Octave 3.8, jcobi/4

Average Error: 54.1 → 10.4

Time: 10.6s

Precision: binary64

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

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \leq 7.2 \cdot 10^{+144}:\\ \;\;\;\;\left(\frac{i}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \frac{i \cdot \frac{\beta}{\mathsf{fma}\left(i, 2, \beta\right)} + \frac{i \cdot i}{\mathsf{fma}\left(i, 2, \beta\right)}}{\mathsf{fma}\left(\beta, \beta, \mathsf{fma}\left(4, i \cdot \left(i + \beta\right), -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

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 (<= i 7.2e+144)
   (*
    (* (/ i (+ alpha (fma i 2.0 beta))) (+ i (+ alpha beta)))
    (/
     (+ (* i (/ beta (fma i 2.0 beta))) (/ (* i i) (fma i 2.0 beta)))
     (fma beta beta (fma 4.0 (* i (+ i beta)) -1.0))))
   0.0625))

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 (i <= 7.2e+144) {
		tmp = ((i / (alpha + fma(i, 2.0, beta))) * (i + (alpha + beta))) * (((i * (beta / fma(i, 2.0, beta))) + ((i * i) / fma(i, 2.0, beta))) / fma(beta, beta, fma(4.0, (i * (i + beta)), -1.0)));
	} else {
		tmp = 0.0625;
	}
	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 (i <= 7.2e+144)
		tmp = Float64(Float64(Float64(i / Float64(alpha + fma(i, 2.0, beta))) * Float64(i + Float64(alpha + beta))) * Float64(Float64(Float64(i * Float64(beta / fma(i, 2.0, beta))) + Float64(Float64(i * i) / fma(i, 2.0, beta))) / fma(beta, beta, fma(4.0, Float64(i * Float64(i + beta)), -1.0))));
	else
		tmp = 0.0625;
	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_] := If[LessEqual[i, 7.2e+144], N[(N[(N[(i / N[(alpha + N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(i * N[(beta / N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(i * i), $MachinePrecision] / N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta * beta + N[(4.0 * N[(i * N[(i + beta), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0625]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if i < 7.1999999999999995e144

   1. Initial program 43.1

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

   2. Simplified 32.1

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

   3. Applied egg-rr 15.8

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

   4. Taylor expanded in alpha around 0 16.3

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

   5. Simplified 11.7

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

   6. Taylor expanded in alpha around 0 16.1

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

   7. Simplified 10.9

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

   if 7.1999999999999995e144 < i

   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 63.9

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

   3. Taylor expanded in i around inf 10.0

      \[\leadsto \color{blue}{0.0625} \]
3. Recombined 2 regimes into one program.

4. Final simplification 10.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 7.2 \cdot 10^{+144}:\\ \;\;\;\;\left(\frac{i}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \frac{i \cdot \frac{\beta}{\mathsf{fma}\left(i, 2, \beta\right)} + \frac{i \cdot i}{\mathsf{fma}\left(i, 2, \beta\right)}}{\mathsf{fma}\left(\beta, \beta, \mathsf{fma}\left(4, i \cdot \left(i + \beta\right), -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Reproduce

herbie shell --seed 2022155 
(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)))