Octave 3.8, jcobi/3

Average Error: 3.6 → 0.1

Time: 12.9s

Precision: binary64

Cost: 1600

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

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

Math

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

↓

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

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 (+ beta (+ alpha 2.0))))
   (/ (* (/ (+ 1.0 alpha) t_0) (/ (+ 1.0 beta) (+ 3.0 (+ alpha beta)))) t_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) {
	double t_0 = beta + (alpha + 2.0);
	return (((1.0 + alpha) / t_0) * ((1.0 + beta) / (3.0 + (alpha + beta)))) / t_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
    real(8) :: t_0
    t_0 = beta + (alpha + 2.0d0)
    code = (((1.0d0 + alpha) / t_0) * ((1.0d0 + beta) / (3.0d0 + (alpha + beta)))) / t_0
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 = beta + (alpha + 2.0);
	return (((1.0 + alpha) / t_0) * ((1.0 + beta) / (3.0 + (alpha + beta)))) / t_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):
	t_0 = beta + (alpha + 2.0)
	return (((1.0 + alpha) / t_0) * ((1.0 + beta) / (3.0 + (alpha + beta)))) / t_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)
	t_0 = Float64(beta + Float64(alpha + 2.0))
	return Float64(Float64(Float64(Float64(1.0 + alpha) / t_0) * Float64(Float64(1.0 + beta) / Float64(3.0 + Float64(alpha + beta)))) / t_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)
	t_0 = beta + (alpha + 2.0);
	tmp = (((1.0 + alpha) / t_0) * ((1.0 + beta) / (3.0 + (alpha + beta)))) / t_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_] := Block[{t$95$0 = N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(1.0 + beta), $MachinePrecision] / N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $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 6.5

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

3. Applied egg-rr 3.6

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

4. Applied egg-rr 0.1

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

5. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.8
Cost	1604

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\beta \leq 292787342029.835:\\ \;\;\;\;\frac{\frac{\frac{1 + \beta}{\beta + 2}}{2 + \left(\alpha + \beta\right)}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t_0} \cdot \left(1 + \frac{-2 - \alpha}{\beta}\right)}{t_0}\\ \end{array} \]

Alternative 2
Error	1.0
Cost	1604

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\beta \leq 4.521737546177817:\\ \;\;\;\;\frac{1 + \beta}{\left(\left(4 - \alpha \cdot -2\right) + \beta \cdot \left(\beta + \left(\alpha + 4\right)\right)\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t_0} \cdot \left(1 + \frac{-2 - \alpha}{\beta}\right)}{t_0}\\ \end{array} \]

Alternative 3
Error	1.0
Cost	1220

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\beta \leq 53333787222600.95:\\ \;\;\;\;\frac{\frac{1 + \beta}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t_0}}{t_0}\\ \end{array} \]

Alternative 4
Error	1.0
Cost	1220

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\beta \leq 53333787222600.95:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 3\right) \cdot \left(t_0 \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t_0}}{t_0}\\ \end{array} \]

Alternative 5
Error	1.7
Cost	1092

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\beta \leq 2.037092589294192:\\ \;\;\;\;\frac{\frac{0.5}{\alpha + 2}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t_0}}{t_0}\\ \end{array} \]

Alternative 6
Error	1.8
Cost	836

\[\begin{array}{l} \mathbf{if}\;\beta \leq 4.521737546177817:\\ \;\;\;\;\frac{\frac{0.5}{\alpha + 2}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 2\right)}\\ \end{array} \]

Alternative 7
Error	1.7
Cost	836

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.037092589294192:\\ \;\;\;\;\frac{\frac{0.5}{\alpha + 2}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 8
Error	4.1
Cost	712

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.037092589294192:\\ \;\;\;\;\alpha \cdot -0.041666666666666664 + 0.08333333333333333\\ \mathbf{elif}\;\beta \leq 6.450129104419427 \cdot 10^{+152}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 9
Error	1.8
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 4.521737546177817:\\ \;\;\;\;\frac{\frac{0.5}{\alpha + 2}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 10
Error	4.1
Cost	584

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.037092589294192:\\ \;\;\;\;\alpha \cdot -0.041666666666666664 + 0.08333333333333333\\ \mathbf{elif}\;\beta \leq 6.450129104419427 \cdot 10^{+152}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 11
Error	2.1
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.037092589294192:\\ \;\;\;\;\alpha \cdot -0.041666666666666664 + 0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 12
Error	33.4
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.037092589294192:\\ \;\;\;\;\alpha \cdot -0.041666666666666664 + 0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\beta}\\ \end{array} \]

Alternative 13
Error	5.3
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.037092589294192:\\ \;\;\;\;\alpha \cdot -0.041666666666666664 + 0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta}\\ \end{array} \]

Alternative 14
Error	33.4
Cost	324

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.037092589294192:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\beta}\\ \end{array} \]

Alternative 15
Error	34.5
Cost	64

\[0.08333333333333333 \]

Reproduce

herbie shell --seed 2022225 
(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)))