Octave 3.8, jcobi/3

Average Accuracy: 94.0% → 99.8%

Time: 23.7s

Precision: binary64

Cost: 1728

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

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

Math

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

↓

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

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

Derivation

1. Initial program 94.0%

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

2. Simplified 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] 94.0	\[ \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

3. Applied egg-rr 99.8%

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

   [Start] 99.8	\[ \frac{\frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \frac{\alpha + \left(\beta + 2\right)}{\beta + 1}}}{\alpha + \left(\beta + 3\right)} \]
   *-un-lft-identity [=>] 99.8	\[ \frac{\frac{\color{blue}{1 \cdot \left(\alpha + 1\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \frac{\alpha + \left(\beta + 2\right)}{\beta + 1}}}{\alpha + \left(\beta + 3\right)} \]
   times-frac [=>] 99.8	\[ \frac{\color{blue}{\frac{1}{\alpha + \left(\beta + 2\right)} \cdot \frac{\alpha + 1}{\frac{\alpha + \left(\beta + 2\right)}{\beta + 1}}}}{\alpha + \left(\beta + 3\right)} \]
   associate-+r+ [=>] 99.8	\[ \frac{\frac{1}{\color{blue}{\left(\alpha + \beta\right) + 2}} \cdot \frac{\alpha + 1}{\frac{\alpha + \left(\beta + 2\right)}{\beta + 1}}}{\alpha + \left(\beta + 3\right)} \]
   +-commutative [=>] 99.8	\[ \frac{\frac{1}{\color{blue}{2 + \left(\alpha + \beta\right)}} \cdot \frac{\alpha + 1}{\frac{\alpha + \left(\beta + 2\right)}{\beta + 1}}}{\alpha + \left(\beta + 3\right)} \]
   frac-2neg [=>] 99.8	\[ \frac{\frac{1}{2 + \left(\alpha + \beta\right)} \cdot \frac{\alpha + 1}{\color{blue}{\frac{-\left(\alpha + \left(\beta + 2\right)\right)}{-\left(\beta + 1\right)}}}}{\alpha + \left(\beta + 3\right)} \]
   neg-sub0 [=>] 99.8	\[ \frac{\frac{1}{2 + \left(\alpha + \beta\right)} \cdot \frac{\alpha + 1}{\frac{\color{blue}{0 - \left(\alpha + \left(\beta + 2\right)\right)}}{-\left(\beta + 1\right)}}}{\alpha + \left(\beta + 3\right)} \]
   metadata-eval [<=] 99.8	\[ \frac{\frac{1}{2 + \left(\alpha + \beta\right)} \cdot \frac{\alpha + 1}{\frac{\color{blue}{\log 1} - \left(\alpha + \left(\beta + 2\right)\right)}{-\left(\beta + 1\right)}}}{\alpha + \left(\beta + 3\right)} \]
   associate-+r+ [=>] 99.8	\[ \frac{\frac{1}{2 + \left(\alpha + \beta\right)} \cdot \frac{\alpha + 1}{\frac{\log 1 - \color{blue}{\left(\left(\alpha + \beta\right) + 2\right)}}{-\left(\beta + 1\right)}}}{\alpha + \left(\beta + 3\right)} \]
   +-commutative [=>] 99.8	\[ \frac{\frac{1}{2 + \left(\alpha + \beta\right)} \cdot \frac{\alpha + 1}{\frac{\log 1 - \color{blue}{\left(2 + \left(\alpha + \beta\right)\right)}}{-\left(\beta + 1\right)}}}{\alpha + \left(\beta + 3\right)} \]
   associate--r+ [=>] 99.8	\[ \frac{\frac{1}{2 + \left(\alpha + \beta\right)} \cdot \frac{\alpha + 1}{\frac{\color{blue}{\left(\log 1 - 2\right) - \left(\alpha + \beta\right)}}{-\left(\beta + 1\right)}}}{\alpha + \left(\beta + 3\right)} \]
   metadata-eval [=>] 99.8	\[ \frac{\frac{1}{2 + \left(\alpha + \beta\right)} \cdot \frac{\alpha + 1}{\frac{\left(\color{blue}{0} - 2\right) - \left(\alpha + \beta\right)}{-\left(\beta + 1\right)}}}{\alpha + \left(\beta + 3\right)} \]
   metadata-eval [=>] 99.8	\[ \frac{\frac{1}{2 + \left(\alpha + \beta\right)} \cdot \frac{\alpha + 1}{\frac{\color{blue}{-2} - \left(\alpha + \beta\right)}{-\left(\beta + 1\right)}}}{\alpha + \left(\beta + 3\right)} \]
   neg-sub0 [=>] 99.8	\[ \frac{\frac{1}{2 + \left(\alpha + \beta\right)} \cdot \frac{\alpha + 1}{\frac{-2 - \left(\alpha + \beta\right)}{\color{blue}{0 - \left(\beta + 1\right)}}}}{\alpha + \left(\beta + 3\right)} \]
   metadata-eval [<=] 99.8	\[ \frac{\frac{1}{2 + \left(\alpha + \beta\right)} \cdot \frac{\alpha + 1}{\frac{-2 - \left(\alpha + \beta\right)}{\color{blue}{\log 1} - \left(\beta + 1\right)}}}{\alpha + \left(\beta + 3\right)} \]
   +-commutative [=>] 99.8	\[ \frac{\frac{1}{2 + \left(\alpha + \beta\right)} \cdot \frac{\alpha + 1}{\frac{-2 - \left(\alpha + \beta\right)}{\log 1 - \color{blue}{\left(1 + \beta\right)}}}}{\alpha + \left(\beta + 3\right)} \]
   associate--r+ [=>] 99.8	\[ \frac{\frac{1}{2 + \left(\alpha + \beta\right)} \cdot \frac{\alpha + 1}{\frac{-2 - \left(\alpha + \beta\right)}{\color{blue}{\left(\log 1 - 1\right) - \beta}}}}{\alpha + \left(\beta + 3\right)} \]
   metadata-eval [=>] 99.8	\[ \frac{\frac{1}{2 + \left(\alpha + \beta\right)} \cdot \frac{\alpha + 1}{\frac{-2 - \left(\alpha + \beta\right)}{\left(\color{blue}{0} - 1\right) - \beta}}}{\alpha + \left(\beta + 3\right)} \]
   metadata-eval [=>] 99.8	\[ \frac{\frac{1}{2 + \left(\alpha + \beta\right)} \cdot \frac{\alpha + 1}{\frac{-2 - \left(\alpha + \beta\right)}{\color{blue}{-1} - \beta}}}{\alpha + \left(\beta + 3\right)} \]

4. Final simplification 99.8%

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

Alternatives

Alternative 1
Accuracy	99.6%
Cost	1732

\[\begin{array}{l} t_0 := \alpha + \left(2 + \beta\right)\\ \mathbf{if}\;\beta \leq 4.2 \cdot 10^{+75}:\\ \;\;\;\;\frac{\frac{\left(1 + \alpha\right) \cdot \left(1 + \beta\right)}{\left(\alpha + \beta\right) + 3}}{t_0 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\beta + 3\right) + 2 \cdot \alpha}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 2
Accuracy	99.4%
Cost	1604

\[\begin{array}{l} t_0 := \alpha + \left(2 + \beta\right)\\ t_1 := \alpha + \left(\beta + 3\right)\\ \mathbf{if}\;\beta \leq 62000:\\ \;\;\;\;\frac{\beta + \left(1 + \alpha\right)}{t_1 \cdot \left(t_0 \cdot t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\beta + 3\right) + 2 \cdot \alpha}}{t_1}\\ \end{array} \]

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	98.6%
Cost	1220

\[\begin{array}{l} \mathbf{if}\;\beta \leq 62000:\\ \;\;\;\;\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(\left(\beta + 3\right) \cdot \left(2 + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\beta + 3\right) + 2 \cdot \alpha}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 5
Accuracy	98.1%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.6 \cdot 10^{+36}:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 3\right) \cdot \left(\left(2 + \beta\right) \cdot \left(2 + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\beta + 3\right) + 2 \cdot \alpha}}{\beta}\\ \end{array} \]

Alternative 6
Accuracy	98.2%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.6 \cdot 10^{+36}:\\ \;\;\;\;\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(\left(\beta + 3\right) \cdot \left(2 + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\beta + 3\right) + 2 \cdot \alpha}}{\beta}\\ \end{array} \]

Alternative 7
Accuracy	98.3%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;\beta \leq 62000:\\ \;\;\;\;\frac{1 + \beta}{\left(2 + \beta\right) \cdot \left(\left(\beta + 3\right) \cdot \left(2 + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\beta + 3\right) + 2 \cdot \alpha}}{\beta + 3}\\ \end{array} \]

Alternative 8
Accuracy	96.9%
Cost	964

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.45:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\beta + 3\right) + 2 \cdot \alpha}}{\beta}\\ \end{array} \]

Alternative 9
Accuracy	96.9%
Cost	836

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.45:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 10
Accuracy	93.8%
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.8:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \end{array} \]

Alternative 11
Accuracy	96.8%
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.8:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 12
Accuracy	46.6%
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.85:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\beta}\\ \end{array} \]

Alternative 13
Accuracy	91.2%
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.8:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \]

Alternative 14
Accuracy	44.4%
Cost	64

\[0.08333333333333333 \]

Reproduce

herbie shell --seed 2023129 
(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)))