Octave 3.8, jcobi/1

Average Error: 15.8 → 0.2

Time: 7.1s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\beta + \alpha, -2, -4\right)\\ t_1 := \frac{\mathsf{fma}\left(0.5, \beta, \mathsf{fma}\left(\beta, -2, -4\right) \cdot -0.25\right)}{\alpha}\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99:\\ \;\;\;\;t_1 + 0.5 \cdot \left(\frac{\mathsf{fma}\left(\beta, -2, -4\right)}{\alpha} \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\alpha, \frac{1}{t_0}, 0.5 - \frac{\beta}{t_0}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (fma (+ beta alpha) -2.0 -4.0))
        (t_1 (/ (fma 0.5 beta (* (fma beta -2.0 -4.0) -0.25)) alpha)))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.99)
     (+ t_1 (* 0.5 (* (/ (fma beta -2.0 -4.0) alpha) t_1)))
     (fma alpha (/ 1.0 t_0) (- 0.5 (/ beta t_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 = fma((beta + alpha), -2.0, -4.0);
	double t_1 = fma(0.5, beta, (fma(beta, -2.0, -4.0) * -0.25)) / alpha;
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99) {
		tmp = t_1 + (0.5 * ((fma(beta, -2.0, -4.0) / alpha) * t_1));
	} else {
		tmp = fma(alpha, (1.0 / t_0), (0.5 - (beta / t_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 = fma(Float64(beta + alpha), -2.0, -4.0)
	t_1 = Float64(fma(0.5, beta, Float64(fma(beta, -2.0, -4.0) * -0.25)) / alpha)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.99)
		tmp = Float64(t_1 + Float64(0.5 * Float64(Float64(fma(beta, -2.0, -4.0) / alpha) * t_1)));
	else
		tmp = fma(alpha, Float64(1.0 / t_0), Float64(0.5 - Float64(beta / t_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] * -2.0 + -4.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.5 * beta + N[(N[(beta * -2.0 + -4.0), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.99], N[(t$95$1 + N[(0.5 * N[(N[(N[(beta * -2.0 + -4.0), $MachinePrecision] / alpha), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(alpha * N[(1.0 / t$95$0), $MachinePrecision] + N[(0.5 - N[(beta / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.98999999999999999

   1. Initial program 58.7

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

   2. Simplified 58.7

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

   3. Taylor expanded in alpha around -inf 3.4

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

   4. Simplified 0.5

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

   if -0.98999999999999999 < (/.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. 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

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \beta, \mathsf{fma}\left(\beta, -2, -4\right) \cdot -0.25\right)}{\alpha} + 0.5 \cdot \left(\frac{\mathsf{fma}\left(\beta, -2, -4\right)}{\alpha} \cdot \frac{\mathsf{fma}\left(0.5, \beta, \mathsf{fma}\left(\beta, -2, -4\right) \cdot -0.25\right)}{\alpha}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\alpha, \frac{1}{\mathsf{fma}\left(\beta + \alpha, -2, -4\right)}, 0.5 - \frac{\beta}{\mathsf{fma}\left(\beta + \alpha, -2, -4\right)}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022202 
(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))