Octave 3.8, jcobi/4

Average Error: 54.0 → 9.5

Time: 23.0s

Precision: binary64

Cost: 708

\[\left(\alpha > -1 \land \beta > -1\right) \land i > 1\]

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\beta \leq 6.5 \cdot 10^{+129}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

FPCore

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

↓

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 6.5e+129) 0.0625 (* (/ i beta) (/ (+ i alpha) beta))))

C

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

↓

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 6.5e+129) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * ((i + alpha) / beta);
	}
	return tmp;
}

Fortran

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

↓

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 6.5d+129) then
        tmp = 0.0625d0
    else
        tmp = (i / beta) * ((i + alpha) / beta)
    end if
    code = tmp
end function

Java

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

↓

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 6.5e+129) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * ((i + alpha) / beta);
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	return (((i * ((alpha + beta) + i)) * ((beta * alpha) + (i * ((alpha + beta) + i)))) / (((alpha + beta) + (2.0 * i)) * ((alpha + beta) + (2.0 * i)))) / ((((alpha + beta) + (2.0 * i)) * ((alpha + beta) + (2.0 * i))) - 1.0)

↓

def code(alpha, beta, i):
	tmp = 0
	if beta <= 6.5e+129:
		tmp = 0.0625
	else:
		tmp = (i / beta) * ((i + alpha) / beta)
	return tmp

Julia

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

↓

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 6.5e+129)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / beta) * Float64(Float64(i + alpha) / beta));
	end
	return tmp
end

MATLAB

function tmp = code(alpha, beta, i)
	tmp = (((i * ((alpha + beta) + i)) * ((beta * alpha) + (i * ((alpha + beta) + i)))) / (((alpha + beta) + (2.0 * i)) * ((alpha + beta) + (2.0 * i)))) / ((((alpha + beta) + (2.0 * i)) * ((alpha + beta) + (2.0 * i))) - 1.0);
end

↓

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 6.5e+129)
		tmp = 0.0625;
	else
		tmp = (i / beta) * ((i + alpha) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[alpha_, beta_, i_] := If[LessEqual[beta, 6.5e+129], 0.0625, N[(N[(i / beta), $MachinePrecision] * N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.4999999999999995e129

   1. Initial program 48.7

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

   2. Simplified 32.4

      \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\beta + \mathsf{fma}\left(i, 2, \alpha\right), \beta + \mathsf{fma}\left(i, 2, \alpha\right), -1\right)} \cdot \frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\frac{\left(\beta + \mathsf{fma}\left(i, 2, \alpha\right)\right) \cdot \left(\beta + \mathsf{fma}\left(i, 2, \alpha\right)\right)}{i + \left(\alpha + \beta\right)}}} \]
      Proof
      (*.f64 (/.f64 i (fma.f64 (+.f64 beta (fma.f64 i 2 alpha)) (+.f64 beta (fma.f64 i 2 alpha)) -1)) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (/.f64 (*.f64 (+.f64 beta (fma.f64 i 2 alpha)) (+.f64 beta (fma.f64 i 2 alpha))) (+.f64 i (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (fma.f64 (+.f64 beta (Rewrite<= fma-def_binary64 (+.f64 (*.f64 i 2) alpha))) (+.f64 beta (fma.f64 i 2 alpha)) -1)) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (/.f64 (*.f64 (+.f64 beta (fma.f64 i 2 alpha)) (+.f64 beta (fma.f64 i 2 alpha))) (+.f64 i (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (fma.f64 (+.f64 beta (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 i)) alpha)) (+.f64 beta (fma.f64 i 2 alpha)) -1)) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (/.f64 (*.f64 (+.f64 beta (fma.f64 i 2 alpha)) (+.f64 beta (fma.f64 i 2 alpha))) (+.f64 i (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (fma.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 beta (*.f64 2 i)) alpha)) (+.f64 beta (fma.f64 i 2 alpha)) -1)) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (/.f64 (*.f64 (+.f64 beta (fma.f64 i 2 alpha)) (+.f64 beta (fma.f64 i 2 alpha))) (+.f64 i (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (fma.f64 (Rewrite<= +-commutative_binary64 (+.f64 alpha (+.f64 beta (*.f64 2 i)))) (+.f64 beta (fma.f64 i 2 alpha)) -1)) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (/.f64 (*.f64 (+.f64 beta (fma.f64 i 2 alpha)) (+.f64 beta (fma.f64 i 2 alpha))) (+.f64 i (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (fma.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 beta (fma.f64 i 2 alpha)) -1)) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (/.f64 (*.f64 (+.f64 beta (fma.f64 i 2 alpha)) (+.f64 beta (fma.f64 i 2 alpha))) (+.f64 i (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (fma.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 beta (Rewrite<= fma-def_binary64 (+.f64 (*.f64 i 2) alpha))) -1)) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (/.f64 (*.f64 (+.f64 beta (fma.f64 i 2 alpha)) (+.f64 beta (fma.f64 i 2 alpha))) (+.f64 i (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (fma.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 beta (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 i)) alpha)) -1)) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (/.f64 (*.f64 (+.f64 beta (fma.f64 i 2 alpha)) (+.f64 beta (fma.f64 i 2 alpha))) (+.f64 i (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (fma.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 beta (*.f64 2 i)) alpha)) -1)) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (/.f64 (*.f64 (+.f64 beta (fma.f64 i 2 alpha)) (+.f64 beta (fma.f64 i 2 alpha))) (+.f64 i (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (fma.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (Rewrite<= +-commutative_binary64 (+.f64 alpha (+.f64 beta (*.f64 2 i)))) -1)) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (/.f64 (*.f64 (+.f64 beta (fma.f64 i 2 alpha)) (+.f64 beta (fma.f64 i 2 alpha))) (+.f64 i (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (fma.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 alpha beta) (*.f64 2 i))) -1)) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (/.f64 (*.f64 (+.f64 beta (fma.f64 i 2 alpha)) (+.f64 beta (fma.f64 i 2 alpha))) (+.f64 i (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (fma.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (Rewrite<= metadata-eval (neg.f64 1)))) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (/.f64 (*.f64 (+.f64 beta (fma.f64 i 2 alpha)) (+.f64 beta (fma.f64 i 2 alpha))) (+.f64 i (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1))) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (/.f64 (*.f64 (+.f64 beta (fma.f64 i 2 alpha)) (+.f64 beta (fma.f64 i 2 alpha))) (+.f64 i (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) (/.f64 (fma.f64 i (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 alpha beta) i)) (*.f64 alpha beta)) (/.f64 (*.f64 (+.f64 beta (fma.f64 i 2 alpha)) (+.f64 beta (fma.f64 i 2 alpha))) (+.f64 i (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) (/.f64 (fma.f64 i (+.f64 (+.f64 alpha beta) i) (Rewrite<= *-commutative_binary64 (*.f64 beta alpha))) (/.f64 (*.f64 (+.f64 beta (fma.f64 i 2 alpha)) (+.f64 beta (fma.f64 i 2 alpha))) (+.f64 i (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (*.f64 beta alpha))) (/.f64 (*.f64 (+.f64 beta (fma.f64 i 2 alpha)) (+.f64 beta (fma.f64 i 2 alpha))) (+.f64 i (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (/.f64 (*.f64 (+.f64 beta (fma.f64 i 2 alpha)) (+.f64 beta (fma.f64 i 2 alpha))) (+.f64 i (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) (/.f64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (/.f64 (*.f64 (+.f64 beta (Rewrite<= fma-def_binary64 (+.f64 (*.f64 i 2) alpha))) (+.f64 beta (fma.f64 i 2 alpha))) (+.f64 i (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) (/.f64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (/.f64 (*.f64 (+.f64 beta (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 i)) alpha)) (+.f64 beta (fma.f64 i 2 alpha))) (+.f64 i (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) (/.f64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (/.f64 (*.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 beta (*.f64 2 i)) alpha)) (+.f64 beta (fma.f64 i 2 alpha))) (+.f64 i (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) (/.f64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (/.f64 (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 alpha (+.f64 beta (*.f64 2 i)))) (+.f64 beta (fma.f64 i 2 alpha))) (+.f64 i (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) (/.f64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (/.f64 (*.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 beta (fma.f64 i 2 alpha))) (+.f64 i (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) (/.f64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (/.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 beta (Rewrite<= fma-def_binary64 (+.f64 (*.f64 i 2) alpha)))) (+.f64 i (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) (/.f64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (/.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 beta (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 i)) alpha))) (+.f64 i (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) (/.f64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (/.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 beta (*.f64 2 i)) alpha))) (+.f64 i (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) (/.f64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (/.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (Rewrite<= +-commutative_binary64 (+.f64 alpha (+.f64 beta (*.f64 2 i))))) (+.f64 i (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) (/.f64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (/.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (+.f64 i (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) (/.f64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (/.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 alpha beta) i))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (+.f64 (+.f64 alpha beta) i)) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))))): 59 points increase in error, 3 points decrease in error
      (*.f64 (/.f64 i (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 (+.f64 alpha beta) i) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 i (*.f64 (+.f64 (+.f64 alpha beta) i) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))))) (*.f64 (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))))): 31 points increase in error, 8 points decrease in error
      (/.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))))) (*.f64 (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))))): 2 points increase in error, 15 points decrease in error
      (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1))): 1 points increase in error, 6 points decrease in error

   3. Taylor expanded in i around inf 4.6

      \[\leadsto \color{blue}{0.0625} \]

   if 6.4999999999999995e129 < beta

   1. Initial program 63.8

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

   2. Taylor expanded in i around 0 63.8

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

   3. Simplified 63.8

      \[\leadsto \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\color{blue}{\left(\beta + \alpha\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 4\right) + \left(i \cdot i\right) \cdot 4}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      Proof
      (+.f64 (*.f64 (+.f64 beta alpha) (+.f64 (+.f64 beta alpha) (*.f64 i 4))) (*.f64 (*.f64 i i) 4)): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 (+.f64 beta alpha) (+.f64 (+.f64 beta alpha) (Rewrite<= *-commutative_binary64 (*.f64 4 i)))) (*.f64 (*.f64 i i) 4)): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (+.f64 beta alpha) (+.f64 beta alpha)) (*.f64 (*.f64 4 i) (+.f64 beta alpha)))) (*.f64 (*.f64 i i) 4)): 1 points increase in error, 1 points decrease in error
      (+.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 (+.f64 beta alpha) 2)) (*.f64 (*.f64 4 i) (+.f64 beta alpha))) (*.f64 (*.f64 i i) 4)): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (pow.f64 (+.f64 beta alpha) 2) (Rewrite<= associate-*r*_binary64 (*.f64 4 (*.f64 i (+.f64 beta alpha))))) (*.f64 (*.f64 i i) 4)): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 4 (*.f64 i (+.f64 beta alpha))) (pow.f64 (+.f64 beta alpha) 2))) (*.f64 (*.f64 i i) 4)): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (*.f64 4 (*.f64 i (+.f64 beta alpha))) (pow.f64 (+.f64 beta alpha) 2)) (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 i 2)) 4)): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (*.f64 4 (*.f64 i (+.f64 beta alpha))) (pow.f64 (+.f64 beta alpha) 2)) (Rewrite<= *-commutative_binary64 (*.f64 4 (pow.f64 i 2)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-+r+_binary64 (+.f64 (*.f64 4 (*.f64 i (+.f64 beta alpha))) (+.f64 (pow.f64 (+.f64 beta alpha) 2) (*.f64 4 (pow.f64 i 2))))): 1 points increase in error, 0 points decrease in error

   4. Taylor expanded in beta around inf 46.8

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

   5. Simplified 18.3

      \[\leadsto \color{blue}{\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}} \]
      Proof
      (*.f64 (/.f64 i beta) (/.f64 (+.f64 i alpha) beta)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 i (+.f64 i alpha)) (*.f64 beta beta))): 93 points increase in error, 10 points decrease in error
      (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 (+.f64 i alpha) i)) (*.f64 beta beta)): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (+.f64 i alpha) i) (Rewrite<= unpow2_binary64 (pow.f64 beta 2))): 0 points increase in error, 0 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 9.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.5 \cdot 10^{+129}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternatives

Alternative 1
Error	17.2
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.5 \cdot 10^{+271}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 2
Error	57.3
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022332 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :precision binary64
  :pre (and (and (> alpha -1.0) (> beta -1.0)) (> i 1.0))
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i)))) (- (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i))) 1.0)))