Octave 3.8, jcobi/4

Average Error: 53.9 → 14.8

Time: 12.7s

Precision: binary64

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

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

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \alpha + \beta\right)\\ t_1 := i \cdot i + i \cdot \beta\\ t_2 := \mathsf{fma}\left(t_0, t_0, -1\right)\\ t_3 := i \cdot \left(i + \left(\alpha + \beta\right)\right)\\ \mathbf{if}\;i \leq 2.4951242434289404 \cdot 10^{+64}:\\ \;\;\;\;\frac{t_3 \cdot \frac{1}{{\left(\frac{t_0}{\sqrt{t_1}}\right)}^{2}}}{t_2}\\ \mathbf{elif}\;i \leq 5.87149832893767 \cdot 10^{+80}:\\ \;\;\;\;\frac{i \cdot \left(i + \alpha\right)}{{\beta}^{2}}\\ \mathbf{elif}\;i \leq 2.2365987727338013 \cdot 10^{+104}:\\ \;\;\;\;\frac{t_3 \cdot \frac{t_1}{{\left(\beta + i \cdot 2\right)}^{2}}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

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 i) (* i beta)))
        (t_2 (fma t_0 t_0 -1.0))
        (t_3 (* i (+ i (+ alpha beta)))))
   (if (<= i 2.4951242434289404e+64)
     (/ (* t_3 (/ 1.0 (pow (/ t_0 (sqrt t_1)) 2.0))) t_2)
     (if (<= i 5.87149832893767e+80)
       (/ (* i (+ i alpha)) (pow beta 2.0))
       (if (<= i 2.2365987727338013e+104)
         (/ (* t_3 (/ t_1 (pow (+ beta (* i 2.0)) 2.0))) t_2)
         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 * i) + (i * beta);
	double t_2 = fma(t_0, t_0, -1.0);
	double t_3 = i * (i + (alpha + beta));
	double tmp;
	if (i <= 2.4951242434289404e+64) {
		tmp = (t_3 * (1.0 / pow((t_0 / sqrt(t_1)), 2.0))) / t_2;
	} else if (i <= 5.87149832893767e+80) {
		tmp = (i * (i + alpha)) / pow(beta, 2.0);
	} else if (i <= 2.2365987727338013e+104) {
		tmp = (t_3 * (t_1 / pow((beta + (i * 2.0)), 2.0))) / t_2;
	} 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(Float64(i * i) + Float64(i * beta))
	t_2 = fma(t_0, t_0, -1.0)
	t_3 = Float64(i * Float64(i + Float64(alpha + beta)))
	tmp = 0.0
	if (i <= 2.4951242434289404e+64)
		tmp = Float64(Float64(t_3 * Float64(1.0 / (Float64(t_0 / sqrt(t_1)) ^ 2.0))) / t_2);
	elseif (i <= 5.87149832893767e+80)
		tmp = Float64(Float64(i * Float64(i + alpha)) / (beta ^ 2.0));
	elseif (i <= 2.2365987727338013e+104)
		tmp = Float64(Float64(t_3 * Float64(t_1 / (Float64(beta + Float64(i * 2.0)) ^ 2.0))) / t_2);
	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[(N[(i * i), $MachinePrecision] + N[(i * beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * t$95$0 + -1.0), $MachinePrecision]}, Block[{t$95$3 = N[(i * N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, 2.4951242434289404e+64], N[(N[(t$95$3 * N[(1.0 / N[Power[N[(t$95$0 / N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[i, 5.87149832893767e+80], N[(N[(i * N[(i + alpha), $MachinePrecision]), $MachinePrecision] / N[Power[beta, 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 2.2365987727338013e+104], N[(N[(t$95$3 * N[(t$95$1 / N[Power[N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], 0.0625]]]]]]]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 4 regimes

2. if i < 2.4951242434289404e64

   1. Initial program 23.2

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

   2. Simplified 23.2

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

   3. Applied egg-rr 11.0

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

   4. Applied egg-rr 11.0

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

   5. Taylor expanded in alpha around 0 11.3

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

   6. Simplified 11.3

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

   if 2.4951242434289404e64 < i < 5.8714983289376704e80

   1. Initial program 35.5

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

   2. Simplified 35.5

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

   3. Taylor expanded in beta around inf 43.5

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

   if 5.8714983289376704e80 < i < 2.2365987727338013e104

   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. Applied egg-rr 20.0

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

   4. Taylor expanded in alpha around 0 20.3

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

   5. Simplified 20.3

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

   if 2.2365987727338013e104 < 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 13.0

      \[\leadsto \color{blue}{0.0625} \]
3. Recombined 4 regimes into one program.

4. Final simplification 14.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 2.4951242434289404 \cdot 10^{+64}:\\ \;\;\;\;\frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \frac{1}{{\left(\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\sqrt{i \cdot i + i \cdot \beta}}\right)}^{2}}}{\mathsf{fma}\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right), \mathsf{fma}\left(i, 2, \alpha + \beta\right), -1\right)}\\ \mathbf{elif}\;i \leq 5.87149832893767 \cdot 10^{+80}:\\ \;\;\;\;\frac{i \cdot \left(i + \alpha\right)}{{\beta}^{2}}\\ \mathbf{elif}\;i \leq 2.2365987727338013 \cdot 10^{+104}:\\ \;\;\;\;\frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \frac{i \cdot i + i \cdot \beta}{{\left(\beta + i \cdot 2\right)}^{2}}}{\mathsf{fma}\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right), \mathsf{fma}\left(i, 2, \alpha + \beta\right), -1\right)}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Reproduce

herbie shell --seed 2022133 
(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)))