Octave 3.8, jcobi/4

Average Error: 54.2 → 1.3

Time: 25.7s

Precision: binary64

Cost: 27972

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

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

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \alpha + \beta\right)\\ t_1 := 1 + t_0\\ t_2 := \frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}\\ \mathbf{if}\;\alpha \leq 6.2 \cdot 10^{+112}:\\ \;\;\;\;\frac{i \cdot \frac{i}{t_2 \cdot t_1}}{t_2 \cdot \left(\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{t_1} \cdot \frac{i}{-1 + t_0}\\ \end{array} \]

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 (+ 1.0 t_0))
        (t_2 (/ (fma i 2.0 beta) (+ i beta))))
   (if (<= alpha 6.2e+112)
     (/ (* i (/ i (* t_2 t_1))) (* t_2 (+ (fma i 2.0 beta) (+ alpha -1.0))))
     (* (/ (+ alpha i) t_1) (/ i (+ -1.0 t_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 = fma(i, 2.0, (alpha + beta));
	double t_1 = 1.0 + t_0;
	double t_2 = fma(i, 2.0, beta) / (i + beta);
	double tmp;
	if (alpha <= 6.2e+112) {
		tmp = (i * (i / (t_2 * t_1))) / (t_2 * (fma(i, 2.0, beta) + (alpha + -1.0)));
	} else {
		tmp = ((alpha + i) / t_1) * (i / (-1.0 + t_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 = fma(i, 2.0, Float64(alpha + beta))
	t_1 = Float64(1.0 + t_0)
	t_2 = Float64(fma(i, 2.0, beta) / Float64(i + beta))
	tmp = 0.0
	if (alpha <= 6.2e+112)
		tmp = Float64(Float64(i * Float64(i / Float64(t_2 * t_1))) / Float64(t_2 * Float64(fma(i, 2.0, beta) + Float64(alpha + -1.0))));
	else
		tmp = Float64(Float64(Float64(alpha + i) / t_1) * Float64(i / Float64(-1.0 + t_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[(i * 2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(i * 2.0 + beta), $MachinePrecision] / N[(i + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[alpha, 6.2e+112], N[(N[(i * N[(i / N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 * N[(N[(i * 2.0 + beta), $MachinePrecision] + N[(alpha + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + i), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(i / N[(-1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if alpha < 6.19999999999999965e112

   1. Initial program 53.7

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

   2. Taylor expanded in alpha around 0 53.8

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

   3. Simplified 41.4

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

      [Start] 53.8	\[ \frac{\frac{{i}^{2} \cdot {\left(\beta + i\right)}^{2}}{{\left(\beta + 2 \cdot i\right)}^{2}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      associate-/l* [=>] 41.4	\[ \frac{\color{blue}{\frac{{i}^{2}}{\frac{{\left(\beta + 2 \cdot i\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      unpow2 [=>] 41.4	\[ \frac{\frac{\color{blue}{i \cdot i}}{\frac{{\left(\beta + 2 \cdot i\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      *-commutative [=>] 41.4	\[ \frac{\frac{i \cdot i}{\frac{{\left(\beta + \color{blue}{i \cdot 2}\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   4. Applied egg-rr 0.6

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

   5. Applied egg-rr 0.7

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

   if 6.19999999999999965e112 < alpha

   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. Taylor expanded in beta around inf 56.2

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

   3. Applied egg-rr 16.4

      \[\leadsto \color{blue}{\frac{i + \alpha}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \frac{i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 6.2 \cdot 10^{+112}:\\ \;\;\;\;\frac{i \cdot \frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta} \cdot \left(1 + \mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta} \cdot \left(\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{1 + \mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{i}{-1 + \mathsf{fma}\left(i, 2, \alpha + \beta\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.3
Cost	21572

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \alpha + \beta\right)\\ \mathbf{if}\;\alpha \leq 6.2 \cdot 10^{+112}:\\ \;\;\;\;\frac{i \cdot \frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\frac{i + \beta}{\beta + \left(1 + i \cdot 2\right)}}}}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta} \cdot \left(\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{1 + t_0} \cdot \frac{i}{-1 + t_0}\\ \end{array} \]

Alternative 2
Error	8.9
Cost	14532

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \alpha + \beta\right)\\ t_1 := \beta + \mathsf{fma}\left(i, 2, \alpha\right)\\ \mathbf{if}\;\beta \leq 1.5 \cdot 10^{+176}:\\ \;\;\;\;\frac{\frac{i}{-1 + t_1} \cdot \left(0.25 \cdot \left(i + \left(\alpha + \beta\right)\right)\right)}{1 + t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\frac{-1 + t_0}{\alpha + i}}}{1 + t_0}\\ \end{array} \]

Alternative 3
Error	9.2
Cost	14276

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 6.2 \cdot 10^{+175}:\\ \;\;\;\;0.25 \cdot \left(\frac{i}{t_0} \cdot \frac{i + \beta}{\beta + i \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{1 + t_0} \cdot \frac{i}{-1 + t_0}\\ \end{array} \]

Alternative 4
Error	9.2
Cost	14276

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 1.1 \cdot 10^{+176}:\\ \;\;\;\;0.25 \cdot \left(\frac{i}{t_0} \cdot \frac{i + \beta}{\beta + i \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\frac{-1 + t_0}{\alpha + i}}}{1 + t_0}\\ \end{array} \]

Alternative 5
Error	10.7
Cost	12420

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + i \cdot 2\\ t_1 := t_0 \cdot t_0\\ t_2 := -1 + t_1\\ t_3 := i \cdot \left(i + \left(\alpha + \beta\right)\right)\\ \mathbf{if}\;\frac{\frac{t_3 \cdot \left(t_3 + \alpha \cdot \beta\right)}{t_1}}{t_2} \leq \infty:\\ \;\;\;\;\frac{\frac{i \cdot i}{\frac{\beta \cdot \beta + 4 \cdot \left(i \cdot i + i \cdot \beta\right)}{{\left(i + \beta\right)}^{2}}}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \beta + \alpha \cdot 2}{i}\right) + \frac{\alpha + \beta}{i} \cdot -0.125\\ \end{array} \]

Alternative 6
Error	10.7
Cost	5956

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + i \cdot 2\\ t_1 := t_0 \cdot t_0\\ t_2 := i \cdot \left(i + \left(\alpha + \beta\right)\right)\\ t_3 := i \cdot \left(i + \beta\right)\\ t_4 := \beta + i \cdot 2\\ \mathbf{if}\;\frac{\frac{t_2 \cdot \left(t_2 + \alpha \cdot \beta\right)}{t_1}}{-1 + t_1} \leq \infty:\\ \;\;\;\;\frac{t_3}{t_4 \cdot \left(\beta + \left(1 + i \cdot 2\right)\right)} \cdot \frac{t_3}{t_4 \cdot \left(-1 + t_4\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \beta + \alpha \cdot 2}{i}\right) + \frac{\alpha + \beta}{i} \cdot -0.125\\ \end{array} \]

Alternative 7
Error	14.6
Cost	1344

\[\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \beta + \alpha \cdot 2}{i}\right) + \frac{\alpha + \beta}{i} \cdot -0.125 \]

Alternative 8
Error	14.6
Cost	1088

\[\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \beta}{i}\right) + \frac{\alpha + \beta}{i} \cdot -0.125 \]

Alternative 9
Error	15.4
Cost	964

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.6 \cdot 10^{+179}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i \cdot i}{\beta + \left(\alpha + 1\right)}}{\beta + -1}\\ \end{array} \]

Alternative 10
Error	17.2
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 8 \cdot 10^{+202}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 11
Error	57.9
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023002 
(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)))