Octave 3.8, jcobi/1

Average Error: 15.7 → 0.3

Time: 5.0s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\frac{\beta - \alpha}{t_0} \leq -0.9999999999988214:\\ \;\;\;\;\left(\frac{1}{\alpha} + \frac{\beta}{\alpha}\right) - \left(\frac{2}{\alpha \cdot \alpha} + \left(3 \cdot \frac{\beta}{\alpha \cdot \alpha} + {\left(\frac{\beta}{\alpha}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\mathsf{fma}\left(\frac{\alpha - \beta}{t_0}, -0.5, 0.5\right)\right)}\\ \end{array} \]

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) -0.9999999999988214)
     (-
      (+ (/ 1.0 alpha) (/ beta alpha))
      (+
       (/ 2.0 (* alpha alpha))
       (+ (* 3.0 (/ beta (* alpha alpha))) (pow (/ beta alpha) 2.0))))
     (exp (log (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) <= -0.9999999999988214) {
		tmp = ((1.0 / alpha) + (beta / alpha)) - ((2.0 / (alpha * alpha)) + ((3.0 * (beta / (alpha * alpha))) + pow((beta / alpha), 2.0)));
	} else {
		tmp = exp(log(fma(((alpha - beta) / t_0), -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) <= -0.9999999999988214)
		tmp = Float64(Float64(Float64(1.0 / alpha) + Float64(beta / alpha)) - Float64(Float64(2.0 / Float64(alpha * alpha)) + Float64(Float64(3.0 * Float64(beta / Float64(alpha * alpha))) + (Float64(beta / alpha) ^ 2.0))));
	else
		tmp = exp(log(fma(Float64(Float64(alpha - beta) / t_0), -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], -0.9999999999988214], N[(N[(N[(1.0 / alpha), $MachinePrecision] + N[(beta / alpha), $MachinePrecision]), $MachinePrecision] - N[(N[(2.0 / N[(alpha * alpha), $MachinePrecision]), $MachinePrecision] + N[(N[(3.0 * N[(beta / N[(alpha * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[(beta / alpha), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Exp[N[Log[N[(N[(N[(alpha - beta), $MachinePrecision] / t$95$0), $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]], $MachinePrecision]], $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)) < -0.99999999999882139

   1. Initial program 60.3

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

   2. Simplified 60.3

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

   3. Taylor expanded in alpha around inf 2.6

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

   4. Simplified 0.0

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

   5. Applied egg-rr 0.0

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

   if -0.99999999999882139 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 0.3

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

   2. Simplified 0.3

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

   3. Applied egg-rr 0.3

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999999988214:\\ \;\;\;\;\left(\frac{1}{\alpha} + \frac{\beta}{\alpha}\right) - \left(\frac{2}{\alpha \cdot \alpha} + \left(3 \cdot \frac{\beta}{\alpha \cdot \alpha} + {\left(\frac{\beta}{\alpha}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\mathsf{fma}\left(\frac{\alpha - \beta}{\left(\beta + \alpha\right) + 2}, -0.5, 0.5\right)\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022134 
(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))