Octave 3.8, jcobi/1

Average Error: 16.7 → 0.3

Time: 6.4s

Precision: binary64

Cost: 1476

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\\ \mathbf{if}\;t_0 \leq -0.99996:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \end{array} \]

FPCore

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

↓

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ alpha beta) 2.0))))
   (if (<= t_0 -0.99996) (/ (+ beta 1.0) 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) / ((alpha + beta) + 2.0);
	double tmp;
	if (t_0 <= -0.99996) {
		tmp = (beta + 1.0) / alpha;
	} else {
		tmp = (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) / ((alpha + beta) + 2.0d0)
    if (t_0 <= (-0.99996d0)) then
        tmp = (beta + 1.0d0) / alpha
    else
        tmp = (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) / ((alpha + beta) + 2.0);
	double tmp;
	if (t_0 <= -0.99996) {
		tmp = (beta + 1.0) / alpha;
	} else {
		tmp = (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) / ((alpha + beta) + 2.0)
	tmp = 0
	if t_0 <= -0.99996:
		tmp = (beta + 1.0) / alpha
	else:
		tmp = (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(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.99996)
		tmp = Float64(Float64(beta + 1.0) / alpha);
	else
		tmp = Float64(Float64(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) / ((alpha + beta) + 2.0);
	tmp = 0.0;
	if (t_0 <= -0.99996)
		tmp = (beta + 1.0) / alpha;
	else
		tmp = (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[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.99996], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(t$95$0 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.99995999999999996

   1. Initial program 59.3

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

   2. Simplified 59.3

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

      [Start] 59.3	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
      rational_best_45_simplify-20 [=>] 59.3	\[ \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Taylor expanded in alpha around inf 0.9

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

   4. Simplified 0.9

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

      [Start] 0.9	\[ \frac{\frac{2 + 2 \cdot \beta}{\alpha}}{2} \]
      rational_best_45_simplify-18 [=>] 0.9	\[ \frac{\frac{2 + \color{blue}{\beta \cdot 2}}{\alpha}}{2} \]
      metadata-eval [<=] 0.9	\[ \frac{\frac{2 + \beta \cdot \color{blue}{\left(1 + 1\right)}}{\alpha}}{2} \]
      rational_best_45_simplify-14 [<=] 0.9	\[ \frac{\frac{2 + \color{blue}{\left(1 \cdot \beta + \beta \cdot 1\right)}}{\alpha}}{2} \]
      rational_best_45_simplify-18 [<=] 0.9	\[ \frac{\frac{2 + \left(\color{blue}{\beta \cdot 1} + \beta \cdot 1\right)}{\alpha}}{2} \]
      rational_best_45_simplify-57 [=>] 0.9	\[ \frac{\frac{2 + \left(\color{blue}{\beta} + \beta \cdot 1\right)}{\alpha}}{2} \]
      rational_best_45_simplify-57 [=>] 0.9	\[ \frac{\frac{2 + \left(\beta + \color{blue}{\beta}\right)}{\alpha}}{2} \]

   5. Taylor expanded in beta around 0 0.9

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

   6. Taylor expanded in alpha around 0 0.9

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

   if -0.99995999999999996 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 0.1

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \leq -0.99996:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\\ \end{array} \]

Alternatives

Alternative 1
Error	22.1
Cost	1116

\[\begin{array}{l} \mathbf{if}\;\beta \leq -2.5 \cdot 10^{-201}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq -1.22 \cdot 10^{-222}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\beta \leq 32:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 6.2 \cdot 10^{+43}:\\ \;\;\;\;1\\ \mathbf{elif}\;\beta \leq 4.1 \cdot 10^{+113}:\\ \;\;\;\;\frac{\beta}{\alpha}\\ \mathbf{elif}\;\beta \leq 3.8 \cdot 10^{+191}:\\ \;\;\;\;1\\ \mathbf{elif}\;\beta \leq 1.3 \cdot 10^{+203}:\\ \;\;\;\;\frac{\beta}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 2
Error	16.3
Cost	844

\[\begin{array}{l} t_0 := \frac{1 - \alpha \cdot 0.5}{2}\\ \mathbf{if}\;\alpha \leq -1.5 \cdot 10^{-130}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq -1.1 \cdot 10^{-162}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 1.8:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \]

Alternative 3
Error	16.6
Cost	716

\[\begin{array}{l} \mathbf{if}\;\alpha \leq -1.5 \cdot 10^{-130}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq -2.1 \cdot 10^{-162}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 54:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \]

Alternative 4
Error	4.2
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 8200:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \]

Alternative 5
Error	20.2
Cost	588

\[\begin{array}{l} \mathbf{if}\;\alpha \leq -1.5 \cdot 10^{-130}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq -1.1 \cdot 10^{-162}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 53:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \]

Alternative 6
Error	19.9
Cost	324

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 53:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \]

Alternative 7
Error	32.7
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023102 
(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))