Octave 3.8, jcobi/3

Average Error: 3.8 → 0.1

Time: 1.0min

Precision: binary64

Cost: 1600

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

Math

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

↓

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

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 (+ (+ alpha beta) 2.0)))
   (/
    (* (/ (- -1.0 alpha) t_0) (/ (+ beta 1.0) t_0))
    (- -3.0 (+ alpha beta)))))

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 = (alpha + beta) + 2.0;
	return (((-1.0 - alpha) / t_0) * ((beta + 1.0) / t_0)) / (-3.0 - (alpha + beta));
}

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 = (alpha + beta) + 2.0d0
    code = ((((-1.0d0) - alpha) / t_0) * ((beta + 1.0d0) / t_0)) / ((-3.0d0) - (alpha + beta))
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 = (alpha + beta) + 2.0;
	return (((-1.0 - alpha) / t_0) * ((beta + 1.0) / t_0)) / (-3.0 - (alpha + beta));
}

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 = (alpha + beta) + 2.0
	return (((-1.0 - alpha) / t_0) * ((beta + 1.0) / t_0)) / (-3.0 - (alpha + beta))

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(Float64(alpha + beta) + 2.0)
	return Float64(Float64(Float64(Float64(-1.0 - alpha) / t_0) * Float64(Float64(beta + 1.0) / t_0)) / Float64(-3.0 - Float64(alpha + beta)))
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 = (alpha + beta) + 2.0;
	tmp = (((-1.0 - alpha) / t_0) * ((beta + 1.0) / t_0)) / (-3.0 - (alpha + beta));
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[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]}, N[(N[(N[(N[(-1.0 - alpha), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(beta + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(-3.0 - N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 3.8

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

2. Simplified 4.7

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

3. Applied egg-rr 0.1

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

4. Applied egg-rr 0.1

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

Alternatives

Alternative 1
Error	4.5
Cost	1668

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2\\ \mathbf{if}\;\beta \leq 180:\\ \;\;\;\;\frac{\beta + 1}{t_0} \cdot \frac{\frac{1 + \alpha}{3 + \alpha}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(-\frac{1 + \alpha}{\beta}\right)\right) \cdot \frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 3}}{t_0}\\ \end{array} \]

Alternative 2
Error	2.7
Cost	1608

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2\\ t_1 := \frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 3}}{t_0}\\ \mathbf{if}\;\alpha \leq 8.2 \cdot 10^{-19}:\\ \;\;\;\;\frac{\beta + 1}{t_0} \cdot \frac{1}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{elif}\;\alpha \leq 2.5 \cdot 10^{+42}:\\ \;\;\;\;\frac{1}{2 + \alpha} \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha} \cdot t_1\\ \end{array} \]

Alternative 3
Error	4.5
Cost	1604

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2\\ \mathbf{if}\;\beta \leq 160:\\ \;\;\;\;\frac{\beta + 1}{t_0} \cdot \frac{\frac{1 + \alpha}{3 + \alpha}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 3}}{t_0}\\ \end{array} \]

Alternative 4
Error	0.1
Cost	1600

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

Alternative 5
Error	4.7
Cost	1476

\[\begin{array}{l} t_0 := \frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 3}}{\left(\alpha + \beta\right) + 2}\\ \mathbf{if}\;\beta \leq 175:\\ \;\;\;\;\frac{1}{2 + \alpha} \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot t_0\\ \end{array} \]

Alternative 6
Error	4.7
Cost	1476

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2\\ \mathbf{if}\;\beta \leq 180:\\ \;\;\;\;\frac{\beta + 1}{t_0} \cdot \frac{1 + \alpha}{\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 3}}{t_0}\\ \end{array} \]

Alternative 7
Error	4.6
Cost	1476

\[\begin{array}{l} \mathbf{if}\;\beta \leq 180:\\ \;\;\;\;\frac{\frac{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}{-3 - \alpha}}{\left(\left(-2 - \alpha\right) - \beta\right) \cdot \left(2 + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 3}}{\left(\alpha + \beta\right) + 2}\\ \end{array} \]

Alternative 8
Error	19.7
Cost	1428

\[\begin{array}{l} t_0 := \frac{1}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ t_1 := \frac{0.16666666666666666 \cdot \left(\beta + 1\right)}{\beta + 2}\\ \mathbf{if}\;\alpha \leq -1.6 \cdot 10^{-308}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\alpha \leq 7.2 \cdot 10^{-274}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 6.5 \cdot 10^{-172}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\alpha \leq 2.05 \cdot 10^{-112}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 2.3:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-\frac{\beta + 1}{\alpha}}{-3 - \left(\alpha + \beta\right)}\\ \end{array} \]

Alternative 9
Error	4.3
Cost	1348

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2\\ \mathbf{if}\;\alpha \leq 260:\\ \;\;\;\;\frac{\beta + 1}{\beta + 2} \cdot \frac{\frac{1}{\beta + 3}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot \frac{\beta + 1}{t_0}}{-3 - \left(\alpha + \beta\right)}\\ \end{array} \]

Alternative 10
Error	4.4
Cost	1348

\[\begin{array}{l} t_0 := \frac{\beta + 1}{\left(\alpha + \beta\right) + 2}\\ \mathbf{if}\;\alpha \leq 242:\\ \;\;\;\;t_0 \cdot \frac{1}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot t_0}{-3 - \left(\alpha + \beta\right)}\\ \end{array} \]

Alternative 11
Error	11.0
Cost	1220

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2\\ \mathbf{if}\;\beta \leq 80:\\ \;\;\;\;\frac{\beta + 1}{t_0} \cdot \frac{0.3333333333333333}{2 + \alpha}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 3}}{t_0}\\ \end{array} \]

Alternative 12
Error	11.0
Cost	1220

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2\\ \mathbf{if}\;\beta \leq 53:\\ \;\;\;\;\frac{\beta + 1}{t_0} \cdot \frac{0.3333333333333333}{t_0}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 3}}{t_0}\\ \end{array} \]

Alternative 13
Error	11.1
Cost	1092

\[\begin{array}{l} \mathbf{if}\;\beta \leq 70:\\ \;\;\;\;\frac{\beta + 1}{\left(\alpha + \beta\right) + 2} \cdot \frac{0.3333333333333333}{2 + \alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + \left(\alpha + 3\right)}}{\beta}\\ \end{array} \]

Alternative 14
Error	11.0
Cost	1092

\[\begin{array}{l} \mathbf{if}\;\beta \leq 35:\\ \;\;\;\;\frac{\beta + 1}{\left(\alpha + \beta\right) + 2} \cdot \frac{0.3333333333333333}{2 + \alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 - \alpha}{-2 - \left(\alpha + \beta\right)}}{\left(\alpha + \beta\right) + 3}\\ \end{array} \]

Alternative 15
Error	18.1
Cost	968

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.72:\\ \;\;\;\;\frac{0.16666666666666666 \cdot \left(\beta + 1\right)}{\beta + 2}\\ \mathbf{elif}\;\beta \leq 5.4 \cdot 10^{+146}:\\ \;\;\;\;\frac{-1 - \alpha}{\left(-3 - \left(\alpha + \beta\right)\right) \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\left(\alpha + \beta\right) + 3}\\ \end{array} \]

Alternative 16
Error	19.0
Cost	904

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.95:\\ \;\;\;\;\frac{0.16666666666666666 \cdot \left(\beta + 1\right)}{\beta + 2}\\ \mathbf{elif}\;\beta \leq 5.4 \cdot 10^{+146}:\\ \;\;\;\;\frac{-1 - \alpha}{-\beta \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\left(\alpha + \beta\right) + 3}\\ \end{array} \]

Alternative 17
Error	19.7
Cost	840

\[\begin{array}{l} \mathbf{if}\;\beta \leq 0.64:\\ \;\;\;\;\frac{0.16666666666666666 \cdot \left(\beta + 1\right)}{\beta + 2}\\ \mathbf{elif}\;\beta \leq 5.4 \cdot 10^{+146}:\\ \;\;\;\;\frac{1}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\left(\alpha + \beta\right) + 3}\\ \end{array} \]

Alternative 18
Error	18.5
Cost	836

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.72:\\ \;\;\;\;\frac{0.16666666666666666 \cdot \left(\beta + 1\right)}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + \left(\alpha + 3\right)}}{\beta}\\ \end{array} \]

Alternative 19
Error	19.6
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 0.64:\\ \;\;\;\;\frac{0.16666666666666666 \cdot \left(\beta + 1\right)}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \end{array} \]

Alternative 20
Error	42.2
Cost	576

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

Alternative 21
Error	47.0
Cost	448

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

Alternative 22
Error	61.3
Cost	192

\[\frac{0.3333333333333333}{\beta} \]

Alternative 23
Error	61.3
Cost	192

\[\frac{1}{\alpha} \]

Reproduce

herbie shell --seed 2023010 
(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)))