}\]

   5. Applied times-frac 38.7

      \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha + \beta}{1} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}\right) \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} + 1}{2}\]

   6. Applied times-frac 38.7

      \[\leadsto \frac{\color{blue}{\frac{\frac{\alpha + \beta}{1}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}} + 1}{2}\]

   7. Applied fma-def 38.7

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

4. Final simplification 11.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 3.24781343515455499 \cdot 10^{83}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\mathsf{fma}\left(\alpha + \beta, \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)\right)}^{3}}}{2}\\ \mathbf{elif}\;\alpha \le 2.2059526387587152 \cdot 10^{119}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}{2}\\ \mathbf{elif}\;\alpha \le 6.1120094823181713 \cdot 10^{200}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\frac{\alpha + \beta}{1}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}, \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}, 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2)) 1) 2))