Octave 3.8, jcobi/4

Average Error: 53.7 → 12.7

Time: 16.2s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := i + \left(\alpha + \beta\right)\\ t_1 := \mathsf{fma}\left(i, t_0, \alpha \cdot \beta\right)\\ t_2 := \mathsf{fma}\left(i, 2, \alpha + \beta\right)\\ t_3 := {t_2}^{2}\\ t_4 := \sqrt{t_3 + -1}\\ t_5 := \mathsf{fma}\left(t_2, t_2, -1\right)\\ \mathbf{if}\;i \leq 3.1189906262604905 \cdot 10^{+69}:\\ \;\;\;\;\frac{i}{t_4} \cdot \frac{t_0 \cdot \frac{t_1}{t_3}}{t_4}\\ \mathbf{elif}\;i \leq 3.1729621267394858 \cdot 10^{+100}:\\ \;\;\;\;\frac{0.25 \cdot \left(i \cdot i\right)}{t_5}\\ \mathbf{elif}\;i \leq 2.587302964673011 \cdot 10^{+146}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt{i \cdot t_0}}{\frac{t_2}{\sqrt{t_1}}}\right)}^{2}}{t_5}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

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 (+ i (+ alpha beta)))
        (t_1 (fma i t_0 (* alpha beta)))
        (t_2 (fma i 2.0 (+ alpha beta)))
        (t_3 (pow t_2 2.0))
        (t_4 (sqrt (+ t_3 -1.0)))
        (t_5 (fma t_2 t_2 -1.0)))
   (if (<= i 3.1189906262604905e+69)
     (* (/ i t_4) (/ (* t_0 (/ t_1 t_3)) t_4))
     (if (<= i 3.1729621267394858e+100)
       (/ (* 0.25 (* i i)) t_5)
       (if (<= i 2.587302964673011e+146)
         (/ (pow (/ (sqrt (* i t_0)) (/ t_2 (sqrt t_1))) 2.0) t_5)
         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 = i + (alpha + beta);
	double t_1 = fma(i, t_0, (alpha * beta));
	double t_2 = fma(i, 2.0, (alpha + beta));
	double t_3 = pow(t_2, 2.0);
	double t_4 = sqrt((t_3 + -1.0));
	double t_5 = fma(t_2, t_2, -1.0);
	double tmp;
	if (i <= 3.1189906262604905e+69) {
		tmp = (i / t_4) * ((t_0 * (t_1 / t_3)) / t_4);
	} else if (i <= 3.1729621267394858e+100) {
		tmp = (0.25 * (i * i)) / t_5;
	} else if (i <= 2.587302964673011e+146) {
		tmp = pow((sqrt((i * t_0)) / (t_2 / sqrt(t_1))), 2.0) / t_5;
	} 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 = Float64(i + Float64(alpha + beta))
	t_1 = fma(i, t_0, Float64(alpha * beta))
	t_2 = fma(i, 2.0, Float64(alpha + beta))
	t_3 = t_2 ^ 2.0
	t_4 = sqrt(Float64(t_3 + -1.0))
	t_5 = fma(t_2, t_2, -1.0)
	tmp = 0.0
	if (i <= 3.1189906262604905e+69)
		tmp = Float64(Float64(i / t_4) * Float64(Float64(t_0 * Float64(t_1 / t_3)) / t_4));
	elseif (i <= 3.1729621267394858e+100)
		tmp = Float64(Float64(0.25 * Float64(i * i)) / t_5);
	elseif (i <= 2.587302964673011e+146)
		tmp = Float64((Float64(sqrt(Float64(i * t_0)) / Float64(t_2 / sqrt(t_1))) ^ 2.0) / t_5);
	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 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(i * t$95$0 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(i * 2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[t$95$2, 2.0], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(t$95$3 + -1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$2 * t$95$2 + -1.0), $MachinePrecision]}, If[LessEqual[i, 3.1189906262604905e+69], N[(N[(i / t$95$4), $MachinePrecision] * N[(N[(t$95$0 * N[(t$95$1 / t$95$3), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 3.1729621267394858e+100], N[(N[(0.25 * N[(i * i), $MachinePrecision]), $MachinePrecision] / t$95$5), $MachinePrecision], If[LessEqual[i, 2.587302964673011e+146], N[(N[Power[N[(N[Sqrt[N[(i * t$95$0), $MachinePrecision]], $MachinePrecision] / N[(t$95$2 / N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t$95$5), $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 < 3.1189906262604905e69

   1. Initial program 23.7

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

   2. Simplified 23.7

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

   3. Applied egg-rr 10.7

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

   4. Applied egg-rr 10.7

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

   if 3.1189906262604905e69 < i < 3.1729621267394858e100

   1. Initial program 55.0

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

   2. Simplified 55.0

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

   3. Applied egg-rr 20.1

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

   4. Taylor expanded in i around inf 19.3

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

   5. Simplified 19.3

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

   if 3.1729621267394858e100 < i < 2.58730296467301078e146

   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 21.6

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

   4. Applied egg-rr 21.6

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

   if 2.58730296467301078e146 < 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 9.8

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

4. Final simplification 12.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 3.1189906262604905 \cdot 10^{+69}:\\ \;\;\;\;\frac{i}{\sqrt{{\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}^{2} + -1}} \cdot \frac{\left(i + \left(\alpha + \beta\right)\right) \cdot \frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{{\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}^{2}}}{\sqrt{{\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}^{2} + -1}}\\ \mathbf{elif}\;i \leq 3.1729621267394858 \cdot 10^{+100}:\\ \;\;\;\;\frac{0.25 \cdot \left(i \cdot i\right)}{\mathsf{fma}\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right), \mathsf{fma}\left(i, 2, \alpha + \beta\right), -1\right)}\\ \mathbf{elif}\;i \leq 2.587302964673011 \cdot 10^{+146}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt{i \cdot \left(i + \left(\alpha + \beta\right)\right)}}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\sqrt{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}}}\right)}^{2}}{\mathsf{fma}\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right), \mathsf{fma}\left(i, 2, \alpha + \beta\right), -1\right)}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Reproduce

herbie shell --seed 2022134 
(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)))