Average Error: 16.3 → 0.1
Time: 2.9s
Precision: binary64
\[\alpha > -1 \land \beta > -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.5:\\ \;\;\;\;\left(\frac{1}{\alpha} + \left(\frac{\beta}{\alpha} + \mathsf{fma}\left(8, \frac{\beta}{{\alpha}^{3}}, \frac{4}{{\alpha}^{3}}\right)\right)\right) + \left({\left(\frac{\beta}{\alpha}\right)}^{3} + \left(\frac{-2}{\alpha \cdot \alpha} + \frac{\beta}{\alpha} \cdot \left(\frac{-3}{\alpha} - \frac{\beta}{\alpha}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 + \frac{\alpha - \beta}{\mathsf{fma}\left(\beta + \alpha, -2, -4\right)}\\ \end{array} \]
(FPCore (alpha beta)
 :precision binary64
 (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))
↓
(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.5)
   (+
    (+
     (/ 1.0 alpha)
     (+
      (/ beta alpha)
      (fma 8.0 (/ beta (pow alpha 3.0)) (/ 4.0 (pow alpha 3.0)))))
    (+
     (pow (/ beta alpha) 3.0)
     (+
      (/ -2.0 (* alpha alpha))
      (* (/ beta alpha) (- (/ -3.0 alpha) (/ beta alpha))))))
   (+ 0.5 (/ (- alpha beta) (fma (+ beta alpha) -2.0 -4.0)))))
double code(double alpha, double beta) {
	return (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
}
↓
double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.5) {
		tmp = ((1.0 / alpha) + ((beta / alpha) + fma(8.0, (beta / pow(alpha, 3.0)), (4.0 / pow(alpha, 3.0))))) + (pow((beta / alpha), 3.0) + ((-2.0 / (alpha * alpha)) + ((beta / alpha) * ((-3.0 / alpha) - (beta / alpha)))));
	} else {
		tmp = 0.5 + ((alpha - beta) / fma((beta + alpha), -2.0, -4.0));
	}
	return tmp;
}

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)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.5)
		tmp = Float64(Float64(Float64(1.0 / alpha) + Float64(Float64(beta / alpha) + fma(8.0, Float64(beta / (alpha ^ 3.0)), Float64(4.0 / (alpha ^ 3.0))))) + Float64((Float64(beta / alpha) ^ 3.0) + Float64(Float64(-2.0 / Float64(alpha * alpha)) + Float64(Float64(beta / alpha) * Float64(Float64(-3.0 / alpha) - Float64(beta / alpha))))));
	else
		tmp = Float64(0.5 + Float64(Float64(alpha - beta) / fma(Float64(beta + alpha), -2.0, -4.0)));
	end
	return tmp
end

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_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], N[(N[(N[(1.0 / alpha), $MachinePrecision] + N[(N[(beta / alpha), $MachinePrecision] + N[(8.0 * N[(beta / N[Power[alpha, 3.0], $MachinePrecision]), $MachinePrecision] + N[(4.0 / N[Power[alpha, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[(beta / alpha), $MachinePrecision], 3.0], $MachinePrecision] + N[(N[(-2.0 / N[(alpha * alpha), $MachinePrecision]), $MachinePrecision] + N[(N[(beta / alpha), $MachinePrecision] * N[(N[(-3.0 / alpha), $MachinePrecision] - N[(beta / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 + N[(N[(alpha - beta), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] * -2.0 + -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

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

\mathbf{else}:\\
\;\;\;\;0.5 + \frac{\alpha - \beta}{\mathsf{fma}\left(\beta + \alpha, -2, -4\right)}\\


\end{array}

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)) < -0.5

    1. Initial program 58.7

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

    2. Simplified58.7

      \[\leadsto \color{blue}{0.5 + \frac{\alpha - \beta}{\mathsf{fma}\left(\beta + \alpha, -2, -4\right)}} \]

    3. Taylor expanded in alpha around inf 5.6

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

    4. Simplified0.3

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

    5. Taylor expanded in beta around 0 0.3

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

    6. Simplified0.3

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

    if -0.5 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

    1. Initial program 0.0

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

    2. Simplified0.0

      \[\leadsto \color{blue}{0.5 + \frac{\alpha - \beta}{\mathsf{fma}\left(\beta + \alpha, -2, -4\right)}} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

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

Reproduce

herbie shell --seed 2022162 
(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))