Octave 3.8, jcobi/2

Percentage Accurate: 63.4% → 97.8%

Time: 24.1s

Alternatives: 12

Speedup: 4.8×

Specification

\[\left(\alpha > -1 \land \beta > -1\right) \land i > 0\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2} + 1}{2} \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ t_0 2.0)) 1.0) 2.0)))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	return (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    t_0 = (alpha + beta) + (2.0d0 * i)
    code = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0d0)) + 1.0d0) / 2.0d0
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	return (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	return (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	return Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(t_0 + 2.0)) + 1.0) / 2.0)
end

MATLAB

function tmp = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	tmp = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]

Initial Program: 63.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2} + 1}{2} \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ t_0 2.0)) 1.0) 2.0)))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	return (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    t_0 = (alpha + beta) + (2.0d0 * i)
    code = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0d0)) + 1.0d0) / 2.0d0
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	return (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	return (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	return Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(t_0 + 2.0)) + 1.0) / 2.0)
end

MATLAB

function tmp = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	tmp = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]

Alternative 1: 97.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq -0.999999995:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\alpha + \beta}{2 \cdot \frac{i}{\beta - \alpha} + \left(\frac{\alpha}{\beta - \alpha} + \frac{\beta}{\beta - \alpha}\right)}}{\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)} + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (if (<=
        (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0))
        -0.999999995)
     (/ (/ (+ 2.0 (+ (* beta 2.0) (* i 4.0))) alpha) 2.0)
     (/
      (+
       (/
        (/
         (+ alpha beta)
         (+
          (* 2.0 (/ i (- beta alpha)))
          (+ (/ alpha (- beta alpha)) (/ beta (- beta alpha)))))
        (+ (+ alpha beta) (+ 2.0 (* 2.0 i))))
       1.0)
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.999999995) {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	} else {
		tmp = ((((alpha + beta) / ((2.0 * (i / (beta - alpha))) + ((alpha / (beta - alpha)) + (beta / (beta - alpha))))) / ((alpha + beta) + (2.0 + (2.0 * i)))) + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0d0 + t_0)) <= (-0.999999995d0)) then
        tmp = ((2.0d0 + ((beta * 2.0d0) + (i * 4.0d0))) / alpha) / 2.0d0
    else
        tmp = ((((alpha + beta) / ((2.0d0 * (i / (beta - alpha))) + ((alpha / (beta - alpha)) + (beta / (beta - alpha))))) / ((alpha + beta) + (2.0d0 + (2.0d0 * i)))) + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.999999995) {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	} else {
		tmp = ((((alpha + beta) / ((2.0 * (i / (beta - alpha))) + ((alpha / (beta - alpha)) + (beta / (beta - alpha))))) / ((alpha + beta) + (2.0 + (2.0 * i)))) + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	tmp = 0
	if ((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.999999995:
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0
	else:
		tmp = ((((alpha + beta) / ((2.0 * (i / (beta - alpha))) + ((alpha / (beta - alpha)) + (beta / (beta - alpha))))) / ((alpha + beta) + (2.0 + (2.0 * i)))) + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0)) <= -0.999999995)
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(beta * 2.0) + Float64(i * 4.0))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(alpha + beta) / Float64(Float64(2.0 * Float64(i / Float64(beta - alpha))) + Float64(Float64(alpha / Float64(beta - alpha)) + Float64(beta / Float64(beta - alpha))))) / Float64(Float64(alpha + beta) + Float64(2.0 + Float64(2.0 * i)))) + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	tmp = 0.0;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.999999995)
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	else
		tmp = ((((alpha + beta) / ((2.0 * (i / (beta - alpha))) + ((alpha / (beta - alpha)) + (beta / (beta - alpha))))) / ((alpha + beta) + (2.0 + (2.0 * i)))) + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision], -0.999999995], N[(N[(N[(2.0 + N[(N[(beta * 2.0), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] / N[(N[(2.0 * N[(i / N[(beta - alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(alpha / N[(beta - alpha), $MachinePrecision]), $MachinePrecision] + N[(beta / N[(beta - alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2)) < -0.99999999500000003

   1. Initial program 2.5%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

   3. Simplified 21.4%

      \[\leadsto \color{blue}{\frac{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in beta around 0 21.4%

      \[\leadsto \frac{\color{blue}{\frac{\alpha}{2 + \left(\alpha + 2 \cdot i\right)}} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1}{2} \]
   6. Step-by-step derivation

      1. associate-+r+ 21.4%

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

   7. Simplified 21.4%

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

   8. Taylor expanded in alpha around inf 84.3%

      \[\leadsto \frac{\color{blue}{\frac{2 + \left(2 \cdot \beta + 4 \cdot i\right)}{\alpha}}}{2} \]

   if -0.99999999500000003 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2))

   1. Initial program 77.0%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. associate-/l* 99.6%

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

      2. associate-+l+ 99.6%

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

      3. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in i around 0 99.6%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.999999995:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\alpha + \beta}{2 \cdot \frac{i}{\beta - \alpha} + \left(\frac{\alpha}{\beta - \alpha} + \frac{\beta}{\beta - \alpha}\right)}}{\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq -0.999999995:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\frac{\alpha + \beta}{\frac{\alpha + \left(\beta + 2 \cdot i\right)}{\beta - \alpha}}}{\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (if (<=
        (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0))
        -0.999999995)
     (/ (/ (+ 2.0 (+ (* beta 2.0) (* i 4.0))) alpha) 2.0)
     (/
      (+
       1.0
       (/
        (/ (+ alpha beta) (/ (+ alpha (+ beta (* 2.0 i))) (- beta alpha)))
        (+ (+ alpha beta) (+ 2.0 (* 2.0 i)))))
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.999999995) {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (((alpha + beta) / ((alpha + (beta + (2.0 * i))) / (beta - alpha))) / ((alpha + beta) + (2.0 + (2.0 * i))))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0d0 + t_0)) <= (-0.999999995d0)) then
        tmp = ((2.0d0 + ((beta * 2.0d0) + (i * 4.0d0))) / alpha) / 2.0d0
    else
        tmp = (1.0d0 + (((alpha + beta) / ((alpha + (beta + (2.0d0 * i))) / (beta - alpha))) / ((alpha + beta) + (2.0d0 + (2.0d0 * i))))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.999999995) {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (((alpha + beta) / ((alpha + (beta + (2.0 * i))) / (beta - alpha))) / ((alpha + beta) + (2.0 + (2.0 * i))))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	tmp = 0
	if ((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.999999995:
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0
	else:
		tmp = (1.0 + (((alpha + beta) / ((alpha + (beta + (2.0 * i))) / (beta - alpha))) / ((alpha + beta) + (2.0 + (2.0 * i))))) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0)) <= -0.999999995)
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(beta * 2.0) + Float64(i * 4.0))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(alpha + beta) / Float64(Float64(alpha + Float64(beta + Float64(2.0 * i))) / Float64(beta - alpha))) / Float64(Float64(alpha + beta) + Float64(2.0 + Float64(2.0 * i))))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	tmp = 0.0;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.999999995)
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	else
		tmp = (1.0 + (((alpha + beta) / ((alpha + (beta + (2.0 * i))) / (beta - alpha))) / ((alpha + beta) + (2.0 + (2.0 * i))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision], -0.999999995], N[(N[(N[(2.0 + N[(N[(beta * 2.0), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(N[(N[(alpha + beta), $MachinePrecision] / N[(N[(alpha + N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta - alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2)) < -0.99999999500000003

   1. Initial program 2.5%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

   3. Simplified 21.4%

      \[\leadsto \color{blue}{\frac{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in beta around 0 21.4%

      \[\leadsto \frac{\color{blue}{\frac{\alpha}{2 + \left(\alpha + 2 \cdot i\right)}} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1}{2} \]
   6. Step-by-step derivation

      1. associate-+r+ 21.4%

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

   7. Simplified 21.4%

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

   8. Taylor expanded in alpha around inf 84.3%

      \[\leadsto \frac{\color{blue}{\frac{2 + \left(2 \cdot \beta + 4 \cdot i\right)}{\alpha}}}{2} \]

   if -0.99999999500000003 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2))

   1. Initial program 77.0%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. associate-/l* 99.6%

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

      2. associate-+l+ 99.6%

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

      3. associate-+l+ 99.6%

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

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\alpha + \beta}{\frac{\alpha + \left(\beta + 2 \cdot i\right)}{\beta - \alpha}}}{\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)} + 1}{2}} \]
   4. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.999999995:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\frac{\alpha + \beta}{\frac{\alpha + \left(\beta + 2 \cdot i\right)}{\beta - \alpha}}}{\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq -0.999999995:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\alpha + \left(\beta - \left(\alpha + \alpha\right)\right)}{\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (if (<=
        (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0))
        -0.999999995)
     (/ (/ (+ 2.0 (+ (* beta 2.0) (* i 4.0))) alpha) 2.0)
     (/
      (+
       1.0
       (/
        (+ alpha (- beta (+ alpha alpha)))
        (+ (+ alpha beta) (+ 2.0 (* 2.0 i)))))
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.999999995) {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	} else {
		tmp = (1.0 + ((alpha + (beta - (alpha + alpha))) / ((alpha + beta) + (2.0 + (2.0 * i))))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0d0 + t_0)) <= (-0.999999995d0)) then
        tmp = ((2.0d0 + ((beta * 2.0d0) + (i * 4.0d0))) / alpha) / 2.0d0
    else
        tmp = (1.0d0 + ((alpha + (beta - (alpha + alpha))) / ((alpha + beta) + (2.0d0 + (2.0d0 * i))))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.999999995) {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	} else {
		tmp = (1.0 + ((alpha + (beta - (alpha + alpha))) / ((alpha + beta) + (2.0 + (2.0 * i))))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	tmp = 0
	if ((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.999999995:
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0
	else:
		tmp = (1.0 + ((alpha + (beta - (alpha + alpha))) / ((alpha + beta) + (2.0 + (2.0 * i))))) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0)) <= -0.999999995)
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(beta * 2.0) + Float64(i * 4.0))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(alpha + Float64(beta - Float64(alpha + alpha))) / Float64(Float64(alpha + beta) + Float64(2.0 + Float64(2.0 * i))))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	tmp = 0.0;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.999999995)
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	else
		tmp = (1.0 + ((alpha + (beta - (alpha + alpha))) / ((alpha + beta) + (2.0 + (2.0 * i))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision], -0.999999995], N[(N[(N[(2.0 + N[(N[(beta * 2.0), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(N[(alpha + N[(beta - N[(alpha + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2)) < -0.99999999500000003

   1. Initial program 2.5%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

   3. Simplified 21.4%

      \[\leadsto \color{blue}{\frac{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in beta around 0 21.4%

      \[\leadsto \frac{\color{blue}{\frac{\alpha}{2 + \left(\alpha + 2 \cdot i\right)}} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1}{2} \]
   6. Step-by-step derivation

      1. associate-+r+ 21.4%

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

   7. Simplified 21.4%

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

   8. Taylor expanded in alpha around inf 84.3%

      \[\leadsto \frac{\color{blue}{\frac{2 + \left(2 \cdot \beta + 4 \cdot i\right)}{\alpha}}}{2} \]

   if -0.99999999500000003 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2))

   1. Initial program 77.0%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. associate-/l* 99.6%

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

      2. associate-+l+ 99.6%

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

      3. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in i around 0 99.6%

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

   6. Taylor expanded in i around 0 98.9%

      \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha + \beta}{\frac{\alpha}{\beta - \alpha} + \frac{\beta}{\beta - \alpha}}}}{\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)} + 1}{2} \]
   7. Step-by-step derivation

      1. +-commutative 98.9%

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

   8. Simplified 98.9%

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

   9. Taylor expanded in beta around -inf 98.9%

      \[\leadsto \frac{\frac{\color{blue}{\left(\alpha + \beta\right) - -1 \cdot \left(-1 \cdot \alpha - \alpha\right)}}{\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)} + 1}{2} \]
   10. Step-by-step derivation

       1. associate--l+ 98.9%

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

       2. mul-1-neg 98.9%

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

       3. mul-1-neg 98.9%

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

   11. Simplified 98.9%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.999999995:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\alpha + \left(\beta - \left(\alpha + \alpha\right)\right)}{\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 5.4 \cdot 10^{+82}:\\ \;\;\;\;\frac{1 + \frac{\beta}{i \cdot 4 + \left(\beta + 2\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 2.3 \cdot 10^{+155} \lor \neg \left(\alpha \leq 10^{+176}\right):\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\alpha}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 5.4e+82)
   (/ (+ 1.0 (/ beta (+ (* i 4.0) (+ beta 2.0)))) 2.0)
   (if (or (<= alpha 2.3e+155) (not (<= alpha 1e+176)))
     (/ (/ (+ 2.0 (+ (* beta 2.0) (* i 4.0))) alpha) 2.0)
     (/ (- 1.0 (/ alpha (+ 2.0 (+ (+ alpha beta) (* 2.0 i))))) 2.0))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 5.4e+82) {
		tmp = (1.0 + (beta / ((i * 4.0) + (beta + 2.0)))) / 2.0;
	} else if ((alpha <= 2.3e+155) || !(alpha <= 1e+176)) {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	} else {
		tmp = (1.0 - (alpha / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 5.4d+82) then
        tmp = (1.0d0 + (beta / ((i * 4.0d0) + (beta + 2.0d0)))) / 2.0d0
    else if ((alpha <= 2.3d+155) .or. (.not. (alpha <= 1d+176))) then
        tmp = ((2.0d0 + ((beta * 2.0d0) + (i * 4.0d0))) / alpha) / 2.0d0
    else
        tmp = (1.0d0 - (alpha / (2.0d0 + ((alpha + beta) + (2.0d0 * i))))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 5.4e+82) {
		tmp = (1.0 + (beta / ((i * 4.0) + (beta + 2.0)))) / 2.0;
	} else if ((alpha <= 2.3e+155) || !(alpha <= 1e+176)) {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	} else {
		tmp = (1.0 - (alpha / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 5.4e+82:
		tmp = (1.0 + (beta / ((i * 4.0) + (beta + 2.0)))) / 2.0
	elif (alpha <= 2.3e+155) or not (alpha <= 1e+176):
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0
	else:
		tmp = (1.0 - (alpha / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 5.4e+82)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(Float64(i * 4.0) + Float64(beta + 2.0)))) / 2.0);
	elseif ((alpha <= 2.3e+155) || !(alpha <= 1e+176))
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(beta * 2.0) + Float64(i * 4.0))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(1.0 - Float64(alpha / Float64(2.0 + Float64(Float64(alpha + beta) + Float64(2.0 * i))))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 5.4e+82)
		tmp = (1.0 + (beta / ((i * 4.0) + (beta + 2.0)))) / 2.0;
	elseif ((alpha <= 2.3e+155) || ~((alpha <= 1e+176)))
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	else
		tmp = (1.0 - (alpha / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 5.4e+82], N[(N[(1.0 + N[(beta / N[(N[(i * 4.0), $MachinePrecision] + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[alpha, 2.3e+155], N[Not[LessEqual[alpha, 1e+176]], $MachinePrecision]], N[(N[(N[(2.0 + N[(N[(beta * 2.0), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 - N[(alpha / N[(2.0 + N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 5.3999999999999999e82

   1. Initial program 75.4%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. associate-/l* 94.7%

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

      2. associate-+l+ 94.7%

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

      3. associate-+l+ 94.7%

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

   3. Simplified 94.7%

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

   5. Taylor expanded in i around 0 94.7%

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

   6. Taylor expanded in alpha around 0 93.4%

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

   7. Taylor expanded in beta around inf 93.0%

      \[\leadsto \frac{\frac{\beta}{\color{blue}{2 + \left(\beta + 4 \cdot i\right)}} + 1}{2} \]
   8. Step-by-step derivation

      1. associate-+r+ 93.0%

         \[\leadsto \frac{\frac{\beta}{\color{blue}{\left(2 + \beta\right) + 4 \cdot i}} + 1}{2} \]

      2. *-commutative 93.0%

         \[\leadsto \frac{\frac{\beta}{\left(2 + \beta\right) + \color{blue}{i \cdot 4}} + 1}{2} \]

   9. Simplified 93.0%

      \[\leadsto \frac{\frac{\beta}{\color{blue}{\left(2 + \beta\right) + i \cdot 4}} + 1}{2} \]

   if 5.3999999999999999e82 < alpha < 2.29999999999999998e155 or 1e176 < alpha

   1. Initial program 5.7%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

   3. Simplified 27.1%

      \[\leadsto \color{blue}{\frac{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in beta around 0 27.1%

      \[\leadsto \frac{\color{blue}{\frac{\alpha}{2 + \left(\alpha + 2 \cdot i\right)}} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1}{2} \]
   6. Step-by-step derivation

      1. associate-+r+ 27.1%

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

   7. Simplified 27.1%

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

   8. Taylor expanded in alpha around inf 78.7%

      \[\leadsto \frac{\color{blue}{\frac{2 + \left(2 \cdot \beta + 4 \cdot i\right)}{\alpha}}}{2} \]

   if 2.29999999999999998e155 < alpha < 1e176

   1. Initial program 1.0%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around inf 88.4%

      \[\leadsto \frac{\frac{\color{blue}{-1 \cdot \alpha}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   4. Step-by-step derivation

      1. neg-mul-1 88.4%

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

   5. Simplified 88.4%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 5.4 \cdot 10^{+82}:\\ \;\;\;\;\frac{1 + \frac{\beta}{i \cdot 4 + \left(\beta + 2\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 2.3 \cdot 10^{+155} \lor \neg \left(\alpha \leq 10^{+176}\right):\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\alpha}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 5.4 \cdot 10^{+82}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\left(1 + 2 \cdot \frac{i}{\beta}\right) \cdot \left(2 + \left(\beta + 2 \cdot i\right)\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 2.9 \cdot 10^{+155} \lor \neg \left(\alpha \leq 10^{+176}\right):\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\alpha}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 5.4e+82)
   (/
    (+ 1.0 (/ beta (* (+ 1.0 (* 2.0 (/ i beta))) (+ 2.0 (+ beta (* 2.0 i))))))
    2.0)
   (if (or (<= alpha 2.9e+155) (not (<= alpha 1e+176)))
     (/ (/ (+ 2.0 (+ (* beta 2.0) (* i 4.0))) alpha) 2.0)
     (/ (- 1.0 (/ alpha (+ 2.0 (+ (+ alpha beta) (* 2.0 i))))) 2.0))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 5.4e+82) {
		tmp = (1.0 + (beta / ((1.0 + (2.0 * (i / beta))) * (2.0 + (beta + (2.0 * i)))))) / 2.0;
	} else if ((alpha <= 2.9e+155) || !(alpha <= 1e+176)) {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	} else {
		tmp = (1.0 - (alpha / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 5.4d+82) then
        tmp = (1.0d0 + (beta / ((1.0d0 + (2.0d0 * (i / beta))) * (2.0d0 + (beta + (2.0d0 * i)))))) / 2.0d0
    else if ((alpha <= 2.9d+155) .or. (.not. (alpha <= 1d+176))) then
        tmp = ((2.0d0 + ((beta * 2.0d0) + (i * 4.0d0))) / alpha) / 2.0d0
    else
        tmp = (1.0d0 - (alpha / (2.0d0 + ((alpha + beta) + (2.0d0 * i))))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 5.4e+82) {
		tmp = (1.0 + (beta / ((1.0 + (2.0 * (i / beta))) * (2.0 + (beta + (2.0 * i)))))) / 2.0;
	} else if ((alpha <= 2.9e+155) || !(alpha <= 1e+176)) {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	} else {
		tmp = (1.0 - (alpha / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 5.4e+82:
		tmp = (1.0 + (beta / ((1.0 + (2.0 * (i / beta))) * (2.0 + (beta + (2.0 * i)))))) / 2.0
	elif (alpha <= 2.9e+155) or not (alpha <= 1e+176):
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0
	else:
		tmp = (1.0 - (alpha / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 5.4e+82)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(Float64(1.0 + Float64(2.0 * Float64(i / beta))) * Float64(2.0 + Float64(beta + Float64(2.0 * i)))))) / 2.0);
	elseif ((alpha <= 2.9e+155) || !(alpha <= 1e+176))
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(beta * 2.0) + Float64(i * 4.0))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(1.0 - Float64(alpha / Float64(2.0 + Float64(Float64(alpha + beta) + Float64(2.0 * i))))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 5.4e+82)
		tmp = (1.0 + (beta / ((1.0 + (2.0 * (i / beta))) * (2.0 + (beta + (2.0 * i)))))) / 2.0;
	elseif ((alpha <= 2.9e+155) || ~((alpha <= 1e+176)))
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	else
		tmp = (1.0 - (alpha / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 5.4e+82], N[(N[(1.0 + N[(beta / N[(N[(1.0 + N[(2.0 * N[(i / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[alpha, 2.9e+155], N[Not[LessEqual[alpha, 1e+176]], $MachinePrecision]], N[(N[(N[(2.0 + N[(N[(beta * 2.0), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 - N[(alpha / N[(2.0 + N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 5.3999999999999999e82

   1. Initial program 75.4%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. associate-/l* 94.7%

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

      2. associate-+l+ 94.7%

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

      3. associate-+l+ 94.7%

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

   3. Simplified 94.7%

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

   5. Taylor expanded in i around 0 94.7%

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

   6. Taylor expanded in alpha around 0 93.4%

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

   if 5.3999999999999999e82 < alpha < 2.8999999999999999e155 or 1e176 < alpha

   1. Initial program 5.7%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

   3. Simplified 27.1%

      \[\leadsto \color{blue}{\frac{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in beta around 0 27.1%

      \[\leadsto \frac{\color{blue}{\frac{\alpha}{2 + \left(\alpha + 2 \cdot i\right)}} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1}{2} \]
   6. Step-by-step derivation

      1. associate-+r+ 27.1%

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

   7. Simplified 27.1%

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

   8. Taylor expanded in alpha around inf 78.7%

      \[\leadsto \frac{\color{blue}{\frac{2 + \left(2 \cdot \beta + 4 \cdot i\right)}{\alpha}}}{2} \]

   if 2.8999999999999999e155 < alpha < 1e176

   1. Initial program 1.0%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around inf 88.4%

      \[\leadsto \frac{\frac{\color{blue}{-1 \cdot \alpha}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   4. Step-by-step derivation

      1. neg-mul-1 88.4%

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

   5. Simplified 88.4%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 5.4 \cdot 10^{+82}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\left(1 + 2 \cdot \frac{i}{\beta}\right) \cdot \left(2 + \left(\beta + 2 \cdot i\right)\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 2.9 \cdot 10^{+155} \lor \neg \left(\alpha \leq 10^{+176}\right):\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\alpha}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 4.6 \cdot 10^{+82}:\\ \;\;\;\;\frac{1 + \frac{\beta}{i \cdot 4 + \left(\beta + 2\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 3.75 \cdot 10^{+155} \lor \neg \left(\alpha \leq 10^{+176}\right):\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 4.6e+82)
   (/ (+ 1.0 (/ beta (+ (* i 4.0) (+ beta 2.0)))) 2.0)
   (if (or (<= alpha 3.75e+155) (not (<= alpha 1e+176)))
     (/ (/ (+ 2.0 (+ (* beta 2.0) (* i 4.0))) alpha) 2.0)
     0.5)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 4.6e+82) {
		tmp = (1.0 + (beta / ((i * 4.0) + (beta + 2.0)))) / 2.0;
	} else if ((alpha <= 3.75e+155) || !(alpha <= 1e+176)) {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 4.6d+82) then
        tmp = (1.0d0 + (beta / ((i * 4.0d0) + (beta + 2.0d0)))) / 2.0d0
    else if ((alpha <= 3.75d+155) .or. (.not. (alpha <= 1d+176))) then
        tmp = ((2.0d0 + ((beta * 2.0d0) + (i * 4.0d0))) / alpha) / 2.0d0
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 4.6e+82) {
		tmp = (1.0 + (beta / ((i * 4.0) + (beta + 2.0)))) / 2.0;
	} else if ((alpha <= 3.75e+155) || !(alpha <= 1e+176)) {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 4.6e+82:
		tmp = (1.0 + (beta / ((i * 4.0) + (beta + 2.0)))) / 2.0
	elif (alpha <= 3.75e+155) or not (alpha <= 1e+176):
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0
	else:
		tmp = 0.5
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 4.6e+82)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(Float64(i * 4.0) + Float64(beta + 2.0)))) / 2.0);
	elseif ((alpha <= 3.75e+155) || !(alpha <= 1e+176))
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(beta * 2.0) + Float64(i * 4.0))) / alpha) / 2.0);
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 4.6e+82)
		tmp = (1.0 + (beta / ((i * 4.0) + (beta + 2.0)))) / 2.0;
	elseif ((alpha <= 3.75e+155) || ~((alpha <= 1e+176)))
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 4.6e+82], N[(N[(1.0 + N[(beta / N[(N[(i * 4.0), $MachinePrecision] + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[alpha, 3.75e+155], N[Not[LessEqual[alpha, 1e+176]], $MachinePrecision]], N[(N[(N[(2.0 + N[(N[(beta * 2.0), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], 0.5]]

Derivation

1. Split input into 3 regimes

2. if alpha < 4.59999999999999976e82

   1. Initial program 75.4%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. associate-/l* 94.7%

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

      2. associate-+l+ 94.7%

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

      3. associate-+l+ 94.7%

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

   3. Simplified 94.7%

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

   5. Taylor expanded in i around 0 94.7%

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

   6. Taylor expanded in alpha around 0 93.4%

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

   7. Taylor expanded in beta around inf 93.0%

      \[\leadsto \frac{\frac{\beta}{\color{blue}{2 + \left(\beta + 4 \cdot i\right)}} + 1}{2} \]
   8. Step-by-step derivation

      1. associate-+r+ 93.0%

         \[\leadsto \frac{\frac{\beta}{\color{blue}{\left(2 + \beta\right) + 4 \cdot i}} + 1}{2} \]

      2. *-commutative 93.0%

         \[\leadsto \frac{\frac{\beta}{\left(2 + \beta\right) + \color{blue}{i \cdot 4}} + 1}{2} \]

   9. Simplified 93.0%

      \[\leadsto \frac{\frac{\beta}{\color{blue}{\left(2 + \beta\right) + i \cdot 4}} + 1}{2} \]

   if 4.59999999999999976e82 < alpha < 3.7499999999999999e155 or 1e176 < alpha

   1. Initial program 5.7%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

   3. Simplified 27.1%

      \[\leadsto \color{blue}{\frac{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in beta around 0 27.1%

      \[\leadsto \frac{\color{blue}{\frac{\alpha}{2 + \left(\alpha + 2 \cdot i\right)}} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1}{2} \]
   6. Step-by-step derivation

      1. associate-+r+ 27.1%

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

   7. Simplified 27.1%

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

   8. Taylor expanded in alpha around inf 78.7%

      \[\leadsto \frac{\color{blue}{\frac{2 + \left(2 \cdot \beta + 4 \cdot i\right)}{\alpha}}}{2} \]

   if 3.7499999999999999e155 < alpha < 1e176

   1. Initial program 1.0%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

   3. Simplified 90.3%

      \[\leadsto \color{blue}{\frac{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in i around inf 87.2%

      \[\leadsto \frac{\color{blue}{1}}{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 4.6 \cdot 10^{+82}:\\ \;\;\;\;\frac{1 + \frac{\beta}{i \cdot 4 + \left(\beta + 2\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 3.75 \cdot 10^{+155} \lor \neg \left(\alpha \leq 10^{+176}\right):\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot i \leq 4 \cdot 10^{+30}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\left(\alpha + \beta\right) + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2 \cdot i}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= (* 2.0 i) 4e+30)
   (/ (+ 1.0 (/ beta (+ (+ alpha beta) 2.0))) 2.0)
   (/ (+ 1.0 (/ beta (+ beta (* 2.0 i)))) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if ((2.0 * i) <= 4e+30) {
		tmp = (1.0 + (beta / ((alpha + beta) + 2.0))) / 2.0;
	} else {
		tmp = (1.0 + (beta / (beta + (2.0 * i)))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if ((2.0d0 * i) <= 4d+30) then
        tmp = (1.0d0 + (beta / ((alpha + beta) + 2.0d0))) / 2.0d0
    else
        tmp = (1.0d0 + (beta / (beta + (2.0d0 * i)))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if ((2.0 * i) <= 4e+30) {
		tmp = (1.0 + (beta / ((alpha + beta) + 2.0))) / 2.0;
	} else {
		tmp = (1.0 + (beta / (beta + (2.0 * i)))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if (2.0 * i) <= 4e+30:
		tmp = (1.0 + (beta / ((alpha + beta) + 2.0))) / 2.0
	else:
		tmp = (1.0 + (beta / (beta + (2.0 * i)))) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (Float64(2.0 * i) <= 4e+30)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(Float64(alpha + beta) + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + Float64(2.0 * i)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if ((2.0 * i) <= 4e+30)
		tmp = (1.0 + (beta / ((alpha + beta) + 2.0))) / 2.0;
	else
		tmp = (1.0 + (beta / (beta + (2.0 * i)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[N[(2.0 * i), $MachinePrecision], 4e+30], N[(N[(1.0 + N[(beta / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(beta / N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 2 i) < 4.0000000000000001e30

   1. Initial program 55.2%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in beta around inf 72.7%

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

   4. Taylor expanded in i around 0 72.7%

      \[\leadsto \frac{\color{blue}{\frac{\beta}{2 + \left(\alpha + \beta\right)}} + 1}{2} \]
   5. Step-by-step derivation

      1. +-commutative 72.7%

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

   6. Simplified 72.7%

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

   if 4.0000000000000001e30 < (*.f64 2 i)

   1. Initial program 62.8%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. associate-/l* 86.4%

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

      2. associate-+l+ 86.4%

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

      3. associate-+l+ 86.4%

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

   3. Simplified 86.4%

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

   5. Taylor expanded in alpha around 0 86.3%

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

   6. Taylor expanded in alpha around inf 85.2%

      \[\leadsto \frac{\color{blue}{\frac{\beta}{\beta + 2 \cdot i}} + 1}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot i \leq 4 \cdot 10^{+30}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\left(\alpha + \beta\right) + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2 \cdot i}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 10^{+156}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\left(\alpha + \beta\right) + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= i 1e+156) (/ (+ 1.0 (/ beta (+ (+ alpha beta) 2.0))) 2.0) 0.5))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 1e+156) {
		tmp = (1.0 + (beta / ((alpha + beta) + 2.0))) / 2.0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (i <= 1d+156) then
        tmp = (1.0d0 + (beta / ((alpha + beta) + 2.0d0))) / 2.0d0
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 1e+156) {
		tmp = (1.0 + (beta / ((alpha + beta) + 2.0))) / 2.0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if i <= 1e+156:
		tmp = (1.0 + (beta / ((alpha + beta) + 2.0))) / 2.0
	else:
		tmp = 0.5
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (i <= 1e+156)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(Float64(alpha + beta) + 2.0))) / 2.0);
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (i <= 1e+156)
		tmp = (1.0 + (beta / ((alpha + beta) + 2.0))) / 2.0;
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[i, 1e+156], N[(N[(1.0 + N[(beta / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if i < 9.9999999999999998e155

   1. Initial program 56.7%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in beta around inf 74.1%

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

   4. Taylor expanded in i around 0 73.2%

      \[\leadsto \frac{\color{blue}{\frac{\beta}{2 + \left(\alpha + \beta\right)}} + 1}{2} \]
   5. Step-by-step derivation

      1. +-commutative 73.2%

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

   6. Simplified 73.2%

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

   if 9.9999999999999998e155 < i

   1. Initial program 64.2%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

   3. Simplified 92.2%

      \[\leadsto \color{blue}{\frac{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in i around inf 88.4%

      \[\leadsto \frac{\color{blue}{1}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 10^{+156}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\left(\alpha + \beta\right) + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 1.05 \cdot 10^{+156}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= i 1.05e+156) (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0) 0.5))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 1.05e+156) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (i <= 1.05d+156) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 1.05e+156) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if i <= 1.05e+156:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = 0.5
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (i <= 1.05e+156)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (i <= 1.05e+156)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[i, 1.05e+156], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if i < 1.04999999999999991e156

   1. Initial program 56.7%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. associate-/l* 76.2%

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

      2. associate-+l+ 76.2%

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

      3. associate-+l+ 76.2%

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

   3. Simplified 76.2%

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

   5. Taylor expanded in i around 0 76.2%

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

   6. Taylor expanded in alpha around 0 74.4%

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

   7. Taylor expanded in i around 0 73.1%

      \[\leadsto \frac{\color{blue}{\frac{\beta}{2 + \beta}} + 1}{2} \]

   if 1.04999999999999991e156 < i

   1. Initial program 64.2%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

   3. Simplified 92.2%

      \[\leadsto \color{blue}{\frac{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in i around inf 88.4%

      \[\leadsto \frac{\color{blue}{1}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 1.05 \cdot 10^{+156}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 78.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \frac{1 + \frac{\beta}{i \cdot 4 + \left(\beta + 2\right)}}{2} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (/ (+ 1.0 (/ beta (+ (* i 4.0) (+ beta 2.0)))) 2.0))

C

double code(double alpha, double beta, double i) {
	return (1.0 + (beta / ((i * 4.0) + (beta + 2.0)))) / 2.0;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    code = (1.0d0 + (beta / ((i * 4.0d0) + (beta + 2.0d0)))) / 2.0d0
end function

Java

public static double code(double alpha, double beta, double i) {
	return (1.0 + (beta / ((i * 4.0) + (beta + 2.0)))) / 2.0;
}

Python

def code(alpha, beta, i):
	return (1.0 + (beta / ((i * 4.0) + (beta + 2.0)))) / 2.0

Julia

function code(alpha, beta, i)
	return Float64(Float64(1.0 + Float64(beta / Float64(Float64(i * 4.0) + Float64(beta + 2.0)))) / 2.0)
end

MATLAB

function tmp = code(alpha, beta, i)
	tmp = (1.0 + (beta / ((i * 4.0) + (beta + 2.0)))) / 2.0;
end

Wolfram

code[alpha_, beta_, i_] := N[(N[(1.0 + N[(beta / N[(N[(i * 4.0), $MachinePrecision] + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 58.7%

   \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
2. Step-by-step derivation

   1. associate-/l* 80.3%

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

   2. associate-+l+ 80.3%

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

   3. associate-+l+ 80.3%

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

3. Simplified 80.3%

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

5. Taylor expanded in i around 0 80.3%

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

6. Taylor expanded in alpha around 0 78.8%

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

7. Taylor expanded in beta around inf 78.5%

   \[\leadsto \frac{\frac{\beta}{\color{blue}{2 + \left(\beta + 4 \cdot i\right)}} + 1}{2} \]
8. Step-by-step derivation

   1. associate-+r+ 78.5%

      \[\leadsto \frac{\frac{\beta}{\color{blue}{\left(2 + \beta\right) + 4 \cdot i}} + 1}{2} \]

   2. *-commutative 78.5%

      \[\leadsto \frac{\frac{\beta}{\left(2 + \beta\right) + \color{blue}{i \cdot 4}} + 1}{2} \]

9. Simplified 78.5%

   \[\leadsto \frac{\frac{\beta}{\color{blue}{\left(2 + \beta\right) + i \cdot 4}} + 1}{2} \]

10. Final simplification 78.5%

    \[\leadsto \frac{1 + \frac{\beta}{i \cdot 4 + \left(\beta + 2\right)}}{2} \]
11. Add Preprocessing

Alternative 11: 72.6% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.6 \cdot 10^{+52}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 (if (<= beta 4.6e+52) 0.5 1.0))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 4.6e+52) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 4.6d+52) then
        tmp = 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 4.6e+52) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 4.6e+52:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 4.6e+52)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 4.6e+52)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 4.6e+52], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 4.6e52

   1. Initial program 68.5%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

   3. Simplified 74.7%

      \[\leadsto \color{blue}{\frac{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in i around inf 70.9%

      \[\leadsto \frac{\color{blue}{1}}{2} \]

   if 4.6e52 < beta

   1. Initial program 40.6%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

   3. Simplified 90.7%

      \[\leadsto \color{blue}{\frac{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in beta around inf 73.8%

      \[\leadsto \frac{\color{blue}{2}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.6 \cdot 10^{+52}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.3% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 0.5)

C

double code(double alpha, double beta, double i) {
	return 0.5;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    code = 0.5d0
end function

Java

public static double code(double alpha, double beta, double i) {
	return 0.5;
}

Python

def code(alpha, beta, i):
	return 0.5

Julia

function code(alpha, beta, i)
	return 0.5
end

MATLAB

function tmp = code(alpha, beta, i)
	tmp = 0.5;
end

Wolfram

code[alpha_, beta_, i_] := 0.5

Derivation

1. Initial program 58.7%

   \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
2. Step-by-step derivation

3. Simplified 80.3%

   \[\leadsto \color{blue}{\frac{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1}{2}} \]
4. Add Preprocessing

5. Taylor expanded in i around inf 57.3%

   \[\leadsto \frac{\color{blue}{1}}{2} \]

6. Final simplification 57.3%

   \[\leadsto 0.5 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024026 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :precision binary64
  :pre (and (and (> alpha -1.0) (> beta -1.0)) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i))) (+ (+ (+ alpha beta) (* 2.0 i)) 2.0)) 1.0) 2.0))