Octave 3.8, jcobi/2

Average Error: 23.6 → 1.3

Time: 24.8s

Precision: binary64

\[\alpha > -1 \land \beta > -1 \land i > 0\]

Math

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

↓

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := 2 + t_0\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{t_1} \leq -0.9999999996245981:\\ \;\;\;\;\frac{\frac{\beta \cdot 2 + \left(2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(\alpha + \beta\right) \cdot \frac{\frac{1}{\frac{t_0}{\beta - \alpha}}}{t_1}}{2}\\ \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (/
  (+
   (/
    (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i)))
    (+ (+ (+ alpha beta) (* 2.0 i)) 2.0))
   1.0)
  2.0))

↓

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))) (t_1 (+ 2.0 t_0)))
   (if (<=
        (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) t_1)
        -0.9999999996245981)
     (/ (/ (+ (* beta 2.0) (+ 2.0 (* i 4.0))) alpha) 2.0)
     (/
      (+ 1.0 (* (+ alpha beta) (/ (/ 1.0 (/ t_0 (- beta alpha))) t_1)))
      2.0))))

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 t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = 2.0 + t_0;
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= -0.9999999996245981) {
		tmp = (((beta * 2.0) + (2.0 + (i * 4.0))) / alpha) / 2.0;
	} else {
		tmp = (1.0 + ((alpha + beta) * ((1.0 / (t_0 / (beta - alpha))) / t_1))) / 2.0;
	}
	return tmp;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2)) < -0.99999999962459807

   1. Initial program 62.8

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

   2. Taylor expanded around inf 5.3

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

   3. Simplified 5.3

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

   if -0.99999999962459807 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2))

   1. Initial program 12.1

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

   3. Applied *-un-lft-identity_binary64 12.1

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

   4. Applied *-un-lft-identity_binary64 12.1

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

   5. Applied times-frac_binary64 0.1

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

   6. Applied times-frac_binary64 0.1

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

   7. Simplified 0.1

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

   8. Simplified 0.1

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

   10. Applied clear-num_binary64 0.1

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

   11. Simplified 0.1

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

   13. Applied *-un-lft-identity_binary64 0.1

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

   14. Applied associate-/r*_binary64 0.1

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

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.9999999996245981:\\ \;\;\;\;\frac{\frac{\beta \cdot 2 + \left(2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(\alpha + \beta\right) \cdot \frac{\frac{1}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \end{array} \]

Reproduce

herbie shell --seed 2021204 
(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))