Octave 3.8, jcobi/1

Average Error: 16.2 → 0.0

Time: 13.6s

Precision: binary64

Cost: 832

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

Math

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

↓

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

FPCore

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

↓

(FPCore (alpha beta)
 :precision binary64
 (/ (/ (- (+ beta beta) -2.0) (+ beta (+ 2.0 alpha))) 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) - -2.0) / (beta + (2.0 + alpha))) / 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) - (-2.0d0)) / (beta + (2.0d0 + alpha))) / 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) - -2.0) / (beta + (2.0 + alpha))) / 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) - -2.0) / (beta + (2.0 + alpha))) / 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(Float64(beta + beta) - -2.0) / Float64(beta + Float64(2.0 + alpha))) / 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) - -2.0) / (beta + (2.0 + alpha))) / 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[(N[(beta + beta), $MachinePrecision] - -2.0), $MachinePrecision] / N[(beta + N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 16.2

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

2. Applied egg-rr 15.9

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

3. Simplified 3.7

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

4. Applied egg-rr 0.0

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

5. Simplified 0.0

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

Alternatives

Alternative 1
Error	0.4
Cost	1476

\[\begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\\ \mathbf{if}\;t_0 \leq -0.995:\\ \;\;\;\;\frac{\frac{2 + 2 \cdot \beta}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \end{array} \]

Alternative 2
Error	21.1
Cost	1108

\[\begin{array}{l} t_0 := \frac{1 + -0.5 \cdot \alpha}{2}\\ \mathbf{if}\;\alpha \leq 4 \cdot 10^{-309}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 1.45 \cdot 10^{-273}:\\ \;\;\;\;\frac{2}{2}\\ \mathbf{elif}\;\alpha \leq 1.15 \cdot 10^{-171}:\\ \;\;\;\;\frac{1}{2}\\ \mathbf{elif}\;\alpha \leq 3.9 \cdot 10^{-122}:\\ \;\;\;\;\frac{2}{2}\\ \mathbf{elif}\;\alpha \leq 1.15:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]

Alternative 3
Error	21.4
Cost	980

\[\begin{array}{l} \mathbf{if}\;\alpha \leq -2 \cdot 10^{-308}:\\ \;\;\;\;\frac{1}{2}\\ \mathbf{elif}\;\alpha \leq 7.2 \cdot 10^{-274}:\\ \;\;\;\;\frac{2}{2}\\ \mathbf{elif}\;\alpha \leq 1.15 \cdot 10^{-171}:\\ \;\;\;\;\frac{1}{2}\\ \mathbf{elif}\;\alpha \leq 3.9 \cdot 10^{-122}:\\ \;\;\;\;\frac{2}{2}\\ \mathbf{elif}\;\alpha \leq 2:\\ \;\;\;\;\frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]

Alternative 4
Error	4.6
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 8.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{2}{2 + \alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \end{array} \]

Alternative 5
Error	4.7
Cost	644

\[\begin{array}{l} \mathbf{if}\;\beta \leq 10.8:\\ \;\;\;\;\frac{\frac{2}{2 + \alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-\frac{2}{\beta}\right) + 2}{2}\\ \end{array} \]

Alternative 6
Error	4.8
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 29.5:\\ \;\;\;\;\frac{\frac{2}{2 + \alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2}\\ \end{array} \]

Alternative 7
Error	18.2
Cost	324

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.1:\\ \;\;\;\;\frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2}\\ \end{array} \]

Alternative 8
Error	31.7
Cost	192

\[\frac{1}{2} \]

Reproduce

herbie shell --seed 2023010 
(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))