Octave 3.8, jcobi/2

Percentage Accurate: 62.5% → 97.7%

Time: 21.4s

Alternatives: 13

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: 62.5% 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: 97.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.9999998:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(2 + \left(\beta \cdot 2 + i \cdot 4\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1}{2}\\ \end{array} \end{array} \]

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))
        -0.9999998)
     (/ (/ (+ (- beta beta) (+ 2.0 (+ (* beta 2.0) (* i 4.0)))) alpha) 2.0)
     (/
      (+
       (*
        (/ (+ alpha beta) (+ (+ alpha beta) (fma 2.0 i 2.0)))
        (/ (- beta alpha) (fma 2.0 i (+ alpha beta))))
       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)) <= -0.9999998) {
		tmp = (((beta - beta) + (2.0 + ((beta * 2.0) + (i * 4.0)))) / alpha) / 2.0;
	} else {
		tmp = ((((alpha + beta) / ((alpha + beta) + fma(2.0, i, 2.0))) * ((beta - alpha) / fma(2.0, i, (alpha + beta)))) + 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)) <= -0.9999998)
		tmp = Float64(Float64(Float64(Float64(beta - beta) + 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(alpha + beta) + fma(2.0, i, 2.0))) * Float64(Float64(beta - alpha) / fma(2.0, i, Float64(alpha + beta)))) + 1.0) / 2.0);
	end
	return 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], -0.9999998], N[(N[(N[(N[(beta - beta), $MachinePrecision] + N[(2.0 + N[(N[(beta * 2.0), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 * i + N[(alpha + beta), $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)) < -0.999999799999999994

   1. Initial program 2.4%

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

   3. Simplified 15.6%

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

   5. Taylor expanded in alpha around inf 90.2%

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

   if -0.999999799999999994 < (/.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 81.0%

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

   3. Simplified 99.8%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.9999998:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(2 + \left(\beta \cdot 2 + i \cdot 4\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.9999998:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(2 + \left(\beta \cdot 2 + i \cdot 4\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\frac{\alpha + \beta}{\frac{\alpha + \left(\beta + 2 \cdot i\right)}{\beta - \alpha}}}{\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)}}{2}\\ \end{array} \end{array} \]

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))
        -0.9999998)
     (/ (/ (+ (- beta beta) (+ 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)) <= -0.9999998) {
		tmp = (((beta - beta) + (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)) <= (-0.9999998d0)) then
        tmp = (((beta - beta) + (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)) <= -0.9999998) {
		tmp = (((beta - beta) + (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)) <= -0.9999998:
		tmp = (((beta - beta) + (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)) <= -0.9999998)
		tmp = Float64(Float64(Float64(Float64(beta - beta) + 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)) <= -0.9999998)
		tmp = (((beta - beta) + (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], -0.9999998], N[(N[(N[(N[(beta - beta), $MachinePrecision] + N[(2.0 + N[(N[(beta * 2.0), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $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)) < -0.999999799999999994

   1. Initial program 2.4%

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

   3. Simplified 15.6%

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

   5. Taylor expanded in alpha around inf 90.2%

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

   if -0.999999799999999994 < (/.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 81.0%

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

      1. associate-/l* 99.8%

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

      2. associate-+l+ 99.8%

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

      3. associate-+l+ 99.8%

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

   3. Simplified 99.8%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.9999998:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(2 + \left(\beta \cdot 2 + i \cdot 4\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\frac{\alpha + \beta}{\frac{\alpha + \left(\beta + 2 \cdot i\right)}{\beta - \alpha}}}{\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.7% accurate, 0.6× speedup

Math

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

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)) -0.5)
     (/ (/ (+ (- beta beta) (+ 2.0 (+ (* beta 2.0) (* i 4.0)))) alpha) 2.0)
     (/
      (+
       1.0
       (/
        1.0
        (* (/ (+ 2.0 (+ beta (* 2.0 i))) beta) (+ 1.0 (* 2.0 (/ i beta))))))
      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)) <= -0.5) {
		tmp = (((beta - beta) + (2.0 + ((beta * 2.0) + (i * 4.0)))) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (1.0 / (((2.0 + (beta + (2.0 * i))) / beta) * (1.0 + (2.0 * (i / beta)))))) / 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)) <= (-0.5d0)) then
        tmp = (((beta - beta) + (2.0d0 + ((beta * 2.0d0) + (i * 4.0d0)))) / alpha) / 2.0d0
    else
        tmp = (1.0d0 + (1.0d0 / (((2.0d0 + (beta + (2.0d0 * i))) / beta) * (1.0d0 + (2.0d0 * (i / beta)))))) / 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)) <= -0.5) {
		tmp = (((beta - beta) + (2.0 + ((beta * 2.0) + (i * 4.0)))) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (1.0 / (((2.0 + (beta + (2.0 * i))) / beta) * (1.0 + (2.0 * (i / beta)))))) / 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)) <= -0.5:
		tmp = (((beta - beta) + (2.0 + ((beta * 2.0) + (i * 4.0)))) / alpha) / 2.0
	else:
		tmp = (1.0 + (1.0 / (((2.0 + (beta + (2.0 * i))) / beta) * (1.0 + (2.0 * (i / beta)))))) / 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)) <= -0.5)
		tmp = Float64(Float64(Float64(Float64(beta - beta) + Float64(2.0 + Float64(Float64(beta * 2.0) + Float64(i * 4.0)))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(1.0 / Float64(Float64(Float64(2.0 + Float64(beta + Float64(2.0 * i))) / beta) * Float64(1.0 + Float64(2.0 * Float64(i / beta)))))) / 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)) <= -0.5)
		tmp = (((beta - beta) + (2.0 + ((beta * 2.0) + (i * 4.0)))) / alpha) / 2.0;
	else
		tmp = (1.0 + (1.0 / (((2.0 + (beta + (2.0 * i))) / beta) * (1.0 + (2.0 * (i / beta)))))) / 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], -0.5], N[(N[(N[(N[(beta - beta), $MachinePrecision] + N[(2.0 + N[(N[(beta * 2.0), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(1.0 / N[(N[(N[(2.0 + N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision] * N[(1.0 + N[(2.0 * N[(i / beta), $MachinePrecision]), $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)) < -0.5

   1. Initial program 3.7%

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

   3. Simplified 16.6%

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

   5. Taylor expanded in alpha around inf 89.5%

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

   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 81.0%

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

   3. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. clear-num 100.0%

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

      3. fma-udef 100.0%

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

      4. +-commutative 100.0%

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

      5. associate-+r+ 100.0%

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

      6. frac-times 100.0%

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

      7. metadata-eval 100.0%

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

      8. +-commutative 100.0%

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

      9. fma-def 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in alpha around 0 99.3%

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

   8. Taylor expanded in alpha around 0 99.3%

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

   9. Taylor expanded in beta around 0 99.3%

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

4. Final simplification 97.3%

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

Alternative 4: 77.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.3 \cdot 10^{+23}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 9.5 \cdot 10^{+60}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 7.6 \cdot 10^{+71}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 7.8 \cdot 10^{+250}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1.3e+23)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (if (<= alpha 9.5e+60)
     (/ (/ (+ 2.0 (* i 4.0)) alpha) 2.0)
     (if (<= alpha 7.6e+71)
       1.0
       (if (<= alpha 7.8e+250)
         (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)
         (/ (+ (* 4.0 (/ i alpha)) (* 2.0 (/ 1.0 alpha))) 2.0))))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.3e+23) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else if (alpha <= 9.5e+60) {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	} else if (alpha <= 7.6e+71) {
		tmp = 1.0;
	} else if (alpha <= 7.8e+250) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = ((4.0 * (i / alpha)) + (2.0 * (1.0 / 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 <= 1.3d+23) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else if (alpha <= 9.5d+60) then
        tmp = ((2.0d0 + (i * 4.0d0)) / alpha) / 2.0d0
    else if (alpha <= 7.6d+71) then
        tmp = 1.0d0
    else if (alpha <= 7.8d+250) then
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    else
        tmp = ((4.0d0 * (i / alpha)) + (2.0d0 * (1.0d0 / alpha))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.3e+23) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else if (alpha <= 9.5e+60) {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	} else if (alpha <= 7.6e+71) {
		tmp = 1.0;
	} else if (alpha <= 7.8e+250) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = ((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 1.3e+23:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	elif alpha <= 9.5e+60:
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0
	elif alpha <= 7.6e+71:
		tmp = 1.0
	elif alpha <= 7.8e+250:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	else:
		tmp = ((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha))) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 1.3e+23)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	elseif (alpha <= 9.5e+60)
		tmp = Float64(Float64(Float64(2.0 + Float64(i * 4.0)) / alpha) / 2.0);
	elseif (alpha <= 7.6e+71)
		tmp = 1.0;
	elseif (alpha <= 7.8e+250)
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(4.0 * Float64(i / alpha)) + Float64(2.0 * Float64(1.0 / alpha))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 1.3e+23)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	elseif (alpha <= 9.5e+60)
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	elseif (alpha <= 7.6e+71)
		tmp = 1.0;
	elseif (alpha <= 7.8e+250)
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	else
		tmp = ((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 1.3e+23], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 9.5e+60], N[(N[(N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 7.6e+71], 1.0, If[LessEqual[alpha, 7.8e+250], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(4.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if alpha < 1.29999999999999996e23

   1. Initial program 82.6%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in i around 0 87.9%

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

      1. +-commutative 87.9%

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

   7. Simplified 87.9%

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

   8. Taylor expanded in alpha around 0 88.4%

      \[\leadsto \frac{\color{blue}{\frac{\beta}{2 + \beta}} + 1}{2} \]
   9. Step-by-step derivation

      1. +-commutative 88.4%

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

   10. Simplified 88.4%

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

   if 1.29999999999999996e23 < alpha < 9.49999999999999988e60

   1. Initial program 17.9%

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

      1. associate-/l* 17.2%

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

      2. associate-+l+ 17.2%

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

      3. associate-+l+ 17.2%

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

   3. Simplified 17.2%

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

   5. Taylor expanded in beta around 0 17.2%

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

      1. associate-*r/ 17.2%

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

      2. mul-1-neg 17.2%

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

      3. +-commutative 17.2%

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

   7. Simplified 17.2%

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

   8. Taylor expanded in alpha around inf 87.6%

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

      1. *-commutative 87.6%

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

   10. Simplified 87.6%

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

   if 9.49999999999999988e60 < alpha < 7.6000000000000001e71

   1. Initial program 35.4%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in beta around inf 100.0%

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

   if 7.6000000000000001e71 < alpha < 7.8e250

   1. Initial program 19.3%

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

   3. Simplified 41.4%

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

   5. Taylor expanded in i around 0 13.8%

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

      1. +-commutative 13.8%

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

   7. Simplified 13.8%

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

   8. Taylor expanded in alpha around inf 58.0%

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

      1. *-commutative 58.0%

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

   10. Simplified 58.0%

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

   if 7.8e250 < alpha

   1. Initial program 1.2%

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

      1. associate-/l* 6.4%

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

      2. associate-+l+ 6.4%

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

      3. associate-+l+ 6.4%

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

   3. Simplified 6.4%

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

   5. Taylor expanded in beta around 0 6.4%

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

      1. associate-*r/ 6.4%

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

      2. mul-1-neg 6.4%

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

      3. +-commutative 6.4%

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

   7. Simplified 6.4%

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

   8. Taylor expanded in alpha around inf 83.9%

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

      1. *-commutative 83.9%

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

   10. Simplified 83.9%

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

   11. Taylor expanded in i around 0 84.0%

       \[\leadsto \frac{\color{blue}{4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}}}{2} \]
3. Recombined 5 regimes into one program.

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.3 \cdot 10^{+23}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 9.5 \cdot 10^{+60}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 7.6 \cdot 10^{+71}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 7.8 \cdot 10^{+250}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 5.5 \cdot 10^{+24}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\beta + 2 \cdot i\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 2.1 \cdot 10^{+110}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 3.1 \cdot 10^{+124}:\\ \;\;\;\;\frac{1 + \frac{1}{1 + \frac{i \cdot 4}{\beta}}}{2}\\ \mathbf{elif}\;\alpha \leq 4.9 \cdot 10^{+249}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 5.5e+24)
   (/ (+ 1.0 (/ beta (+ 2.0 (+ beta (* 2.0 i))))) 2.0)
   (if (<= alpha 2.1e+110)
     (/ (/ (+ 2.0 (* i 4.0)) alpha) 2.0)
     (if (<= alpha 3.1e+124)
       (/ (+ 1.0 (/ 1.0 (+ 1.0 (/ (* i 4.0) beta)))) 2.0)
       (if (<= alpha 4.9e+249)
         (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)
         (/ (+ (* 4.0 (/ i alpha)) (* 2.0 (/ 1.0 alpha))) 2.0))))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 5.5e+24) {
		tmp = (1.0 + (beta / (2.0 + (beta + (2.0 * i))))) / 2.0;
	} else if (alpha <= 2.1e+110) {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	} else if (alpha <= 3.1e+124) {
		tmp = (1.0 + (1.0 / (1.0 + ((i * 4.0) / beta)))) / 2.0;
	} else if (alpha <= 4.9e+249) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = ((4.0 * (i / alpha)) + (2.0 * (1.0 / 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 <= 5.5d+24) then
        tmp = (1.0d0 + (beta / (2.0d0 + (beta + (2.0d0 * i))))) / 2.0d0
    else if (alpha <= 2.1d+110) then
        tmp = ((2.0d0 + (i * 4.0d0)) / alpha) / 2.0d0
    else if (alpha <= 3.1d+124) then
        tmp = (1.0d0 + (1.0d0 / (1.0d0 + ((i * 4.0d0) / beta)))) / 2.0d0
    else if (alpha <= 4.9d+249) then
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    else
        tmp = ((4.0d0 * (i / alpha)) + (2.0d0 * (1.0d0 / alpha))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 5.5e+24) {
		tmp = (1.0 + (beta / (2.0 + (beta + (2.0 * i))))) / 2.0;
	} else if (alpha <= 2.1e+110) {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	} else if (alpha <= 3.1e+124) {
		tmp = (1.0 + (1.0 / (1.0 + ((i * 4.0) / beta)))) / 2.0;
	} else if (alpha <= 4.9e+249) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = ((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 5.5e+24:
		tmp = (1.0 + (beta / (2.0 + (beta + (2.0 * i))))) / 2.0
	elif alpha <= 2.1e+110:
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0
	elif alpha <= 3.1e+124:
		tmp = (1.0 + (1.0 / (1.0 + ((i * 4.0) / beta)))) / 2.0
	elif alpha <= 4.9e+249:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	else:
		tmp = ((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha))) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 5.5e+24)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(2.0 + Float64(beta + Float64(2.0 * i))))) / 2.0);
	elseif (alpha <= 2.1e+110)
		tmp = Float64(Float64(Float64(2.0 + Float64(i * 4.0)) / alpha) / 2.0);
	elseif (alpha <= 3.1e+124)
		tmp = Float64(Float64(1.0 + Float64(1.0 / Float64(1.0 + Float64(Float64(i * 4.0) / beta)))) / 2.0);
	elseif (alpha <= 4.9e+249)
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(4.0 * Float64(i / alpha)) + Float64(2.0 * Float64(1.0 / alpha))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 5.5e+24)
		tmp = (1.0 + (beta / (2.0 + (beta + (2.0 * i))))) / 2.0;
	elseif (alpha <= 2.1e+110)
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	elseif (alpha <= 3.1e+124)
		tmp = (1.0 + (1.0 / (1.0 + ((i * 4.0) / beta)))) / 2.0;
	elseif (alpha <= 4.9e+249)
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	else
		tmp = ((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 5.5e+24], N[(N[(1.0 + N[(beta / N[(2.0 + N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 2.1e+110], N[(N[(N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 3.1e+124], N[(N[(1.0 + N[(1.0 / N[(1.0 + N[(N[(i * 4.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 4.9e+249], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(4.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if alpha < 5.5000000000000002e24

   1. Initial program 82.6%

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

   3. Taylor expanded in beta around inf 98.4%

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

   4. Taylor expanded in alpha around 0 98.4%

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

   if 5.5000000000000002e24 < alpha < 2.10000000000000015e110

   1. Initial program 22.5%

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

      1. associate-/l* 32.4%

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

      2. associate-+l+ 32.4%

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

      3. associate-+l+ 32.4%

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

   3. Simplified 32.4%

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

   5. Taylor expanded in beta around 0 13.7%

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

      1. associate-*r/ 13.7%

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

      2. mul-1-neg 13.7%

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

      3. +-commutative 13.7%

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

   7. Simplified 13.7%

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

   8. Taylor expanded in alpha around inf 66.1%

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

      1. *-commutative 66.1%

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

   10. Simplified 66.1%

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

   if 2.10000000000000015e110 < alpha < 3.1000000000000002e124

   1. Initial program 51.7%

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

   3. Simplified 84.0%

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

      1. clear-num 84.0%

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

      2. clear-num 84.0%

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

      3. fma-udef 84.0%

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

      4. +-commutative 84.0%

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

      5. associate-+r+ 84.0%

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

      6. frac-times 84.0%

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

      7. metadata-eval 84.0%

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

      8. +-commutative 84.0%

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

      9. fma-def 84.0%

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

   6. Applied egg-rr 84.0%

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

   7. Taylor expanded in alpha around 0 84.2%

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

   8. Taylor expanded in beta around inf 76.8%

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

   9. Taylor expanded in i around inf 76.8%

      \[\leadsto \frac{\frac{1}{1 + \color{blue}{4 \cdot \frac{i}{\beta}}} + 1}{2} \]
   10. Step-by-step derivation

       1. associate-*r/ 76.8%

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

       2. *-commutative 76.8%

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

   11. Simplified 76.8%

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

   if 3.1000000000000002e124 < alpha < 4.8999999999999996e249

   1. Initial program 12.0%

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

   3. Simplified 38.8%

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

   5. Taylor expanded in i around 0 13.0%

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

      1. +-commutative 13.0%

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

   7. Simplified 13.0%

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

   8. Taylor expanded in alpha around inf 60.4%

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

      1. *-commutative 60.4%

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

   10. Simplified 60.4%

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

   if 4.8999999999999996e249 < alpha

   1. Initial program 1.2%

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

      1. associate-/l* 6.4%

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

      2. associate-+l+ 6.4%

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

      3. associate-+l+ 6.4%

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

   3. Simplified 6.4%

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

   5. Taylor expanded in beta around 0 6.4%

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

      1. associate-*r/ 6.4%

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

      2. mul-1-neg 6.4%

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

      3. +-commutative 6.4%

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

   7. Simplified 6.4%

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

   8. Taylor expanded in alpha around inf 83.9%

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

      1. *-commutative 83.9%

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

   10. Simplified 83.9%

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

   11. Taylor expanded in i around 0 84.0%

       \[\leadsto \frac{\color{blue}{4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}}}{2} \]
3. Recombined 5 regimes into one program.

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 5.5 \cdot 10^{+24}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\beta + 2 \cdot i\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 2.1 \cdot 10^{+110}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 3.1 \cdot 10^{+124}:\\ \;\;\;\;\frac{1 + \frac{1}{1 + \frac{i \cdot 4}{\beta}}}{2}\\ \mathbf{elif}\;\alpha \leq 4.9 \cdot 10^{+249}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + i \cdot 4\\ \mathbf{if}\;\alpha \leq 5.5 \cdot 10^{+24}:\\ \;\;\;\;\frac{1 + \frac{1}{1 + \frac{t_0}{\beta}}}{2}\\ \mathbf{elif}\;\alpha \leq 4.4 \cdot 10^{+110}:\\ \;\;\;\;\frac{\frac{t_0}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 5.5 \cdot 10^{+124}:\\ \;\;\;\;\frac{1 + \frac{1}{1 + \frac{i \cdot 4}{\beta}}}{2}\\ \mathbf{elif}\;\alpha \leq 4.9 \cdot 10^{+249}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ 2.0 (* i 4.0))))
   (if (<= alpha 5.5e+24)
     (/ (+ 1.0 (/ 1.0 (+ 1.0 (/ t_0 beta)))) 2.0)
     (if (<= alpha 4.4e+110)
       (/ (/ t_0 alpha) 2.0)
       (if (<= alpha 5.5e+124)
         (/ (+ 1.0 (/ 1.0 (+ 1.0 (/ (* i 4.0) beta)))) 2.0)
         (if (<= alpha 4.9e+249)
           (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)
           (/ (+ (* 4.0 (/ i alpha)) (* 2.0 (/ 1.0 alpha))) 2.0)))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = 2.0 + (i * 4.0);
	double tmp;
	if (alpha <= 5.5e+24) {
		tmp = (1.0 + (1.0 / (1.0 + (t_0 / beta)))) / 2.0;
	} else if (alpha <= 4.4e+110) {
		tmp = (t_0 / alpha) / 2.0;
	} else if (alpha <= 5.5e+124) {
		tmp = (1.0 + (1.0 / (1.0 + ((i * 4.0) / beta)))) / 2.0;
	} else if (alpha <= 4.9e+249) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = ((4.0 * (i / alpha)) + (2.0 * (1.0 / 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) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 + (i * 4.0d0)
    if (alpha <= 5.5d+24) then
        tmp = (1.0d0 + (1.0d0 / (1.0d0 + (t_0 / beta)))) / 2.0d0
    else if (alpha <= 4.4d+110) then
        tmp = (t_0 / alpha) / 2.0d0
    else if (alpha <= 5.5d+124) then
        tmp = (1.0d0 + (1.0d0 / (1.0d0 + ((i * 4.0d0) / beta)))) / 2.0d0
    else if (alpha <= 4.9d+249) then
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    else
        tmp = ((4.0d0 * (i / alpha)) + (2.0d0 * (1.0d0 / alpha))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = 2.0 + (i * 4.0);
	double tmp;
	if (alpha <= 5.5e+24) {
		tmp = (1.0 + (1.0 / (1.0 + (t_0 / beta)))) / 2.0;
	} else if (alpha <= 4.4e+110) {
		tmp = (t_0 / alpha) / 2.0;
	} else if (alpha <= 5.5e+124) {
		tmp = (1.0 + (1.0 / (1.0 + ((i * 4.0) / beta)))) / 2.0;
	} else if (alpha <= 4.9e+249) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = ((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = 2.0 + (i * 4.0)
	tmp = 0
	if alpha <= 5.5e+24:
		tmp = (1.0 + (1.0 / (1.0 + (t_0 / beta)))) / 2.0
	elif alpha <= 4.4e+110:
		tmp = (t_0 / alpha) / 2.0
	elif alpha <= 5.5e+124:
		tmp = (1.0 + (1.0 / (1.0 + ((i * 4.0) / beta)))) / 2.0
	elif alpha <= 4.9e+249:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	else:
		tmp = ((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha))) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(2.0 + Float64(i * 4.0))
	tmp = 0.0
	if (alpha <= 5.5e+24)
		tmp = Float64(Float64(1.0 + Float64(1.0 / Float64(1.0 + Float64(t_0 / beta)))) / 2.0);
	elseif (alpha <= 4.4e+110)
		tmp = Float64(Float64(t_0 / alpha) / 2.0);
	elseif (alpha <= 5.5e+124)
		tmp = Float64(Float64(1.0 + Float64(1.0 / Float64(1.0 + Float64(Float64(i * 4.0) / beta)))) / 2.0);
	elseif (alpha <= 4.9e+249)
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(4.0 * Float64(i / alpha)) + Float64(2.0 * Float64(1.0 / alpha))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = 2.0 + (i * 4.0);
	tmp = 0.0;
	if (alpha <= 5.5e+24)
		tmp = (1.0 + (1.0 / (1.0 + (t_0 / beta)))) / 2.0;
	elseif (alpha <= 4.4e+110)
		tmp = (t_0 / alpha) / 2.0;
	elseif (alpha <= 5.5e+124)
		tmp = (1.0 + (1.0 / (1.0 + ((i * 4.0) / beta)))) / 2.0;
	elseif (alpha <= 4.9e+249)
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	else
		tmp = ((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[alpha, 5.5e+24], N[(N[(1.0 + N[(1.0 / N[(1.0 + N[(t$95$0 / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 4.4e+110], N[(N[(t$95$0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 5.5e+124], N[(N[(1.0 + N[(1.0 / N[(1.0 + N[(N[(i * 4.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 4.9e+249], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(4.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if alpha < 5.5000000000000002e24

   1. Initial program 82.6%

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

   3. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. clear-num 100.0%

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

      3. fma-udef 100.0%

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

      4. +-commutative 100.0%

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

      5. associate-+r+ 100.0%

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

      6. frac-times 100.0%

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

      7. metadata-eval 100.0%

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

      8. +-commutative 100.0%

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

      9. fma-def 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in alpha around 0 99.5%

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

   8. Taylor expanded in beta around inf 98.7%

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

   9. Taylor expanded in beta around 0 98.7%

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

   if 5.5000000000000002e24 < alpha < 4.39999999999999984e110

   1. Initial program 22.5%

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

      1. associate-/l* 32.4%

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

      2. associate-+l+ 32.4%

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

      3. associate-+l+ 32.4%

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

   3. Simplified 32.4%

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

   5. Taylor expanded in beta around 0 13.7%

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

      1. associate-*r/ 13.7%

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

      2. mul-1-neg 13.7%

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

      3. +-commutative 13.7%

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

   7. Simplified 13.7%

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

   8. Taylor expanded in alpha around inf 66.1%

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

      1. *-commutative 66.1%

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

   10. Simplified 66.1%

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

   if 4.39999999999999984e110 < alpha < 5.49999999999999977e124

   1. Initial program 51.7%

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

   3. Simplified 84.0%

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

      1. clear-num 84.0%

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

      2. clear-num 84.0%

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

      3. fma-udef 84.0%

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

      4. +-commutative 84.0%

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

      5. associate-+r+ 84.0%

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

      6. frac-times 84.0%

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

      7. metadata-eval 84.0%

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

      8. +-commutative 84.0%

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

      9. fma-def 84.0%

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

   6. Applied egg-rr 84.0%

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

   7. Taylor expanded in alpha around 0 84.2%

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

   8. Taylor expanded in beta around inf 76.8%

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

   9. Taylor expanded in i around inf 76.8%

      \[\leadsto \frac{\frac{1}{1 + \color{blue}{4 \cdot \frac{i}{\beta}}} + 1}{2} \]
   10. Step-by-step derivation

       1. associate-*r/ 76.8%

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

       2. *-commutative 76.8%

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

   11. Simplified 76.8%

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

   if 5.49999999999999977e124 < alpha < 4.8999999999999996e249

   1. Initial program 12.0%

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

   3. Simplified 38.8%

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

   5. Taylor expanded in i around 0 13.0%

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

      1. +-commutative 13.0%

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

   7. Simplified 13.0%

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

   8. Taylor expanded in alpha around inf 60.4%

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

      1. *-commutative 60.4%

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

   10. Simplified 60.4%

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

   if 4.8999999999999996e249 < alpha

   1. Initial program 1.2%

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

      1. associate-/l* 6.4%

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

      2. associate-+l+ 6.4%

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

      3. associate-+l+ 6.4%

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

   3. Simplified 6.4%

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

   5. Taylor expanded in beta around 0 6.4%

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

      1. associate-*r/ 6.4%

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

      2. mul-1-neg 6.4%

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

      3. +-commutative 6.4%

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

   7. Simplified 6.4%

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

   8. Taylor expanded in alpha around inf 83.9%

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

      1. *-commutative 83.9%

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

   10. Simplified 83.9%

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

   11. Taylor expanded in i around 0 84.0%

       \[\leadsto \frac{\color{blue}{4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}}}{2} \]
3. Recombined 5 regimes into one program.

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 5.5 \cdot 10^{+24}:\\ \;\;\;\;\frac{1 + \frac{1}{1 + \frac{2 + i \cdot 4}{\beta}}}{2}\\ \mathbf{elif}\;\alpha \leq 4.4 \cdot 10^{+110}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 5.5 \cdot 10^{+124}:\\ \;\;\;\;\frac{1 + \frac{1}{1 + \frac{i \cdot 4}{\beta}}}{2}\\ \mathbf{elif}\;\alpha \leq 4.9 \cdot 10^{+249}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \mathbf{if}\;\alpha \leq 3.5 \cdot 10^{+24}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 6.5 \cdot 10^{+59}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 7.2 \cdot 10^{+71}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 5.1 \cdot 10^{+249}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (/ (/ (+ 2.0 (* i 4.0)) alpha) 2.0)))
   (if (<= alpha 3.5e+24)
     (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
     (if (<= alpha 6.5e+59)
       t_0
       (if (<= alpha 7.2e+71)
         1.0
         (if (<= alpha 5.1e+249)
           (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)
           t_0))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	double tmp;
	if (alpha <= 3.5e+24) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else if (alpha <= 6.5e+59) {
		tmp = t_0;
	} else if (alpha <= 7.2e+71) {
		tmp = 1.0;
	} else if (alpha <= 5.1e+249) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = t_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 = ((2.0d0 + (i * 4.0d0)) / alpha) / 2.0d0
    if (alpha <= 3.5d+24) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else if (alpha <= 6.5d+59) then
        tmp = t_0
    else if (alpha <= 7.2d+71) then
        tmp = 1.0d0
    else if (alpha <= 5.1d+249) then
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	double tmp;
	if (alpha <= 3.5e+24) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else if (alpha <= 6.5e+59) {
		tmp = t_0;
	} else if (alpha <= 7.2e+71) {
		tmp = 1.0;
	} else if (alpha <= 5.1e+249) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = ((2.0 + (i * 4.0)) / alpha) / 2.0
	tmp = 0
	if alpha <= 3.5e+24:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	elif alpha <= 6.5e+59:
		tmp = t_0
	elif alpha <= 7.2e+71:
		tmp = 1.0
	elif alpha <= 5.1e+249:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	else:
		tmp = t_0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(Float64(2.0 + Float64(i * 4.0)) / alpha) / 2.0)
	tmp = 0.0
	if (alpha <= 3.5e+24)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	elseif (alpha <= 6.5e+59)
		tmp = t_0;
	elseif (alpha <= 7.2e+71)
		tmp = 1.0;
	elseif (alpha <= 5.1e+249)
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	tmp = 0.0;
	if (alpha <= 3.5e+24)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	elseif (alpha <= 6.5e+59)
		tmp = t_0;
	elseif (alpha <= 7.2e+71)
		tmp = 1.0;
	elseif (alpha <= 5.1e+249)
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[alpha, 3.5e+24], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 6.5e+59], t$95$0, If[LessEqual[alpha, 7.2e+71], 1.0, If[LessEqual[alpha, 5.1e+249], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if alpha < 3.5000000000000002e24

   1. Initial program 82.6%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in i around 0 87.9%

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

      1. +-commutative 87.9%

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

   7. Simplified 87.9%

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

   8. Taylor expanded in alpha around 0 88.4%

      \[\leadsto \frac{\color{blue}{\frac{\beta}{2 + \beta}} + 1}{2} \]
   9. Step-by-step derivation

      1. +-commutative 88.4%

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

   10. Simplified 88.4%

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

   if 3.5000000000000002e24 < alpha < 6.50000000000000021e59 or 5.10000000000000039e249 < alpha

   1. Initial program 7.4%

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

      1. associate-/l* 10.4%

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

      2. associate-+l+ 10.4%

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

      3. associate-+l+ 10.4%

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

   3. Simplified 10.4%

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

   5. Taylor expanded in beta around 0 10.4%

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

      1. associate-*r/ 10.4%

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

      2. mul-1-neg 10.4%

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

      3. +-commutative 10.4%

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

   7. Simplified 10.4%

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

   8. Taylor expanded in alpha around inf 85.3%

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

      1. *-commutative 85.3%

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

   10. Simplified 85.3%

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

   if 6.50000000000000021e59 < alpha < 7.1999999999999999e71

   1. Initial program 35.4%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in beta around inf 100.0%

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

   if 7.1999999999999999e71 < alpha < 5.10000000000000039e249

   1. Initial program 19.3%

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

   3. Simplified 41.4%

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

   5. Taylor expanded in i around 0 13.8%

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

      1. +-commutative 13.8%

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

   7. Simplified 13.8%

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

   8. Taylor expanded in alpha around inf 58.0%

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

      1. *-commutative 58.0%

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

   10. Simplified 58.0%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 3.5 \cdot 10^{+24}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 6.5 \cdot 10^{+59}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 7.2 \cdot 10^{+71}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 5.1 \cdot 10^{+249}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 87.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 5.3 \cdot 10^{+23}:\\ \;\;\;\;\frac{1 + \frac{1}{1 + \frac{2 + i \cdot 4}{\beta}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(2 + \left(\beta \cdot 2 + i \cdot 4\right)\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 5.3e+23)
   (/ (+ 1.0 (/ 1.0 (+ 1.0 (/ (+ 2.0 (* i 4.0)) beta)))) 2.0)
   (/ (/ (+ (- beta beta) (+ 2.0 (+ (* beta 2.0) (* i 4.0)))) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 5.3e+23) {
		tmp = (1.0 + (1.0 / (1.0 + ((2.0 + (i * 4.0)) / beta)))) / 2.0;
	} else {
		tmp = (((beta - beta) + (2.0 + ((beta * 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 <= 5.3d+23) then
        tmp = (1.0d0 + (1.0d0 / (1.0d0 + ((2.0d0 + (i * 4.0d0)) / beta)))) / 2.0d0
    else
        tmp = (((beta - beta) + (2.0d0 + ((beta * 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 <= 5.3e+23) {
		tmp = (1.0 + (1.0 / (1.0 + ((2.0 + (i * 4.0)) / beta)))) / 2.0;
	} else {
		tmp = (((beta - beta) + (2.0 + ((beta * 2.0) + (i * 4.0)))) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 5.3e+23:
		tmp = (1.0 + (1.0 / (1.0 + ((2.0 + (i * 4.0)) / beta)))) / 2.0
	else:
		tmp = (((beta - beta) + (2.0 + ((beta * 2.0) + (i * 4.0)))) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 5.3e+23)
		tmp = Float64(Float64(1.0 + Float64(1.0 / Float64(1.0 + Float64(Float64(2.0 + Float64(i * 4.0)) / beta)))) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(beta - beta) + Float64(2.0 + Float64(Float64(beta * 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 <= 5.3e+23)
		tmp = (1.0 + (1.0 / (1.0 + ((2.0 + (i * 4.0)) / beta)))) / 2.0;
	else
		tmp = (((beta - beta) + (2.0 + ((beta * 2.0) + (i * 4.0)))) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 5.3e+23], N[(N[(1.0 + N[(1.0 / N[(1.0 + N[(N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(beta - beta), $MachinePrecision] + N[(2.0 + N[(N[(beta * 2.0), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 5.3000000000000001e23

   1. Initial program 82.6%

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

   3. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. clear-num 100.0%

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

      3. fma-udef 100.0%

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

      4. +-commutative 100.0%

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

      5. associate-+r+ 100.0%

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

      6. frac-times 100.0%

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

      7. metadata-eval 100.0%

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

      8. +-commutative 100.0%

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

      9. fma-def 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in alpha around 0 99.5%

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

   8. Taylor expanded in beta around inf 98.7%

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

   9. Taylor expanded in beta around 0 98.7%

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

   if 5.3000000000000001e23 < alpha

   1. Initial program 16.6%

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

   3. Simplified 35.2%

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

   5. Taylor expanded in alpha around inf 70.9%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 5.3 \cdot 10^{+23}:\\ \;\;\;\;\frac{1 + \frac{1}{1 + \frac{2 + i \cdot 4}{\beta}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(2 + \left(\beta \cdot 2 + i \cdot 4\right)\right)}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 6.2 \cdot 10^{-25}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\left(\alpha + \beta\right) + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{1}{1 + \frac{i \cdot 4}{\beta}}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= i 6.2e-25)
   (/ (+ 1.0 (/ beta (+ (+ alpha beta) 2.0))) 2.0)
   (/ (+ 1.0 (/ 1.0 (+ 1.0 (/ (* i 4.0) beta)))) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 6.2e-25) {
		tmp = (1.0 + (beta / ((alpha + beta) + 2.0))) / 2.0;
	} else {
		tmp = (1.0 + (1.0 / (1.0 + ((i * 4.0) / beta)))) / 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 (i <= 6.2d-25) then
        tmp = (1.0d0 + (beta / ((alpha + beta) + 2.0d0))) / 2.0d0
    else
        tmp = (1.0d0 + (1.0d0 / (1.0d0 + ((i * 4.0d0) / beta)))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 6.2e-25) {
		tmp = (1.0 + (beta / ((alpha + beta) + 2.0))) / 2.0;
	} else {
		tmp = (1.0 + (1.0 / (1.0 + ((i * 4.0) / beta)))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if i <= 6.2e-25:
		tmp = (1.0 + (beta / ((alpha + beta) + 2.0))) / 2.0
	else:
		tmp = (1.0 + (1.0 / (1.0 + ((i * 4.0) / beta)))) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (i <= 6.2e-25)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(Float64(alpha + beta) + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(1.0 / Float64(1.0 + Float64(Float64(i * 4.0) / beta)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (i <= 6.2e-25)
		tmp = (1.0 + (beta / ((alpha + beta) + 2.0))) / 2.0;
	else
		tmp = (1.0 + (1.0 / (1.0 + ((i * 4.0) / beta)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[i, 6.2e-25], N[(N[(1.0 + N[(beta / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(1.0 / N[(1.0 + N[(N[(i * 4.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 6.19999999999999989e-25

   1. Initial program 56.2%

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

   3. Taylor expanded in beta around inf 71.0%

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

   4. Taylor expanded in i around 0 71.0%

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

      1. +-commutative 71.0%

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

   6. Simplified 71.0%

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

   if 6.19999999999999989e-25 < i

   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

   3. Simplified 91.2%

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

      1. clear-num 91.2%

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

      2. clear-num 91.2%

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

      3. fma-udef 91.2%

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

      4. +-commutative 91.2%

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

      5. associate-+r+ 91.2%

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

      6. frac-times 91.2%

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

      7. metadata-eval 91.2%

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

      8. +-commutative 91.2%

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

      9. fma-def 91.2%

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

   6. Applied egg-rr 91.2%

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

   7. Taylor expanded in alpha around 0 90.1%

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

   8. Taylor expanded in beta around inf 88.8%

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

   9. Taylor expanded in i around inf 88.8%

      \[\leadsto \frac{\frac{1}{1 + \color{blue}{4 \cdot \frac{i}{\beta}}} + 1}{2} \]
   10. Step-by-step derivation

       1. associate-*r/ 88.8%

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

       2. *-commutative 88.8%

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

   11. Simplified 88.8%

       \[\leadsto \frac{\frac{1}{1 + \color{blue}{\frac{i \cdot 4}{\beta}}} + 1}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 6.2 \cdot 10^{-25}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\left(\alpha + \beta\right) + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{1}{1 + \frac{i \cdot 4}{\beta}}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 2 \cdot 10^{+74}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= i 2e+74) (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0) 0.5))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 2e+74) {
		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+74) 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+74) {
		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+74:
		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+74)
		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+74)
		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+74], 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 < 1.9999999999999999e74

   1. Initial program 56.6%

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

   3. Simplified 75.1%

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

   5. Taylor expanded in i around 0 72.4%

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

      1. +-commutative 72.4%

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

   7. Simplified 72.4%

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

   8. Taylor expanded in alpha around 0 71.5%

      \[\leadsto \frac{\color{blue}{\frac{\beta}{2 + \beta}} + 1}{2} \]
   9. Step-by-step derivation

      1. +-commutative 71.5%

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

   10. Simplified 71.5%

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

   if 1.9999999999999999e74 < i

   1. Initial program 77.8%

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

   3. Simplified 94.5%

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

   5. Taylor expanded in i around inf 87.3%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 2 \cdot 10^{+74}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 77.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 5.5 \cdot 10^{+24}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 5.5e+24)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 5.5e+24) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (beta * 2.0)) / 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 <= 5.5d+24) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 5.5e+24) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 5.5e+24:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 5.5e+24)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 5.5e+24)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 5.5e+24], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 5.5000000000000002e24

   1. Initial program 82.6%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in i around 0 87.9%

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

      1. +-commutative 87.9%

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

   7. Simplified 87.9%

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

   8. Taylor expanded in alpha around 0 88.4%

      \[\leadsto \frac{\color{blue}{\frac{\beta}{2 + \beta}} + 1}{2} \]
   9. Step-by-step derivation

      1. +-commutative 88.4%

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

   10. Simplified 88.4%

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

   if 5.5000000000000002e24 < alpha

   1. Initial program 16.6%

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

   3. Simplified 35.2%

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

   5. Taylor expanded in i around 0 15.2%

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

      1. +-commutative 15.2%

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

   7. Simplified 15.2%

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

   8. Taylor expanded in alpha around inf 55.9%

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

      1. *-commutative 55.9%

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

   10. Simplified 55.9%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 5.5 \cdot 10^{+24}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 72.1% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.08 \cdot 10^{+86}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 (if (<= beta 1.08e+86) 0.5 1.0))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.08e+86) {
		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 <= 1.08d+86) 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 <= 1.08e+86) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.08e+86:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.08e+86)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1.08e+86)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 1.08e+86], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 1.07999999999999993e86

   1. Initial program 77.6%

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

   3. Simplified 80.1%

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

   5. Taylor expanded in i around inf 75.2%

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

   if 1.07999999999999993e86 < beta

   1. Initial program 25.1%

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

   3. Simplified 92.5%

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

   5. Taylor expanded in beta around inf 79.6%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.08 \cdot 10^{+86}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 61.0% 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 65.3%

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

3. Simplified 83.0%

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

5. Taylor expanded in i around inf 64.1%

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

6. Final simplification 64.1%

   \[\leadsto 0.5 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024021 
(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))