Octave 3.8, jcobi/2

Percentage Accurate: 63.2% → 97.1%

Time: 13.6s

Alternatives: 9

Speedup: 9.5×

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: 63.2% 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: 97.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_2 := 2 + t_1\\ \mathbf{if}\;\frac{\frac{t_0}{t_1}}{t_2} \leq -1:\\ \;\;\;\;\frac{\frac{\beta}{\alpha} + \left(4 \cdot \frac{i}{\alpha} + \left(2 \cdot \frac{1}{\alpha} + \frac{\beta}{\alpha}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\frac{1}{2 \cdot \frac{i}{t_0} + \frac{1}{\beta - \alpha}}}{t_2}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* (+ alpha beta) (- beta alpha)))
        (t_1 (+ (+ alpha beta) (* 2.0 i)))
        (t_2 (+ 2.0 t_1)))
   (if (<= (/ (/ t_0 t_1) t_2) -1.0)
     (/
      (+
       (/ beta alpha)
       (+ (* 4.0 (/ i alpha)) (+ (* 2.0 (/ 1.0 alpha)) (/ beta alpha))))
      2.0)
     (/
      (+ 1.0 (/ (/ 1.0 (+ (* 2.0 (/ i t_0)) (/ 1.0 (- beta alpha)))) t_2))
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) * (beta - alpha);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = 2.0 + t_1;
	double tmp;
	if (((t_0 / t_1) / t_2) <= -1.0) {
		tmp = ((beta / alpha) + ((4.0 * (i / alpha)) + ((2.0 * (1.0 / alpha)) + (beta / alpha)))) / 2.0;
	} else {
		tmp = (1.0 + ((1.0 / ((2.0 * (i / t_0)) + (1.0 / (beta - alpha)))) / t_2)) / 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (alpha + beta) * (beta - alpha)
    t_1 = (alpha + beta) + (2.0d0 * i)
    t_2 = 2.0d0 + t_1
    if (((t_0 / t_1) / t_2) <= (-1.0d0)) then
        tmp = ((beta / alpha) + ((4.0d0 * (i / alpha)) + ((2.0d0 * (1.0d0 / alpha)) + (beta / alpha)))) / 2.0d0
    else
        tmp = (1.0d0 + ((1.0d0 / ((2.0d0 * (i / t_0)) + (1.0d0 / (beta - alpha)))) / t_2)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) * (beta - alpha);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = 2.0 + t_1;
	double tmp;
	if (((t_0 / t_1) / t_2) <= -1.0) {
		tmp = ((beta / alpha) + ((4.0 * (i / alpha)) + ((2.0 * (1.0 / alpha)) + (beta / alpha)))) / 2.0;
	} else {
		tmp = (1.0 + ((1.0 / ((2.0 * (i / t_0)) + (1.0 / (beta - alpha)))) / t_2)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) * (beta - alpha)
	t_1 = (alpha + beta) + (2.0 * i)
	t_2 = 2.0 + t_1
	tmp = 0
	if ((t_0 / t_1) / t_2) <= -1.0:
		tmp = ((beta / alpha) + ((4.0 * (i / alpha)) + ((2.0 * (1.0 / alpha)) + (beta / alpha)))) / 2.0
	else:
		tmp = (1.0 + ((1.0 / ((2.0 * (i / t_0)) + (1.0 / (beta - alpha)))) / t_2)) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) * Float64(beta - alpha))
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_2 = Float64(2.0 + t_1)
	tmp = 0.0
	if (Float64(Float64(t_0 / t_1) / t_2) <= -1.0)
		tmp = Float64(Float64(Float64(beta / alpha) + Float64(Float64(4.0 * Float64(i / alpha)) + Float64(Float64(2.0 * Float64(1.0 / alpha)) + Float64(beta / alpha)))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(1.0 / Float64(Float64(2.0 * Float64(i / t_0)) + Float64(1.0 / Float64(beta - alpha)))) / t_2)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) * (beta - alpha);
	t_1 = (alpha + beta) + (2.0 * i);
	t_2 = 2.0 + t_1;
	tmp = 0.0;
	if (((t_0 / t_1) / t_2) <= -1.0)
		tmp = ((beta / alpha) + ((4.0 * (i / alpha)) + ((2.0 * (1.0 / alpha)) + (beta / alpha)))) / 2.0;
	else
		tmp = (1.0 + ((1.0 / ((2.0 * (i / t_0)) + (1.0 / (beta - alpha)))) / t_2)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 + t$95$1), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$0 / t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision], -1.0], N[(N[(N[(beta / alpha), $MachinePrecision] + N[(N[(4.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision] + N[(beta / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(N[(1.0 / N[(N[(2.0 * N[(i / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(beta - alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $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)) < -1

   1. Initial program 1.7%

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

   3. Simplified 13.2%

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

   4. Taylor expanded in alpha around inf 92.0%

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

   5. Taylor expanded in i around 0 92.0%

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

   if -1 < (/.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 75.9%

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

      1. clear-num 76.0%

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

      2. inv-pow 76.0%

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

      3. associate-+r+ 76.0%

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

      4. +-commutative 76.0%

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

      5. fma-udef 76.0%

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

   3. Applied egg-rr 76.0%

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

      1. unpow-1 76.0%

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

      2. +-commutative 76.0%

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

      3. *-commutative 76.0%

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

      4. +-commutative 76.0%

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

   5. Simplified 76.0%

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

   6. Taylor expanded in i around 0 99.9%

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

4. Final simplification 98.1%

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

Alternative 2: 96.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := 2 + t_0\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{t_1} \leq -0.5:\\ \;\;\;\;\frac{\frac{\beta}{\alpha} + \left(4 \cdot \frac{i}{\alpha} + \left(2 \cdot \frac{1}{\alpha} + \frac{\beta}{\alpha}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{t_1}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))) (t_1 (+ 2.0 t_0)))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) t_1) -0.5)
     (/
      (+
       (/ beta alpha)
       (+ (* 4.0 (/ i alpha)) (+ (* 2.0 (/ 1.0 alpha)) (/ beta alpha))))
      2.0)
     (/ (+ 1.0 (/ (- beta alpha) t_1)) 2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = 2.0 + t_0;
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= -0.5) {
		tmp = ((beta / alpha) + ((4.0 * (i / alpha)) + ((2.0 * (1.0 / alpha)) + (beta / alpha)))) / 2.0;
	} else {
		tmp = (1.0 + ((beta - alpha) / t_1)) / 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) :: t_1
    real(8) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    t_1 = 2.0d0 + t_0
    if (((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= (-0.5d0)) then
        tmp = ((beta / alpha) + ((4.0d0 * (i / alpha)) + ((2.0d0 * (1.0d0 / alpha)) + (beta / alpha)))) / 2.0d0
    else
        tmp = (1.0d0 + ((beta - alpha) / t_1)) / 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 t_1 = 2.0 + t_0;
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= -0.5) {
		tmp = ((beta / alpha) + ((4.0 * (i / alpha)) + ((2.0 * (1.0 / alpha)) + (beta / alpha)))) / 2.0;
	} else {
		tmp = (1.0 + ((beta - alpha) / t_1)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	t_1 = 2.0 + t_0
	tmp = 0
	if ((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= -0.5:
		tmp = ((beta / alpha) + ((4.0 * (i / alpha)) + ((2.0 * (1.0 / alpha)) + (beta / alpha)))) / 2.0
	else:
		tmp = (1.0 + ((beta - alpha) / t_1)) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(2.0 + t_0)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / t_1) <= -0.5)
		tmp = Float64(Float64(Float64(beta / alpha) + Float64(Float64(4.0 * Float64(i / alpha)) + Float64(Float64(2.0 * Float64(1.0 / alpha)) + Float64(beta / alpha)))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / t_1)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	t_1 = 2.0 + t_0;
	tmp = 0.0;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= -0.5)
		tmp = ((beta / alpha) + ((4.0 * (i / alpha)) + ((2.0 * (1.0 / alpha)) + (beta / alpha)))) / 2.0;
	else
		tmp = (1.0 + ((beta - alpha) / t_1)) / 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]}, Block[{t$95$1 = N[(2.0 + t$95$0), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision], -0.5], N[(N[(N[(beta / alpha), $MachinePrecision] + N[(N[(4.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision] + N[(beta / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / t$95$1), $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)) < -0.5

   1. Initial program 3.1%

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

   3. Simplified 14.4%

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

   4. Taylor expanded in alpha around inf 91.0%

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

   5. Taylor expanded in i around 0 91.0%

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

   if -0.5 < (/.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 75.9%

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

   2. Taylor expanded in i around 0 99.2%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.5:\\ \;\;\;\;\frac{\frac{\beta}{\alpha} + \left(4 \cdot \frac{i}{\alpha} + \left(2 \cdot \frac{1}{\alpha} + \frac{\beta}{\alpha}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \end{array} \]

Alternative 3: 89.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 7.4 \cdot 10^{+70}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 1.6 \cdot 10^{+89}:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha} + \left(4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}\right)}{2}\\ \mathbf{elif}\;\alpha \leq 1.35 \cdot 10^{+111}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta + \left(\beta + i \cdot 4\right)\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 7.4e+70)
   (/ (+ 1.0 (/ (- beta alpha) (+ 2.0 (+ (+ alpha beta) (* 2.0 i))))) 2.0)
   (if (<= alpha 1.6e+89)
     (/
      (+ (* 2.0 (/ beta alpha)) (+ (* 4.0 (/ i alpha)) (* 2.0 (/ 1.0 alpha))))
      2.0)
     (if (<= alpha 1.35e+111)
       (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
       (/ (/ (+ 2.0 (+ beta (+ beta (* i 4.0)))) alpha) 2.0)))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 7.4e+70) {
		tmp = (1.0 + ((beta - alpha) / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else if (alpha <= 1.6e+89) {
		tmp = ((2.0 * (beta / alpha)) + ((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha)))) / 2.0;
	} else if (alpha <= 1.35e+111) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (beta + (beta + (i * 4.0)))) / alpha) / 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 <= 7.4d+70) then
        tmp = (1.0d0 + ((beta - alpha) / (2.0d0 + ((alpha + beta) + (2.0d0 * i))))) / 2.0d0
    else if (alpha <= 1.6d+89) then
        tmp = ((2.0d0 * (beta / alpha)) + ((4.0d0 * (i / alpha)) + (2.0d0 * (1.0d0 / alpha)))) / 2.0d0
    else if (alpha <= 1.35d+111) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((2.0d0 + (beta + (beta + (i * 4.0d0)))) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 7.4e+70) {
		tmp = (1.0 + ((beta - alpha) / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else if (alpha <= 1.6e+89) {
		tmp = ((2.0 * (beta / alpha)) + ((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha)))) / 2.0;
	} else if (alpha <= 1.35e+111) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (beta + (beta + (i * 4.0)))) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 7.4e+70:
		tmp = (1.0 + ((beta - alpha) / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0
	elif alpha <= 1.6e+89:
		tmp = ((2.0 * (beta / alpha)) + ((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha)))) / 2.0
	elif alpha <= 1.35e+111:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((2.0 + (beta + (beta + (i * 4.0)))) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 7.4e+70)
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / Float64(2.0 + Float64(Float64(alpha + beta) + Float64(2.0 * i))))) / 2.0);
	elseif (alpha <= 1.6e+89)
		tmp = Float64(Float64(Float64(2.0 * Float64(beta / alpha)) + Float64(Float64(4.0 * Float64(i / alpha)) + Float64(2.0 * Float64(1.0 / alpha)))) / 2.0);
	elseif (alpha <= 1.35e+111)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(beta + Float64(beta + Float64(i * 4.0)))) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 7.4e+70)
		tmp = (1.0 + ((beta - alpha) / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	elseif (alpha <= 1.6e+89)
		tmp = ((2.0 * (beta / alpha)) + ((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha)))) / 2.0;
	elseif (alpha <= 1.35e+111)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((2.0 + (beta + (beta + (i * 4.0)))) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 7.4e+70], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 1.6e+89], N[(N[(N[(2.0 * N[(beta / alpha), $MachinePrecision]), $MachinePrecision] + N[(N[(4.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 1.35e+111], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(beta + N[(beta + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if alpha < 7.39999999999999977e70

   1. Initial program 76.4%

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

   2. Taylor expanded in i around 0 97.2%

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

   if 7.39999999999999977e70 < alpha < 1.59999999999999994e89

   1. Initial program 21.2%

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

   3. Simplified 31.0%

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

   4. Taylor expanded in alpha around inf 74.3%

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

   5. Taylor expanded in i around 0 74.3%

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

      1. associate--l+ 74.3%

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

      2. sub-neg 74.3%

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

      3. *-commutative 74.3%

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

      4. mul-1-neg 74.3%

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

      5. remove-double-neg 74.3%

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

   7. Simplified 74.3%

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

   8. Taylor expanded in beta around 0 74.4%

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

   if 1.59999999999999994e89 < alpha < 1.3499999999999999e111

   1. Initial program 44.8%

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

      1. associate-/l/ 43.4%

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

      2. associate-+l+ 43.4%

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

      3. associate-+l+ 43.4%

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

   3. Simplified 43.4%

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

   4. Taylor expanded in i around 0 58.6%

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

      1. associate-+r+ 58.6%

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

   6. Simplified 58.6%

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

   7. Taylor expanded in alpha around 0 86.5%

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

   if 1.3499999999999999e111 < alpha

   1. Initial program 3.3%

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

   3. Simplified 22.4%

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

   4. Taylor expanded in alpha around inf 83.3%

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

   5. Taylor expanded in i around 0 83.4%

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

      1. associate--l+ 83.4%

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

      2. sub-neg 83.4%

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

      3. *-commutative 83.4%

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

      4. mul-1-neg 83.4%

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

      5. remove-double-neg 83.4%

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

   7. Simplified 83.4%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 7.4 \cdot 10^{+70}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 1.6 \cdot 10^{+89}:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha} + \left(4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}\right)}{2}\\ \mathbf{elif}\;\alpha \leq 1.35 \cdot 10^{+111}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta + \left(\beta + i \cdot 4\right)\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 4: 82.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.2 \cdot 10^{+71} \lor \neg \left(\alpha \leq 5.2 \cdot 10^{+87}\right) \land \alpha \leq 3.2 \cdot 10^{+112}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta + \left(\beta + i \cdot 4\right)\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (or (<= alpha 1.2e+71) (and (not (<= alpha 5.2e+87)) (<= alpha 3.2e+112)))
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (+ 2.0 (+ beta (+ beta (* i 4.0)))) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if ((alpha <= 1.2e+71) || (!(alpha <= 5.2e+87) && (alpha <= 3.2e+112))) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (beta + (beta + (i * 4.0)))) / alpha) / 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 <= 1.2d+71) .or. (.not. (alpha <= 5.2d+87)) .and. (alpha <= 3.2d+112)) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((2.0d0 + (beta + (beta + (i * 4.0d0)))) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if ((alpha <= 1.2e+71) || (!(alpha <= 5.2e+87) && (alpha <= 3.2e+112))) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (beta + (beta + (i * 4.0)))) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if (alpha <= 1.2e+71) or (not (alpha <= 5.2e+87) and (alpha <= 3.2e+112)):
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((2.0 + (beta + (beta + (i * 4.0)))) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if ((alpha <= 1.2e+71) || (!(alpha <= 5.2e+87) && (alpha <= 3.2e+112)))
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(beta + Float64(beta + Float64(i * 4.0)))) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if ((alpha <= 1.2e+71) || (~((alpha <= 5.2e+87)) && (alpha <= 3.2e+112)))
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((2.0 + (beta + (beta + (i * 4.0)))) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[Or[LessEqual[alpha, 1.2e+71], And[N[Not[LessEqual[alpha, 5.2e+87]], $MachinePrecision], LessEqual[alpha, 3.2e+112]]], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(beta + N[(beta + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.1999999999999999e71 or 5.19999999999999997e87 < alpha < 3.19999999999999986e112

   1. Initial program 75.3%

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

      1. associate-/l/ 74.6%

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

      2. associate-+l+ 74.6%

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

      3. associate-+l+ 74.6%

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

   3. Simplified 74.6%

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

   4. Taylor expanded in i around 0 81.6%

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

      1. associate-+r+ 81.6%

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

   6. Simplified 81.6%

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

   7. Taylor expanded in alpha around 0 86.7%

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

   if 1.1999999999999999e71 < alpha < 5.19999999999999997e87 or 3.19999999999999986e112 < alpha

   1. Initial program 6.3%

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

   3. Simplified 23.9%

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

   4. Taylor expanded in alpha around inf 81.8%

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

   5. Taylor expanded in i around 0 81.9%

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

      1. associate--l+ 81.9%

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

      2. sub-neg 81.9%

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

      3. *-commutative 81.9%

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

      4. mul-1-neg 81.9%

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

      5. remove-double-neg 81.9%

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

   7. Simplified 81.9%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.2 \cdot 10^{+71} \lor \neg \left(\alpha \leq 5.2 \cdot 10^{+87}\right) \land \alpha \leq 3.2 \cdot 10^{+112}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta + \left(\beta + i \cdot 4\right)\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 5: 89.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 3.2 \cdot 10^{+71}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 2.4 \cdot 10^{+90} \lor \neg \left(\alpha \leq 1.75 \cdot 10^{+109}\right):\\ \;\;\;\;\frac{\frac{2 + \left(\beta + \left(\beta + i \cdot 4\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 3.2e+71)
   (/ (+ 1.0 (/ (- beta alpha) (+ 2.0 (+ (+ alpha beta) (* 2.0 i))))) 2.0)
   (if (or (<= alpha 2.4e+90) (not (<= alpha 1.75e+109)))
     (/ (/ (+ 2.0 (+ beta (+ beta (* i 4.0)))) alpha) 2.0)
     (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 3.2e+71) {
		tmp = (1.0 + ((beta - alpha) / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else if ((alpha <= 2.4e+90) || !(alpha <= 1.75e+109)) {
		tmp = ((2.0 + (beta + (beta + (i * 4.0)))) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (beta / (beta + 2.0))) / 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) :: tmp
    if (alpha <= 3.2d+71) then
        tmp = (1.0d0 + ((beta - alpha) / (2.0d0 + ((alpha + beta) + (2.0d0 * i))))) / 2.0d0
    else if ((alpha <= 2.4d+90) .or. (.not. (alpha <= 1.75d+109))) then
        tmp = ((2.0d0 + (beta + (beta + (i * 4.0d0)))) / alpha) / 2.0d0
    else
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 3.2e+71) {
		tmp = (1.0 + ((beta - alpha) / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else if ((alpha <= 2.4e+90) || !(alpha <= 1.75e+109)) {
		tmp = ((2.0 + (beta + (beta + (i * 4.0)))) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 3.2e+71:
		tmp = (1.0 + ((beta - alpha) / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0
	elif (alpha <= 2.4e+90) or not (alpha <= 1.75e+109):
		tmp = ((2.0 + (beta + (beta + (i * 4.0)))) / alpha) / 2.0
	else:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 3.2e+71)
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / Float64(2.0 + Float64(Float64(alpha + beta) + Float64(2.0 * i))))) / 2.0);
	elseif ((alpha <= 2.4e+90) || !(alpha <= 1.75e+109))
		tmp = Float64(Float64(Float64(2.0 + Float64(beta + Float64(beta + Float64(i * 4.0)))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 3.2e+71)
		tmp = (1.0 + ((beta - alpha) / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	elseif ((alpha <= 2.4e+90) || ~((alpha <= 1.75e+109)))
		tmp = ((2.0 + (beta + (beta + (i * 4.0)))) / alpha) / 2.0;
	else
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 3.2e+71], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[alpha, 2.4e+90], N[Not[LessEqual[alpha, 1.75e+109]], $MachinePrecision]], N[(N[(N[(2.0 + N[(beta + N[(beta + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 3.20000000000000023e71

   1. Initial program 76.4%

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

   2. Taylor expanded in i around 0 97.2%

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

   if 3.20000000000000023e71 < alpha < 2.4000000000000001e90 or 1.74999999999999992e109 < alpha

   1. Initial program 6.3%

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

   3. Simplified 23.9%

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

   4. Taylor expanded in alpha around inf 81.8%

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

   5. Taylor expanded in i around 0 81.9%

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

      1. associate--l+ 81.9%

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

      2. sub-neg 81.9%

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

      3. *-commutative 81.9%

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

      4. mul-1-neg 81.9%

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

      5. remove-double-neg 81.9%

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

   7. Simplified 81.9%

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

   if 2.4000000000000001e90 < alpha < 1.74999999999999992e109

   1. Initial program 44.8%

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

      1. associate-/l/ 43.4%

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

      2. associate-+l+ 43.4%

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

      3. associate-+l+ 43.4%

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

   3. Simplified 43.4%

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

   4. Taylor expanded in i around 0 58.6%

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

      1. associate-+r+ 58.6%

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

   6. Simplified 58.6%

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

   7. Taylor expanded in alpha around 0 86.5%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 3.2 \cdot 10^{+71}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 2.4 \cdot 10^{+90} \lor \neg \left(\alpha \leq 1.75 \cdot 10^{+109}\right):\\ \;\;\;\;\frac{\frac{2 + \left(\beta + \left(\beta + i \cdot 4\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \end{array} \]

Alternative 6: 79.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 4.9 \cdot 10^{+70} \lor \neg \left(\alpha \leq 2.15 \cdot 10^{+92}\right) \land \alpha \leq 5.3 \cdot 10^{+107}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (or (<= alpha 4.9e+70)
         (and (not (<= alpha 2.15e+92)) (<= alpha 5.3e+107)))
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (+ 2.0 (* i 4.0)) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if ((alpha <= 4.9e+70) || (!(alpha <= 2.15e+92) && (alpha <= 5.3e+107))) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 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 <= 4.9d+70) .or. (.not. (alpha <= 2.15d+92)) .and. (alpha <= 5.3d+107)) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((2.0d0 + (i * 4.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if ((alpha <= 4.9e+70) || (!(alpha <= 2.15e+92) && (alpha <= 5.3e+107))) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if (alpha <= 4.9e+70) or (not (alpha <= 2.15e+92) and (alpha <= 5.3e+107)):
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if ((alpha <= 4.9e+70) || (!(alpha <= 2.15e+92) && (alpha <= 5.3e+107)))
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(i * 4.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if ((alpha <= 4.9e+70) || (~((alpha <= 2.15e+92)) && (alpha <= 5.3e+107)))
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[Or[LessEqual[alpha, 4.9e+70], And[N[Not[LessEqual[alpha, 2.15e+92]], $MachinePrecision], LessEqual[alpha, 5.3e+107]]], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 4.90000000000000028e70 or 2.1499999999999999e92 < alpha < 5.3e107

   1. Initial program 75.3%

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

      1. associate-/l/ 74.6%

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

      2. associate-+l+ 74.6%

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

      3. associate-+l+ 74.6%

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

   3. Simplified 74.6%

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

   4. Taylor expanded in i around 0 81.6%

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

      1. associate-+r+ 81.6%

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

   6. Simplified 81.6%

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

   7. Taylor expanded in alpha around 0 86.7%

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

   if 4.90000000000000028e70 < alpha < 2.1499999999999999e92 or 5.3e107 < alpha

   1. Initial program 6.3%

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

   3. Simplified 23.9%

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

   4. Taylor expanded in alpha around inf 81.8%

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

   5. Taylor expanded in i around 0 81.9%

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

      1. associate--l+ 81.9%

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

      2. sub-neg 81.9%

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

      3. *-commutative 81.9%

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

      4. mul-1-neg 81.9%

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

      5. remove-double-neg 81.9%

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

   7. Simplified 81.9%

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

   8. Taylor expanded in beta around 0 69.9%

      \[\leadsto \frac{\frac{2 + \color{blue}{4 \cdot i}}{\alpha}}{2} \]
   9. Step-by-step derivation

      1. *-commutative 69.9%

         \[\leadsto \frac{\frac{2 + \color{blue}{i \cdot 4}}{\alpha}}{2} \]

   10. Simplified 69.9%

       \[\leadsto \frac{\frac{2 + \color{blue}{i \cdot 4}}{\alpha}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 4.9 \cdot 10^{+70} \lor \neg \left(\alpha \leq 2.15 \cdot 10^{+92}\right) \land \alpha \leq 5.3 \cdot 10^{+107}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 7: 75.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 2 \cdot 10^{+98}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= i 2e+98) (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0) 0.5))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 2e+98) {
		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 <= 2d+98) 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 <= 2e+98) {
		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 <= 2e+98:
		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 <= 2e+98)
		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 <= 2e+98)
		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, 2e+98], 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 < 2e98

   1. Initial program 55.7%

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

      1. associate-/l/ 54.9%

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

      2. associate-+l+ 54.9%

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

      3. associate-+l+ 54.9%

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

   3. Simplified 54.9%

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

   4. Taylor expanded in i around 0 69.9%

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

      1. associate-+r+ 69.9%

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

   6. Simplified 69.9%

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

   7. Taylor expanded in alpha around 0 69.7%

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

   if 2e98 < i

   1. Initial program 65.0%

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

      1. associate-/l/ 64.2%

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

      2. associate-+l+ 64.2%

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

      3. associate-+l+ 64.2%

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

   3. Simplified 64.2%

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

   4. Taylor expanded in i around inf 64.2%

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

   5. Taylor expanded in i around inf 81.4%

      \[\leadsto \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 2 \cdot 10^{+98}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 8: 72.2% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.2 \cdot 10^{+75}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 (if (<= beta 4.2e+75) 0.5 1.0))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 4.2e+75) {
		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 <= 4.2d+75) 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 <= 4.2e+75) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 4.2e+75:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 4.2e+75)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 4.2e+75)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 4.2e+75], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 4.19999999999999997e75

   1. Initial program 72.3%

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

      1. associate-/l/ 72.1%

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

      2. associate-+l+ 72.1%

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

      3. associate-+l+ 72.1%

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

   3. Simplified 72.1%

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

   4. Taylor expanded in i around inf 47.8%

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

   5. Taylor expanded in i around inf 70.4%

      \[\leadsto \color{blue}{0.5} \]

   if 4.19999999999999997e75 < beta

   1. Initial program 24.8%

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

      1. associate-/l/ 22.5%

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

      2. associate-+l+ 22.5%

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

      3. associate-+l+ 22.5%

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

   3. Simplified 22.5%

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

   4. Taylor expanded in beta around inf 72.9%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.2 \cdot 10^{+75}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9: 61.7% 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 59.1%

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

   1. associate-/l/ 58.4%

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

   2. associate-+l+ 58.4%

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

   3. associate-+l+ 58.4%

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

3. Simplified 58.4%

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

4. Taylor expanded in i around inf 37.8%

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

5. Taylor expanded in i around inf 60.8%

   \[\leadsto \color{blue}{0.5} \]

6. Final simplification 60.8%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023334 
(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))