Octave 3.8, jcobi/1

Average Error: 16.2 → 0.3

Time: 6.9s

Precision: binary64

Cost: 8068

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

Math

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

↓

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

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) (+ (+ alpha beta) 2.0))))
   (if (<= t_0 -0.5)
     (/
      (+ (* 2.0 (+ (/ beta alpha) (/ 1.0 alpha))) (/ -4.0 (pow alpha 2.0)))
      2.0)
     (/ (+ t_0 1.0) 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) / ((alpha + beta) + 2.0);
	double tmp;
	if (t_0 <= -0.5) {
		tmp = ((2.0 * ((beta / alpha) + (1.0 / alpha))) + (-4.0 / pow(alpha, 2.0))) / 2.0;
	} else {
		tmp = (t_0 + 1.0) / 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) :: tmp
    t_0 = (beta - alpha) / ((alpha + beta) + 2.0d0)
    if (t_0 <= (-0.5d0)) then
        tmp = ((2.0d0 * ((beta / alpha) + (1.0d0 / alpha))) + ((-4.0d0) / (alpha ** 2.0d0))) / 2.0d0
    else
        tmp = (t_0 + 1.0d0) / 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) / ((alpha + beta) + 2.0);
	double tmp;
	if (t_0 <= -0.5) {
		tmp = ((2.0 * ((beta / alpha) + (1.0 / alpha))) + (-4.0 / Math.pow(alpha, 2.0))) / 2.0;
	} else {
		tmp = (t_0 + 1.0) / 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) / ((alpha + beta) + 2.0)
	tmp = 0
	if t_0 <= -0.5:
		tmp = ((2.0 * ((beta / alpha) + (1.0 / alpha))) + (-4.0 / math.pow(alpha, 2.0))) / 2.0
	else:
		tmp = (t_0 + 1.0) / 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(Float64(alpha + beta) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = Float64(Float64(Float64(2.0 * Float64(Float64(beta / alpha) + Float64(1.0 / alpha))) + Float64(-4.0 / (alpha ^ 2.0))) / 2.0);
	else
		tmp = Float64(Float64(t_0 + 1.0) / 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) / ((alpha + beta) + 2.0);
	tmp = 0.0;
	if (t_0 <= -0.5)
		tmp = ((2.0 * ((beta / alpha) + (1.0 / alpha))) + (-4.0 / (alpha ^ 2.0))) / 2.0;
	else
		tmp = (t_0 + 1.0) / 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[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(N[(N[(2.0 * N[(N[(beta / alpha), $MachinePrecision] + N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-4.0 / N[Power[alpha, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$0 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.5

   1. Initial program 58.6

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

   2. Simplified 58.6

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

      [Start] 58.6	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 58.6	\[ \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Taylor expanded in alpha around inf 4.4

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

   4. Simplified 4.4

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

      [Start] 4.4	\[ \frac{\left(2 \cdot \frac{\beta}{\alpha} + \left(-1 \cdot \frac{{\left(\beta + 2\right)}^{2}}{{\alpha}^{2}} + 2 \cdot \frac{1}{\alpha}\right)\right) - \frac{\beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}}{2} \]
      rational_best_oopsla_all_46_json_45_simplify-82 [=>] 4.4	\[ \frac{\color{blue}{\left(-1 \cdot \frac{{\left(\beta + 2\right)}^{2}}{{\alpha}^{2}} + \left(2 \cdot \frac{\beta}{\alpha} + 2 \cdot \frac{1}{\alpha}\right)\right)} - \frac{\beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}}{2} \]
      rational_best_oopsla_all_46_json_45_simplify-107 [=>] 4.4	\[ \frac{\color{blue}{\left(2 \cdot \frac{\beta}{\alpha} + 2 \cdot \frac{1}{\alpha}\right) + \left(-1 \cdot \frac{{\left(\beta + 2\right)}^{2}}{{\alpha}^{2}} - \frac{\beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}\right)}}{2} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 4.4	\[ \frac{\left(\color{blue}{\frac{\beta}{\alpha} \cdot 2} + 2 \cdot \frac{1}{\alpha}\right) + \left(-1 \cdot \frac{{\left(\beta + 2\right)}^{2}}{{\alpha}^{2}} - \frac{\beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}\right)}{2} \]
      rational_best_oopsla_all_46_json_45_simplify-23 [=>] 4.4	\[ \frac{\color{blue}{2 \cdot \left(\frac{\beta}{\alpha} + \frac{1}{\alpha}\right)} + \left(-1 \cdot \frac{{\left(\beta + 2\right)}^{2}}{{\alpha}^{2}} - \frac{\beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}\right)}{2} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 4.4	\[ \frac{2 \cdot \left(\frac{\beta}{\alpha} + \frac{1}{\alpha}\right) + \left(\color{blue}{\frac{{\left(\beta + 2\right)}^{2}}{{\alpha}^{2}} \cdot -1} - \frac{\beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}\right)}{2} \]
      rational_best_oopsla_all_46_json_45_simplify-92 [=>] 4.4	\[ \frac{2 \cdot \left(\frac{\beta}{\alpha} + \frac{1}{\alpha}\right) + \left(\color{blue}{\left(-\frac{{\left(\beta + 2\right)}^{2}}{{\alpha}^{2}}\right)} - \frac{\beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}\right)}{2} \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 4.4	\[ \frac{2 \cdot \left(\frac{\beta}{\alpha} + \frac{1}{\alpha}\right) + \left(\left(-\frac{{\color{blue}{\left(2 + \beta\right)}}^{2}}{{\alpha}^{2}}\right) - \frac{\beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}\right)}{2} \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 4.4	\[ \frac{2 \cdot \left(\frac{\beta}{\alpha} + \frac{1}{\alpha}\right) + \left(\left(-\frac{{\left(2 + \beta\right)}^{2}}{{\alpha}^{2}}\right) - \frac{\beta \cdot \color{blue}{\left(2 + \beta\right)}}{{\alpha}^{2}}\right)}{2} \]

   5. Taylor expanded in beta around 0 1.0

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

   if -0.5 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 0.0

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

4. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.4
Cost	1476

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

Alternative 2
Error	7.6
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 4400000000000:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]

Alternative 3
Error	4.2
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 4200000000000:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + 2 \cdot \beta}{\alpha}}{2}\\ \end{array} \]

Alternative 4
Error	17.9
Cost	644

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

Alternative 5
Error	18.1
Cost	580

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

Alternative 6
Error	18.4
Cost	196

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

Alternative 7
Error	32.7
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023090 
(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))