Octave 3.8, jcobi/1

Average Error: 16.5 → 0.1

Time: 6.2s

Precision: binary64

Cost: 1732

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t_0 \leq -0.9998:\\ \;\;\;\;\frac{\frac{-2 + \beta \cdot -2}{\alpha} \cdot \left(-1 + \frac{\beta + 2}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \end{array} \]

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) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.9998)
     (/
      (* (/ (+ -2.0 (* beta -2.0)) alpha) (+ -1.0 (/ (+ beta 2.0) alpha)))
      2.0)
     (/ (+ 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) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.9998) {
		tmp = (((-2.0 + (beta * -2.0)) / alpha) * (-1.0 + ((beta + 2.0) / alpha))) / 2.0;
	} 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) / ((beta + alpha) + 2.0d0)
    if (t_0 <= (-0.9998d0)) then
        tmp = ((((-2.0d0) + (beta * (-2.0d0))) / alpha) * ((-1.0d0) + ((beta + 2.0d0) / alpha))) / 2.0d0
    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) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.9998) {
		tmp = (((-2.0 + (beta * -2.0)) / alpha) * (-1.0 + ((beta + 2.0) / alpha))) / 2.0;
	} 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) / ((beta + alpha) + 2.0)
	tmp = 0
	if t_0 <= -0.9998:
		tmp = (((-2.0 + (beta * -2.0)) / alpha) * (-1.0 + ((beta + 2.0) / alpha))) / 2.0
	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(beta + alpha) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.9998)
		tmp = Float64(Float64(Float64(Float64(-2.0 + Float64(beta * -2.0)) / alpha) * Float64(-1.0 + Float64(Float64(beta + 2.0) / alpha))) / 2.0);
	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) / ((beta + alpha) + 2.0);
	tmp = 0.0;
	if (t_0 <= -0.9998)
		tmp = (((-2.0 + (beta * -2.0)) / alpha) * (-1.0 + ((beta + 2.0) / alpha))) / 2.0;
	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[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.9998], N[(N[(N[(N[(-2.0 + N[(beta * -2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] * N[(-1.0 + N[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $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)) < -0.99980000000000002

   1. Initial program 59.3

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

   2. Taylor expanded in alpha around -inf 3.6

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

   3. Simplified 0.2

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

   if -0.99980000000000002 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 0.0

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9998:\\ \;\;\;\;\frac{\frac{-2 + \beta \cdot -2}{\alpha} \cdot \left(-1 + \frac{\beta + 2}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	1476

\[\begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t_0 \leq -0.9998:\\ \;\;\;\;\frac{\frac{2 + \left(\beta + \beta\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \end{array} \]

Alternative 2
Error	18.1
Cost	972

\[\begin{array}{l} t_0 := \frac{1 - \frac{\alpha}{\alpha + 2}}{2}\\ \mathbf{if}\;\beta \leq -2.2943922739913888 \cdot 10^{-185}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq -4.356164407919565 \cdot 10^{-206}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\beta \leq 279.8777461370767:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3
Error	15.9
Cost	972

\[\begin{array}{l} t_0 := \frac{1 - \frac{\alpha}{\alpha + 2}}{2}\\ \mathbf{if}\;\alpha \leq -1.1911697505617745 \cdot 10^{-112}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq -1.228394930163902 \cdot 10^{-139}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 5733.610063552057:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta + \beta\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 4
Error	18.9
Cost	844

\[\begin{array}{l} t_0 := \frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{if}\;\beta \leq -4.563535139700859 \cdot 10^{-185}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq -4.356164407919565 \cdot 10^{-206}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\beta \leq 4.000999854152087 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5
Error	4.4
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.2943257189652617 \cdot 10^{+29}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta + \beta\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 6
Error	19.1
Cost	460

\[\begin{array}{l} \mathbf{if}\;\beta \leq -4.563535139700859 \cdot 10^{-185}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq -4.356164407919565 \cdot 10^{-206}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\beta \leq 279.8777461370767:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7
Error	19.8
Cost	324

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 5.930092259289494 \cdot 10^{-6}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \]

Alternative 8
Error	32.3
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2022217 
(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))