Octave 3.8, jcobi/1

Average Error: 25.27% → 0.45%

Time: 6.9s

Precision: binary64

Cost: 2116

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t_0 \leq -0.5:\\ \;\;\;\;\frac{\frac{2}{\alpha} + \left(\beta \cdot \left(\frac{2}{\alpha} - \frac{6}{\alpha \cdot \alpha}\right) + \frac{-4}{\alpha \cdot \alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \end{array} \]

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) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.5)
     (/
      (+
       (/ 2.0 alpha)
       (+
        (* beta (- (/ 2.0 alpha) (/ 6.0 (* alpha alpha))))
        (/ -4.0 (* alpha alpha))))
      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) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.5) {
		tmp = ((2.0 / alpha) + ((beta * ((2.0 / alpha) - (6.0 / (alpha * alpha)))) + (-4.0 / (alpha * alpha)))) / 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) / ((beta + alpha) + 2.0d0)
    if (t_0 <= (-0.5d0)) then
        tmp = ((2.0d0 / alpha) + ((beta * ((2.0d0 / alpha) - (6.0d0 / (alpha * alpha)))) + ((-4.0d0) / (alpha * alpha)))) / 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) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.5) {
		tmp = ((2.0 / alpha) + ((beta * ((2.0 / alpha) - (6.0 / (alpha * alpha)))) + (-4.0 / (alpha * alpha)))) / 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) / ((beta + alpha) + 2.0)
	tmp = 0
	if t_0 <= -0.5:
		tmp = ((2.0 / alpha) + ((beta * ((2.0 / alpha) - (6.0 / (alpha * alpha)))) + (-4.0 / (alpha * alpha)))) / 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(beta + alpha) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = Float64(Float64(Float64(2.0 / alpha) + Float64(Float64(beta * Float64(Float64(2.0 / alpha) - Float64(6.0 / Float64(alpha * alpha)))) + Float64(-4.0 / Float64(alpha * alpha)))) / 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) / ((beta + alpha) + 2.0);
	tmp = 0.0;
	if (t_0 <= -0.5)
		tmp = ((2.0 / alpha) + ((beta * ((2.0 / alpha) - (6.0 / (alpha * alpha)))) + (-4.0 / (alpha * alpha)))) / 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[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(N[(N[(2.0 / alpha), $MachinePrecision] + N[(N[(beta * N[(N[(2.0 / alpha), $MachinePrecision] - N[(6.0 / N[(alpha * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-4.0 / N[(alpha * alpha), $MachinePrecision]), $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 91.53

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

   2. Simplified 91.53

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

      [Start] 91.53	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
      +-commutative [=>] 91.53	\[ \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Taylor expanded in alpha around inf 6.95

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

   4. Simplified 6.95

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

      [Start] 6.95	\[ \frac{\left(2 \cdot \frac{\beta}{\alpha} + \left(-1 \cdot \frac{{\left(\beta + 2\right)}^{2}}{{\alpha}^{2}} + 2 \cdot \frac{1}{\alpha}\right)\right) - \frac{\beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}}{2} \]
      fma-def [=>] 6.95	\[ \frac{\color{blue}{\mathsf{fma}\left(2, \frac{\beta}{\alpha}, -1 \cdot \frac{{\left(\beta + 2\right)}^{2}}{{\alpha}^{2}} + 2 \cdot \frac{1}{\alpha}\right)} - \frac{\beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}}{2} \]
      fma-def [=>] 6.95	\[ \frac{\mathsf{fma}\left(2, \frac{\beta}{\alpha}, \color{blue}{\mathsf{fma}\left(-1, \frac{{\left(\beta + 2\right)}^{2}}{{\alpha}^{2}}, 2 \cdot \frac{1}{\alpha}\right)}\right) - \frac{\beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}}{2} \]
      +-commutative [=>] 6.95	\[ \frac{\mathsf{fma}\left(2, \frac{\beta}{\alpha}, \mathsf{fma}\left(-1, \frac{{\color{blue}{\left(2 + \beta\right)}}^{2}}{{\alpha}^{2}}, 2 \cdot \frac{1}{\alpha}\right)\right) - \frac{\beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}}{2} \]
      unpow2 [=>] 6.95	\[ \frac{\mathsf{fma}\left(2, \frac{\beta}{\alpha}, \mathsf{fma}\left(-1, \frac{{\left(2 + \beta\right)}^{2}}{\color{blue}{\alpha \cdot \alpha}}, 2 \cdot \frac{1}{\alpha}\right)\right) - \frac{\beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}}{2} \]
      associate-*r/ [=>] 6.95	\[ \frac{\mathsf{fma}\left(2, \frac{\beta}{\alpha}, \mathsf{fma}\left(-1, \frac{{\left(2 + \beta\right)}^{2}}{\alpha \cdot \alpha}, \color{blue}{\frac{2 \cdot 1}{\alpha}}\right)\right) - \frac{\beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}}{2} \]
      metadata-eval [=>] 6.95	\[ \frac{\mathsf{fma}\left(2, \frac{\beta}{\alpha}, \mathsf{fma}\left(-1, \frac{{\left(2 + \beta\right)}^{2}}{\alpha \cdot \alpha}, \frac{\color{blue}{2}}{\alpha}\right)\right) - \frac{\beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}}{2} \]
      associate-/l* [=>] 6.95	\[ \frac{\mathsf{fma}\left(2, \frac{\beta}{\alpha}, \mathsf{fma}\left(-1, \frac{{\left(2 + \beta\right)}^{2}}{\alpha \cdot \alpha}, \frac{2}{\alpha}\right)\right) - \color{blue}{\frac{\beta}{\frac{{\alpha}^{2}}{\beta + 2}}}}{2} \]
      unpow2 [=>] 6.95	\[ \frac{\mathsf{fma}\left(2, \frac{\beta}{\alpha}, \mathsf{fma}\left(-1, \frac{{\left(2 + \beta\right)}^{2}}{\alpha \cdot \alpha}, \frac{2}{\alpha}\right)\right) - \frac{\beta}{\frac{\color{blue}{\alpha \cdot \alpha}}{\beta + 2}}}{2} \]
      +-commutative [=>] 6.95	\[ \frac{\mathsf{fma}\left(2, \frac{\beta}{\alpha}, \mathsf{fma}\left(-1, \frac{{\left(2 + \beta\right)}^{2}}{\alpha \cdot \alpha}, \frac{2}{\alpha}\right)\right) - \frac{\beta}{\frac{\alpha \cdot \alpha}{\color{blue}{2 + \beta}}}}{2} \]

   5. Taylor expanded in beta around 0 1.59

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \frac{1}{\alpha} + \beta \cdot \left(2 \cdot \frac{1}{\alpha} - 6 \cdot \frac{1}{{\alpha}^{2}}\right)\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}}{2} \]

   6. Simplified 1.59

      \[\leadsto \frac{\color{blue}{\frac{2}{\alpha} + \left(\beta \cdot \left(\frac{2}{\alpha} - \frac{6}{\alpha \cdot \alpha}\right) - \frac{4}{\alpha \cdot \alpha}\right)}}{2} \]
      Proof

      [Start] 1.59	\[ \frac{\left(2 \cdot \frac{1}{\alpha} + \beta \cdot \left(2 \cdot \frac{1}{\alpha} - 6 \cdot \frac{1}{{\alpha}^{2}}\right)\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}{2} \]
      associate--l+ [=>] 1.59	\[ \frac{\color{blue}{2 \cdot \frac{1}{\alpha} + \left(\beta \cdot \left(2 \cdot \frac{1}{\alpha} - 6 \cdot \frac{1}{{\alpha}^{2}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}}{2} \]
      associate-*r/ [=>] 1.59	\[ \frac{\color{blue}{\frac{2 \cdot 1}{\alpha}} + \left(\beta \cdot \left(2 \cdot \frac{1}{\alpha} - 6 \cdot \frac{1}{{\alpha}^{2}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}{2} \]
      metadata-eval [=>] 1.59	\[ \frac{\frac{\color{blue}{2}}{\alpha} + \left(\beta \cdot \left(2 \cdot \frac{1}{\alpha} - 6 \cdot \frac{1}{{\alpha}^{2}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}{2} \]
      associate-*r/ [=>] 1.59	\[ \frac{\frac{2}{\alpha} + \left(\beta \cdot \left(\color{blue}{\frac{2 \cdot 1}{\alpha}} - 6 \cdot \frac{1}{{\alpha}^{2}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}{2} \]
      metadata-eval [=>] 1.59	\[ \frac{\frac{2}{\alpha} + \left(\beta \cdot \left(\frac{\color{blue}{2}}{\alpha} - 6 \cdot \frac{1}{{\alpha}^{2}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}{2} \]
      associate-*r/ [=>] 1.59	\[ \frac{\frac{2}{\alpha} + \left(\beta \cdot \left(\frac{2}{\alpha} - \color{blue}{\frac{6 \cdot 1}{{\alpha}^{2}}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}{2} \]
      metadata-eval [=>] 1.59	\[ \frac{\frac{2}{\alpha} + \left(\beta \cdot \left(\frac{2}{\alpha} - \frac{\color{blue}{6}}{{\alpha}^{2}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}{2} \]
      unpow2 [=>] 1.59	\[ \frac{\frac{2}{\alpha} + \left(\beta \cdot \left(\frac{2}{\alpha} - \frac{6}{\color{blue}{\alpha \cdot \alpha}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}{2} \]
      associate-*r/ [=>] 1.59	\[ \frac{\frac{2}{\alpha} + \left(\beta \cdot \left(\frac{2}{\alpha} - \frac{6}{\alpha \cdot \alpha}\right) - \color{blue}{\frac{4 \cdot 1}{{\alpha}^{2}}}\right)}{2} \]
      metadata-eval [=>] 1.59	\[ \frac{\frac{2}{\alpha} + \left(\beta \cdot \left(\frac{2}{\alpha} - \frac{6}{\alpha \cdot \alpha}\right) - \frac{\color{blue}{4}}{{\alpha}^{2}}\right)}{2} \]
      unpow2 [=>] 1.59	\[ \frac{\frac{2}{\alpha} + \left(\beta \cdot \left(\frac{2}{\alpha} - \frac{6}{\alpha \cdot \alpha}\right) - \frac{4}{\color{blue}{\alpha \cdot \alpha}}\right)}{2} \]

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

   1. Initial program 0.01

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

4. Final simplification 0.45

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.5:\\ \;\;\;\;\frac{\frac{2}{\alpha} + \left(\beta \cdot \left(\frac{2}{\alpha} - \frac{6}{\alpha \cdot \alpha}\right) + \frac{-4}{\alpha \cdot \alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.62%
Cost	1476

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

Alternative 2
Error	11.95%
Cost	708

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

Alternative 3
Error	6.55%
Cost	708

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

Alternative 4
Error	28.29%
Cost	580

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

Alternative 5
Error	28.03%
Cost	580

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

Alternative 6
Error	28.81%
Cost	196

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

Alternative 7
Error	51.16%
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))