Octave 3.8, jcobi/1

Average Error: 16.2 → 0.2

Time: 19.8s

Precision: binary64

Cost: 1988

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \leq -0.9999995:\\ \;\;\;\;\frac{\frac{2 + 2 \cdot \beta}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - \beta}{\frac{\alpha + \left(2 + \beta\right)}{\alpha - \beta} \cdot \left(\beta - \alpha\right)} + 1}{2}\\ \end{array} \]

FPCore

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

↓

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ alpha beta) 2.0)) -0.9999995)
   (/ (/ (+ 2.0 (* 2.0 beta)) alpha) 2.0)
   (/
    (+
     (/
      (- alpha beta)
      (* (/ (+ alpha (+ 2.0 beta)) (- alpha beta)) (- beta alpha)))
     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 tmp;
	if (((beta - alpha) / ((alpha + beta) + 2.0)) <= -0.9999995) {
		tmp = ((2.0 + (2.0 * beta)) / alpha) / 2.0;
	} else {
		tmp = (((alpha - beta) / (((alpha + (2.0 + beta)) / (alpha - beta)) * (beta - alpha))) + 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) :: tmp
    if (((beta - alpha) / ((alpha + beta) + 2.0d0)) <= (-0.9999995d0)) then
        tmp = ((2.0d0 + (2.0d0 * beta)) / alpha) / 2.0d0
    else
        tmp = (((alpha - beta) / (((alpha + (2.0d0 + beta)) / (alpha - beta)) * (beta - alpha))) + 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 tmp;
	if (((beta - alpha) / ((alpha + beta) + 2.0)) <= -0.9999995) {
		tmp = ((2.0 + (2.0 * beta)) / alpha) / 2.0;
	} else {
		tmp = (((alpha - beta) / (((alpha + (2.0 + beta)) / (alpha - beta)) * (beta - alpha))) + 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):
	tmp = 0
	if ((beta - alpha) / ((alpha + beta) + 2.0)) <= -0.9999995:
		tmp = ((2.0 + (2.0 * beta)) / alpha) / 2.0
	else:
		tmp = (((alpha - beta) / (((alpha + (2.0 + beta)) / (alpha - beta)) * (beta - alpha))) + 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)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) <= -0.9999995)
		tmp = Float64(Float64(Float64(2.0 + Float64(2.0 * beta)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(alpha - beta) / Float64(Float64(Float64(alpha + Float64(2.0 + beta)) / Float64(alpha - beta)) * Float64(beta - alpha))) + 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)
	tmp = 0.0;
	if (((beta - alpha) / ((alpha + beta) + 2.0)) <= -0.9999995)
		tmp = ((2.0 + (2.0 * beta)) / alpha) / 2.0;
	else
		tmp = (((alpha - beta) / (((alpha + (2.0 + beta)) / (alpha - beta)) * (beta - alpha))) + 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_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.9999995], N[(N[(N[(2.0 + N[(2.0 * beta), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(alpha - beta), $MachinePrecision] / N[(N[(N[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(alpha - beta), $MachinePrecision]), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.999999500000000041

   1. Initial program 59.5

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

   2. Simplified 59.5

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

      [Start] 59.5	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
      rational.json-simplify-1 [=>] 59.5	\[ \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Taylor expanded in alpha around inf 0.7

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

   if -0.999999500000000041 < (/.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	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
      rational.json-simplify-1 [=>] 0.1	\[ \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Applied egg-rr 0.1

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

   4. Simplified 0.1

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

      [Start] 0.1	\[ \frac{\frac{1}{\beta - \alpha} \cdot \frac{\beta + \left(\alpha + 2\right)}{\left(\beta + \left(\alpha + 2\right)\right) \cdot \frac{\frac{\beta + \left(\alpha + 2\right)}{\alpha - \beta}}{\alpha - \beta}} + 1}{2} \]
      rational.json-simplify-2 [=>] 0.1	\[ \frac{\color{blue}{\frac{\beta + \left(\alpha + 2\right)}{\left(\beta + \left(\alpha + 2\right)\right) \cdot \frac{\frac{\beta + \left(\alpha + 2\right)}{\alpha - \beta}}{\alpha - \beta}} \cdot \frac{1}{\beta - \alpha}} + 1}{2} \]
      rational.json-simplify-46 [=>] 0.1	\[ \frac{\color{blue}{\frac{\frac{\beta + \left(\alpha + 2\right)}{\beta + \left(\alpha + 2\right)}}{\frac{\frac{\beta + \left(\alpha + 2\right)}{\alpha - \beta}}{\alpha - \beta}}} \cdot \frac{1}{\beta - \alpha} + 1}{2} \]
      rational.json-simplify-61 [=>] 0.1	\[ \frac{\color{blue}{\frac{\alpha - \beta}{\frac{\frac{\beta + \left(\alpha + 2\right)}{\alpha - \beta}}{\frac{\beta + \left(\alpha + 2\right)}{\beta + \left(\alpha + 2\right)}}}} \cdot \frac{1}{\beta - \alpha} + 1}{2} \]
      rational.json-simplify-50 [=>] 0.1	\[ \frac{\frac{\alpha - \beta}{\frac{\frac{\beta + \left(\alpha + 2\right)}{\alpha - \beta}}{\frac{\beta + \left(\alpha + 2\right)}{\beta + \left(\alpha + 2\right)}}} \cdot \color{blue}{\frac{-1}{\alpha - \beta}} + 1}{2} \]
      metadata-eval [=>] 0.1	\[ \frac{\frac{\alpha - \beta}{\frac{\frac{\beta + \left(\alpha + 2\right)}{\alpha - \beta}}{\frac{\beta + \left(\alpha + 2\right)}{\beta + \left(\alpha + 2\right)}}} \cdot \frac{\color{blue}{-1}}{\alpha - \beta} + 1}{2} \]
      rational.json-simplify-55 [=>] 0.1	\[ \frac{\color{blue}{\frac{\frac{-1}{\alpha - \beta}}{\frac{\frac{\frac{\beta + \left(\alpha + 2\right)}{\alpha - \beta}}{\frac{\beta + \left(\alpha + 2\right)}{\beta + \left(\alpha + 2\right)}}}{\alpha - \beta}}} + 1}{2} \]
      rational.json-simplify-60 [=>] 0.1	\[ \frac{\frac{\frac{-1}{\alpha - \beta}}{\frac{\color{blue}{\frac{\beta + \left(\alpha + 2\right)}{\alpha - \beta}}}{\alpha - \beta}} + 1}{2} \]
      metadata-eval [<=] 0.1	\[ \frac{\frac{\frac{\color{blue}{-1}}{\alpha - \beta}}{\frac{\frac{\beta + \left(\alpha + 2\right)}{\alpha - \beta}}{\alpha - \beta}} + 1}{2} \]
      rational.json-simplify-50 [<=] 0.1	\[ \frac{\frac{\color{blue}{\frac{1}{\beta - \alpha}}}{\frac{\frac{\beta + \left(\alpha + 2\right)}{\alpha - \beta}}{\alpha - \beta}} + 1}{2} \]
      rational.json-simplify-44 [=>] 0.1	\[ \frac{\color{blue}{\frac{\frac{1}{\frac{\frac{\beta + \left(\alpha + 2\right)}{\alpha - \beta}}{\alpha - \beta}}}{\beta - \alpha}} + 1}{2} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \leq -0.9999995:\\ \;\;\;\;\frac{\frac{2 + 2 \cdot \beta}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - \beta}{\frac{\alpha + \left(2 + \beta\right)}{\alpha - \beta} \cdot \left(\beta - \alpha\right)} + 1}{2}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	1476

\[\begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\\ \mathbf{if}\;t_0 \leq -0.9999995:\\ \;\;\;\;\frac{\frac{2 + 2 \cdot \beta}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \end{array} \]

Alternative 2
Error	4.0
Cost	900

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 2200000:\\ \;\;\;\;\frac{-\left(\frac{-1}{\frac{\beta + 2}{\beta}} + -1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + 2 \cdot \beta}{\alpha}}{2}\\ \end{array} \]

Alternative 3
Error	19.4
Cost	844

\[\begin{array}{l} \mathbf{if}\;\beta \leq 8.2 \cdot 10^{-198}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 2.3 \cdot 10^{-132}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;\frac{0.5 \cdot \beta + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4
Error	19.3
Cost	844

\[\begin{array}{l} \mathbf{if}\;\beta \leq 9.4 \cdot 10^{-201}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 2.35 \cdot 10^{-132}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;\frac{0.5 \cdot \beta + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - \frac{2}{\beta}}{2}\\ \end{array} \]

Alternative 5
Error	7.1
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1950000:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + 2}{\alpha}}{2}\\ \end{array} \]

Alternative 6
Error	4.0
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 2200000:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + 2 \cdot \beta}{\alpha}}{2}\\ \end{array} \]

Alternative 7
Error	19.6
Cost	460

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.5 \cdot 10^{-202}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 2.3 \cdot 10^{-132}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8
Error	18.3
Cost	196

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

Alternative 9
Error	40.1
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023073 
(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))