Octave 3.8, jcobi/2

Percentage Accurate: 63.5% → 97.6%

Time: 25.1s

Alternatives: 11

Speedup: 4.8×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 63.5% 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: 97.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq -0.9999995:\\ \;\;\;\;\frac{\frac{2 + \left(\beta + \left(\beta + i \cdot 4\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\frac{1}{\frac{\mathsf{fma}\left(2, \frac{i}{\beta - \alpha}, \frac{\alpha}{\beta - \alpha}\right) + \frac{\beta}{\beta - \alpha}}{\alpha + \beta}}}{\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)}}{2}\\ \end{array} \end{array} \]

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))
        -0.9999995)
     (/ (/ (+ 2.0 (+ beta (+ beta (* i 4.0)))) alpha) 2.0)
     (/
      (+
       1.0
       (/
        (/
         1.0
         (/
          (+
           (fma 2.0 (/ i (- beta alpha)) (/ alpha (- beta alpha)))
           (/ beta (- beta alpha)))
          (+ alpha beta)))
        (+ (+ 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)) <= -0.9999995) {
		tmp = ((2.0 + (beta + (beta + (i * 4.0)))) / alpha) / 2.0;
	} else {
		tmp = (1.0 + ((1.0 / ((fma(2.0, (i / (beta - alpha)), (alpha / (beta - alpha))) + (beta / (beta - alpha))) / (alpha + beta))) / ((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)) <= -0.9999995)
		tmp = Float64(Float64(Float64(2.0 + Float64(beta + Float64(beta + Float64(i * 4.0)))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(1.0 / Float64(Float64(fma(2.0, Float64(i / Float64(beta - alpha)), Float64(alpha / Float64(beta - alpha))) + Float64(beta / Float64(beta - alpha))) / Float64(alpha + beta))) / Float64(Float64(alpha + beta) + Float64(2.0 + Float64(2.0 * i))))) / 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], -0.9999995], N[(N[(N[(2.0 + N[(beta + N[(beta + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(N[(1.0 / N[(N[(N[(2.0 * N[(i / N[(beta - alpha), $MachinePrecision]), $MachinePrecision] + N[(alpha / N[(beta - alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(beta / N[(beta - alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + beta), $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)) < -0.999999500000000041

   1. Initial program 3.2%

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

   3. Simplified 19.8%

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

   5. Taylor expanded in alpha around inf 86.5%

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

   6. Taylor expanded in i around 0 86.5%

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

      1. associate--l+ 86.6%

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

      2. sub-neg 86.6%

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

      3. *-commutative 86.6%

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

      4. mul-1-neg 86.6%

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

      5. remove-double-neg 86.6%

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

   8. Simplified 86.6%

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

   if -0.999999500000000041 < (/.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 79.4%

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

      1. associate-/l* 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

   5. Taylor expanded in i around 0 99.8%

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

      1. clear-num 99.8%

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

      2. inv-pow 99.8%

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

      3. associate-+r+ 99.8%

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

      4. fma-def 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. unpow-1 99.8%

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

   9. Simplified 99.8%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.9999995:\\ \;\;\;\;\frac{\frac{2 + \left(\beta + \left(\beta + i \cdot 4\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\frac{1}{\frac{\mathsf{fma}\left(2, \frac{i}{\beta - \alpha}, \frac{\alpha}{\beta - \alpha}\right) + \frac{\beta}{\beta - \alpha}}{\alpha + \beta}}}{\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq -0.9999995:\\ \;\;\;\;\frac{\frac{2 + \left(\beta + \left(\beta + i \cdot 4\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\frac{\alpha + \beta}{\frac{\alpha + \left(\beta + 2 \cdot i\right)}{\beta - \alpha}}}{\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)}}{2}\\ \end{array} \end{array} \]

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))
        -0.9999995)
     (/ (/ (+ 2.0 (+ beta (+ beta (* 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)) <= -0.9999995) {
		tmp = ((2.0 + (beta + (beta + (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)) <= (-0.9999995d0)) then
        tmp = ((2.0d0 + (beta + (beta + (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)) <= -0.9999995) {
		tmp = ((2.0 + (beta + (beta + (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)) <= -0.9999995:
		tmp = ((2.0 + (beta + (beta + (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)) <= -0.9999995)
		tmp = Float64(Float64(Float64(2.0 + Float64(beta + Float64(beta + 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)) <= -0.9999995)
		tmp = ((2.0 + (beta + (beta + (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], -0.9999995], N[(N[(N[(2.0 + N[(beta + N[(beta + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(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)) < -0.999999500000000041

   1. Initial program 3.2%

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

   3. Simplified 19.8%

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

   5. Taylor expanded in alpha around inf 86.5%

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

   6. Taylor expanded in i around 0 86.5%

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

      1. associate--l+ 86.6%

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

      2. sub-neg 86.6%

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

      3. *-commutative 86.6%

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

      4. mul-1-neg 86.6%

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

      5. remove-double-neg 86.6%

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

   8. Simplified 86.6%

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

   if -0.999999500000000041 < (/.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 79.4%

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

      1. associate-/l* 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 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.9999995:\\ \;\;\;\;\frac{\frac{2 + \left(\beta + \left(\beta + i \cdot 4\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\frac{\alpha + \beta}{\frac{\alpha + \left(\beta + 2 \cdot i\right)}{\beta - \alpha}}}{\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq -0.5:\\ \;\;\;\;\frac{\frac{2 + \left(\beta + \left(\beta + i \cdot 4\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\frac{1}{\frac{1 + 2 \cdot \frac{i}{\beta}}{\beta}}}{\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)}}{2}\\ \end{array} \end{array} \]

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

   1. Initial program 4.2%

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

   3. Simplified 20.6%

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

   5. Taylor expanded in alpha around inf 85.9%

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

   6. Taylor expanded in i around 0 86.0%

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

      1. associate--l+ 86.0%

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

      2. sub-neg 86.0%

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

      3. *-commutative 86.0%

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

      4. mul-1-neg 86.0%

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

      5. remove-double-neg 86.0%

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

   8. Simplified 86.0%

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

   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 79.5%

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

      1. associate-/l* 100.0%

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

      2. associate-+l+ 100.0%

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

      3. associate-+l+ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in i around 0 100.0%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

      3. associate-+r+ 100.0%

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

      4. fma-def 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. unpow-1 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in alpha around 0 99.1%

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

4. Final simplification 95.7%

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

Alternative 4: 76.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq -4.2 \cdot 10^{-213}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq -1.15 \cdot 10^{-293}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 1.7 \cdot 10^{+98} \lor \neg \left(\alpha \leq 1.1 \cdot 10^{+166}\right) \land \alpha \leq 9.2 \cdot 10^{+220}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\left(\alpha + \beta\right) + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta + \beta\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha -4.2e-213)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (if (<= alpha -1.15e-293)
     0.5
     (if (or (<= alpha 1.7e+98)
             (and (not (<= alpha 1.1e+166)) (<= alpha 9.2e+220)))
       (/ (+ 1.0 (/ beta (+ (+ alpha beta) 2.0))) 2.0)
       (/ (/ (+ 2.0 (+ beta beta)) alpha) 2.0)))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= -4.2e-213) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else if (alpha <= -1.15e-293) {
		tmp = 0.5;
	} else if ((alpha <= 1.7e+98) || (!(alpha <= 1.1e+166) && (alpha <= 9.2e+220))) {
		tmp = (1.0 + (beta / ((alpha + beta) + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (beta + beta)) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= (-4.2d-213)) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else if (alpha <= (-1.15d-293)) then
        tmp = 0.5d0
    else if ((alpha <= 1.7d+98) .or. (.not. (alpha <= 1.1d+166)) .and. (alpha <= 9.2d+220)) then
        tmp = (1.0d0 + (beta / ((alpha + beta) + 2.0d0))) / 2.0d0
    else
        tmp = ((2.0d0 + (beta + beta)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= -4.2e-213) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else if (alpha <= -1.15e-293) {
		tmp = 0.5;
	} else if ((alpha <= 1.7e+98) || (!(alpha <= 1.1e+166) && (alpha <= 9.2e+220))) {
		tmp = (1.0 + (beta / ((alpha + beta) + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (beta + beta)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= -4.2e-213:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	elif alpha <= -1.15e-293:
		tmp = 0.5
	elif (alpha <= 1.7e+98) or (not (alpha <= 1.1e+166) and (alpha <= 9.2e+220)):
		tmp = (1.0 + (beta / ((alpha + beta) + 2.0))) / 2.0
	else:
		tmp = ((2.0 + (beta + beta)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= -4.2e-213)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	elseif (alpha <= -1.15e-293)
		tmp = 0.5;
	elseif ((alpha <= 1.7e+98) || (!(alpha <= 1.1e+166) && (alpha <= 9.2e+220)))
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(Float64(alpha + beta) + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(beta + beta)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= -4.2e-213)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	elseif (alpha <= -1.15e-293)
		tmp = 0.5;
	elseif ((alpha <= 1.7e+98) || (~((alpha <= 1.1e+166)) && (alpha <= 9.2e+220)))
		tmp = (1.0 + (beta / ((alpha + beta) + 2.0))) / 2.0;
	else
		tmp = ((2.0 + (beta + beta)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, -4.2e-213], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, -1.15e-293], 0.5, If[Or[LessEqual[alpha, 1.7e+98], And[N[Not[LessEqual[alpha, 1.1e+166]], $MachinePrecision], LessEqual[alpha, 9.2e+220]]], N[(N[(1.0 + N[(beta / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(beta + beta), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if alpha < -4.1999999999999997e-213

   1. Initial program 87.1%

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

      1. associate-/l* 100.0%

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

      2. associate-+l+ 100.0%

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

      3. associate-+l+ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in i around 0 100.0%

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

   6. Taylor expanded in alpha around 0 98.1%

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

   7. Taylor expanded in i around 0 92.8%

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

   if -4.1999999999999997e-213 < alpha < -1.14999999999999998e-293

   1. Initial program 83.2%

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

   3. Taylor expanded in i around inf 92.9%

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

   if -1.14999999999999998e-293 < alpha < 1.69999999999999986e98 or 1.1e166 < alpha < 9.19999999999999987e220

   1. Initial program 63.7%

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

      1. associate-/l* 86.2%

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

      2. associate-+l+ 86.2%

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

      3. associate-+l+ 86.2%

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

   3. Simplified 86.2%

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

   5. Taylor expanded in i around 0 86.3%

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

      1. clear-num 86.3%

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

      2. inv-pow 86.3%

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

      3. associate-+r+ 86.3%

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

      4. fma-def 86.3%

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

   7. Applied egg-rr 86.3%

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

      1. unpow-1 86.3%

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

   9. Simplified 86.3%

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

   10. Taylor expanded in alpha around 0 84.5%

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

   11. Taylor expanded in i around 0 81.3%

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

       1. +-commutative 81.3%

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

   13. Simplified 81.3%

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

   if 1.69999999999999986e98 < alpha < 1.1e166 or 9.19999999999999987e220 < alpha

   1. Initial program 4.1%

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

   3. Simplified 24.9%

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

   5. Taylor expanded in alpha around inf 81.2%

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

   6. Taylor expanded in i around 0 64.3%

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

      1. associate--l+ 64.3%

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

      2. sub-neg 64.3%

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

      3. mul-1-neg 64.3%

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

      4. remove-double-neg 64.3%

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

   8. Simplified 64.3%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq -4.2 \cdot 10^{-213}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq -1.15 \cdot 10^{-293}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 1.7 \cdot 10^{+98} \lor \neg \left(\alpha \leq 1.1 \cdot 10^{+166}\right) \land \alpha \leq 9.2 \cdot 10^{+220}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\left(\alpha + \beta\right) + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta + \beta\right)}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{if}\;\alpha \leq -4.2 \cdot 10^{-213}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\alpha \leq -1.04 \cdot 10^{-293}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 1.9 \cdot 10^{+99} \lor \neg \left(\alpha \leq 1.12 \cdot 10^{+166}\right) \land \alpha \leq 2.1 \cdot 10^{+220}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta + \beta\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)))
   (if (<= alpha -4.2e-213)
     t_0
     (if (<= alpha -1.04e-293)
       0.5
       (if (or (<= alpha 1.9e+99)
               (and (not (<= alpha 1.12e+166)) (<= alpha 2.1e+220)))
         t_0
         (/ (/ (+ 2.0 (+ beta beta)) alpha) 2.0))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (1.0 + (beta / (beta + 2.0))) / 2.0;
	double tmp;
	if (alpha <= -4.2e-213) {
		tmp = t_0;
	} else if (alpha <= -1.04e-293) {
		tmp = 0.5;
	} else if ((alpha <= 1.9e+99) || (!(alpha <= 1.12e+166) && (alpha <= 2.1e+220))) {
		tmp = t_0;
	} else {
		tmp = ((2.0 + (beta + beta)) / 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 = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    if (alpha <= (-4.2d-213)) then
        tmp = t_0
    else if (alpha <= (-1.04d-293)) then
        tmp = 0.5d0
    else if ((alpha <= 1.9d+99) .or. (.not. (alpha <= 1.12d+166)) .and. (alpha <= 2.1d+220)) then
        tmp = t_0
    else
        tmp = ((2.0d0 + (beta + beta)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (1.0 + (beta / (beta + 2.0))) / 2.0;
	double tmp;
	if (alpha <= -4.2e-213) {
		tmp = t_0;
	} else if (alpha <= -1.04e-293) {
		tmp = 0.5;
	} else if ((alpha <= 1.9e+99) || (!(alpha <= 1.12e+166) && (alpha <= 2.1e+220))) {
		tmp = t_0;
	} else {
		tmp = ((2.0 + (beta + beta)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (1.0 + (beta / (beta + 2.0))) / 2.0
	tmp = 0
	if alpha <= -4.2e-213:
		tmp = t_0
	elif alpha <= -1.04e-293:
		tmp = 0.5
	elif (alpha <= 1.9e+99) or (not (alpha <= 1.12e+166) and (alpha <= 2.1e+220)):
		tmp = t_0
	else:
		tmp = ((2.0 + (beta + beta)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0)
	tmp = 0.0
	if (alpha <= -4.2e-213)
		tmp = t_0;
	elseif (alpha <= -1.04e-293)
		tmp = 0.5;
	elseif ((alpha <= 1.9e+99) || (!(alpha <= 1.12e+166) && (alpha <= 2.1e+220)))
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(beta + beta)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (1.0 + (beta / (beta + 2.0))) / 2.0;
	tmp = 0.0;
	if (alpha <= -4.2e-213)
		tmp = t_0;
	elseif (alpha <= -1.04e-293)
		tmp = 0.5;
	elseif ((alpha <= 1.9e+99) || (~((alpha <= 1.12e+166)) && (alpha <= 2.1e+220)))
		tmp = t_0;
	else
		tmp = ((2.0 + (beta + beta)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[alpha, -4.2e-213], t$95$0, If[LessEqual[alpha, -1.04e-293], 0.5, If[Or[LessEqual[alpha, 1.9e+99], And[N[Not[LessEqual[alpha, 1.12e+166]], $MachinePrecision], LessEqual[alpha, 2.1e+220]]], t$95$0, N[(N[(N[(2.0 + N[(beta + beta), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if alpha < -4.1999999999999997e-213 or -1.04e-293 < alpha < 1.9e99 or 1.1199999999999999e166 < alpha < 2.10000000000000007e220

   1. Initial program 71.3%

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

      1. associate-/l* 90.7%

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

      2. associate-+l+ 90.7%

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

      3. associate-+l+ 90.7%

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

   3. Simplified 90.7%

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

   5. Taylor expanded in i around 0 90.7%

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

   6. Taylor expanded in alpha around 0 88.9%

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

   7. Taylor expanded in i around 0 84.3%

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

   if -4.1999999999999997e-213 < alpha < -1.04e-293

   1. Initial program 83.2%

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

   3. Taylor expanded in i around inf 92.9%

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

   if 1.9e99 < alpha < 1.1199999999999999e166 or 2.10000000000000007e220 < alpha

   1. Initial program 4.1%

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

   3. Simplified 24.9%

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

   5. Taylor expanded in alpha around inf 81.2%

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

   6. Taylor expanded in i around 0 64.3%

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

      1. associate--l+ 64.3%

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

      2. sub-neg 64.3%

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

      3. mul-1-neg 64.3%

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

      4. remove-double-neg 64.3%

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

   8. Simplified 64.3%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq -4.2 \cdot 10^{-213}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq -1.04 \cdot 10^{-293}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 1.9 \cdot 10^{+99} \lor \neg \left(\alpha \leq 1.12 \cdot 10^{+166}\right) \land \alpha \leq 2.1 \cdot 10^{+220}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta + \beta\right)}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + i \cdot 4\\ \mathbf{if}\;\alpha \leq 6.4 \cdot 10^{+15} \lor \neg \left(\alpha \leq 1.95 \cdot 10^{+26}\right) \land \alpha \leq 1.1 \cdot 10^{+95}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + t\_0}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta + t\_0\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ beta (* i 4.0))))
   (if (or (<= alpha 6.4e+15)
           (and (not (<= alpha 1.95e+26)) (<= alpha 1.1e+95)))
     (/ (+ 1.0 (/ beta (+ 2.0 t_0))) 2.0)
     (/ (/ (+ 2.0 (+ beta t_0)) alpha) 2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = beta + (i * 4.0);
	double tmp;
	if ((alpha <= 6.4e+15) || (!(alpha <= 1.95e+26) && (alpha <= 1.1e+95))) {
		tmp = (1.0 + (beta / (2.0 + t_0))) / 2.0;
	} else {
		tmp = ((2.0 + (beta + t_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 = beta + (i * 4.0d0)
    if ((alpha <= 6.4d+15) .or. (.not. (alpha <= 1.95d+26)) .and. (alpha <= 1.1d+95)) then
        tmp = (1.0d0 + (beta / (2.0d0 + t_0))) / 2.0d0
    else
        tmp = ((2.0d0 + (beta + t_0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = beta + (i * 4.0);
	double tmp;
	if ((alpha <= 6.4e+15) || (!(alpha <= 1.95e+26) && (alpha <= 1.1e+95))) {
		tmp = (1.0 + (beta / (2.0 + t_0))) / 2.0;
	} else {
		tmp = ((2.0 + (beta + t_0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = beta + (i * 4.0)
	tmp = 0
	if (alpha <= 6.4e+15) or (not (alpha <= 1.95e+26) and (alpha <= 1.1e+95)):
		tmp = (1.0 + (beta / (2.0 + t_0))) / 2.0
	else:
		tmp = ((2.0 + (beta + t_0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(beta + Float64(i * 4.0))
	tmp = 0.0
	if ((alpha <= 6.4e+15) || (!(alpha <= 1.95e+26) && (alpha <= 1.1e+95)))
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(2.0 + t_0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(beta + t_0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = beta + (i * 4.0);
	tmp = 0.0;
	if ((alpha <= 6.4e+15) || (~((alpha <= 1.95e+26)) && (alpha <= 1.1e+95)))
		tmp = (1.0 + (beta / (2.0 + t_0))) / 2.0;
	else
		tmp = ((2.0 + (beta + t_0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(beta + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[alpha, 6.4e+15], And[N[Not[LessEqual[alpha, 1.95e+26]], $MachinePrecision], LessEqual[alpha, 1.1e+95]]], N[(N[(1.0 + N[(beta / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(beta + t$95$0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if alpha < 6.4e15 or 1.95e26 < alpha < 1.0999999999999999e95

   1. Initial program 81.7%

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

      1. associate-/l* 97.5%

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

      2. associate-+l+ 97.5%

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

      3. associate-+l+ 97.5%

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

   3. Simplified 97.5%

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

   5. Taylor expanded in i around 0 97.5%

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

   6. Taylor expanded in alpha around 0 96.3%

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

   7. Taylor expanded in beta around inf 95.5%

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

   if 6.4e15 < alpha < 1.95e26 or 1.0999999999999999e95 < alpha

   1. Initial program 3.8%

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

   3. Simplified 32.5%

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

   5. Taylor expanded in alpha around inf 73.9%

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

   6. Taylor expanded in i around 0 74.0%

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

      1. associate--l+ 74.0%

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

      2. sub-neg 74.0%

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

      3. *-commutative 74.0%

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

      4. mul-1-neg 74.0%

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

      5. remove-double-neg 74.0%

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

   8. Simplified 74.0%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 6.4 \cdot 10^{+15} \lor \neg \left(\alpha \leq 1.95 \cdot 10^{+26}\right) \land \alpha \leq 1.1 \cdot 10^{+95}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\beta + i \cdot 4\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta + \left(\beta + i \cdot 4\right)\right)}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + i \cdot 4\\ \mathbf{if}\;\alpha \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\left(1 + 2 \cdot \frac{i}{\beta}\right) \cdot \left(2 + \left(\beta + 2 \cdot i\right)\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 1.05 \cdot 10^{+26} \lor \neg \left(\alpha \leq 5.8 \cdot 10^{+96}\right):\\ \;\;\;\;\frac{\frac{2 + \left(\beta + t\_0\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + t\_0}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ beta (* i 4.0))))
   (if (<= alpha 2e+14)
     (/
      (+
       1.0
       (/ beta (* (+ 1.0 (* 2.0 (/ i beta))) (+ 2.0 (+ beta (* 2.0 i))))))
      2.0)
     (if (or (<= alpha 1.05e+26) (not (<= alpha 5.8e+96)))
       (/ (/ (+ 2.0 (+ beta t_0)) alpha) 2.0)
       (/ (+ 1.0 (/ beta (+ 2.0 t_0))) 2.0)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = beta + (i * 4.0);
	double tmp;
	if (alpha <= 2e+14) {
		tmp = (1.0 + (beta / ((1.0 + (2.0 * (i / beta))) * (2.0 + (beta + (2.0 * i)))))) / 2.0;
	} else if ((alpha <= 1.05e+26) || !(alpha <= 5.8e+96)) {
		tmp = ((2.0 + (beta + t_0)) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (beta / (2.0 + t_0))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: tmp
    t_0 = beta + (i * 4.0d0)
    if (alpha <= 2d+14) then
        tmp = (1.0d0 + (beta / ((1.0d0 + (2.0d0 * (i / beta))) * (2.0d0 + (beta + (2.0d0 * i)))))) / 2.0d0
    else if ((alpha <= 1.05d+26) .or. (.not. (alpha <= 5.8d+96))) then
        tmp = ((2.0d0 + (beta + t_0)) / alpha) / 2.0d0
    else
        tmp = (1.0d0 + (beta / (2.0d0 + t_0))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = beta + (i * 4.0);
	double tmp;
	if (alpha <= 2e+14) {
		tmp = (1.0 + (beta / ((1.0 + (2.0 * (i / beta))) * (2.0 + (beta + (2.0 * i)))))) / 2.0;
	} else if ((alpha <= 1.05e+26) || !(alpha <= 5.8e+96)) {
		tmp = ((2.0 + (beta + t_0)) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (beta / (2.0 + t_0))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = beta + (i * 4.0)
	tmp = 0
	if alpha <= 2e+14:
		tmp = (1.0 + (beta / ((1.0 + (2.0 * (i / beta))) * (2.0 + (beta + (2.0 * i)))))) / 2.0
	elif (alpha <= 1.05e+26) or not (alpha <= 5.8e+96):
		tmp = ((2.0 + (beta + t_0)) / alpha) / 2.0
	else:
		tmp = (1.0 + (beta / (2.0 + t_0))) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(beta + Float64(i * 4.0))
	tmp = 0.0
	if (alpha <= 2e+14)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(Float64(1.0 + Float64(2.0 * Float64(i / beta))) * Float64(2.0 + Float64(beta + Float64(2.0 * i)))))) / 2.0);
	elseif ((alpha <= 1.05e+26) || !(alpha <= 5.8e+96))
		tmp = Float64(Float64(Float64(2.0 + Float64(beta + t_0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(2.0 + t_0))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = beta + (i * 4.0);
	tmp = 0.0;
	if (alpha <= 2e+14)
		tmp = (1.0 + (beta / ((1.0 + (2.0 * (i / beta))) * (2.0 + (beta + (2.0 * i)))))) / 2.0;
	elseif ((alpha <= 1.05e+26) || ~((alpha <= 5.8e+96)))
		tmp = ((2.0 + (beta + t_0)) / alpha) / 2.0;
	else
		tmp = (1.0 + (beta / (2.0 + t_0))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(beta + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[alpha, 2e+14], N[(N[(1.0 + N[(beta / N[(N[(1.0 + N[(2.0 * N[(i / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[alpha, 1.05e+26], N[Not[LessEqual[alpha, 5.8e+96]], $MachinePrecision]], N[(N[(N[(2.0 + N[(beta + t$95$0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(beta / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 2e14

   1. Initial program 85.2%

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

      1. associate-/l* 99.5%

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

      2. associate-+l+ 99.5%

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

      3. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in i around 0 99.5%

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

   6. Taylor expanded in alpha around 0 98.1%

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

   if 2e14 < alpha < 1.05e26 or 5.79999999999999955e96 < alpha

   1. Initial program 3.8%

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

   3. Simplified 32.5%

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

   5. Taylor expanded in alpha around inf 73.9%

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

   6. Taylor expanded in i around 0 74.0%

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

      1. associate--l+ 74.0%

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

      2. sub-neg 74.0%

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

      3. *-commutative 74.0%

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

      4. mul-1-neg 74.0%

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

      5. remove-double-neg 74.0%

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

   8. Simplified 74.0%

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

   if 1.05e26 < alpha < 5.79999999999999955e96

   1. Initial program 56.1%

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

      1. associate-/l* 82.5%

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

      2. associate-+l+ 82.5%

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

      3. associate-+l+ 82.5%

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

   3. Simplified 82.5%

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

   5. Taylor expanded in i around 0 82.5%

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

   6. Taylor expanded in alpha around 0 83.0%

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

   7. Taylor expanded in beta around inf 83.0%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\left(1 + 2 \cdot \frac{i}{\beta}\right) \cdot \left(2 + \left(\beta + 2 \cdot i\right)\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 1.05 \cdot 10^{+26} \lor \neg \left(\alpha \leq 5.8 \cdot 10^{+96}\right):\\ \;\;\;\;\frac{\frac{2 + \left(\beta + \left(\beta + i \cdot 4\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\beta + i \cdot 4\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.8 \cdot 10^{+97}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\beta + i \cdot 4\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 2.4 \cdot 10^{+164} \lor \neg \left(\alpha \leq 2.1 \cdot 10^{+220}\right):\\ \;\;\;\;\frac{\frac{2 + \left(\beta + \beta\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\left(\alpha + \beta\right) + 2}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 2.8e+97)
   (/ (+ 1.0 (/ beta (+ 2.0 (+ beta (* i 4.0))))) 2.0)
   (if (or (<= alpha 2.4e+164) (not (<= alpha 2.1e+220)))
     (/ (/ (+ 2.0 (+ beta beta)) alpha) 2.0)
     (/ (+ 1.0 (/ beta (+ (+ alpha beta) 2.0))) 2.0))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2.8e+97) {
		tmp = (1.0 + (beta / (2.0 + (beta + (i * 4.0))))) / 2.0;
	} else if ((alpha <= 2.4e+164) || !(alpha <= 2.1e+220)) {
		tmp = ((2.0 + (beta + beta)) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (beta / ((alpha + beta) + 2.0))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 2.8d+97) then
        tmp = (1.0d0 + (beta / (2.0d0 + (beta + (i * 4.0d0))))) / 2.0d0
    else if ((alpha <= 2.4d+164) .or. (.not. (alpha <= 2.1d+220))) then
        tmp = ((2.0d0 + (beta + beta)) / alpha) / 2.0d0
    else
        tmp = (1.0d0 + (beta / ((alpha + beta) + 2.0d0))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2.8e+97) {
		tmp = (1.0 + (beta / (2.0 + (beta + (i * 4.0))))) / 2.0;
	} else if ((alpha <= 2.4e+164) || !(alpha <= 2.1e+220)) {
		tmp = ((2.0 + (beta + beta)) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (beta / ((alpha + beta) + 2.0))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 2.8e+97:
		tmp = (1.0 + (beta / (2.0 + (beta + (i * 4.0))))) / 2.0
	elif (alpha <= 2.4e+164) or not (alpha <= 2.1e+220):
		tmp = ((2.0 + (beta + beta)) / alpha) / 2.0
	else:
		tmp = (1.0 + (beta / ((alpha + beta) + 2.0))) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 2.8e+97)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(2.0 + Float64(beta + Float64(i * 4.0))))) / 2.0);
	elseif ((alpha <= 2.4e+164) || !(alpha <= 2.1e+220))
		tmp = Float64(Float64(Float64(2.0 + Float64(beta + beta)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(Float64(alpha + beta) + 2.0))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 2.8e+97)
		tmp = (1.0 + (beta / (2.0 + (beta + (i * 4.0))))) / 2.0;
	elseif ((alpha <= 2.4e+164) || ~((alpha <= 2.1e+220)))
		tmp = ((2.0 + (beta + beta)) / alpha) / 2.0;
	else
		tmp = (1.0 + (beta / ((alpha + beta) + 2.0))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 2.8e+97], N[(N[(1.0 + N[(beta / N[(2.0 + N[(beta + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[alpha, 2.4e+164], N[Not[LessEqual[alpha, 2.1e+220]], $MachinePrecision]], N[(N[(N[(2.0 + N[(beta + beta), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(beta / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 2.7999999999999999e97

   1. Initial program 79.8%

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

      1. associate-/l* 95.1%

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

      2. associate-+l+ 95.1%

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

      3. associate-+l+ 95.1%

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

   3. Simplified 95.1%

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

   5. Taylor expanded in i around 0 95.1%

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

   6. Taylor expanded in alpha around 0 94.0%

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

   7. Taylor expanded in beta around inf 93.2%

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

   if 2.7999999999999999e97 < alpha < 2.40000000000000011e164 or 2.10000000000000007e220 < alpha

   1. Initial program 4.1%

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

   3. Simplified 24.9%

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

   5. Taylor expanded in alpha around inf 81.2%

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

   6. Taylor expanded in i around 0 64.3%

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

      1. associate--l+ 64.3%

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

      2. sub-neg 64.3%

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

      3. mul-1-neg 64.3%

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

      4. remove-double-neg 64.3%

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

   8. Simplified 64.3%

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

   if 2.40000000000000011e164 < alpha < 2.10000000000000007e220

   1. Initial program 1.4%

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

      1. associate-/l* 57.9%

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

      2. associate-+l+ 57.9%

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

      3. associate-+l+ 57.9%

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

   3. Simplified 57.9%

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

   5. Taylor expanded in i around 0 58.2%

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

      1. clear-num 58.4%

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

      2. inv-pow 58.4%

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

      3. associate-+r+ 58.4%

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

      4. fma-def 58.4%

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

   7. Applied egg-rr 58.4%

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

      1. unpow-1 58.4%

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

   9. Simplified 58.4%

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

   10. Taylor expanded in alpha around 0 51.3%

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

   11. Taylor expanded in i around 0 51.3%

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

       1. +-commutative 51.3%

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

   13. Simplified 51.3%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.8 \cdot 10^{+97}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\beta + i \cdot 4\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 2.4 \cdot 10^{+164} \lor \neg \left(\alpha \leq 2.1 \cdot 10^{+220}\right):\\ \;\;\;\;\frac{\frac{2 + \left(\beta + \beta\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\left(\alpha + \beta\right) + 2}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot i \leq 5 \cdot 10^{+169}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= (* 2.0 i) 5e+169) (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0) 0.5))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if ((2.0 * i) <= 5e+169) {
		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 ((2.0d0 * i) <= 5d+169) 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 ((2.0 * i) <= 5e+169) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if (2.0 * i) <= 5e+169:
		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 (Float64(2.0 * i) <= 5e+169)
		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 ((2.0 * i) <= 5e+169)
		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[N[(2.0 * i), $MachinePrecision], 5e+169], 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 (*.f64 2 i) < 5.00000000000000017e169

   1. Initial program 59.1%

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

      1. associate-/l* 75.7%

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

      2. associate-+l+ 75.7%

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

      3. associate-+l+ 75.7%

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

   3. Simplified 75.7%

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

   5. Taylor expanded in i around 0 75.7%

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

   6. Taylor expanded in alpha around 0 73.7%

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

   7. Taylor expanded in i around 0 71.7%

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

   if 5.00000000000000017e169 < (*.f64 2 i)

   1. Initial program 62.7%

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

   3. Taylor expanded in i around inf 86.9%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot i \leq 5 \cdot 10^{+169}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 72.5% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.65 \cdot 10^{+67}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 (if (<= beta 2.65e+67) 0.5 1.0))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.65e+67) {
		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 <= 2.65d+67) 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 <= 2.65e+67) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 2.65e+67:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 2.65e+67)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 2.65e+67)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 2.65e+67], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 2.65e67

   1. Initial program 72.1%

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

   3. Taylor expanded in i around inf 71.7%

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

   if 2.65e67 < beta

   1. Initial program 28.2%

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

   3. Taylor expanded in beta around inf 71.2%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.65 \cdot 10^{+67}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.2% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[alpha_, beta_, i_] := 0.5

Derivation

1. Initial program 59.8%

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

3. Taylor expanded in i around inf 59.6%

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

4. Final simplification 59.6%

   \[\leadsto 0.5 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024031 
(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))