Octave 3.8, jcobi/4

Average Error: 54.0 → 14.3

Time: 20.6s

Precision: binary64

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

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

Math

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

↓

\[\begin{array}{l} t_0 := \alpha + \mathsf{fma}\left(i, 2, \beta\right)\\ \mathbf{if}\;\beta \leq 3.9444340291518456 \cdot 10^{+118}:\\ \;\;\;\;{\left(\left(0.25 + 0.03125 \cdot \frac{1}{i \cdot i}\right) + \left(\frac{\beta \cdot \beta}{i \cdot i} \cdot -0.0625 + \frac{\alpha \cdot \alpha}{i \cdot i} \cdot -0.0625\right)\right)}^{2}\\ \mathbf{elif}\;\beta \leq 1.4424082428851503 \cdot 10^{+139}:\\ \;\;\;\;{\left(\sqrt{\frac{\frac{{i}^{2} + \beta \cdot i}{\beta + i \cdot 2}}{{t_0}^{2} + -1} \cdot \left(\left(\beta + \left(i + \alpha\right)\right) \cdot \frac{i}{t_0}\right)}\right)}^{2}\\ \mathbf{elif}\;\beta \leq 4.095819014891881 \cdot 10^{+181}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{\beta} \cdot \sqrt{i \cdot \left(i + \alpha\right)}\right)}^{2}\\ \end{array} \]

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
 (let* ((t_0 (+ alpha (fma i 2.0 beta))))
   (if (<= beta 3.9444340291518456e+118)
     (pow
      (+
       (+ 0.25 (* 0.03125 (/ 1.0 (* i i))))
       (+
        (* (/ (* beta beta) (* i i)) -0.0625)
        (* (/ (* alpha alpha) (* i i)) -0.0625)))
      2.0)
     (if (<= beta 1.4424082428851503e+139)
       (pow
        (sqrt
         (*
          (/
           (/ (+ (pow i 2.0) (* beta i)) (+ beta (* i 2.0)))
           (+ (pow t_0 2.0) -1.0))
          (* (+ beta (+ i alpha)) (/ i t_0))))
        2.0)
       (if (<= beta 4.095819014891881e+181)
         0.0625
         (pow (* (/ 1.0 beta) (sqrt (* i (+ i alpha)))) 2.0))))))

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 t_0 = alpha + fma(i, 2.0, beta);
	double tmp;
	if (beta <= 3.9444340291518456e+118) {
		tmp = pow(((0.25 + (0.03125 * (1.0 / (i * i)))) + ((((beta * beta) / (i * i)) * -0.0625) + (((alpha * alpha) / (i * i)) * -0.0625))), 2.0);
	} else if (beta <= 1.4424082428851503e+139) {
		tmp = pow(sqrt(((((pow(i, 2.0) + (beta * i)) / (beta + (i * 2.0))) / (pow(t_0, 2.0) + -1.0)) * ((beta + (i + alpha)) * (i / t_0)))), 2.0);
	} else if (beta <= 4.095819014891881e+181) {
		tmp = 0.0625;
	} else {
		tmp = pow(((1.0 / beta) * sqrt((i * (i + alpha)))), 2.0);
	}
	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)
	t_0 = Float64(alpha + fma(i, 2.0, beta))
	tmp = 0.0
	if (beta <= 3.9444340291518456e+118)
		tmp = Float64(Float64(0.25 + Float64(0.03125 * Float64(1.0 / Float64(i * i)))) + Float64(Float64(Float64(Float64(beta * beta) / Float64(i * i)) * -0.0625) + Float64(Float64(Float64(alpha * alpha) / Float64(i * i)) * -0.0625))) ^ 2.0;
	elseif (beta <= 1.4424082428851503e+139)
		tmp = sqrt(Float64(Float64(Float64(Float64((i ^ 2.0) + Float64(beta * i)) / Float64(beta + Float64(i * 2.0))) / Float64((t_0 ^ 2.0) + -1.0)) * Float64(Float64(beta + Float64(i + alpha)) * Float64(i / t_0)))) ^ 2.0;
	elseif (beta <= 4.095819014891881e+181)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(1.0 / beta) * sqrt(Float64(i * Float64(i + alpha)))) ^ 2.0;
	end
	return 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_] := Block[{t$95$0 = N[(alpha + N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 3.9444340291518456e+118], N[Power[N[(N[(0.25 + N[(0.03125 * N[(1.0 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(beta * beta), $MachinePrecision] / N[(i * i), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] + N[(N[(N[(alpha * alpha), $MachinePrecision] / N[(i * i), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[beta, 1.4424082428851503e+139], N[Power[N[Sqrt[N[(N[(N[(N[(N[Power[i, 2.0], $MachinePrecision] + N[(beta * i), $MachinePrecision]), $MachinePrecision] / N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[t$95$0, 2.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(beta + N[(i + alpha), $MachinePrecision]), $MachinePrecision] * N[(i / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[beta, 4.095819014891881e+181], 0.0625, N[Power[N[(N[(1.0 / beta), $MachinePrecision] * N[Sqrt[N[(i * N[(i + alpha), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]]]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 4 regimes

2. if beta < 3.9444340291518456e118

   1. Initial program 48.5

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

   2. Simplified 43.3

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

   3. Applied egg-rr 32.1

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

   4. Taylor expanded in i around inf 3.7

      \[\leadsto {\color{blue}{\left(\left(0.25 + 0.03125 \cdot \frac{1}{{i}^{2}}\right) - \left(0.0625 \cdot \frac{{\beta}^{2}}{{i}^{2}} + 0.0625 \cdot \frac{{\alpha}^{2}}{{i}^{2}}\right)\right)}}^{2} \]

   5. Simplified 3.7

      \[\leadsto {\color{blue}{\left(\left(0.25 + 0.03125 \cdot \frac{1}{i \cdot i}\right) - \left(0.0625 \cdot \frac{\beta \cdot \beta}{i \cdot i} + 0.0625 \cdot \frac{\alpha \cdot \alpha}{i \cdot i}\right)\right)}}^{2} \]

   if 3.9444340291518456e118 < beta < 1.44240824288515029e139

   1. Initial program 57.9

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

   2. Simplified 62.5

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

   3. Applied egg-rr 30.0

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

   4. Taylor expanded in alpha around 0 32.2

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

   if 1.44240824288515029e139 < beta < 4.0958190148918808e181

   1. Initial program 63.6

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

   2. Simplified 61.8

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

   3. Taylor expanded in i around inf 30.7

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

   if 4.0958190148918808e181 < beta

   1. Initial program 64.0

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

   2. Simplified 57.0

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

   3. Applied egg-rr 57.0

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

   4. Taylor expanded in beta around inf 33.2

      \[\leadsto {\color{blue}{\left(\frac{1}{\beta} \cdot \sqrt{\left(i + \alpha\right) \cdot i}\right)}}^{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 14.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.9444340291518456 \cdot 10^{+118}:\\ \;\;\;\;{\left(\left(0.25 + 0.03125 \cdot \frac{1}{i \cdot i}\right) + \left(\frac{\beta \cdot \beta}{i \cdot i} \cdot -0.0625 + \frac{\alpha \cdot \alpha}{i \cdot i} \cdot -0.0625\right)\right)}^{2}\\ \mathbf{elif}\;\beta \leq 1.4424082428851503 \cdot 10^{+139}:\\ \;\;\;\;{\left(\sqrt{\frac{\frac{{i}^{2} + \beta \cdot i}{\beta + i \cdot 2}}{{\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right)}^{2} + -1} \cdot \left(\left(\beta + \left(i + \alpha\right)\right) \cdot \frac{i}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)}\right)}^{2}\\ \mathbf{elif}\;\beta \leq 4.095819014891881 \cdot 10^{+181}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{\beta} \cdot \sqrt{i \cdot \left(i + \alpha\right)}\right)}^{2}\\ \end{array} \]

Reproduce

herbie shell --seed 2022153 
(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)))