Octave 3.8, jcobi/2

Average Error: 24.1 → 1.4

Time: 10.1s

Precision: binary64

\[\left(\alpha > -1 \land \beta > -1\right) \land i > 0\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.9999999997663539 \end{array}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\beta + \mathsf{fma}\left(2, i, 1\right)}{\alpha}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_1 := \mathsf{fma}\left(2, i, \alpha + \beta\right)\\ \frac{1 + \left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{2 + t_1}}{t_1}}{2} \end{array}\\ \end{array} \]

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
 (if (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
       (<=
        (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0))
        -0.9999999997663539))
   (expm1 (log1p (/ (+ beta (fma 2.0 i 1.0)) alpha)))
   (let* ((t_1 (fma 2.0 i (+ alpha beta))))
     (/
      (+ 1.0 (* (+ alpha beta) (/ (/ (- beta alpha) (+ 2.0 t_1)) 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 tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.9999999997663539) {
		tmp = expm1(log1p((beta + fma(2.0, i, 1.0)) / alpha));
	} else {
		double t_1 = fma(2.0, i, (alpha + beta));
		tmp = (1.0 + ((alpha + beta) * (((beta - alpha) / (2.0 + t_1)) / 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)) < -0.999999999766353898

   1. Initial program 62.7

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

   2. Simplified 55.5

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

   3. Taylor expanded in alpha around inf 5.9

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

   4. Simplified 5.9

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

   5. Applied expm1-log1p-u_binary64 5.9

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\mathsf{fma}\left(2, \beta, 2 + i \cdot 4\right)}{\alpha}}{2}\right)\right)} \]

   6. Simplified 5.9

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\frac{\beta + \mathsf{fma}\left(2, i, 1\right)}{\alpha}\right)}\right) \]

   if -0.999999999766353898 < (/.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 13.0

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

   2. Simplified 9.5

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

   3. Applied *-un-lft-identity_binary64 9.5

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

   4. Applied times-frac_binary64 0.1

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

   5. Applied fma-udef_binary64 0.1

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

   6. Simplified 0.1

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

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.9999999997663539:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\beta + \mathsf{fma}\left(2, i, 1\right)}{\alpha}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{2 + \mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{2}\\ \end{array} \]

Reproduce

herbie shell --seed 2022081 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :precision binary64
  :pre (and (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))