Octave 3.8, jcobi/3

Average Error: 3.6 → 0.2

Time: 9.6s

Precision: binary64

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

\[ \begin{array}{c}[alpha, beta] = \mathsf{sort}([alpha, beta])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} t_0 := 1 + \left(\left(\beta + \alpha\right) + 2\right)\\ \mathbf{if}\;\beta \leq 5.542132606452304 \cdot 10^{+90}:\\ \;\;\;\;\frac{\frac{-1 - \mathsf{fma}\left(\alpha, \beta, \beta + \alpha\right)}{\alpha + \left(\beta + 2\right)} \cdot \frac{1}{-2 - \left(\beta + \alpha\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta \cdot \beta} + \left(\frac{-1 - \alpha}{\frac{\beta}{\frac{\mathsf{fma}\left(2, \alpha, 4\right)}{\beta}}} - \left(\frac{-1 - \alpha}{\beta} - \frac{\alpha}{\beta \cdot \beta}\right)\right)}{t_0}\\ \end{array} \]

FPCore

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

↓

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 1.0 (+ (+ beta alpha) 2.0))))
   (if (<= beta 5.542132606452304e+90)
     (/
      (*
       (/ (- -1.0 (fma alpha beta (+ beta alpha))) (+ alpha (+ beta 2.0)))
       (/ 1.0 (- -2.0 (+ beta alpha))))
      t_0)
     (/
      (+
       (/ 1.0 (* beta beta))
       (-
        (/ (- -1.0 alpha) (/ beta (/ (fma 2.0 alpha 4.0) beta)))
        (- (/ (- -1.0 alpha) beta) (/ alpha (* beta beta)))))
      t_0))))

C

double code(double alpha, double beta) {
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / ((alpha + beta) + (2.0 * 1.0))) / ((alpha + beta) + (2.0 * 1.0))) / (((alpha + beta) + (2.0 * 1.0)) + 1.0);
}

↓

double code(double alpha, double beta) {
	double t_0 = 1.0 + ((beta + alpha) + 2.0);
	double tmp;
	if (beta <= 5.542132606452304e+90) {
		tmp = (((-1.0 - fma(alpha, beta, (beta + alpha))) / (alpha + (beta + 2.0))) * (1.0 / (-2.0 - (beta + alpha)))) / t_0;
	} else {
		tmp = ((1.0 / (beta * beta)) + (((-1.0 - alpha) / (beta / (fma(2.0, alpha, 4.0) / beta))) - (((-1.0 - alpha) / beta) - (alpha / (beta * beta))))) / t_0;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	return Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) + Float64(beta * alpha)) + 1.0) / Float64(Float64(alpha + beta) + Float64(2.0 * 1.0))) / Float64(Float64(alpha + beta) + Float64(2.0 * 1.0))) / Float64(Float64(Float64(alpha + beta) + Float64(2.0 * 1.0)) + 1.0))
end

↓

function code(alpha, beta)
	t_0 = Float64(1.0 + Float64(Float64(beta + alpha) + 2.0))
	tmp = 0.0
	if (beta <= 5.542132606452304e+90)
		tmp = Float64(Float64(Float64(Float64(-1.0 - fma(alpha, beta, Float64(beta + alpha))) / Float64(alpha + Float64(beta + 2.0))) * Float64(1.0 / Float64(-2.0 - Float64(beta + alpha)))) / t_0);
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(beta * beta)) + Float64(Float64(Float64(-1.0 - alpha) / Float64(beta / Float64(fma(2.0, alpha, 4.0) / beta))) - Float64(Float64(Float64(-1.0 - alpha) / beta) - Float64(alpha / Float64(beta * beta))))) / t_0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

↓

code[alpha_, beta_] := Block[{t$95$0 = N[(1.0 + N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 5.542132606452304e+90], N[(N[(N[(N[(-1.0 - N[(alpha * beta + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(-2.0 - N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-1.0 - alpha), $MachinePrecision] / N[(beta / N[(N[(2.0 * alpha + 4.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(-1.0 - alpha), $MachinePrecision] / beta), $MachinePrecision] - N[(alpha / N[(beta * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.542132606452304e90

   1. Initial program 0.1

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

   2. Applied egg-rr 0.1

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

   if 5.542132606452304e90 < beta

   1. Initial program 8.5

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

   2. Applied egg-rr 8.5

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

   3. Applied egg-rr 8.5

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\frac{\mathsf{fma}\left(\alpha, \beta, 1\right) + \left(\alpha + \beta\right)}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   4. Taylor expanded in beta around -inf 4.3

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

   5. Simplified 0.4

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.542132606452304 \cdot 10^{+90}:\\ \;\;\;\;\frac{\frac{-1 - \mathsf{fma}\left(\alpha, \beta, \beta + \alpha\right)}{\alpha + \left(\beta + 2\right)} \cdot \frac{1}{-2 - \left(\beta + \alpha\right)}}{1 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta \cdot \beta} + \left(\frac{-1 - \alpha}{\frac{\beta}{\frac{\mathsf{fma}\left(2, \alpha, 4\right)}{\beta}}} - \left(\frac{-1 - \alpha}{\beta} - \frac{\alpha}{\beta \cdot \beta}\right)\right)}{1 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022209 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :precision binary64
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ (+ alpha beta) (* 2.0 1.0)) 1.0)))