Octave 3.8, jcobi/4

Average Error: 54.1 → 12.3

Time: 14.6s

Precision: binary64

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

\[[alpha, beta] = \mathsf{sort}([alpha, beta]) \\]

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \alpha + \beta\right)\\ t_1 := i + \left(\alpha + \beta\right)\\ \mathbf{if}\;i \leq 1.0110780720278652 \cdot 10^{+45}:\\ \;\;\;\;\left(\left(i \cdot t_1\right) \cdot \frac{\mathsf{fma}\left(i, t_1, \alpha \cdot \beta\right)}{{t_0}^{2}}\right) \cdot \frac{1}{\left(\beta \cdot \beta + 4 \cdot \left(i \cdot i + i \cdot \beta\right)\right) - 1}\\ \mathbf{elif}\;i \leq 1.882654528648364 \cdot 10^{+145}:\\ \;\;\;\;\frac{\left(i \cdot i\right) \cdot 0.25}{\mathsf{fma}\left(t_0, t_0, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

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
 (let* ((t_0 (fma i 2.0 (+ alpha beta))) (t_1 (+ i (+ alpha beta))))
   (if (<= i 1.0110780720278652e+45)
     (*
      (* (* i t_1) (/ (fma i t_1 (* alpha beta)) (pow t_0 2.0)))
      (/ 1.0 (- (+ (* beta beta) (* 4.0 (+ (* i i) (* i beta)))) 1.0)))
     (if (<= i 1.882654528648364e+145)
       (/ (* (* i i) 0.25) (fma t_0 t_0 -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 t_0 = fma(i, 2.0, (alpha + beta));
	double t_1 = i + (alpha + beta);
	double tmp;
	if (i <= 1.0110780720278652e+45) {
		tmp = ((i * t_1) * (fma(i, t_1, (alpha * beta)) / pow(t_0, 2.0))) * (1.0 / (((beta * beta) + (4.0 * ((i * i) + (i * beta)))) - 1.0));
	} else if (i <= 1.882654528648364e+145) {
		tmp = ((i * i) * 0.25) / fma(t_0, t_0, -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)
	t_0 = fma(i, 2.0, Float64(alpha + beta))
	t_1 = Float64(i + Float64(alpha + beta))
	tmp = 0.0
	if (i <= 1.0110780720278652e+45)
		tmp = Float64(Float64(Float64(i * t_1) * Float64(fma(i, t_1, Float64(alpha * beta)) / (t_0 ^ 2.0))) * Float64(1.0 / Float64(Float64(Float64(beta * beta) + Float64(4.0 * Float64(Float64(i * i) + Float64(i * beta)))) - 1.0)));
	elseif (i <= 1.882654528648364e+145)
		tmp = Float64(Float64(Float64(i * i) * 0.25) / fma(t_0, t_0, -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_] := Block[{t$95$0 = N[(i * 2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, 1.0110780720278652e+45], N[(N[(N[(i * t$95$1), $MachinePrecision] * N[(N[(i * t$95$1 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision] / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(N[(beta * beta), $MachinePrecision] + N[(4.0 * N[(N[(i * i), $MachinePrecision] + N[(i * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 1.882654528648364e+145], N[(N[(N[(i * i), $MachinePrecision] * 0.25), $MachinePrecision] / N[(t$95$0 * t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision], 0.0625]]]]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 3 regimes

2. if i < 1.01107807202786521e45

   1. Initial program 23.3

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

   2. Simplified 23.3

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

   3. Applied egg-rr 8.8

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

   4. Applied egg-rr 8.8

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

   5. Taylor expanded in alpha around 0 9.1

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

   6. Simplified 9.1

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

   if 1.01107807202786521e45 < i < 1.8826545286483641e145

   1. Initial program 52.0

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

   2. Simplified 52.0

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

   3. Applied egg-rr 17.8

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

   4. Taylor expanded in i around inf 18.5

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

   5. Simplified 18.5

      \[\leadsto \frac{\color{blue}{0.25 \cdot \left(i \cdot i\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right), \mathsf{fma}\left(i, 2, \alpha + \beta\right), -1\right)} \]

   if 1.8826545286483641e145 < 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 64.0

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

   3. Taylor expanded in i around inf 9.6

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

4. Final simplification 12.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 1.0110780720278652 \cdot 10^{+45}:\\ \;\;\;\;\left(\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{{\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}^{2}}\right) \cdot \frac{1}{\left(\beta \cdot \beta + 4 \cdot \left(i \cdot i + i \cdot \beta\right)\right) - 1}\\ \mathbf{elif}\;i \leq 1.882654528648364 \cdot 10^{+145}:\\ \;\;\;\;\frac{\left(i \cdot i\right) \cdot 0.25}{\mathsf{fma}\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right), \mathsf{fma}\left(i, 2, \alpha + \beta\right), -1\right)}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Reproduce

herbie shell --seed 2022131 
(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)))