Octave 3.8, jcobi/1

Average Error: 16.1 → 0.0

Time: 14.3s

Precision: binary64

Cost: 1216

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

Math

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

↓

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

FPCore

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

↓

(FPCore (alpha beta)
 :precision binary64
 (/
  (- (/ beta (+ beta (+ alpha 2.0))) (/ 1.0 (- -1.0 (/ alpha (+ beta 2.0)))))
  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) {
	return ((beta / (beta + (alpha + 2.0))) - (1.0 / (-1.0 - (alpha / (beta + 2.0))))) / 2.0;
}

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
    code = ((beta / (beta + (alpha + 2.0d0))) - (1.0d0 / ((-1.0d0) - (alpha / (beta + 2.0d0))))) / 2.0d0
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) {
	return ((beta / (beta + (alpha + 2.0))) - (1.0 / (-1.0 - (alpha / (beta + 2.0))))) / 2.0;
}

Python

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

↓

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

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

MATLAB

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

↓

function tmp = code(alpha, beta)
	tmp = ((beta / (beta + (alpha + 2.0))) - (1.0 / (-1.0 - (alpha / (beta + 2.0))))) / 2.0;
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_] := N[(N[(N[(beta / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[(-1.0 - N[(alpha / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 16.1

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

2. Simplified 16.1

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

   [Start] 16.1	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   +-commutative [=>] 16.1	\[ \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

3. Applied egg-rr 15.6

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

4. Applied egg-rr 15.6

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

5. Taylor expanded in alpha around 0 0.0

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

6. Simplified 0.0

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

   [Start] 0.0	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{1}{-1 \cdot \frac{\alpha}{\beta + 2} - 1}}{2} \]
   sub-neg [=>] 0.0	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{1}{\color{blue}{-1 \cdot \frac{\alpha}{\beta + 2} + \left(-1\right)}}}{2} \]
   metadata-eval [=>] 0.0	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{1}{-1 \cdot \frac{\alpha}{\beta + 2} + \color{blue}{-1}}}{2} \]
   +-commutative [=>] 0.0	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{1}{\color{blue}{-1 + -1 \cdot \frac{\alpha}{\beta + 2}}}}{2} \]
   mul-1-neg [=>] 0.0	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{1}{-1 + \color{blue}{\left(-\frac{\alpha}{\beta + 2}\right)}}}{2} \]
   neg-sub0 [=>] 0.0	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{1}{-1 + \color{blue}{\left(0 - \frac{\alpha}{\beta + 2}\right)}}}{2} \]
   metadata-eval [<=] 0.0	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{1}{-1 + \left(\color{blue}{\left(-1 + 1\right)} - \frac{\alpha}{\beta + 2}\right)}}{2} \]
   associate-+r- [=>] 0.0	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{1}{\color{blue}{\left(-1 + \left(-1 + 1\right)\right) - \frac{\alpha}{\beta + 2}}}}{2} \]
   metadata-eval [=>] 0.0	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{1}{\left(-1 + \color{blue}{0}\right) - \frac{\alpha}{\beta + 2}}}{2} \]
   metadata-eval [=>] 0.0	\[ \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{1}{\color{blue}{-1} - \frac{\alpha}{\beta + 2}}}{2} \]

7. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.1
Cost	1860

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)} \leq -0.8:\\ \;\;\;\;\frac{\frac{\beta}{\alpha} - \frac{1}{-1 - \frac{\alpha}{\beta + 2}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{t_0} - \left(-1 + \frac{\alpha}{t_0}\right)}{2}\\ \end{array} \]

Alternative 2
Error	0.1
Cost	1604

\[\begin{array}{l} t_0 := \frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)}\\ \mathbf{if}\;t_0 \leq -0.8:\\ \;\;\;\;\frac{\frac{\beta}{\alpha} - \frac{1}{-1 - \frac{\alpha}{\beta + 2}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + t_0}{2}\\ \end{array} \]

Alternative 3
Error	0.2
Cost	1476

\[\begin{array}{l} t_0 := \frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)}\\ \mathbf{if}\;t_0 \leq -0.999995:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + t_0}{2}\\ \end{array} \]

Alternative 4
Error	8.4
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 175000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]

Alternative 5
Error	4.6
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 370000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 6
Error	18.1
Cost	580

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

Alternative 7
Error	18.4
Cost	196

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

Alternative 8
Error	32.3
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023066 
(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))