Average Error: 3.8 → 0.1
Time: 7.7s
Precision: binary64
\[\alpha > -1 \land \beta > -1\]
\[[alpha, beta] = \mathsf{sort}([alpha, beta]) \\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
↓
\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2\\ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(1 + \alpha\right) \cdot \frac{\frac{1 + \beta}{t_0}}{t_0}}{\alpha + \left(\beta + 3\right)}\right)\right) \end{array} \]
(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 (+ (+ alpha beta) 2.0)))
   (expm1
    (log1p
     (/
      (* (+ 1.0 alpha) (/ (/ (+ 1.0 beta) t_0) t_0))
      (+ alpha (+ beta 3.0)))))))
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 = (alpha + beta) + 2.0;
	return expm1(log1p((((1.0 + alpha) * (((1.0 + beta) / t_0) / t_0)) / (alpha + (beta + 3.0)))));
}

public static 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);
}
↓
public static double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 2.0;
	return Math.expm1(Math.log1p((((1.0 + alpha) * (((1.0 + beta) / t_0) / t_0)) / (alpha + (beta + 3.0)))));
}

def code(alpha, 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)
↓
def code(alpha, beta):
	t_0 = (alpha + beta) + 2.0
	return math.expm1(math.log1p((((1.0 + alpha) * (((1.0 + beta) / t_0) / t_0)) / (alpha + (beta + 3.0)))))

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(Float64(alpha + beta) + 2.0)
	return expm1(log1p(Float64(Float64(Float64(1.0 + alpha) * Float64(Float64(Float64(1.0 + beta) / t_0) / t_0)) / Float64(alpha + Float64(beta + 3.0)))))
end

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[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]}, N[(Exp[N[Log[1 + N[(N[(N[(1.0 + alpha), $MachinePrecision] * N[(N[(N[(1.0 + beta), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]]

\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 := \left(\alpha + \beta\right) + 2\\
\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(1 + \alpha\right) \cdot \frac{\frac{1 + \beta}{t_0}}{t_0}}{\alpha + \left(\beta + 3\right)}\right)\right)
\end{array}

Error

Bits error versus alpha

Bits error versus beta

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 3.8

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

  2. Simplified2.3

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

  3. Applied egg-rr0.1

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

  4. Applied egg-rr0.2

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

  5. Applied egg-rr0.1

    \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(1 + \alpha\right) \cdot \frac{\frac{1 + \beta}{\left(\beta + \alpha\right) + 2}}{\left(\beta + \alpha\right) + 2}}{\alpha + \left(\beta + 3\right)}\right)\right)} \]

  6. Final simplification0.1

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(1 + \alpha\right) \cdot \frac{\frac{1 + \beta}{\left(\alpha + \beta\right) + 2}}{\left(\alpha + \beta\right) + 2}}{\alpha + \left(\beta + 3\right)}\right)\right) \]

Reproduce

herbie shell --seed 2022134 
(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)))