Octave 3.8, jcobi/1

Average Error: 16.0 → 0.4

Time: 3.5s

Precision: binary64

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

Math

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

↓

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

FPCore

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

↓

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -1.0)
   (+ (/ beta alpha) (/ 1.0 alpha))
   (fma (- alpha beta) (/ 1.0 (fma (+ beta alpha) -2.0 -4.0)) 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 tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -1.0) {
		tmp = (beta / alpha) + (1.0 / alpha);
	} else {
		tmp = fma((alpha - beta), (1.0 / fma((beta + alpha), -2.0, -4.0)), 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)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -1.0)
		tmp = Float64(Float64(beta / alpha) + Float64(1.0 / alpha));
	else
		tmp = fma(Float64(alpha - beta), Float64(1.0 / fma(Float64(beta + alpha), -2.0, -4.0)), 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_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -1.0], N[(N[(beta / alpha), $MachinePrecision] + N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision], N[(N[(alpha - beta), $MachinePrecision] * N[(1.0 / N[(N[(beta + alpha), $MachinePrecision] * -2.0 + -4.0), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -1

   1. Initial program 60.6

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

   2. Simplified 60.7

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

   3. Taylor expanded in alpha around -inf 0.0

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

   4. Simplified 0.0

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

   5. Taylor expanded in beta around 0 0.0

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

   if -1 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 0.5

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

   2. Simplified 0.5

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

   3. Applied egg-rr 0.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(\alpha - \beta, \frac{1}{\mathsf{fma}\left(\alpha + \beta, -2, -4\right)}, 0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.4

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

Reproduce

herbie shell --seed 2022192 
(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))