Octave 3.8, jcobi/3

Average Accuracy: 93.8% → 99.8%

Time: 21.6s

Precision: binary64

Cost: 1728

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

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

Math

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

↓

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

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
 (/
  (/
   (* (/ 1.0 (+ (+ 2.0 alpha) beta)) (+ 1.0 alpha))
   (/ (- -2.0 (+ alpha beta)) (- -1.0 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 (((1.0 / ((2.0 + alpha) + beta)) * (1.0 + alpha)) / ((-2.0 - (alpha + beta)) / (-1.0 - 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 = (((1.0d0 / ((2.0d0 + alpha) + beta)) * (1.0d0 + alpha)) / (((-2.0d0) - (alpha + beta)) / ((-1.0d0) - 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 (((1.0 / ((2.0 + alpha) + beta)) * (1.0 + alpha)) / ((-2.0 - (alpha + beta)) / (-1.0 - 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 (((1.0 / ((2.0 + alpha) + beta)) * (1.0 + alpha)) / ((-2.0 - (alpha + beta)) / (-1.0 - 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(Float64(1.0 / Float64(Float64(2.0 + alpha) + beta)) * Float64(1.0 + alpha)) / Float64(Float64(-2.0 - Float64(alpha + beta)) / Float64(-1.0 - 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 = (((1.0 / ((2.0 + alpha) + beta)) * (1.0 + alpha)) / ((-2.0 - (alpha + beta)) / (-1.0 - 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[(N[(1.0 / N[(N[(2.0 + alpha), $MachinePrecision] + beta), $MachinePrecision]), $MachinePrecision] * N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision] / N[(N[(-2.0 - N[(alpha + beta), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.8%

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

2. Simplified 99.8%

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

   [Start] 93.8	\[ \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

3. Applied egg-rr 99.8%

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

4. Simplified 99.8%

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

   [Start] 99.8	\[ \frac{\frac{1}{2 + \left(\alpha + \beta\right)} \cdot \frac{\alpha + 1}{\frac{-2 - \left(\alpha + \beta\right)}{-1 - \beta}}}{\alpha + \left(\beta + 3\right)} \]
   +-commutative [=>] 99.8	\[ \frac{\frac{1}{2 + \left(\alpha + \beta\right)} \cdot \frac{\color{blue}{1 + \alpha}}{\frac{-2 - \left(\alpha + \beta\right)}{-1 - \beta}}}{\alpha + \left(\beta + 3\right)} \]
   associate-*r/ [=>] 99.8	\[ \frac{\color{blue}{\frac{\frac{1}{2 + \left(\alpha + \beta\right)} \cdot \left(1 + \alpha\right)}{\frac{-2 - \left(\alpha + \beta\right)}{-1 - \beta}}}}{\alpha + \left(\beta + 3\right)} \]
   associate-+r+ [=>] 99.8	\[ \frac{\frac{\frac{1}{\color{blue}{\left(2 + \alpha\right) + \beta}} \cdot \left(1 + \alpha\right)}{\frac{-2 - \left(\alpha + \beta\right)}{-1 - \beta}}}{\alpha + \left(\beta + 3\right)} \]
   +-commutative [=>] 99.8	\[ \frac{\frac{\frac{1}{\left(2 + \alpha\right) + \beta} \cdot \left(1 + \alpha\right)}{\frac{-2 - \color{blue}{\left(\beta + \alpha\right)}}{-1 - \beta}}}{\alpha + \left(\beta + 3\right)} \]

5. Final simplification 99.8%

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

Alternatives

Alternative 1
Accuracy	99.6%
Cost	1732

\[\begin{array}{l} t_0 := \alpha + \left(2 + \beta\right)\\ \mathbf{if}\;\beta \leq 3.4 \cdot 10^{+75}:\\ \;\;\;\;\frac{\frac{\left(1 + \alpha\right) \cdot \left(1 + \beta\right)}{3 + \left(\alpha + \beta\right)}}{t_0 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\beta + 3\right) + 2 \cdot \alpha}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 2
Accuracy	99.5%
Cost	1604

\[\begin{array}{l} t_0 := 2 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 62000000000:\\ \;\;\;\;\frac{\alpha + \left(1 + \beta\right)}{t_0 \cdot \left(t_0 \cdot \left(\beta + \left(\alpha + 3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\beta + 3\right) + 2 \cdot \alpha}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 3
Accuracy	99.8%
Cost	1600

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

Alternative 4
Accuracy	97.5%
Cost	1220

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

Alternative 5
Accuracy	98.3%
Cost	1220

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.9:\\ \;\;\;\;\frac{1 + \alpha}{\left(2 + \left(\alpha + \beta\right)\right) \cdot \left(\left(2 + \alpha\right) \cdot \left(\alpha + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\beta + 3\right) + 2 \cdot \alpha}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 6
Accuracy	99.2%
Cost	1220

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1920000000000:\\ \;\;\;\;\frac{\frac{1 + \beta}{2 + \beta}}{\left(2 + \beta\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\beta + 3\right) + 2 \cdot \alpha}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 7
Accuracy	97.1%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;\beta \leq 4.6:\\ \;\;\;\;\frac{\frac{0.5}{2 + \alpha}}{\alpha + \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 - \alpha}{\left(2 + \alpha\right) + \beta}}{\left(-3 - \beta\right) - \alpha}\\ \end{array} \]

Alternative 8
Accuracy	96.9%
Cost	836

\[\begin{array}{l} \mathbf{if}\;\beta \leq 13.2:\\ \;\;\;\;\frac{\frac{0.5}{2 + \alpha}}{\alpha + \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \]

Alternative 9
Accuracy	97.1%
Cost	836

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ \mathbf{if}\;\beta \leq 4.8:\\ \;\;\;\;\frac{\frac{0.5}{2 + \alpha}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{t_0}\\ \end{array} \]

Alternative 10
Accuracy	96.5%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.5:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \]

Alternative 11
Accuracy	93.7%
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.5:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \end{array} \]

Alternative 12
Accuracy	45.3%
Cost	452

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 2.5:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\alpha \cdot \alpha}\\ \end{array} \]

Alternative 13
Accuracy	45.3%
Cost	452

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 3.5:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha \cdot \alpha}\\ \end{array} \]

Alternative 14
Accuracy	90.9%
Cost	452

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

Alternative 15
Accuracy	91.4%
Cost	452

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

Alternative 16
Accuracy	43.6%
Cost	64

\[0.08333333333333333 \]

Reproduce

herbie shell --seed 2023125 
(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)))