Octave 3.8, jcobi/3

Average Error: 3.6 → 1.5

Time: 2.4min

Precision: binary64

Cost: 4548

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

Math

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

↓

\[\begin{array}{l} t_0 := 2 + \left(\alpha + \beta\right)\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot 1\\ t_2 := \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{t_1}}{t_1}}{t_1 + 1}\\ \mathbf{if}\;t_2 \leq 0.0833333348:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\left(t_0 \cdot t_0\right) \cdot \left(3 + \left(\alpha + \beta\right)\right)}\\ \end{array} \]

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 (+ 2.0 (+ alpha beta)))
        (t_1 (+ (+ alpha beta) (* 2.0 1.0)))
        (t_2
         (/
          (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) t_1) t_1)
          (+ t_1 1.0))))
   (if (<= t_2 0.0833333348)
     t_2
     (/ (+ beta 1.0) (* (* t_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 = 2.0 + (alpha + beta);
	double t_1 = (alpha + beta) + (2.0 * 1.0);
	double t_2 = (((((alpha + beta) + (beta * alpha)) + 1.0) / t_1) / t_1) / (t_1 + 1.0);
	double tmp;
	if (t_2 <= 0.0833333348) {
		tmp = t_2;
	} else {
		tmp = (beta + 1.0) / ((t_0 * t_0) * (3.0 + (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) :: t_2
    real(8) :: tmp
    t_0 = 2.0d0 + (alpha + beta)
    t_1 = (alpha + beta) + (2.0d0 * 1.0d0)
    t_2 = (((((alpha + beta) + (beta * alpha)) + 1.0d0) / t_1) / t_1) / (t_1 + 1.0d0)
    if (t_2 <= 0.0833333348d0) then
        tmp = t_2
    else
        tmp = (beta + 1.0d0) / ((t_0 * t_0) * (3.0d0 + (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 = 2.0 + (alpha + beta);
	double t_1 = (alpha + beta) + (2.0 * 1.0);
	double t_2 = (((((alpha + beta) + (beta * alpha)) + 1.0) / t_1) / t_1) / (t_1 + 1.0);
	double tmp;
	if (t_2 <= 0.0833333348) {
		tmp = t_2;
	} else {
		tmp = (beta + 1.0) / ((t_0 * t_0) * (3.0 + (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 = 2.0 + (alpha + beta)
	t_1 = (alpha + beta) + (2.0 * 1.0)
	t_2 = (((((alpha + beta) + (beta * alpha)) + 1.0) / t_1) / t_1) / (t_1 + 1.0)
	tmp = 0
	if t_2 <= 0.0833333348:
		tmp = t_2
	else:
		tmp = (beta + 1.0) / ((t_0 * t_0) * (3.0 + (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(2.0 + Float64(alpha + beta))
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * 1.0))
	t_2 = Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) + Float64(beta * alpha)) + 1.0) / t_1) / t_1) / Float64(t_1 + 1.0))
	tmp = 0.0
	if (t_2 <= 0.0833333348)
		tmp = t_2;
	else
		tmp = Float64(Float64(beta + 1.0) / Float64(Float64(t_0 * t_0) * Float64(3.0 + Float64(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 = 2.0 + (alpha + beta);
	t_1 = (alpha + beta) + (2.0 * 1.0);
	t_2 = (((((alpha + beta) + (beta * alpha)) + 1.0) / t_1) / t_1) / (t_1 + 1.0);
	tmp = 0.0;
	if (t_2 <= 0.0833333348)
		tmp = t_2;
	else
		tmp = (beta + 1.0) / ((t_0 * t_0) * (3.0 + (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[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0833333348], t$95$2, N[(N[(beta + 1.0), $MachinePrecision] / N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[(3.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (/.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 beta alpha)) 1) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) 1)) < 0.0833333348000000057

   1. Initial program 0.1

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

   if 0.0833333348000000057 < (/.f64 (/.f64 (/.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 beta alpha)) 1) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) 1))

   1. Initial program 53.7

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

   2. Simplified 53.7

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

   3. Applied egg-rr 53.7

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

   4. Taylor expanded in alpha around 0 21.2

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

Alternatives

Alternative 1
Error	3.5
Cost	1864

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot 1\\ t_1 := 2 + \left(\alpha + \beta\right)\\ t_2 := 3 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 5 \cdot 10^{-66}:\\ \;\;\;\;\frac{\frac{\frac{-1 + \left(-\alpha\right)}{2 + \alpha}}{t_2}}{-2 - \left(\alpha + \beta\right)}\\ \mathbf{elif}\;\beta \leq 1.05 \cdot 10^{+69}:\\ \;\;\;\;\frac{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}{\left(t_1 \cdot t_1\right) \cdot t_2}\\ \mathbf{elif}\;\beta \leq 1.85 \cdot 10^{+183}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t_0}}{t_0 + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \end{array} \]

Alternative 2
Error	2.5
Cost	1732

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot 1\\ t_1 := 2 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\alpha \leq 5.8 \cdot 10^{+75}:\\ \;\;\;\;\frac{\frac{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}{t_1 \cdot \left(3 + \left(\alpha + \beta\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + 1}{t_0}}{t_0 + 1}\\ \end{array} \]

Alternative 3
Error	2.5
Cost	1732

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot 1\\ t_1 := -2 - \left(\alpha + \beta\right)\\ \mathbf{if}\;\alpha \leq 5.8 \cdot 10^{+75}:\\ \;\;\;\;\frac{\frac{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}{t_1}}{t_1 \cdot \left(3 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + 1}{t_0}}{t_0 + 1}\\ \end{array} \]

Alternative 4
Error	2.0
Cost	1732

\[\begin{array}{l} t_0 := -2 - \left(\alpha + \beta\right)\\ \mathbf{if}\;\alpha \leq 1.46 \cdot 10^{+122}:\\ \;\;\;\;\frac{\frac{\frac{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}{t_0}}{3 + \left(\alpha + \beta\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 - \beta}{\alpha}}{t_0}\\ \end{array} \]

Alternative 5
Error	4.9
Cost	1220

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

Alternative 6
Error	4.1
Cost	1220

\[\begin{array}{l} t_0 := -2 - \left(\alpha + \beta\right)\\ \mathbf{if}\;\alpha \leq 21500:\\ \;\;\;\;\frac{\frac{-1 - \beta}{\left(3 + \beta\right) \cdot \left(2 + \beta\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\left(\beta - -1\right)}{t_0 \cdot \left(3 + \left(\alpha + \beta\right)\right)}\\ \end{array} \]

Alternative 7
Error	10.4
Cost	1156

\[\begin{array}{l} t_0 := -2 - \left(\alpha + \beta\right)\\ \mathbf{if}\;\alpha \leq 21500:\\ \;\;\;\;\frac{\frac{-0.5}{\beta + 3}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\left(\beta - -1\right)}{t_0 \cdot \left(3 + \left(\alpha + \beta\right)\right)}\\ \end{array} \]

Alternative 8
Error	11.1
Cost	1092

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 30500:\\ \;\;\;\;\frac{\frac{-0.5}{\beta + 3}}{-2 - \left(\alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + 1}{\alpha}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \end{array} \]

Alternative 9
Error	11.8
Cost	836

\[\begin{array}{l} t_0 := -2 - \left(\alpha + \beta\right)\\ \mathbf{if}\;\alpha \leq 28000:\\ \;\;\;\;\frac{\frac{-0.5}{\beta + 3}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{\alpha}}{t_0}\\ \end{array} \]

Alternative 10
Error	11.1
Cost	836

\[\begin{array}{l} t_0 := -2 - \left(\alpha + \beta\right)\\ \mathbf{if}\;\alpha \leq 27000:\\ \;\;\;\;\frac{\frac{-0.5}{\beta + 3}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 - \beta}{\alpha}}{t_0}\\ \end{array} \]

Alternative 11
Error	18.7
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 130:\\ \;\;\;\;\frac{0.5}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 12
Error	11.8
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 98000:\\ \;\;\;\;\frac{0.5}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{\alpha}}{-2 - \left(\alpha + \beta\right)}\\ \end{array} \]

Alternative 13
Error	39.9
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 130:\\ \;\;\;\;\frac{0.5}{3 + \alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 14
Error	41.3
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 130:\\ \;\;\;\;\frac{0.5}{3 + \alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \]

Alternative 15
Error	41.2
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 130:\\ \;\;\;\;\frac{0.5}{3 + \alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta}\\ \end{array} \]

Alternative 16
Error	56.6
Cost	320

\[\frac{0.5}{3 + \alpha} \]

Alternative 17
Error	61.3
Cost	192

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

Alternative 18
Error	61.3
Cost	192

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

Reproduce

herbie shell --seed 2023033 
(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)))