Octave 3.8, jcobi/1

Average Error: 16.1 → 0.1

Time: 4.3s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\frac{\beta - \alpha}{t_0} \leq -0.9999997685984842:\\ \;\;\;\;\left(\frac{1}{\alpha} + \left(\frac{\beta}{\alpha} + \mathsf{fma}\left(8, \frac{\beta}{{\alpha}^{3}}, \frac{4}{{\alpha}^{3}}\right)\right)\right) - \mathsf{fma}\left(3, \frac{\beta}{\alpha \cdot \alpha}, \frac{2}{\alpha \cdot \alpha}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({\left(\frac{t_0}{\alpha - \beta}\right)}^{-1}, -0.5, 0.5\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)))
   (if (<= (/ (- beta alpha) t_0) -0.9999997685984842)
     (-
      (+
       (/ 1.0 alpha)
       (+
        (/ beta alpha)
        (fma 8.0 (/ beta (pow alpha 3.0)) (/ 4.0 (pow alpha 3.0)))))
      (fma 3.0 (/ beta (* alpha alpha)) (/ 2.0 (* alpha alpha))))
     (fma (pow (/ t_0 (- alpha beta)) -1.0) -0.5 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 t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (((beta - alpha) / t_0) <= -0.9999997685984842) {
		tmp = ((1.0 / alpha) + ((beta / alpha) + fma(8.0, (beta / pow(alpha, 3.0)), (4.0 / pow(alpha, 3.0))))) - fma(3.0, (beta / (alpha * alpha)), (2.0 / (alpha * alpha)));
	} else {
		tmp = fma(pow((t_0 / (alpha - beta)), -1.0), -0.5, 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)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / t_0) <= -0.9999997685984842)
		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))))) - fma(3.0, Float64(beta / Float64(alpha * alpha)), Float64(2.0 / Float64(alpha * alpha))));
	else
		tmp = fma((Float64(t_0 / Float64(alpha - beta)) ^ -1.0), -0.5, 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_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / t$95$0), $MachinePrecision], -0.9999997685984842], 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[(3.0 * N[(beta / N[(alpha * alpha), $MachinePrecision]), $MachinePrecision] + N[(2.0 / N[(alpha * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(t$95$0 / N[(alpha - beta), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * -0.5 + 0.5), $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)) < -0.999999768598484162

   1. Initial program 59.8

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

   2. Simplified 59.8

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

   3. Taylor expanded in alpha around inf 5.1

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

   4. Simplified 2.5

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

   5. Taylor expanded in beta around 0 0.2

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

   6. Simplified 0.2

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

   if -0.999999768598484162 < (/.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

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

   3. Applied egg-rr 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999997685984842:\\ \;\;\;\;\left(\frac{1}{\alpha} + \left(\frac{\beta}{\alpha} + \mathsf{fma}\left(8, \frac{\beta}{{\alpha}^{3}}, \frac{4}{{\alpha}^{3}}\right)\right)\right) - \mathsf{fma}\left(3, \frac{\beta}{\alpha \cdot \alpha}, \frac{2}{\alpha \cdot \alpha}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({\left(\frac{\left(\beta + \alpha\right) + 2}{\alpha - \beta}\right)}^{-1}, -0.5, 0.5\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022133 
(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))