Octave 3.8, jcobi/1

Average Error: 16.1 → 0.3

Time: 5.5s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99998:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(0.5 - \frac{\alpha - \beta}{\mathsf{fma}\left(\beta + \alpha, 2, 4\right)}\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.99998)
   (/ (+ beta 1.0) alpha)
   (expm1 (log1p (- 0.5 (/ (- alpha beta) (fma (+ beta alpha) 2.0 4.0)))))))

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 tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99998) {
		tmp = (beta + 1.0) / alpha;
	} else {
		tmp = expm1(log1p((0.5 - ((alpha - beta) / fma((beta + alpha), 2.0, 4.0)))));
	}
	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)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.99998)
		tmp = Float64(Float64(beta + 1.0) / alpha);
	else
		tmp = expm1(log1p(Float64(0.5 - Float64(Float64(alpha - beta) / fma(Float64(beta + alpha), 2.0, 4.0)))));
	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_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.99998], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision], N[(Exp[N[Log[1 + N[(0.5 - N[(N[(alpha - beta), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] * 2.0 + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.99997999999999998

   1. Initial program 59.3

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

   2. Simplified 59.3

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

   3. Taylor expanded in alpha around inf 0.8

      \[\leadsto \color{blue}{\frac{0.5 \cdot \beta + 0.25 \cdot \left(4 + 2 \cdot \beta\right)}{\alpha}} \]

   4. Simplified 0.8

      \[\leadsto \color{blue}{\frac{1 + \beta}{\alpha}} \]

   if -0.99997999999999998 < (/.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

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

   3. Applied egg-rr 0.1

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.5 - \frac{\alpha - \beta}{\mathsf{fma}\left(\alpha + \beta, 2, 4\right)}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99998:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(0.5 - \frac{\alpha - \beta}{\mathsf{fma}\left(\beta + \alpha, 2, 4\right)}\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022209 
(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))