Octave 3.8, jcobi/2

Average Error: 24.0 → 11.0

Time: 6.6s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \le 8.1176783251492383 \cdot 10^{131}:\\ \;\;\;\;\frac{\frac{{\left(\frac{\beta - \alpha}{\left(\beta + \left(\alpha + 2 \cdot i\right)\right) \cdot \left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right)}\right)}^{3} + {1}^{3}}{1 \cdot 1 - \frac{\beta - \alpha}{\left(\beta + \left(\alpha + 2 \cdot i\right)\right) \cdot \left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right)} \cdot \left(1 - \frac{\beta - \alpha}{\left(\beta + \left(\alpha + 2 \cdot i\right)\right) \cdot \left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right)}\right)}}{2}\\ \mathbf{elif}\;\alpha \le 4.03001152902504068 \cdot 10^{155} \lor \neg \left(\alpha \le 8.88082771297024187 \cdot 10^{187}\right):\\ \;\;\;\;\frac{\frac{\beta}{\left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{\frac{\beta}{\left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)}}\right) - \sqrt[3]{{\left(\frac{\alpha}{\left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)} - 1\right)}^{3}}}{2}\\ \end{array}\]

C

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

↓

double code(double alpha, double beta, double i) {
	double VAR;
	if ((alpha <= 8.117678325149238e+131)) {
		VAR = ((double) (((double) (((double) (((double) pow(((double) (((double) (beta - alpha)) / ((double) (((double) (beta + ((double) (alpha + ((double) (2.0 * i)))))) * ((double) (((double) (((double) (2.0 + ((double) (2.0 * i)))) / ((double) (alpha + beta)))) + 1.0)))))), 3.0)) + ((double) pow(1.0, 3.0)))) / ((double) (((double) (1.0 * 1.0)) - ((double) (((double) (((double) (beta - alpha)) / ((double) (((double) (beta + ((double) (alpha + ((double) (2.0 * i)))))) * ((double) (((double) (((double) (2.0 + ((double) (2.0 * i)))) / ((double) (alpha + beta)))) + 1.0)))))) * ((double) (1.0 - ((double) (((double) (beta - alpha)) / ((double) (((double) (beta + ((double) (alpha + ((double) (2.0 * i)))))) * ((double) (((double) (((double) (2.0 + ((double) (2.0 * i)))) / ((double) (alpha + beta)))) + 1.0)))))))))))))) / 2.0));
	} else {
		double VAR_1;
		if (((alpha <= 4.0300115290250407e+155) || !(alpha <= 8.880827712970242e+187))) {
			VAR_1 = ((double) (((double) (((double) (beta / ((double) (((double) (((double) (((double) (2.0 + ((double) (2.0 * i)))) / ((double) (alpha + beta)))) + 1.0)) * ((double) (alpha + ((double) (beta + ((double) (2.0 * i)))))))))) - ((double) (((double) (4.0 / ((double) (alpha * alpha)))) - ((double) (((double) (2.0 / alpha)) + ((double) (8.0 / ((double) pow(alpha, 3.0)))))))))) / 2.0));
		} else {
			VAR_1 = ((double) (((double) (((double) log(((double) exp(((double) (beta / ((double) (((double) (((double) (((double) (2.0 + ((double) (2.0 * i)))) / ((double) (alpha + beta)))) + 1.0)) * ((double) (alpha + ((double) (beta + ((double) (2.0 * i)))))))))))))) - ((double) cbrt(((double) pow(((double) (((double) (alpha / ((double) (((double) (((double) (((double) (2.0 + ((double) (2.0 * i)))) / ((double) (alpha + beta)))) + 1.0)) * ((double) (alpha + ((double) (beta + ((double) (2.0 * i)))))))))) - 1.0)), 3.0)))))) / 2.0));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 3 regimes

2. if alpha < 8.1176783251492383e131

   1. Initial program 14.8

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

   2. Simplified 4.2

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

   4. Applied flip3-+ 4.2

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

   5. Simplified 4.2

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

   6. Simplified 4.2

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

   if 8.1176783251492383e131 < alpha < 4.03001152902504068e155 or 8.88082771297024187e187 < alpha

   1. Initial program 61.2

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

   2. Simplified 48.0

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

   4. Applied div-sub 48.0

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

   5. Applied associate-+l- 46.4

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

   6. Simplified 46.4

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

   7. Taylor expanded around inf 38.7

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

   8. Simplified 38.7

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

   if 4.03001152902504068e155 < alpha < 8.88082771297024187e187

   1. Initial program 64.0

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

   2. Simplified 39.8

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

   4. Applied div-sub 39.8

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

   5. Applied associate-+l- 38.9

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

   6. Simplified 38.9

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

   8. Applied add-log-exp 39.8

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

   10. Applied add-cbrt-cube 39.9

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

   11. Simplified 39.9

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

4. Final simplification 11.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 8.1176783251492383 \cdot 10^{131}:\\ \;\;\;\;\frac{\frac{{\left(\frac{\beta - \alpha}{\left(\beta + \left(\alpha + 2 \cdot i\right)\right) \cdot \left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right)}\right)}^{3} + {1}^{3}}{1 \cdot 1 - \frac{\beta - \alpha}{\left(\beta + \left(\alpha + 2 \cdot i\right)\right) \cdot \left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right)} \cdot \left(1 - \frac{\beta - \alpha}{\left(\beta + \left(\alpha + 2 \cdot i\right)\right) \cdot \left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right)}\right)}}{2}\\ \mathbf{elif}\;\alpha \le 4.03001152902504068 \cdot 10^{155} \lor \neg \left(\alpha \le 8.88082771297024187 \cdot 10^{187}\right):\\ \;\;\;\;\frac{\frac{\beta}{\left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{\frac{\beta}{\left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)}}\right) - \sqrt[3]{{\left(\frac{\alpha}{\left(\frac{2 + 2 \cdot i}{\alpha + \beta} + 1\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)} - 1\right)}^{3}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :precision binary64
  :pre (and (> alpha -1.0) (> beta -1.0) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i))) (+ (+ (+ alpha beta) (* 2.0 i)) 2.0)) 1.0) 2.0))