Octave 3.8, jcobi/3

Average Error: 5.9% → 0.53%

Time: 18.8s

Precision: binary64

Cost: 1732

\[\alpha > -1 \land \beta > -1\]

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

Math

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

↓

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 3.2 \cdot 10^{+56}:\\ \;\;\;\;\frac{\frac{\left(\alpha + 1\right) \cdot \left(1 + \beta\right)}{3 + \left(\alpha + \beta\right)}}{t_0 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta} + \frac{\alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (/
  (/
   (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2.0 1.0)))
   (+ (+ alpha beta) (* 2.0 1.0)))
  (+ (+ (+ alpha beta) (* 2.0 1.0)) 1.0)))

↓

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 2.0))))
   (if (<= beta 3.2e+56)
     (/ (/ (* (+ alpha 1.0) (+ 1.0 beta)) (+ 3.0 (+ alpha beta))) (* t_0 t_0))
     (/ (+ (/ 1.0 beta) (/ alpha beta)) (+ alpha (+ beta 3.0))))))

C

double code(double alpha, double beta) {
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / ((alpha + beta) + (2.0 * 1.0))) / ((alpha + beta) + (2.0 * 1.0))) / (((alpha + beta) + (2.0 * 1.0)) + 1.0);
}

↓

double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 3.2e+56) {
		tmp = (((alpha + 1.0) * (1.0 + beta)) / (3.0 + (alpha + beta))) / (t_0 * t_0);
	} else {
		tmp = ((1.0 / beta) + (alpha / beta)) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = (((((alpha + beta) + (beta * alpha)) + 1.0d0) / ((alpha + beta) + (2.0d0 * 1.0d0))) / ((alpha + beta) + (2.0d0 * 1.0d0))) / (((alpha + beta) + (2.0d0 * 1.0d0)) + 1.0d0)
end function

↓

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    if (beta <= 3.2d+56) then
        tmp = (((alpha + 1.0d0) * (1.0d0 + beta)) / (3.0d0 + (alpha + beta))) / (t_0 * t_0)
    else
        tmp = ((1.0d0 / beta) + (alpha / beta)) / (alpha + (beta + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / ((alpha + beta) + (2.0 * 1.0))) / ((alpha + beta) + (2.0 * 1.0))) / (((alpha + beta) + (2.0 * 1.0)) + 1.0);
}

↓

public static double code(double alpha, double beta) {
	double t_0 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 3.2e+56) {
		tmp = (((alpha + 1.0) * (1.0 + beta)) / (3.0 + (alpha + beta))) / (t_0 * t_0);
	} else {
		tmp = ((1.0 / beta) + (alpha / beta)) / (alpha + (beta + 3.0));
	}
	return tmp;
}

Python

def code(alpha, beta):
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / ((alpha + beta) + (2.0 * 1.0))) / ((alpha + beta) + (2.0 * 1.0))) / (((alpha + beta) + (2.0 * 1.0)) + 1.0)

↓

def code(alpha, beta):
	t_0 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 3.2e+56:
		tmp = (((alpha + 1.0) * (1.0 + beta)) / (3.0 + (alpha + beta))) / (t_0 * t_0)
	else:
		tmp = ((1.0 / beta) + (alpha / beta)) / (alpha + (beta + 3.0))
	return tmp

Julia

function code(alpha, beta)
	return Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) + Float64(beta * alpha)) + 1.0) / Float64(Float64(alpha + beta) + Float64(2.0 * 1.0))) / Float64(Float64(alpha + beta) + Float64(2.0 * 1.0))) / Float64(Float64(Float64(alpha + beta) + Float64(2.0 * 1.0)) + 1.0))
end

↓

function code(alpha, beta)
	t_0 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 3.2e+56)
		tmp = Float64(Float64(Float64(Float64(alpha + 1.0) * Float64(1.0 + beta)) / Float64(3.0 + Float64(alpha + beta))) / Float64(t_0 * t_0));
	else
		tmp = Float64(Float64(Float64(1.0 / beta) + Float64(alpha / beta)) / Float64(alpha + Float64(beta + 3.0)));
	end
	return tmp
end

MATLAB

function tmp = code(alpha, beta)
	tmp = (((((alpha + beta) + (beta * alpha)) + 1.0) / ((alpha + beta) + (2.0 * 1.0))) / ((alpha + beta) + (2.0 * 1.0))) / (((alpha + beta) + (2.0 * 1.0)) + 1.0);
end

↓

function tmp_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 3.2e+56)
		tmp = (((alpha + 1.0) * (1.0 + beta)) / (3.0 + (alpha + beta))) / (t_0 * t_0);
	else
		tmp = ((1.0 / beta) + (alpha / beta)) / (alpha + (beta + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

↓

code[alpha_, beta_] := Block[{t$95$0 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 3.2e+56], N[(N[(N[(N[(alpha + 1.0), $MachinePrecision] * N[(1.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / beta), $MachinePrecision] + N[(alpha / beta), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.20000000000000003e56

   1. Initial program 0.2

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

   2. Simplified 0.18

      \[\leadsto \color{blue}{\frac{\frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{\left(\alpha + \beta\right) + 3}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}} \]
      Proof

      [Start] 0.2	\[ \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      associate-/l/ [=>] 0.21	\[ \frac{\color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      associate-/r* [<=] 0.2	\[ \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)}} \]
      associate-/l/ [<=] 0.19	\[ \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   if 3.20000000000000003e56 < beta

   1. Initial program 12.19

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

   2. Simplified 0.18

      \[\leadsto \color{blue}{\frac{\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \frac{\alpha + \left(\beta + 2\right)}{\beta + 1}}}{\alpha + \left(\beta + 3\right)}} \]
      Proof

      [Start] 12.19	\[ \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

   3. Taylor expanded in beta around inf 0.92

      \[\leadsto \frac{\frac{\alpha + 1}{\color{blue}{\beta}}}{\alpha + \left(\beta + 3\right)} \]

   4. Taylor expanded in alpha around 0 0.92

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

4. Final simplification 0.53

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.2 \cdot 10^{+56}:\\ \;\;\;\;\frac{\frac{\left(\alpha + 1\right) \cdot \left(1 + \beta\right)}{3 + \left(\alpha + \beta\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta} + \frac{\alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.18%
Cost	1600

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \frac{\frac{\alpha + 1}{t_0 \cdot \frac{t_0}{1 + \beta}}}{\alpha + \left(\beta + 3\right)} \end{array} \]

Alternative 2
Error	1.4%
Cost	1348

\[\begin{array}{l} \mathbf{if}\;\beta \leq 10000000000:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 3}}{\beta + 2} \cdot \frac{1}{\beta + \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\beta + 3\right) + \alpha \cdot 2}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 3
Error	1.72%
Cost	1348

\[\begin{array}{l} \mathbf{if}\;\beta \leq 0.82:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + 2}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(3 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\beta + 3\right) + \alpha \cdot 2}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 4
Error	1.66%
Cost	1220

\[\begin{array}{l} \mathbf{if}\;\beta \leq 22000000000000:\\ \;\;\;\;\frac{-1 - \beta}{\left(\beta + 3\right) \cdot \left(\left(\beta + 2\right) \cdot \left(-2 - \left(\alpha + \beta\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta} + \frac{\alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 5
Error	1.38%
Cost	1220

\[\begin{array}{l} \mathbf{if}\;\beta \leq 54000000:\\ \;\;\;\;\frac{-1 - \beta}{\left(\beta + 3\right) \cdot \left(\left(\beta + 2\right) \cdot \left(-2 - \left(\alpha + \beta\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\beta + 3\right) + \alpha \cdot 2}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 6
Error	2.91%
Cost	964

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.4:\\ \;\;\;\;\frac{1}{\beta + \left(\alpha + 2\right)} \cdot \left(0.16666666666666666 + \beta \cdot 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 7
Error	2.91%
Cost	964

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.3:\\ \;\;\;\;\frac{1}{\beta + \left(\alpha + 2\right)} \cdot \left(0.16666666666666666 + \beta \cdot 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta} + \frac{\alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 8
Error	2.94%
Cost	836

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.25:\\ \;\;\;\;\frac{0.16666666666666666 + \beta \cdot 0.027777777777777776}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 9
Error	3.38%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 7.5:\\ \;\;\;\;\frac{1}{\beta + \left(\alpha + 2\right)} \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]

Alternative 10
Error	2.98%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 5:\\ \;\;\;\;\frac{0.16666666666666666 + \beta \cdot 0.027777777777777776}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]

Alternative 11
Error	8.44%
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.7:\\ \;\;\;\;0.16666666666666666 \cdot \frac{1}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta}\\ \end{array} \]

Alternative 12
Error	6.36%
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 4:\\ \;\;\;\;0.16666666666666666 \cdot \frac{1}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \end{array} \]

Alternative 13
Error	3.43%
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.6:\\ \;\;\;\;0.16666666666666666 \cdot \frac{1}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]

Alternative 14
Error	8.87%
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 4:\\ \;\;\;\;\frac{0.16666666666666666}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \]

Alternative 15
Error	8.44%
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.9:\\ \;\;\;\;\frac{0.16666666666666666}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta}\\ \end{array} \]

Alternative 16
Error	55.42%
Cost	320

\[\frac{0.16666666666666666}{\alpha + 2} \]

Alternative 17
Error	53.82%
Cost	320

\[\frac{0.16666666666666666}{\beta + 2} \]

Alternative 18
Error	89.87%
Cost	64

\[0.027777777777777776 \]

Reproduce

herbie shell --seed 2023090 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :precision binary64
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ (+ alpha beta) (* 2.0 1.0)) 1.0)))