Octave 3.8, jcobi/3

Average Error: 3.6 → 0.2

Time: 15.6s

Precision: binary64

Cost: 1732

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

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

Math

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

↓

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ t_1 := \frac{1 + \alpha}{t_0}\\ \mathbf{if}\;\beta \leq 1.3489613858521963 \cdot 10^{+106}:\\ \;\;\;\;\frac{t_1 \cdot \left(1 + \beta\right)}{t_0 \cdot \left(3 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{\alpha + \left(\beta + 3\right)} \cdot \left(1 + \frac{-1 - \alpha}{\beta}\right)\\ \end{array} \]

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))) (t_1 (/ (+ 1.0 alpha) t_0)))
   (if (<= beta 1.3489613858521963e+106)
     (/ (* t_1 (+ 1.0 beta)) (* t_0 (+ 3.0 (+ alpha beta))))
     (* (/ t_1 (+ alpha (+ beta 3.0))) (+ 1.0 (/ (- -1.0 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);
	double t_1 = (1.0 + alpha) / t_0;
	double tmp;
	if (beta <= 1.3489613858521963e+106) {
		tmp = (t_1 * (1.0 + beta)) / (t_0 * (3.0 + (alpha + beta)));
	} else {
		tmp = (t_1 / (alpha + (beta + 3.0))) * (1.0 + ((-1.0 - alpha) / beta));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_0 = alpha + (beta + 2.0d0)
    t_1 = (1.0d0 + alpha) / t_0
    if (beta <= 1.3489613858521963d+106) then
        tmp = (t_1 * (1.0d0 + beta)) / (t_0 * (3.0d0 + (alpha + beta)))
    else
        tmp = (t_1 / (alpha + (beta + 3.0d0))) * (1.0d0 + (((-1.0d0) - alpha) / beta))
    end if
    code = tmp
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);
	double t_1 = (1.0 + alpha) / t_0;
	double tmp;
	if (beta <= 1.3489613858521963e+106) {
		tmp = (t_1 * (1.0 + beta)) / (t_0 * (3.0 + (alpha + beta)));
	} else {
		tmp = (t_1 / (alpha + (beta + 3.0))) * (1.0 + ((-1.0 - alpha) / beta));
	}
	return tmp;
}

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)
	t_1 = (1.0 + alpha) / t_0
	tmp = 0
	if beta <= 1.3489613858521963e+106:
		tmp = (t_1 * (1.0 + beta)) / (t_0 * (3.0 + (alpha + beta)))
	else:
		tmp = (t_1 / (alpha + (beta + 3.0))) * (1.0 + ((-1.0 - alpha) / beta))
	return tmp

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(alpha + Float64(beta + 2.0))
	t_1 = Float64(Float64(1.0 + alpha) / t_0)
	tmp = 0.0
	if (beta <= 1.3489613858521963e+106)
		tmp = Float64(Float64(t_1 * Float64(1.0 + beta)) / Float64(t_0 * Float64(3.0 + Float64(alpha + beta))));
	else
		tmp = Float64(Float64(t_1 / Float64(alpha + Float64(beta + 3.0))) * Float64(1.0 + Float64(Float64(-1.0 - alpha) / beta)));
	end
	return tmp
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_2 = code(alpha, beta)
	t_0 = alpha + (beta + 2.0);
	t_1 = (1.0 + alpha) / t_0;
	tmp = 0.0;
	if (beta <= 1.3489613858521963e+106)
		tmp = (t_1 * (1.0 + beta)) / (t_0 * (3.0 + (alpha + beta)));
	else
		tmp = (t_1 / (alpha + (beta + 3.0))) * (1.0 + ((-1.0 - alpha) / beta));
	end
	tmp_2 = tmp;
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[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[beta, 1.3489613858521963e+106], N[(N[(t$95$1 * N[(1.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(-1.0 - alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.3489613858521963e106

   1. Initial program 0.1

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

   2. Simplified 0.3

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

   3. Applied egg-rr 0.1

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

   4. Applied egg-rr 0.1

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

   5. Applied egg-rr 0.1

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

   6. Applied egg-rr 0.1

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

   if 1.3489613858521963e106 < beta

   1. Initial program 8.9

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

   2. Simplified 16.6

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

   3. Applied egg-rr 5.4

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

   4. Applied egg-rr 0.1

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

   5. Applied egg-rr 0.1

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

   6. Taylor expanded in beta around inf 0.4

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

   7. Simplified 0.4

      \[\leadsto \color{blue}{\left(1 - \frac{1 + \alpha}{\beta}\right)} \cdot \frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.3489613858521963 \cdot 10^{+106}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \left(1 + \beta\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(3 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)} \cdot \left(1 + \frac{-1 - \alpha}{\beta}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	1.1
Cost	1604

\[\begin{array}{l} \mathbf{if}\;\beta \leq 0.04432448353856024:\\ \;\;\;\;\frac{\frac{1 + \alpha}{6 + \alpha \cdot \left(\alpha + 5\right)}}{\beta + \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)} \cdot \left(1 + \frac{-1 - \alpha}{\beta}\right)\\ \end{array} \]

Alternative 2
Error	0.1
Cost	1600

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

Alternative 3
Error	0.1
Cost	1600

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

Alternative 4
Error	1.5
Cost	1220

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\beta \leq 0.04432448353856024:\\ \;\;\;\;\frac{\frac{1 + \alpha}{6 + \alpha \cdot \left(\alpha + 5\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 3\right)}}{t_0}\\ \end{array} \]

Alternative 5
Error	1.8
Cost	1092

\[\begin{array}{l} \mathbf{if}\;\beta \leq 0.04432448353856024:\\ \;\;\;\;\frac{0.16666666666666666 + \alpha \cdot \left(0.027777777777777776 + \alpha \cdot -0.05092592592592592\right)}{\beta + \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]

Alternative 6
Error	1.7
Cost	1092

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\beta \leq 0.04432448353856024:\\ \;\;\;\;\frac{0.16666666666666666 + \alpha \cdot \left(0.027777777777777776 + \alpha \cdot -0.05092592592592592\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 3\right)}}{t_0}\\ \end{array} \]

Alternative 7
Error	2.0
Cost	836

\[\begin{array}{l} \mathbf{if}\;\beta \leq 0.04432448353856024:\\ \;\;\;\;\frac{0.16666666666666666 + \alpha \cdot 0.027777777777777776}{\beta + \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \]

Alternative 8
Error	1.9
Cost	836

\[\begin{array}{l} \mathbf{if}\;\beta \leq 0.04432448353856024:\\ \;\;\;\;\frac{0.16666666666666666 + \alpha \cdot 0.027777777777777776}{\beta + \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]

Alternative 9
Error	3.9
Cost	712

\[\begin{array}{l} \mathbf{if}\;\beta \leq 0.04432448353856024:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + \left(\alpha + 2\right)}\\ \mathbf{elif}\;\beta \leq 2.3781021609824397 \cdot 10^{+158}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 10
Error	2.4
Cost	712

\[\begin{array}{l} \mathbf{if}\;\beta \leq 0.04432448353856024:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + \left(\alpha + 2\right)}\\ \mathbf{elif}\;\beta \leq 5.053039017544123 \cdot 10^{+153}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 11
Error	2.2
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 0.04432448353856024:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \]

Alternative 12
Error	3.9
Cost	584

\[\begin{array}{l} \mathbf{if}\;\beta \leq 0.04432448353856024:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{elif}\;\beta \leq 2.3781021609824397 \cdot 10^{+158}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 13
Error	3.9
Cost	584

\[\begin{array}{l} \mathbf{if}\;\beta \leq 0.04432448353856024:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + \left(\alpha + 2\right)}\\ \mathbf{elif}\;\beta \leq 2.3781021609824397 \cdot 10^{+158}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 14
Error	27.9
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 0.04432448353856024:\\ \;\;\;\;0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta}\\ \end{array} \]

Alternative 15
Error	5.4
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 0.04432448353856024:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta}\\ \end{array} \]

Alternative 16
Error	56.4
Cost	324

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.0072098152202787 \cdot 10^{+130}:\\ \;\;\;\;0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\beta}\\ \end{array} \]

Alternative 17
Error	56.1
Cost	324

\[\begin{array}{l} \mathbf{if}\;\beta \leq 0.04432448353856024:\\ \;\;\;\;0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta}\\ \end{array} \]

Alternative 18
Error	57.1
Cost	64

\[0.16666666666666666 \]

Reproduce

herbie shell --seed 2022284 
(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)))