Octave 3.8, jcobi/1

Average Error: 16.4 → 0.1

Time: 5.4s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\frac{\beta - \alpha}{t_0} \leq -0.9999938945134544:\\ \;\;\;\;\left(\frac{1}{\alpha} + \frac{\beta}{\alpha}\right) - \left(\frac{2}{\alpha \cdot \alpha} + \mathsf{fma}\left(3, \frac{\beta}{\alpha \cdot \alpha}, \frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{t_0}{\alpha - \beta}}, -0.5, 0.5\right)\\ \end{array} \]

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) -0.9999938945134544)
     (-
      (+ (/ 1.0 alpha) (/ beta alpha))
      (+
       (/ 2.0 (* alpha alpha))
       (fma 3.0 (/ beta (* alpha alpha)) (* (/ beta alpha) (/ beta alpha)))))
     (fma (/ 1.0 (/ t_0 (- alpha beta))) -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) <= -0.9999938945134544) {
		tmp = ((1.0 / alpha) + (beta / alpha)) - ((2.0 / (alpha * alpha)) + fma(3.0, (beta / (alpha * alpha)), ((beta / alpha) * (beta / alpha))));
	} else {
		tmp = fma((1.0 / (t_0 / (alpha - beta))), -0.5, 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta)
	return Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) + 1.0) / 2.0)
end

↓

function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / t_0) <= -0.9999938945134544)
		tmp = Float64(Float64(Float64(1.0 / alpha) + Float64(beta / alpha)) - Float64(Float64(2.0 / Float64(alpha * alpha)) + fma(3.0, Float64(beta / Float64(alpha * alpha)), Float64(Float64(beta / alpha) * Float64(beta / alpha)))));
	else
		tmp = fma(Float64(1.0 / Float64(t_0 / Float64(alpha - beta))), -0.5, 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]

↓

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / t$95$0), $MachinePrecision], -0.9999938945134544], N[(N[(N[(1.0 / alpha), $MachinePrecision] + N[(beta / alpha), $MachinePrecision]), $MachinePrecision] - N[(N[(2.0 / N[(alpha * alpha), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(beta / N[(alpha * alpha), $MachinePrecision]), $MachinePrecision] + N[(N[(beta / alpha), $MachinePrecision] * N[(beta / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(t$95$0 / N[(alpha - beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]]]

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)) < -0.99999389451345444

   1. Initial program 59.3

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

   2. Simplified 59.3

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

   3. Taylor expanded in alpha around inf 3.4

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

   4. Simplified 0.1

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

   if -0.99999389451345444 < (/.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. Applied clear-num_binary64 0.1

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{\left(\beta + \alpha\right) + 2}{\alpha - \beta}}}, -0.5, 0.5\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999938945134544:\\ \;\;\;\;\left(\frac{1}{\alpha} + \frac{\beta}{\alpha}\right) - \left(\frac{2}{\alpha \cdot \alpha} + \mathsf{fma}\left(3, \frac{\beta}{\alpha \cdot \alpha}, \frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{\left(\beta + \alpha\right) + 2}{\alpha - \beta}}, -0.5, 0.5\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022137 
(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))