Octave 3.8, jcobi/3

Average Error: 4.0 → 0.1

Time: 24.4s

Precision: binary64

Cost: 1600

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

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

Math

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

↓

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

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 (+ beta (+ alpha 2.0))))
   (/ (/ (+ 1.0 alpha) (* (/ t_0 (+ 1.0 beta)) (+ alpha (+ beta 3.0)))) t_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 = beta + (alpha + 2.0);
	return ((1.0 + alpha) / ((t_0 / (1.0 + beta)) * (alpha + (beta + 3.0)))) / t_0;
}

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
    t_0 = beta + (alpha + 2.0d0)
    code = ((1.0d0 + alpha) / ((t_0 / (1.0d0 + beta)) * (alpha + (beta + 3.0d0)))) / t_0
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 = beta + (alpha + 2.0);
	return ((1.0 + alpha) / ((t_0 / (1.0 + beta)) * (alpha + (beta + 3.0)))) / t_0;
}

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 = beta + (alpha + 2.0)
	return ((1.0 + alpha) / ((t_0 / (1.0 + beta)) * (alpha + (beta + 3.0)))) / t_0

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(beta + Float64(alpha + 2.0))
	return Float64(Float64(Float64(1.0 + alpha) / Float64(Float64(t_0 / Float64(1.0 + beta)) * Float64(alpha + Float64(beta + 3.0)))) / t_0)
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 = code(alpha, beta)
	t_0 = beta + (alpha + 2.0);
	tmp = ((1.0 + alpha) / ((t_0 / (1.0 + beta)) * (alpha + (beta + 3.0)))) / t_0;
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[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(N[(t$95$0 / N[(1.0 + beta), $MachinePrecision]), $MachinePrecision] * N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 4.0

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

2. Simplified 2.3

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

   [Start] 4.0	\[ \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   associate-/l/ [=>] 4.9	\[ \frac{\color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   associate-/r* [<=] 10.2	\[ \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)}} \]
   +-commutative [=>] 10.2	\[ \frac{\color{blue}{1 + \left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \]
   associate-+l+ [=>] 10.2	\[ \frac{1 + \color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \]
   associate-+r+ [=>] 10.2	\[ \frac{\color{blue}{\left(1 + \alpha\right) + \left(\beta + \beta \cdot \alpha\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \]
   *-lft-identity [<=] 10.2	\[ \frac{\left(1 + \alpha\right) + \left(\color{blue}{1 \cdot \beta} + \beta \cdot \alpha\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \]
   *-commutative [=>] 10.2	\[ \frac{\left(1 + \alpha\right) + \left(1 \cdot \beta + \color{blue}{\alpha \cdot \beta}\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \]
   distribute-rgt-in [<=] 10.2	\[ \frac{\left(1 + \alpha\right) + \color{blue}{\beta \cdot \left(1 + \alpha\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \]
   distribute-rgt1-in [=>] 10.2	\[ \frac{\color{blue}{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \]
   times-frac [=>] 2.4	\[ \color{blue}{\frac{\beta + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \cdot \frac{1 + \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}} \]
   *-commutative [=>] 2.4	\[ \color{blue}{\frac{1 + \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \cdot \frac{\beta + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]
   associate-*r/ [=>] 2.3	\[ \color{blue}{\frac{\frac{1 + \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \cdot \left(\beta + 1\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

3. Applied egg-rr 0.1

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

4. Simplified 0.1

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

   [Start] 0.1	\[ \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 3\right)}}{\beta + \left(\alpha + 2\right)} \cdot \frac{1 + \beta}{\beta + \left(\alpha + 2\right)} \]
   associate-*r/ [=>] 0.1	\[ \color{blue}{\frac{\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 3\right)}}{\beta + \left(\alpha + 2\right)} \cdot \left(1 + \beta\right)}{\beta + \left(\alpha + 2\right)}} \]
   associate-/r/ [<=] 0.1	\[ \frac{\color{blue}{\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 3\right)}}{\frac{\beta + \left(\alpha + 2\right)}{1 + \beta}}}}{\beta + \left(\alpha + 2\right)} \]
   associate-/l/ [=>] 0.1	\[ \frac{\color{blue}{\frac{\alpha + 1}{\frac{\beta + \left(\alpha + 2\right)}{1 + \beta} \cdot \left(\alpha + \left(\beta + 3\right)\right)}}}{\beta + \left(\alpha + 2\right)} \]
   +-commutative [=>] 0.1	\[ \frac{\frac{\color{blue}{1 + \alpha}}{\frac{\beta + \left(\alpha + 2\right)}{1 + \beta} \cdot \left(\alpha + \left(\beta + 3\right)\right)}}{\beta + \left(\alpha + 2\right)} \]
   +-commutative [<=] 0.1	\[ \frac{\frac{1 + \alpha}{\frac{\beta + \color{blue}{\left(2 + \alpha\right)}}{1 + \beta} \cdot \left(\alpha + \left(\beta + 3\right)\right)}}{\beta + \left(\alpha + 2\right)} \]
   +-commutative [<=] 0.1	\[ \frac{\frac{1 + \alpha}{\frac{\beta + \left(2 + \alpha\right)}{\color{blue}{\beta + 1}} \cdot \left(\alpha + \left(\beta + 3\right)\right)}}{\beta + \left(\alpha + 2\right)} \]
   +-commutative [=>] 0.1	\[ \frac{\frac{1 + \alpha}{\frac{\beta + \left(2 + \alpha\right)}{\beta + 1} \cdot \color{blue}{\left(\left(\beta + 3\right) + \alpha\right)}}}{\beta + \left(\alpha + 2\right)} \]
   +-commutative [<=] 0.1	\[ \frac{\frac{1 + \alpha}{\frac{\beta + \left(2 + \alpha\right)}{\beta + 1} \cdot \left(\left(\beta + 3\right) + \alpha\right)}}{\beta + \color{blue}{\left(2 + \alpha\right)}} \]

5. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	1732

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 3 \cdot 10^{+16}:\\ \;\;\;\;\frac{\left(1 + \beta\right) \cdot \frac{1 + \alpha}{3 + \left(\alpha + \beta\right)}}{t_0 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 + \alpha\right) \cdot \frac{1}{\beta + \left(\alpha + 3\right)}}{1 + \frac{\alpha}{\beta}}}{\beta + \left(\alpha + 2\right)}\\ \end{array} \]

Alternative 2
Error	0.1
Cost	1600

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

Alternative 3
Error	0.7
Cost	1476

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\beta \leq 14500000000000:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(1 + \frac{\alpha}{\beta}\right)}}{t_0}\\ \end{array} \]

Alternative 4
Error	1.0
Cost	1472

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

Alternative 5
Error	1.0
Cost	1344

\[\frac{\frac{1 + \alpha}{\frac{\beta + 3}{\frac{1 + \beta}{\beta + 2}}}}{\beta + \left(\alpha + 2\right)} \]

Alternative 6
Error	1.2
Cost	1220

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\beta \leq 3.85:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + \left(\alpha + 4\right)}}{t_0}\\ \end{array} \]

Alternative 7
Error	1.1
Cost	1220

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

Alternative 8
Error	0.8
Cost	1220

\[\begin{array}{l} \mathbf{if}\;\beta \leq 500000000:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}}{\beta + \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\beta + 3\right) + \alpha \cdot 2}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 9
Error	1.5
Cost	1092

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.85:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + 2}}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 10
Error	1.2
Cost	1092

\[\begin{array}{l} \mathbf{if}\;\beta \leq 0.78:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + 2}}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + \left(\alpha + 4\right)}}{\beta + \left(\alpha + 2\right)}\\ \end{array} \]

Alternative 11
Error	2.1
Cost	836

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 12
Error	4.2
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.7:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \end{array} \]

Alternative 13
Error	2.1
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.7:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 14
Error	5.9
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.45:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \]

Alternative 15
Error	35.3
Cost	64

\[0.08333333333333333 \]

Reproduce

herbie shell --seed 2023025 
(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)))