Octave 3.8, jcobi/3

Average Error: 3.8 → 0.1

Time: 18.4s

Precision: binary64

Cost: 1600

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

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

Math

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

↓

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

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))))
   (/ (/ (/ (+ 1.0 beta) t_0) (/ t_0 (+ 1.0 alpha))) (+ 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);
	return (((1.0 + beta) / t_0) / (t_0 / (1.0 + alpha))) / (alpha + (beta + 3.0));
}

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

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

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

Derivation

1. Initial program 3.8

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

2. Simplified 0.1

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

   [Start] 3.8	\[ \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   +-commutative [=>] 3.8	\[ \frac{\frac{\frac{\color{blue}{\left(\beta \cdot \alpha + \left(\alpha + \beta\right)\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   associate-+l+ [=>] 3.8	\[ \frac{\frac{\frac{\color{blue}{\beta \cdot \alpha + \left(\left(\alpha + \beta\right) + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   associate-+l+ [=>] 3.8	\[ \frac{\frac{\frac{\beta \cdot \alpha + \color{blue}{\left(\alpha + \left(\beta + 1\right)\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   associate-+r+ [=>] 3.8	\[ \frac{\frac{\frac{\color{blue}{\left(\beta \cdot \alpha + \alpha\right) + \left(\beta + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   distribute-lft1-in [=>] 3.8	\[ \frac{\frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \alpha} + \left(\beta + 1\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   *-commutative [=>] 3.8	\[ \frac{\frac{\frac{\color{blue}{\alpha \cdot \left(\beta + 1\right)} + \left(\beta + 1\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   distribute-lft1-in [=>] 3.8	\[ \frac{\frac{\frac{\color{blue}{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   +-commutative [<=] 3.8	\[ \frac{\frac{\frac{\color{blue}{\left(1 + \alpha\right)} \cdot \left(\beta + 1\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   associate-/l* [=>] 0.1	\[ \frac{\frac{\color{blue}{\frac{1 + \alpha}{\frac{\left(\alpha + \beta\right) + 2 \cdot 1}{\beta + 1}}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   associate-/l/ [=>] 0.1	\[ \frac{\color{blue}{\frac{1 + \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \frac{\left(\alpha + \beta\right) + 2 \cdot 1}{\beta + 1}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   +-commutative [=>] 0.1	\[ \frac{\frac{\color{blue}{\alpha + 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \frac{\left(\alpha + \beta\right) + 2 \cdot 1}{\beta + 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   metadata-eval [=>] 0.1	\[ \frac{\frac{\alpha + 1}{\left(\left(\alpha + \beta\right) + \color{blue}{2}\right) \cdot \frac{\left(\alpha + \beta\right) + 2 \cdot 1}{\beta + 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   associate-+l+ [=>] 0.1	\[ \frac{\frac{\alpha + 1}{\color{blue}{\left(\alpha + \left(\beta + 2\right)\right)} \cdot \frac{\left(\alpha + \beta\right) + 2 \cdot 1}{\beta + 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   metadata-eval [=>] 0.1	\[ \frac{\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \frac{\left(\alpha + \beta\right) + \color{blue}{2}}{\beta + 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   associate-+l+ [=>] 0.1	\[ \frac{\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \frac{\color{blue}{\alpha + \left(\beta + 2\right)}}{\beta + 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   metadata-eval [=>] 0.1	\[ \frac{\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \frac{\alpha + \left(\beta + 2\right)}{\beta + 1}}}{\left(\left(\alpha + \beta\right) + \color{blue}{2}\right) + 1} \]
   associate-+l+ [=>] 0.1	\[ \frac{\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \frac{\alpha + \left(\beta + 2\right)}{\beta + 1}}}{\color{blue}{\left(\alpha + \beta\right) + \left(2 + 1\right)}} \]
   associate-+l+ [=>] 0.1	\[ \frac{\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \frac{\alpha + \left(\beta + 2\right)}{\beta + 1}}}{\color{blue}{\alpha + \left(\beta + \left(2 + 1\right)\right)}} \]
   metadata-eval [=>] 0.1	\[ \frac{\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \frac{\alpha + \left(\beta + 2\right)}{\beta + 1}}}{\alpha + \left(\beta + \color{blue}{3}\right)} \]

3. Applied egg-rr 0.1

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

4. Applied egg-rr 0.1

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

5. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.9
Cost	1604

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ t_1 := \alpha + \left(\beta + 3\right)\\ \mathbf{if}\;\beta \leq 2.4 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{\frac{1 + \beta}{t_0}}{\beta + 2}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t_0} \cdot \left(1 + \frac{-1 - \alpha}{\beta}\right)}{t_1}\\ \end{array} \]

Alternative 2
Error	0.2
Cost	1600

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

Alternative 3
Error	0.1
Cost	1600

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

Alternative 4
Error	1.0
Cost	1472

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

Alternative 5
Error	0.7
Cost	1348

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ \mathbf{if}\;\beta \leq 3.7:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + 2} \cdot \frac{1}{\alpha + 2}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\beta + 3\right) + \alpha \cdot 2}}{t_0}\\ \end{array} \]

Alternative 6
Error	0.9
Cost	1348

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 11200000000:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 3}}{t_0 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\beta + 3\right) + \alpha \cdot 2}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 7
Error	0.8
Cost	1348

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ \mathbf{if}\;\beta \leq 17000000000:\\ \;\;\;\;\frac{\frac{\frac{1 + \beta}{\alpha + \left(\beta + 2\right)}}{\beta + 2}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\beta + 3\right) + \alpha \cdot 2}}{t_0}\\ \end{array} \]

Alternative 8
Error	1.3
Cost	1220

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ \mathbf{if}\;\beta \leq 0.44:\\ \;\;\;\;\frac{\frac{0.5}{\beta + 2}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\beta + 3\right) + \alpha \cdot 2}}{t_0}\\ \end{array} \]

Alternative 9
Error	1.4
Cost	1092

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ \mathbf{if}\;\beta \leq 0.44:\\ \;\;\;\;\frac{\frac{0.5}{\beta + 2}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + \left(\alpha + 3\right)}}{t_0}\\ \end{array} \]

Alternative 10
Error	1.8
Cost	836

\[\begin{array}{l} \mathbf{if}\;\beta \leq 9.5:\\ \;\;\;\;\frac{\frac{0.5}{\beta + 2}}{\alpha + \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\frac{\beta}{1 + \alpha}}\\ \end{array} \]

Alternative 11
Error	1.7
Cost	836

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ \mathbf{if}\;\beta \leq 4.5:\\ \;\;\;\;\frac{\frac{0.5}{\beta + 2}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{t_0}\\ \end{array} \]

Alternative 12
Error	24.9
Cost	712

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.8:\\ \;\;\;\;\frac{0.5}{\alpha + \left(\beta + 3\right)}\\ \mathbf{elif}\;\beta \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 13
Error	2.0
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 10.5:\\ \;\;\;\;\frac{0.5}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 14
Error	2.0
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 10:\\ \;\;\;\;\frac{\frac{0.5}{\beta + 3}}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 15
Error	2.1
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 8.2:\\ \;\;\;\;\frac{\frac{0.5}{\beta + 3}}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\frac{\beta}{1 + \alpha}}\\ \end{array} \]

Alternative 16
Error	26.5
Cost	584

\[\begin{array}{l} \mathbf{if}\;\beta \leq 4.2:\\ \;\;\;\;\frac{0.5}{\alpha + \left(\beta + 3\right)}\\ \mathbf{elif}\;\beta \leq 3.5 \cdot 10^{+159}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 17
Error	24.6
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 4.2:\\ \;\;\;\;\frac{0.5}{\alpha + \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 18
Error	29.7
Cost	452

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

Alternative 19
Error	32.8
Cost	320

\[\frac{1}{\beta \cdot \beta} \]

Alternative 20
Error	32.6
Cost	320

\[\frac{\frac{1}{\beta}}{\beta} \]

Alternative 21
Error	61.7
Cost	192

\[\frac{-1}{\beta} \]

Alternative 22
Error	61.6
Cost	192

\[\frac{-0.16666666666666666}{\beta} \]

Reproduce

herbie shell --seed 2023010 
(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)))