Octave 3.8, jcobi/1

Average Error: 16.6 → 0.2

Time: 8.2s

Precision: binary64

Cost: 1860

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

Math

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

↓

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

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))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.999999995)
     (/ (+ beta 1.0) alpha)
     (/ (- (/ beta t_0) (+ (/ alpha 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 + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999999995) {
		tmp = (beta + 1.0) / alpha;
	} else {
		tmp = ((beta / t_0) - ((alpha / 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 + 2.0d0)
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.999999995d0)) then
        tmp = (beta + 1.0d0) / alpha
    else
        tmp = ((beta / t_0) - ((alpha / 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 + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999999995) {
		tmp = (beta + 1.0) / alpha;
	} else {
		tmp = ((beta / t_0) - ((alpha / 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 + 2.0)
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999999995:
		tmp = (beta + 1.0) / alpha
	else:
		tmp = ((beta / t_0) - ((alpha / 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(beta + Float64(alpha + 2.0))
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.999999995)
		tmp = Float64(Float64(beta + 1.0) / alpha);
	else
		tmp = Float64(Float64(Float64(beta / t_0) - Float64(Float64(alpha / 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 + 2.0);
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999999995)
		tmp = (beta + 1.0) / alpha;
	else
		tmp = ((beta / t_0) - ((alpha / 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[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.999999995], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(N[(beta / t$95$0), $MachinePrecision] - N[(N[(alpha / t$95$0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.99999999500000003

   1. Initial program 59.8

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

   2. Simplified 59.8

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

      [Start] 59.8	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
      +-commutative [=>] 59.8	\[ \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Taylor expanded in alpha around inf 0.4

      \[\leadsto \frac{\color{blue}{\frac{2 + 2 \cdot \beta}{\alpha}}}{2} \]

   4. Taylor expanded in alpha around 0 0.4

      \[\leadsto \color{blue}{0.5 \cdot \frac{2 + 2 \cdot \beta}{\alpha}} \]

   5. Simplified 0.4

      \[\leadsto \color{blue}{\frac{1 + \beta}{\alpha}} \]
      Proof

      [Start] 0.4	\[ 0.5 \cdot \frac{2 + 2 \cdot \beta}{\alpha} \]
      +-commutative [=>] 0.4	\[ 0.5 \cdot \frac{\color{blue}{2 \cdot \beta + 2}}{\alpha} \]
      fma-udef [<=] 0.4	\[ 0.5 \cdot \frac{\color{blue}{\mathsf{fma}\left(2, \beta, 2\right)}}{\alpha} \]
      metadata-eval [<=] 0.4	\[ \color{blue}{\frac{1}{2}} \cdot \frac{\mathsf{fma}\left(2, \beta, 2\right)}{\alpha} \]
      associate-/r/ [<=] 0.4	\[ \color{blue}{\frac{1}{\frac{2}{\frac{\mathsf{fma}\left(2, \beta, 2\right)}{\alpha}}}} \]
      *-lft-identity [<=] 0.4	\[ \frac{1}{\frac{2}{\color{blue}{1 \cdot \frac{\mathsf{fma}\left(2, \beta, 2\right)}{\alpha}}}} \]
      associate-*r/ [=>] 0.4	\[ \frac{1}{\frac{2}{\color{blue}{\frac{1 \cdot \mathsf{fma}\left(2, \beta, 2\right)}{\alpha}}}} \]
      associate-*l/ [<=] 0.5	\[ \frac{1}{\frac{2}{\color{blue}{\frac{1}{\alpha} \cdot \mathsf{fma}\left(2, \beta, 2\right)}}} \]
      fma-udef [=>] 0.5	\[ \frac{1}{\frac{2}{\frac{1}{\alpha} \cdot \color{blue}{\left(2 \cdot \beta + 2\right)}}} \]
      distribute-rgt-out [<=] 0.5	\[ \frac{1}{\frac{2}{\color{blue}{\left(2 \cdot \beta\right) \cdot \frac{1}{\alpha} + 2 \cdot \frac{1}{\alpha}}}} \]
      associate-*l* [=>] 0.5	\[ \frac{1}{\frac{2}{\color{blue}{2 \cdot \left(\beta \cdot \frac{1}{\alpha}\right)} + 2 \cdot \frac{1}{\alpha}}} \]
      distribute-lft-out [=>] 0.5	\[ \frac{1}{\frac{2}{\color{blue}{2 \cdot \left(\beta \cdot \frac{1}{\alpha} + \frac{1}{\alpha}\right)}}} \]
      distribute-lft1-in [=>] 0.5	\[ \frac{1}{\frac{2}{2 \cdot \color{blue}{\left(\left(\beta + 1\right) \cdot \frac{1}{\alpha}\right)}}} \]
      /-rgt-identity [<=] 0.5	\[ \frac{1}{\frac{2}{2 \cdot \left(\color{blue}{\frac{\beta + 1}{1}} \cdot \frac{1}{\alpha}\right)}} \]
      associate-*r/ [=>] 0.4	\[ \frac{1}{\frac{2}{2 \cdot \color{blue}{\frac{\frac{\beta + 1}{1} \cdot 1}{\alpha}}}} \]
      associate-*l/ [<=] 0.4	\[ \frac{1}{\frac{2}{2 \cdot \color{blue}{\left(\frac{\frac{\beta + 1}{1}}{\alpha} \cdot 1\right)}}} \]
      *-rgt-identity [=>] 0.4	\[ \frac{1}{\frac{2}{2 \cdot \color{blue}{\frac{\frac{\beta + 1}{1}}{\alpha}}}} \]
      associate-/r* [=>] 0.4	\[ \frac{1}{\color{blue}{\frac{\frac{2}{2}}{\frac{\frac{\beta + 1}{1}}{\alpha}}}} \]
      metadata-eval [=>] 0.4	\[ \frac{1}{\frac{\color{blue}{1}}{\frac{\frac{\beta + 1}{1}}{\alpha}}} \]
      associate-/l* [<=] 0.4	\[ \color{blue}{\frac{1 \cdot \frac{\frac{\beta + 1}{1}}{\alpha}}{1}} \]
      *-lft-identity [=>] 0.4	\[ \frac{\color{blue}{\frac{\frac{\beta + 1}{1}}{\alpha}}}{1} \]
      /-rgt-identity [=>] 0.4	\[ \color{blue}{\frac{\frac{\beta + 1}{1}}{\alpha}} \]
      /-rgt-identity [=>] 0.4	\[ \frac{\color{blue}{\beta + 1}}{\alpha} \]
      +-commutative [=>] 0.4	\[ \frac{\color{blue}{1 + \beta}}{\alpha} \]

   if -0.99999999500000003 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 0.1

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

   2. Simplified 0.1

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

      [Start] 0.1	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
      +-commutative [=>] 0.1	\[ \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Applied egg-rr 0.1

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

4. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.2
Cost	1476

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

Alternative 2
Error	6.9
Cost	836

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 4.6 \cdot 10^{+100}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \]

Alternative 3
Error	6.9
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 4.6 \cdot 10^{+100}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \]

Alternative 4
Error	18.6
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	19.7
Cost	324

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 520:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \]

Alternative 6
Error	18.9
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 14.6:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7
Error	32.5
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2022356 
(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))