Octave 3.8, jcobi/3

Average Error: 3.5 → 0.1

Time: 21.5s

Precision: binary64

Cost: 1600

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

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

Math

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

↓

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

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
 (/
  (/
   (+ alpha 1.0)
   (/ (+ 2.0 (+ alpha beta)) (/ (- -1.0 beta) (- -2.0 (+ 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) {
	return ((alpha + 1.0) / ((2.0 + (alpha + beta)) / ((-1.0 - beta) / (-2.0 - (alpha + beta))))) / (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
    code = ((alpha + 1.0d0) / ((2.0d0 + (alpha + beta)) / (((-1.0d0) - beta) / ((-2.0d0) - (alpha + beta))))) / (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) {
	return ((alpha + 1.0) / ((2.0 + (alpha + beta)) / ((-1.0 - beta) / (-2.0 - (alpha + beta))))) / (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):
	return ((alpha + 1.0) / ((2.0 + (alpha + beta)) / ((-1.0 - beta) / (-2.0 - (alpha + beta))))) / (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)
	return Float64(Float64(Float64(alpha + 1.0) / Float64(Float64(2.0 + Float64(alpha + beta)) / Float64(Float64(-1.0 - beta) / Float64(-2.0 - Float64(alpha + beta))))) / 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)
	tmp = ((alpha + 1.0) / ((2.0 + (alpha + beta)) / ((-1.0 - beta) / (-2.0 - (alpha + beta))))) / (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_] := N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] / N[(N[(-1.0 - beta), $MachinePrecision] / N[(-2.0 - N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 3.5

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

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

3. Applied egg-rr 0.1

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

4. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.2
Cost	1732

\[\begin{array}{l} t_0 := \alpha + \left(2 + \beta\right)\\ \mathbf{if}\;\beta \leq 2 \cdot 10^{+128}:\\ \;\;\;\;\frac{\left(1 + \beta\right) \cdot \frac{\alpha + 1}{\left(\alpha + \beta\right) + 3}}{t_0 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\beta + 3\right) + \alpha \cdot 2}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 2
Error	0.2
Cost	1732

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ \mathbf{if}\;\beta \leq 6 \cdot 10^{+128}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \frac{-1 - \beta}{\left(-2 - \left(\alpha + \beta\right)\right) \cdot \left(\alpha + \left(2 + \beta\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\beta + 3\right) + \alpha \cdot 2}}{t_0}\\ \end{array} \]

Alternative 3
Error	0.1
Cost	1600

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

Alternative 4
Error	1.0
Cost	1472

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

Alternative 5
Error	1.0
Cost	1348

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.55:\\ \;\;\;\;\frac{1}{\left(\beta + \left(\alpha + 2\right)\right) \cdot \frac{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}{\alpha + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\beta + 3\right) + \alpha \cdot 2}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 6
Error	1.0
Cost	1220

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.2:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + 2}}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\beta + 3\right) + \alpha \cdot 2}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 7
Error	1.6
Cost	1092

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

Alternative 8
Error	1.5
Cost	1092

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.8:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + 2}}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 - \alpha}{\beta + \left(\alpha + 2\right)}}{\left(-3 - \beta\right) - \alpha}\\ \end{array} \]

Alternative 9
Error	1.8
Cost	964

\[\begin{array}{l} \mathbf{if}\;\beta \leq 24:\\ \;\;\;\;\frac{\alpha + 1}{\left(\alpha + 2\right) \cdot \left(6 + \alpha \cdot 5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 10
Error	2.1
Cost	836

\[\begin{array}{l} \mathbf{if}\;\beta \leq 5.9:\\ \;\;\;\;\frac{1}{\left(\beta + \left(\alpha + 2\right)\right) \cdot 6}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 11
Error	1.8
Cost	836

\[\begin{array}{l} \mathbf{if}\;\beta \leq 24.5:\\ \;\;\;\;\frac{1}{\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(6 - \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 12
Error	4.3
Cost	712

\[\begin{array}{l} \mathbf{if}\;\beta \leq 7.6:\\ \;\;\;\;\frac{0.16666666666666666}{2 + \beta}\\ \mathbf{elif}\;\beta \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \]

Alternative 13
Error	4.3
Cost	712

\[\begin{array}{l} \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;\frac{0.16666666666666666}{2 + \beta}\\ \mathbf{elif}\;\beta \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(2 + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \]

Alternative 14
Error	2.5
Cost	712

\[\begin{array}{l} \mathbf{if}\;\beta \leq 8.5:\\ \;\;\;\;\frac{0.16666666666666666}{2 + \beta}\\ \mathbf{elif}\;\beta \leq 10^{+154}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \]

Alternative 15
Error	2.1
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 7.6:\\ \;\;\;\;\frac{1}{\left(\beta + \left(\alpha + 2\right)\right) \cdot 6}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]

Alternative 16
Error	2.1
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 7.9:\\ \;\;\;\;\frac{0.16666666666666666}{2 + \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]

Alternative 17
Error	56.2
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.2:\\ \;\;\;\;0.16666666666666666 + \beta \cdot -0.1388888888888889\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta}\\ \end{array} \]

Alternative 18
Error	5.8
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 7.6:\\ \;\;\;\;\frac{0.16666666666666666}{2 + \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \]

Alternative 19
Error	56.2
Cost	324

\[\begin{array}{l} \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta}\\ \end{array} \]

Alternative 20
Error	34.6
Cost	320

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

Alternative 21
Error	57.3
Cost	64

\[0.16666666666666666 \]

Reproduce

herbie shell --seed 2023046 
(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)))