Octave 3.8, jcobi/1

Average Error: 16.2 → 0.1

Time: 8.6s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\frac{\beta - \alpha}{t_0} \leq -0.9999999560433918:\\ \;\;\;\;\left({\left(\frac{\beta}{\alpha}\right)}^{3} + \left(\frac{1}{\alpha} + \left(\frac{\beta}{\alpha} + \mathsf{fma}\left(8, \frac{\beta}{{\alpha}^{3}}, \frac{4}{{\alpha}^{3}}\right)\right)\right)\right) - \left(\frac{2}{\alpha \cdot \alpha} + \mathsf{fma}\left(3, \frac{\beta}{\alpha \cdot \alpha}, \frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\alpha - \beta}{t_0}, -0.5, 0.5\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)))
   (if (<= (/ (- beta alpha) t_0) -0.9999999560433918)
     (-
      (+
       (pow (/ beta alpha) 3.0)
       (+
        (/ 1.0 alpha)
        (+
         (/ beta alpha)
         (fma 8.0 (/ beta (pow alpha 3.0)) (/ 4.0 (pow alpha 3.0))))))
      (+
       (/ 2.0 (* alpha alpha))
       (fma 3.0 (/ beta (* alpha alpha)) (* (/ beta alpha) (/ beta alpha)))))
     (fma (/ (- alpha beta) t_0) -0.5 0.5))))

C

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

↓

double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (((beta - alpha) / t_0) <= -0.9999999560433918) {
		tmp = (pow((beta / alpha), 3.0) + ((1.0 / alpha) + ((beta / alpha) + fma(8.0, (beta / pow(alpha, 3.0)), (4.0 / pow(alpha, 3.0)))))) - ((2.0 / (alpha * alpha)) + fma(3.0, (beta / (alpha * alpha)), ((beta / alpha) * (beta / alpha))));
	} else {
		tmp = fma(((alpha - beta) / t_0), -0.5, 0.5);
	}
	return tmp;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.999999956043391758

   1. Initial program 59.6

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

   2. Simplified 59.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\alpha - \beta}{\left(\beta + \alpha\right) + 2}, -0.5, 0.5\right)} \]

   3. Taylor expanded in alpha around inf 5.4

      \[\leadsto \color{blue}{\left(\frac{{\beta}^{3}}{{\alpha}^{3}} + \left(\frac{1}{\alpha} + \left(4 \cdot \frac{1}{{\alpha}^{3}} + \left(\frac{\beta}{\alpha} + \left(8 \cdot \frac{\beta}{{\alpha}^{3}} + 5 \cdot \frac{{\beta}^{2}}{{\alpha}^{3}}\right)\right)\right)\right)\right) - \left(2 \cdot \frac{1}{{\alpha}^{2}} + \left(3 \cdot \frac{\beta}{{\alpha}^{2}} + \frac{{\beta}^{2}}{{\alpha}^{2}}\right)\right)} \]

   4. Simplified 2.9

      \[\leadsto \color{blue}{\left({\left(\frac{\beta}{\alpha}\right)}^{3} + \left(\frac{1}{\alpha} + \left(\frac{4}{{\alpha}^{3}} + \left(\frac{\beta}{\alpha} + \mathsf{fma}\left(8, \frac{\beta}{{\alpha}^{3}}, 5 \cdot \frac{\beta \cdot \beta}{{\alpha}^{3}}\right)\right)\right)\right)\right) - \left(\frac{2}{\alpha \cdot \alpha} + \mathsf{fma}\left(3, \frac{\beta}{\alpha \cdot \alpha}, \frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right)\right)} \]

   5. Taylor expanded in beta around 0 0.0

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

   6. Simplified 0.0

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

   if -0.999999956043391758 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 0.1

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

   2. Simplified 0.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\alpha - \beta}{\left(\beta + \alpha\right) + 2}, -0.5, 0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999560433918:\\ \;\;\;\;\left({\left(\frac{\beta}{\alpha}\right)}^{3} + \left(\frac{1}{\alpha} + \left(\frac{\beta}{\alpha} + \mathsf{fma}\left(8, \frac{\beta}{{\alpha}^{3}}, \frac{4}{{\alpha}^{3}}\right)\right)\right)\right) - \left(\frac{2}{\alpha \cdot \alpha} + \mathsf{fma}\left(3, \frac{\beta}{\alpha \cdot \alpha}, \frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\alpha - \beta}{\left(\beta + \alpha\right) + 2}, -0.5, 0.5\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022067 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))