Octave 3.8, jcobi/2

Average Error: 23.6 → 11.5

Time: 28.3min

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.40874699682471261 \cdot 10^{210}:\\ \;\;\;\;\frac{{\left({\left(\left(\alpha + \beta\right) \cdot \left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}\right) + 1\right)}^{3}\right)}^{\frac{1}{3}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\frac{1}{\alpha} \cdot \beta\right) \cdot \sqrt[3]{8}\right) \cdot \left(\left(\frac{1}{\beta} + 1\right) - \frac{3}{\alpha}\right)}{2}\\ \end{array}\]

C

double code(double alpha, double beta, double i) {
	return (((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 <= 1.4087469968247126e+210)) {
		VAR = (((double) pow(((double) pow(((double) (((double) (((double) (alpha + beta)) * ((double) ((((double) (beta - alpha)) / ((double) (((double) (alpha + beta)) + ((double) (2.0 * i))))) * (1.0 / ((double) (((double) (((double) (alpha + beta)) + ((double) (2.0 * i)))) + 2.0))))))) + 1.0)), 3.0)), 0.3333333333333333)) / 2.0);
	} else {
		VAR = (((double) (((double) (((double) ((1.0 / alpha) * beta)) * ((double) cbrt(8.0)))) * ((double) (((double) ((1.0 / beta) + 1.0)) - (3.0 / alpha))))) / 2.0);
	}
	return VAR;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if alpha < 1.40874699682471261e210

   1. Initial program 18.8

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

   3. Applied *-un-lft-identity 18.8

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

   4. Applied *-un-lft-identity 18.8

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

   5. Applied times-frac 7.7

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

   6. Applied times-frac 7.6

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

   7. Simplified 7.6

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

   9. Applied add-cbrt-cube 7.7

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

   10. Simplified 7.7

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

   12. Applied pow1/3 7.6

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

   14. Applied div-inv 7.6

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

   if 1.40874699682471261e210 < alpha

   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. Using strategy rm

   3. Applied *-un-lft-identity 64.0

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

   4. Applied *-un-lft-identity 64.0

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

   5. Applied times-frac 51.2

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

   6. Applied times-frac 51.3

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

   7. Simplified 51.3

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

   9. Applied add-cbrt-cube 51.3

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

   10. Simplified 51.3

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

   12. Applied pow1/3 51.3

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

   13. Taylor expanded around inf 48.7

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

   14. Simplified 43.6

       \[\leadsto \frac{\color{blue}{\left(\left(\frac{1}{\alpha} \cdot \beta\right) \cdot \sqrt[3]{8}\right) \cdot \left(\left(\frac{1}{\beta} + 1\right) - \frac{3}{\alpha}\right)}}{2}\]
3. Recombined 2 regimes into one program.

4. Final simplification 11.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.40874699682471261 \cdot 10^{210}:\\ \;\;\;\;\frac{{\left({\left(\left(\alpha + \beta\right) \cdot \left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}\right) + 1\right)}^{3}\right)}^{\frac{1}{3}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\frac{1}{\alpha} \cdot \beta\right) \cdot \sqrt[3]{8}\right) \cdot \left(\left(\frac{1}{\beta} + 1\right) - \frac{3}{\alpha}\right)}{2}\\ \end{array}\]

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))