Octave 3.8, jcobi/3

Average Error: 3.6 → 0.1

Time: 22.2s

Precision: binary64

Cost: 1600

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

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

Math

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

↓

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

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) (+ beta (+ alpha 2.0)))))
  (+ 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) / (beta + (alpha + 2.0))))) / (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) / (beta + (alpha + 2.0d0))))) / (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) / (beta + (alpha + 2.0))))) / (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) / (beta + (alpha + 2.0))))) / (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(beta + Float64(alpha + 2.0))))) / 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) / (beta + (alpha + 2.0))))) / (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[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 3.6

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

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

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. Simplified 0.1

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

   [Start] 0.1	\[ \frac{\frac{\alpha + 1}{\frac{2 + \left(\alpha + \beta\right)}{\frac{-1 - \beta}{-2 - \left(\alpha + \beta\right)}}}}{\alpha + \left(\beta + 3\right)} \]
   associate-/l* [<=] 2.1	\[ \frac{\frac{\alpha + 1}{\color{blue}{\frac{\left(2 + \left(\alpha + \beta\right)\right) \cdot \left(-2 - \left(\alpha + \beta\right)\right)}{-1 - \beta}}}}{\alpha + \left(\beta + 3\right)} \]
   *-commutative [=>] 2.1	\[ \frac{\frac{\alpha + 1}{\frac{\color{blue}{\left(-2 - \left(\alpha + \beta\right)\right) \cdot \left(2 + \left(\alpha + \beta\right)\right)}}{-1 - \beta}}}{\alpha + \left(\beta + 3\right)} \]
   associate-/l* [=>] 0.1	\[ \frac{\frac{\alpha + 1}{\color{blue}{\frac{-2 - \left(\alpha + \beta\right)}{\frac{-1 - \beta}{2 + \left(\alpha + \beta\right)}}}}}{\alpha + \left(\beta + 3\right)} \]
   +-commutative [=>] 0.1	\[ \frac{\frac{\alpha + 1}{\frac{-2 - \color{blue}{\left(\beta + \alpha\right)}}{\frac{-1 - \beta}{2 + \left(\alpha + \beta\right)}}}}{\alpha + \left(\beta + 3\right)} \]
   associate-+r+ [=>] 0.1	\[ \frac{\frac{\alpha + 1}{\frac{-2 - \left(\beta + \alpha\right)}{\frac{-1 - \beta}{\color{blue}{\left(2 + \alpha\right) + \beta}}}}}{\alpha + \left(\beta + 3\right)} \]

5. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.3
Cost	1732

\[\begin{array}{l} \mathbf{if}\;\beta \leq 8 \cdot 10^{+15}:\\ \;\;\;\;\frac{\alpha + 1}{\left(-2 - \beta\right) - \alpha} \cdot \frac{-1 - \beta}{\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\beta + 3\right) + \alpha \cdot 2}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 2
Error	0.2
Cost	1732

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 10^{+116}:\\ \;\;\;\;\frac{\frac{\left(\alpha + 1\right) \cdot \left(1 + \beta\right)}{3 + \left(\alpha + \beta\right)}}{t_0 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\beta + 3\right) + \alpha \cdot 2}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 3
Error	0.3
Cost	1604

\[\begin{array}{l} t_0 := 2 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 100000000:\\ \;\;\;\;\frac{\alpha + \left(1 + \beta\right)}{t_0 \cdot \left(t_0 \cdot \left(\beta + \left(\alpha + 3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\beta + 3\right) + \alpha \cdot 2}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 4
Error	0.1
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 5
Error	0.8
Cost	1348

\[\begin{array}{l} \mathbf{if}\;\beta \leq 100000000:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)} \cdot \frac{1}{\alpha + \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\beta + 3\right) + \alpha \cdot 2}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 6
Error	22.8
Cost	1088

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

Alternative 7
Error	22.9
Cost	960

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

Alternative 8
Error	22.9
Cost	836

\[\begin{array}{l} \mathbf{if}\;\beta \leq 5.8 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{1}{\beta + 3}}{\alpha + \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]

Alternative 9
Error	22.9
Cost	836

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ \mathbf{if}\;\beta \leq 5.8 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{1}{\beta + 3}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{t_0}\\ \end{array} \]

Alternative 10
Error	25.1
Cost	712

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3:\\ \;\;\;\;0.1111111111111111\\ \mathbf{elif}\;\beta \leq 2.4 \cdot 10^{+161}:\\ \;\;\;\;\frac{1}{\beta} \cdot \frac{1}{\beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \]

Alternative 11
Error	23.5
Cost	712

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3:\\ \;\;\;\;0.1111111111111111\\ \mathbf{elif}\;\beta \leq 10^{+154}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \]

Alternative 12
Error	26.5
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3:\\ \;\;\;\;0.1111111111111111\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \]

Alternative 13
Error	23.2
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.2:\\ \;\;\;\;0.1111111111111111\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]

Alternative 14
Error	26.7
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3:\\ \;\;\;\;0.1111111111111111\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \]

Alternative 15
Error	55.7
Cost	324

\[\begin{array}{l} \mathbf{if}\;\beta \leq 4.5:\\ \;\;\;\;0.1111111111111111\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\beta}\\ \end{array} \]

Alternative 16
Error	56.7
Cost	64

\[0.1111111111111111 \]

Reproduce

herbie shell --seed 2023067 
(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)))