Octave 3.8, jcobi/1

Average Error: 16.3 → 0.0

Time: 5.4s

Precision: binary64

Cost: 832

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

Math

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

↓

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

FPCore

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

↓

(FPCore (alpha beta)
 :precision binary64
 (/ (/ (+ 2.0 (+ beta beta)) (+ alpha (+ 2.0 beta))) 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 ((2.0 + (beta + beta)) / (alpha + (2.0 + beta))) / 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 = ((2.0d0 + (beta + beta)) / (alpha + (2.0d0 + beta))) / 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 ((2.0 + (beta + beta)) / (alpha + (2.0 + beta))) / 2.0;
}

Python

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

↓

def code(alpha, beta):
	return ((2.0 + (beta + beta)) / (alpha + (2.0 + beta))) / 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(2.0 + Float64(beta + beta)) / Float64(alpha + Float64(2.0 + beta))) / 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 = ((2.0 + (beta + beta)) / (alpha + (2.0 + beta))) / 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[(2.0 + N[(beta + beta), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 16.3

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

2. Simplified 16.3

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

   [Start] 16.3	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   rational.json-simplify-1 [=>] 16.3	\[ \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

3. Applied egg-rr 0.0

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

4. Simplified 0.0

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

   [Start] 0.0	\[ \frac{\frac{2 \cdot \left(\beta + 1\right)}{\left(\beta + \alpha\right) + 2}}{2} \]
   rational.json-simplify-1 [=>] 0.0	\[ \frac{\frac{2 \cdot \color{blue}{\left(1 + \beta\right)}}{\left(\beta + \alpha\right) + 2}}{2} \]
   rational.json-simplify-19 [<=] 0.0	\[ \frac{\frac{\color{blue}{2 + 2 \cdot \beta}}{\left(\beta + \alpha\right) + 2}}{2} \]
   rational.json-simplify-2 [=>] 0.0	\[ \frac{\frac{2 + \color{blue}{\beta \cdot 2}}{\left(\beta + \alpha\right) + 2}}{2} \]
   metadata-eval [<=] 0.0	\[ \frac{\frac{2 + \beta \cdot \color{blue}{\left(1 - -1\right)}}{\left(\beta + \alpha\right) + 2}}{2} \]
   rational.json-simplify-22 [<=] 0.0	\[ \frac{\frac{2 + \color{blue}{\left(\beta - \beta \cdot -1\right)}}{\left(\beta + \alpha\right) + 2}}{2} \]
   rational.json-simplify-2 [<=] 0.0	\[ \frac{\frac{2 + \left(\beta - \color{blue}{-1 \cdot \beta}\right)}{\left(\beta + \alpha\right) + 2}}{2} \]
   rational.json-simplify-35 [<=] 0.0	\[ \frac{\frac{\color{blue}{2 - \left(-1 \cdot \beta - \beta\right)}}{\left(\beta + \alpha\right) + 2}}{2} \]
   rational.json-simplify-29 [<=] 0.0	\[ \frac{\frac{\color{blue}{\beta - \left(-1 \cdot \beta - 2\right)}}{\left(\beta + \alpha\right) + 2}}{2} \]
   rational.json-simplify-2 [=>] 0.0	\[ \frac{\frac{\beta - \left(\color{blue}{\beta \cdot -1} - 2\right)}{\left(\beta + \alpha\right) + 2}}{2} \]
   rational.json-simplify-9 [<=] 0.0	\[ \frac{\frac{\beta - \left(\color{blue}{\left(-\beta\right)} - 2\right)}{\left(\beta + \alpha\right) + 2}}{2} \]
   rational.json-simplify-27 [<=] 0.0	\[ \frac{\frac{\beta - \color{blue}{\left(-\left(\beta + 2\right)\right)}}{\left(\beta + \alpha\right) + 2}}{2} \]
   rational.json-simplify-1 [=>] 0.0	\[ \frac{\frac{\beta - \left(-\color{blue}{\left(2 + \beta\right)}\right)}{\left(\beta + \alpha\right) + 2}}{2} \]
   rational.json-simplify-27 [=>] 0.0	\[ \frac{\frac{\beta - \color{blue}{\left(\left(-2\right) - \beta\right)}}{\left(\beta + \alpha\right) + 2}}{2} \]
   metadata-eval [=>] 0.0	\[ \frac{\frac{\beta - \left(\color{blue}{-2} - \beta\right)}{\left(\beta + \alpha\right) + 2}}{2} \]
   metadata-eval [<=] 0.0	\[ \frac{\frac{\beta - \left(\color{blue}{\left(-1 + -1\right)} - \beta\right)}{\left(\beta + \alpha\right) + 2}}{2} \]
   rational.json-simplify-34 [=>] 0.0	\[ \frac{\frac{\color{blue}{\left(\beta + \beta\right) - \left(-1 + -1\right)}}{\left(\beta + \alpha\right) + 2}}{2} \]
   rational.json-simplify-41 [=>] 0.0	\[ \frac{\frac{\color{blue}{\left(-\left(-1 + -1\right)\right) + \left(\beta + \beta\right)}}{\left(\beta + \alpha\right) + 2}}{2} \]
   metadata-eval [=>] 0.0	\[ \frac{\frac{\left(-\color{blue}{-2}\right) + \left(\beta + \beta\right)}{\left(\beta + \alpha\right) + 2}}{2} \]
   metadata-eval [=>] 0.0	\[ \frac{\frac{\color{blue}{2} + \left(\beta + \beta\right)}{\left(\beta + \alpha\right) + 2}}{2} \]
   rational.json-simplify-1 [=>] 0.0	\[ \frac{\frac{2 + \left(\beta + \beta\right)}{\color{blue}{2 + \left(\beta + \alpha\right)}}}{2} \]
   rational.json-simplify-33 [=>] 0.0	\[ \frac{\frac{2 + \left(\beta + \beta\right)}{\color{blue}{\left(2 + \beta\right) + \alpha}}}{2} \]
   rational.json-simplify-1 [<=] 0.0	\[ \frac{\frac{2 + \left(\beta + \beta\right)}{\color{blue}{\left(\beta + 2\right)} + \alpha}}{2} \]
   rational.json-simplify-1 [=>] 0.0	\[ \frac{\frac{2 + \left(\beta + \beta\right)}{\color{blue}{\alpha + \left(\beta + 2\right)}}}{2} \]
   rational.json-simplify-1 [=>] 0.0	\[ \frac{\frac{2 + \left(\beta + \beta\right)}{\alpha + \color{blue}{\left(2 + \beta\right)}}}{2} \]

5. Final simplification 0.0

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

Alternatives

Alternative 1
Error	1.3
Cost	836

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.7:\\ \;\;\;\;\frac{1}{2 + \alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \beta}{\left(\alpha + \left(\beta + 2\right)\right) \cdot 2}\\ \end{array} \]

Alternative 2
Error	1.1
Cost	836

\[\begin{array}{l} \mathbf{if}\;\beta \leq 9.5:\\ \;\;\;\;\frac{\beta + \left(\beta + 2\right)}{\left(2 + \alpha\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \beta}{\left(\alpha + \left(\beta + 2\right)\right) \cdot 2}\\ \end{array} \]

Alternative 3
Error	1.0
Cost	836

\[\begin{array}{l} \mathbf{if}\;\beta \leq 9.5:\\ \;\;\;\;\frac{\frac{2 + \left(\beta + \beta\right)}{2 + \alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \beta}{\left(\alpha + \left(\beta + 2\right)\right) \cdot 2}\\ \end{array} \]

Alternative 4
Error	0.1
Cost	832

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

Alternative 5
Error	4.1
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 5.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \alpha} + \frac{\beta}{\alpha}\\ \end{array} \]

Alternative 6
Error	18.2
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;0.5 + 0.25 \cdot \beta\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7
Error	4.6
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 8.8:\\ \;\;\;\;\frac{1}{2 + \alpha}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8
Error	18.4
Cost	196

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

Alternative 9
Error	32.7
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023053 
(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))