Average Error: 3.5 → 0.2
Time: 7.5s
Precision: binary64
\[\alpha > -1 \land \beta > -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
↓
\[\frac{\left(\alpha + 1\right) \cdot {\left(\frac{\sqrt{1 + \beta}}{\left(\alpha + \beta\right) + 2}\right)}^{2}}{\alpha + \left(\beta + 3\right)} \]
(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
 (/
  (* (+ alpha 1.0) (pow (/ (sqrt (+ 1.0 beta)) (+ (+ alpha beta) 2.0)) 2.0))
  (+ 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) {
	return ((alpha + 1.0) * pow((sqrt((1.0 + beta)) / ((alpha + beta) + 2.0)), 2.0)) / (alpha + (beta + 3.0));
}

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = (((((alpha + beta) + (beta * alpha)) + 1.0d0) / ((alpha + beta) + (2.0d0 * 1.0d0))) / ((alpha + beta) + (2.0d0 * 1.0d0))) / (((alpha + beta) + (2.0d0 * 1.0d0)) + 1.0d0)
end function
↓
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = ((alpha + 1.0d0) * ((sqrt((1.0d0 + beta)) / ((alpha + beta) + 2.0d0)) ** 2.0d0)) / (alpha + (beta + 3.0d0))
end function

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) {
	return ((alpha + 1.0) * Math.pow((Math.sqrt((1.0 + beta)) / ((alpha + beta) + 2.0)), 2.0)) / (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):
	return ((alpha + 1.0) * math.pow((math.sqrt((1.0 + beta)) / ((alpha + beta) + 2.0)), 2.0)) / (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)
	return Float64(Float64(Float64(alpha + 1.0) * (Float64(sqrt(Float64(1.0 + beta)) / Float64(Float64(alpha + beta) + 2.0)) ^ 2.0)) / Float64(alpha + Float64(beta + 3.0)))
end

function tmp = code(alpha, beta)
	tmp = (((((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);
end
↓
function tmp = code(alpha, beta)
	tmp = ((alpha + 1.0) * ((sqrt((1.0 + beta)) / ((alpha + beta) + 2.0)) ^ 2.0)) / (alpha + (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_] := N[(N[(N[(alpha + 1.0), $MachinePrecision] * N[Power[N[(N[Sqrt[N[(1.0 + beta), $MachinePrecision]], $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $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}
↓
\frac{\left(\alpha + 1\right) \cdot {\left(\frac{\sqrt{1 + \beta}}{\left(\alpha + \beta\right) + 2}\right)}^{2}}{\alpha + \left(\beta + 3\right)}

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 3.5

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

  2. Simplified2.1

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

  3. Applied egg-rr0.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2022145 
(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)))