Octave 3.8, jcobi/1

Average Error: 16.3 → 0.1

Time: 20.7s

Precision: binary64

Cost: 22276

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)}\\ t_1 := {\left(1 + t_0\right)}^{2}\\ \mathbf{if}\;t_0 \leq -0.99995:\\ \;\;\;\;0.5 \cdot \left(\frac{2 \cdot \left(\beta + 1\right)}{\alpha} + \left(\frac{8}{{\alpha}^{3}} + \frac{-4}{\alpha \cdot \alpha}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(t_1 + t_0 \cdot t_1\right)}^{0.3333333333333333}}{2}\\ \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))

↓

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ 2.0 (+ beta alpha))))
        (t_1 (pow (+ 1.0 t_0) 2.0)))
   (if (<= t_0 -0.99995)
     (*
      0.5
      (+
       (/ (* 2.0 (+ beta 1.0)) alpha)
       (+ (/ 8.0 (pow alpha 3.0)) (/ -4.0 (* alpha alpha)))))
     (/ (pow (+ t_1 (* t_0 t_1)) 0.3333333333333333) 2.0))))

C

double code(double alpha, double beta) {
	return (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
}

↓

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / (2.0 + (beta + alpha));
	double t_1 = pow((1.0 + t_0), 2.0);
	double tmp;
	if (t_0 <= -0.99995) {
		tmp = 0.5 * (((2.0 * (beta + 1.0)) / alpha) + ((8.0 / pow(alpha, 3.0)) + (-4.0 / (alpha * alpha))));
	} else {
		tmp = pow((t_1 + (t_0 * t_1)), 0.3333333333333333) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = (((beta - alpha) / ((alpha + beta) + 2.0d0)) + 1.0d0) / 2.0d0
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 = (beta - alpha) / (2.0d0 + (beta + alpha))
    t_1 = (1.0d0 + t_0) ** 2.0d0
    if (t_0 <= (-0.99995d0)) then
        tmp = 0.5d0 * (((2.0d0 * (beta + 1.0d0)) / alpha) + ((8.0d0 / (alpha ** 3.0d0)) + ((-4.0d0) / (alpha * alpha))))
    else
        tmp = ((t_1 + (t_0 * t_1)) ** 0.3333333333333333d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	return (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
}

↓

public static double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / (2.0 + (beta + alpha));
	double t_1 = Math.pow((1.0 + t_0), 2.0);
	double tmp;
	if (t_0 <= -0.99995) {
		tmp = 0.5 * (((2.0 * (beta + 1.0)) / alpha) + ((8.0 / Math.pow(alpha, 3.0)) + (-4.0 / (alpha * alpha))));
	} else {
		tmp = Math.pow((t_1 + (t_0 * t_1)), 0.3333333333333333) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	return (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0

↓

def code(alpha, beta):
	t_0 = (beta - alpha) / (2.0 + (beta + alpha))
	t_1 = math.pow((1.0 + t_0), 2.0)
	tmp = 0
	if t_0 <= -0.99995:
		tmp = 0.5 * (((2.0 * (beta + 1.0)) / alpha) + ((8.0 / math.pow(alpha, 3.0)) + (-4.0 / (alpha * alpha))))
	else:
		tmp = math.pow((t_1 + (t_0 * t_1)), 0.3333333333333333) / 2.0
	return tmp

Julia

function code(alpha, beta)
	return Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) + 1.0) / 2.0)
end

↓

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(2.0 + Float64(beta + alpha)))
	t_1 = Float64(1.0 + t_0) ^ 2.0
	tmp = 0.0
	if (t_0 <= -0.99995)
		tmp = Float64(0.5 * Float64(Float64(Float64(2.0 * Float64(beta + 1.0)) / alpha) + Float64(Float64(8.0 / (alpha ^ 3.0)) + Float64(-4.0 / Float64(alpha * alpha)))));
	else
		tmp = Float64((Float64(t_1 + Float64(t_0 * t_1)) ^ 0.3333333333333333) / 2.0);
	end
	return tmp
end

MATLAB

function tmp = code(alpha, beta)
	tmp = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
end

↓

function tmp_2 = code(alpha, beta)
	t_0 = (beta - alpha) / (2.0 + (beta + alpha));
	t_1 = (1.0 + t_0) ^ 2.0;
	tmp = 0.0;
	if (t_0 <= -0.99995)
		tmp = 0.5 * (((2.0 * (beta + 1.0)) / alpha) + ((8.0 / (alpha ^ 3.0)) + (-4.0 / (alpha * alpha))));
	else
		tmp = ((t_1 + (t_0 * t_1)) ^ 0.3333333333333333) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]

↓

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(1.0 + t$95$0), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t$95$0, -0.99995], N[(0.5 * N[(N[(N[(2.0 * N[(beta + 1.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] + N[(N[(8.0 / N[Power[alpha, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-4.0 / N[(alpha * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(t$95$1 + N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision], 0.3333333333333333], $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.999950000000000006

   1. Initial program 59.3

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

   2. Simplified 59.3

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

      [Start] 59.3	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
      +-commutative [=>] 59.3	\[ \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Applied egg-rr 57.4

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

   4. Applied egg-rr 57.4

      \[\leadsto \frac{\frac{\beta}{\left(\beta + \alpha\right) + 2} - \color{blue}{\log \left(e^{\frac{\alpha}{\left(\beta + \alpha\right) + 2} - 1}\right)}}{2} \]

   5. Taylor expanded in alpha around inf 6.0

      \[\leadsto \color{blue}{0.5 \cdot \frac{2 + 2 \cdot \beta}{\alpha} + \left(0.5 \cdot \frac{-1 \cdot {\left(\beta + 2\right)}^{2} - \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}} + 0.5 \cdot \frac{{\left(\beta + 2\right)}^{3} - -1 \cdot \left(\beta \cdot {\left(\beta + 2\right)}^{2}\right)}{{\alpha}^{3}}\right)} \]

   6. Simplified 6.0

      \[\leadsto \color{blue}{0.5 \cdot \left(\frac{\left(\beta + 1\right) \cdot 2}{\alpha} + \left(\frac{-\left(\beta + 2\right) \cdot \left(\left(\beta + 2\right) + \beta\right)}{\alpha \cdot \alpha} + \frac{{\left(\beta + 2\right)}^{2} \cdot \left(\beta + \left(\beta + 2\right)\right)}{{\alpha}^{3}}\right)\right)} \]
      Proof

      [Start] 6.0	\[ 0.5 \cdot \frac{2 + 2 \cdot \beta}{\alpha} + \left(0.5 \cdot \frac{-1 \cdot {\left(\beta + 2\right)}^{2} - \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}} + 0.5 \cdot \frac{{\left(\beta + 2\right)}^{3} - -1 \cdot \left(\beta \cdot {\left(\beta + 2\right)}^{2}\right)}{{\alpha}^{3}}\right) \]
      distribute-lft-out [=>] 6.0	\[ 0.5 \cdot \frac{2 + 2 \cdot \beta}{\alpha} + \color{blue}{0.5 \cdot \left(\frac{-1 \cdot {\left(\beta + 2\right)}^{2} - \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}} + \frac{{\left(\beta + 2\right)}^{3} - -1 \cdot \left(\beta \cdot {\left(\beta + 2\right)}^{2}\right)}{{\alpha}^{3}}\right)} \]
      distribute-lft-out [=>] 6.0	\[ \color{blue}{0.5 \cdot \left(\frac{2 + 2 \cdot \beta}{\alpha} + \left(\frac{-1 \cdot {\left(\beta + 2\right)}^{2} - \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}} + \frac{{\left(\beta + 2\right)}^{3} - -1 \cdot \left(\beta \cdot {\left(\beta + 2\right)}^{2}\right)}{{\alpha}^{3}}\right)\right)} \]
      *-commutative [=>] 6.0	\[ 0.5 \cdot \left(\frac{2 + \color{blue}{\beta \cdot 2}}{\alpha} + \left(\frac{-1 \cdot {\left(\beta + 2\right)}^{2} - \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}} + \frac{{\left(\beta + 2\right)}^{3} - -1 \cdot \left(\beta \cdot {\left(\beta + 2\right)}^{2}\right)}{{\alpha}^{3}}\right)\right) \]
      distribute-rgt1-in [=>] 6.0	\[ 0.5 \cdot \left(\frac{\color{blue}{\left(\beta + 1\right) \cdot 2}}{\alpha} + \left(\frac{-1 \cdot {\left(\beta + 2\right)}^{2} - \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}} + \frac{{\left(\beta + 2\right)}^{3} - -1 \cdot \left(\beta \cdot {\left(\beta + 2\right)}^{2}\right)}{{\alpha}^{3}}\right)\right) \]

   7. Taylor expanded in beta around 0 0.3

      \[\leadsto 0.5 \cdot \left(\frac{\left(\beta + 1\right) \cdot 2}{\alpha} + \color{blue}{\left(8 \cdot \frac{1}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}\right) \]

   8. Simplified 0.3

      \[\leadsto 0.5 \cdot \left(\frac{\left(\beta + 1\right) \cdot 2}{\alpha} + \color{blue}{\left(\frac{8}{{\alpha}^{3}} - \frac{4}{\alpha \cdot \alpha}\right)}\right) \]
      Proof

      [Start] 0.3	\[ 0.5 \cdot \left(\frac{\left(\beta + 1\right) \cdot 2}{\alpha} + \left(8 \cdot \frac{1}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\right)\right) \]
      associate-*r/ [=>] 0.3	\[ 0.5 \cdot \left(\frac{\left(\beta + 1\right) \cdot 2}{\alpha} + \left(\color{blue}{\frac{8 \cdot 1}{{\alpha}^{3}}} - 4 \cdot \frac{1}{{\alpha}^{2}}\right)\right) \]
      metadata-eval [=>] 0.3	\[ 0.5 \cdot \left(\frac{\left(\beta + 1\right) \cdot 2}{\alpha} + \left(\frac{\color{blue}{8}}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\right)\right) \]
      associate-*r/ [=>] 0.3	\[ 0.5 \cdot \left(\frac{\left(\beta + 1\right) \cdot 2}{\alpha} + \left(\frac{8}{{\alpha}^{3}} - \color{blue}{\frac{4 \cdot 1}{{\alpha}^{2}}}\right)\right) \]
      metadata-eval [=>] 0.3	\[ 0.5 \cdot \left(\frac{\left(\beta + 1\right) \cdot 2}{\alpha} + \left(\frac{8}{{\alpha}^{3}} - \frac{\color{blue}{4}}{{\alpha}^{2}}\right)\right) \]
      unpow2 [=>] 0.3	\[ 0.5 \cdot \left(\frac{\left(\beta + 1\right) \cdot 2}{\alpha} + \left(\frac{8}{{\alpha}^{3}} - \frac{4}{\color{blue}{\alpha \cdot \alpha}}\right)\right) \]

   if -0.999950000000000006 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 0.0

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

   2. Simplified 0.0

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

      [Start] 0.0	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
      +-commutative [=>] 0.0	\[ \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Applied egg-rr 0.0

      \[\leadsto \frac{\color{blue}{{\left({\left(\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1\right)}^{3}\right)}^{0.3333333333333333}}}{2} \]

   4. Applied egg-rr 0.0

      \[\leadsto \frac{{\color{blue}{\left({\left(\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1\right)}^{2} + \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \cdot {\left(\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1\right)}^{2}\right)}}^{0.3333333333333333}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)} \leq -0.99995:\\ \;\;\;\;0.5 \cdot \left(\frac{2 \cdot \left(\beta + 1\right)}{\alpha} + \left(\frac{8}{{\alpha}^{3}} + \frac{-4}{\alpha \cdot \alpha}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({\left(1 + \frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)}\right)}^{2} + \frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)} \cdot {\left(1 + \frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)}\right)}^{2}\right)}^{0.3333333333333333}}{2}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	8324

\[\begin{array}{l} t_0 := 2 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{t_0} \leq -0.99995:\\ \;\;\;\;0.5 \cdot \left(\frac{2 \cdot \left(\beta + 1\right)}{\alpha} + \left(\frac{8}{{\alpha}^{3}} + \frac{-4}{\alpha \cdot \alpha}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{t_0} - \mathsf{fma}\left(\alpha, \frac{1}{t_0}, -1\right)}{2}\\ \end{array} \]

Alternative 2
Error	0.1
Cost	8260

\[\begin{array}{l} t_0 := 2 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{t_0} \leq -0.99995:\\ \;\;\;\;0.5 \cdot \left(\frac{2 \cdot \left(\beta + 1\right)}{\alpha} + \frac{\beta + \left(\beta + 2\right)}{\alpha} \cdot \frac{-2 - \beta}{\alpha}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{t_0} - \mathsf{fma}\left(\alpha, \frac{1}{t_0}, -1\right)}{2}\\ \end{array} \]

Alternative 3
Error	0.1
Cost	2116

\[\begin{array}{l} t_0 := \frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)}\\ \mathbf{if}\;t_0 \leq -0.99995:\\ \;\;\;\;0.5 \cdot \left(\frac{2 \cdot \left(\beta + 1\right)}{\alpha} + \frac{\beta + \left(\beta + 2\right)}{\alpha} \cdot \frac{-2 - \beta}{\alpha}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + t_0}{2}\\ \end{array} \]

Alternative 4
Error	0.2
Cost	1476

\[\begin{array}{l} t_0 := \frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)}\\ \mathbf{if}\;t_0 \leq -0.999999995:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + t_0}{2}\\ \end{array} \]

Alternative 5
Error	20.9
Cost	1244

\[\begin{array}{l} t_0 := \frac{\frac{2}{\alpha}}{2}\\ \mathbf{if}\;\beta \leq -2.4 \cdot 10^{-57}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq -1.8 \cdot 10^{-112}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq -5.1 \cdot 10^{-222}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq -1.2 \cdot 10^{-258}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq 8.5 \cdot 10^{-273}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 1.4 \cdot 10^{-231}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\beta}\\ \end{array} \]

Alternative 6
Error	20.7
Cost	1244

\[\begin{array}{l} t_0 := \frac{\frac{2}{\alpha}}{2}\\ \mathbf{if}\;\beta \leq -2.4 \cdot 10^{-57}:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{elif}\;\beta \leq -3.9 \cdot 10^{-112}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq -4.8 \cdot 10^{-222}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq -9 \cdot 10^{-259}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq 2.1 \cdot 10^{-273}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 1.45 \cdot 10^{-231}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\beta}\\ \end{array} \]

Alternative 7
Error	8.4
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 2.5 \cdot 10^{+35}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]

Alternative 8
Error	4.8
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 2.65 \cdot 10^{+35}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 9
Error	18.2
Cost	452

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

Alternative 10
Error	18.3
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.96:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11
Error	40.0
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023187 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))