Octave 3.8, jcobi/1

Average Error: 16.0 → 0.1

Time: 3.3s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t_0 \leq -0.9999988829058588:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{\beta}{\alpha}, \frac{2}{\alpha}\right) - \mathsf{fma}\left(2, \frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}, \mathsf{fma}\left(\frac{6}{\alpha}, \frac{\beta}{\alpha}, \frac{\frac{4}{\alpha}}{\alpha}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \end{array} \]

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) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.9999988829058588)
     (/
      (-
       (fma 2.0 (/ beta alpha) (/ 2.0 alpha))
       (fma
        2.0
        (* (/ beta alpha) (/ beta alpha))
        (fma (/ 6.0 alpha) (/ beta alpha) (/ (/ 4.0 alpha) alpha))))
      2.0)
     (/ (+ t_0 1.0) 2.0))))

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) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.9999988829058588) {
		tmp = (fma(2.0, (beta / alpha), (2.0 / alpha)) - fma(2.0, ((beta / alpha) * (beta / alpha)), fma((6.0 / alpha), (beta / alpha), ((4.0 / alpha) / alpha)))) / 2.0;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	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) / Float64(Float64(beta + alpha) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.9999988829058588)
		tmp = Float64(Float64(fma(2.0, Float64(beta / alpha), Float64(2.0 / alpha)) - fma(2.0, Float64(Float64(beta / alpha) * Float64(beta / alpha)), fma(Float64(6.0 / alpha), Float64(beta / alpha), Float64(Float64(4.0 / alpha) / alpha)))) / 2.0);
	else
		tmp = Float64(Float64(t_0 + 1.0) / 2.0);
	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] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.9999988829058588], N[(N[(N[(2.0 * N[(beta / alpha), $MachinePrecision] + N[(2.0 / alpha), $MachinePrecision]), $MachinePrecision] - N[(2.0 * N[(N[(beta / alpha), $MachinePrecision] * N[(beta / alpha), $MachinePrecision]), $MachinePrecision] + N[(N[(6.0 / alpha), $MachinePrecision] * N[(beta / alpha), $MachinePrecision] + N[(N[(4.0 / alpha), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$0 + 1.0), $MachinePrecision] / 2.0), $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)) < -0.99999888290585881

   1. Initial program 59.5

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

   2. Taylor expanded in alpha around inf 3.0

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \frac{1}{\alpha} + 2 \cdot \frac{\beta}{\alpha}\right) - \left(4 \cdot \frac{1}{{\alpha}^{2}} + \left(6 \cdot \frac{\beta}{{\alpha}^{2}} + 2 \cdot \frac{{\beta}^{2}}{{\alpha}^{2}}\right)\right)}}{2} \]

   3. Simplified 0.1

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(2, \frac{\beta}{\alpha}, \frac{2}{\alpha}\right) - \mathsf{fma}\left(2, \frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}, \mathsf{fma}\left(\frac{6}{\alpha}, \frac{\beta}{\alpha}, \frac{\frac{4}{\alpha}}{\alpha}\right)\right)}}{2} \]

   if -0.99999888290585881 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999988829058588:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{\beta}{\alpha}, \frac{2}{\alpha}\right) - \mathsf{fma}\left(2, \frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}, \mathsf{fma}\left(\frac{6}{\alpha}, \frac{\beta}{\alpha}, \frac{\frac{4}{\alpha}}{\alpha}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]

Reproduce

herbie shell --seed 2022152 
(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))