Octave 3.8, jcobi/3

Average Error: 4.0 → 0.1

Time: 23.8s

Precision: binary64

Cost: 1600

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

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

Math

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

↓

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

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

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

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

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

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(2.0 + Float64(beta + alpha))
	return Float64(Float64(Float64(Float64(beta + 1.0) / t_0) / t_0) / Float64(Float64(alpha + Float64(beta + 3.0)) / Float64(1.0 + alpha)))
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 = 2.0 + (beta + alpha);
	tmp = (((beta + 1.0) / t_0) / t_0) / ((alpha + (beta + 3.0)) / (1.0 + alpha));
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[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(beta + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision]), $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 4.8

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

   [Start] 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/ [=>] 4.8	\[ \frac{\color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   associate-/r* [<=] 10.0	\[ \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)}} \]
   associate-/l/ [<=] 4.8	\[ \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

3. Applied egg-rr 0.5

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

4. Applied egg-rr 0.1

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

5. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.8
Cost	1732

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\alpha \leq 5.1 \cdot 10^{+17}:\\ \;\;\;\;\frac{\beta + 1}{\beta + 2} \cdot \left(\frac{1}{t_0} \cdot \frac{1 + \alpha}{\beta + \left(\alpha + 3\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha + \left(\beta + 3\right)} \cdot \frac{\frac{\alpha}{\frac{t_0}{\beta}}}{t_0}\\ \end{array} \]

Alternative 2
Error	0.4
Cost	1732

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}{3 + \left(\beta + \alpha\right)}}{t_0 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha + \left(\beta + 3\right)} \cdot \frac{\frac{\alpha}{\frac{t_0}{\beta}}}{t_0}\\ \end{array} \]

Alternative 3
Error	0.4
Cost	1732

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{\beta + 1}{\frac{-3 - \left(\beta + \alpha\right)}{-1 - \alpha}}}{t_0 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha + \left(\beta + 3\right)} \cdot \frac{\frac{\alpha}{\frac{t_0}{\beta}}}{t_0}\\ \end{array} \]

Alternative 4
Error	1.4
Cost	1604

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\alpha \leq 134000000:\\ \;\;\;\;\frac{\frac{\beta + 1}{\beta + 3}}{t_0 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha + \left(\beta + 3\right)} \cdot \frac{\frac{\alpha}{\frac{t_0}{\beta}}}{t_0}\\ \end{array} \]

Alternative 5
Error	0.1
Cost	1600

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

Alternative 6
Error	0.8
Cost	1348

\[\begin{array}{l} \mathbf{if}\;\beta \leq 106000000:\\ \;\;\;\;\frac{\beta + 1}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)} \cdot \frac{1}{\beta + \left(2 + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\beta + 3\right) + 2 \cdot \alpha}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 7
Error	0.8
Cost	1348

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 250000000:\\ \;\;\;\;\frac{\frac{\beta + 1}{\beta + 3}}{t_0 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\beta + 3\right) + 2 \cdot \alpha}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 8
Error	1.2
Cost	1220

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ \mathbf{if}\;\beta \leq 0.62:\\ \;\;\;\;\frac{\frac{\frac{1}{\beta + 2}}{2 - \beta}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\beta + 3\right) + 2 \cdot \alpha}}{t_0}\\ \end{array} \]

Alternative 9
Error	1.9
Cost	1092

\[\begin{array}{l} \mathbf{if}\;\beta \leq 0.88:\\ \;\;\;\;\frac{1}{\left(\beta + 3\right) \cdot \left(\left(\beta + 2\right) \cdot \left(2 - \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 - \alpha}{\beta + \left(2 + \alpha\right)}}{\left(-3 - \beta\right) - \alpha}\\ \end{array} \]

Alternative 10
Error	1.6
Cost	1092

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

Alternative 11
Error	1.9
Cost	964

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.55:\\ \;\;\;\;\frac{1}{\left(\beta + 3\right) \cdot \left(\left(\beta + 2\right) \cdot \left(2 - \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 12
Error	24.2
Cost	836

\[\begin{array}{l} \mathbf{if}\;\beta \leq 7 \cdot 10^{+153}:\\ \;\;\;\;\frac{1 + \alpha}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \]

Alternative 13
Error	24.2
Cost	836

\[\begin{array}{l} \mathbf{if}\;\beta \leq 9.4 \cdot 10^{+114}:\\ \;\;\;\;\frac{1 + \alpha}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 14
Error	24.5
Cost	712

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.16:\\ \;\;\;\;\beta \cdot -0.1388888888888889 + 0.16666666666666666\\ \mathbf{elif}\;\beta \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 15
Error	24.3
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.16:\\ \;\;\;\;\beta \cdot -0.1388888888888889 + 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \]

Alternative 16
Error	24.3
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 6.8 \cdot 10^{+21}:\\ \;\;\;\;\frac{1}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \]

Alternative 17
Error	24.4
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 4.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{1}{\beta + 3}}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \]

Alternative 18
Error	26.4
Cost	584

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.16:\\ \;\;\;\;\beta \cdot -0.1388888888888889 + 0.16666666666666666\\ \mathbf{elif}\;\beta \leq 5 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 19
Error	56.1
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.1:\\ \;\;\;\;\beta \cdot -0.1388888888888889 + 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{0.2222222222222222}{\beta}\\ \end{array} \]

Alternative 20
Error	28.1
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.16:\\ \;\;\;\;\beta \cdot -0.1388888888888889 + 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \]

Alternative 21
Error	56.1
Cost	324

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.35:\\ \;\;\;\;0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{0.2222222222222222}{\beta}\\ \end{array} \]

Alternative 22
Error	62.6
Cost	64

\[-0.1111111111111111 \]

Alternative 23
Error	57.2
Cost	64

\[0.16666666666666666 \]

Reproduce

herbie shell --seed 2023066 
(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)))