Octave 3.8, jcobi/2

Percentage Accurate: 63.3% → 97.9%

Time: 14.1s

Alternatives: 10

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: 63.3% 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: 97.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq -0.99999998:\\ \;\;\;\;\frac{\frac{\left(2 + \beta \cdot 2\right) + i \cdot 4}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\alpha + \beta}{\frac{\beta + \left(\alpha + 2 \cdot i\right)}{\beta - \alpha} \cdot \left(\beta + \left(\alpha + \left(2 + 2 \cdot i\right)\right)\right)}}{2}\\ \end{array} \end{array} \]

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))
        -0.99999998)
     (/ (/ (+ (+ 2.0 (* beta 2.0)) (* i 4.0)) alpha) 2.0)
     (/
      (+
       1.0
       (/
        (+ alpha beta)
        (*
         (/ (+ beta (+ alpha (* 2.0 i))) (- beta alpha))
         (+ beta (+ alpha (+ 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)) <= -0.99999998) {
		tmp = (((2.0 + (beta * 2.0)) + (i * 4.0)) / alpha) / 2.0;
	} else {
		tmp = (1.0 + ((alpha + beta) / (((beta + (alpha + (2.0 * i))) / (beta - alpha)) * (beta + (alpha + (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)) <= (-0.99999998d0)) then
        tmp = (((2.0d0 + (beta * 2.0d0)) + (i * 4.0d0)) / alpha) / 2.0d0
    else
        tmp = (1.0d0 + ((alpha + beta) / (((beta + (alpha + (2.0d0 * i))) / (beta - alpha)) * (beta + (alpha + (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)) <= -0.99999998) {
		tmp = (((2.0 + (beta * 2.0)) + (i * 4.0)) / alpha) / 2.0;
	} else {
		tmp = (1.0 + ((alpha + beta) / (((beta + (alpha + (2.0 * i))) / (beta - alpha)) * (beta + (alpha + (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)) <= -0.99999998:
		tmp = (((2.0 + (beta * 2.0)) + (i * 4.0)) / alpha) / 2.0
	else:
		tmp = (1.0 + ((alpha + beta) / (((beta + (alpha + (2.0 * i))) / (beta - alpha)) * (beta + (alpha + (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)) <= -0.99999998)
		tmp = Float64(Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) + Float64(i * 4.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(alpha + beta) / Float64(Float64(Float64(beta + Float64(alpha + Float64(2.0 * i))) / Float64(beta - alpha)) * Float64(beta + Float64(alpha + 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)) <= -0.99999998)
		tmp = (((2.0 + (beta * 2.0)) + (i * 4.0)) / alpha) / 2.0;
	else
		tmp = (1.0 + ((alpha + beta) / (((beta + (alpha + (2.0 * i))) / (beta - alpha)) * (beta + (alpha + (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], -0.99999998], N[(N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(N[(alpha + beta), $MachinePrecision] / N[(N[(N[(beta + N[(alpha + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta - alpha), $MachinePrecision]), $MachinePrecision] * N[(beta + N[(alpha + N[(2.0 + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.999999980000000011

   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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1\right), \color{blue}{2}\right) \]

   3. Simplified 15.0%

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

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\left(\beta - \alpha\right) \cdot \left(\alpha + \beta\right)}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}\right), 1\right), 2\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}\right), 1\right), 2\right) \]

      3. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\left(\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)\right) \cdot \frac{1}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}\right), 1\right), 2\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)\right), \left(\frac{1}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}\right)\right), 1\right), 2\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)\right), \left(\frac{1}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}\right)\right), 1\right), 2\right) \]

      6. difference-of-squares N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\beta \cdot \beta - \alpha \cdot \alpha\right), \left(\frac{1}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}\right)\right), 1\right), 2\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\beta \cdot \beta\right), \left(\alpha \cdot \alpha\right)\right), \left(\frac{1}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}\right)\right), 1\right), 2\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \left(\alpha \cdot \alpha\right)\right), \left(\frac{1}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}\right)\right), 1\right), 2\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \mathsf{*.f64}\left(\alpha, \alpha\right)\right), \left(\frac{1}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}\right)\right), 1\right), 2\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \mathsf{*.f64}\left(\alpha, \alpha\right)\right), \mathsf{/.f64}\left(1, \left(\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)\right)\right)\right), 1\right), 2\right) \]

      11. associate-+r+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \mathsf{*.f64}\left(\alpha, \alpha\right)\right), \mathsf{/.f64}\left(1, \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)\right)\right)\right), 1\right), 2\right) \]

      12. associate-+r+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \mathsf{*.f64}\left(\alpha, \alpha\right)\right), \mathsf{/.f64}\left(1, \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)\right)\right)\right)\right), 1\right), 2\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \mathsf{*.f64}\left(\alpha, \alpha\right)\right), \mathsf{/.f64}\left(1, \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right)\right)\right)\right), 1\right), 2\right) \]

      14. associate-+l+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \mathsf{*.f64}\left(\alpha, \alpha\right)\right), \mathsf{/.f64}\left(1, \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)\right)\right)\right), 1\right), 2\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \mathsf{*.f64}\left(\alpha, \alpha\right)\right), \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right), \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)\right)\right)\right), 1\right), 2\right) \]

   6. Applied egg-rr 2.6%

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

   7. Taylor expanded in alpha around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{2 + \left(2 \cdot \beta + 4 \cdot i\right)}{\alpha}\right)}, 2\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right), \alpha\right), 2\right) \]

      2. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(2 + 2 \cdot \beta\right) + 4 \cdot i\right), \alpha\right), 2\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(2 + 2 \cdot \beta\right), \left(4 \cdot i\right)\right), \alpha\right), 2\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \left(2 \cdot \beta\right)\right), \left(4 \cdot i\right)\right), \alpha\right), 2\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(2, \beta\right)\right), \left(4 \cdot i\right)\right), \alpha\right), 2\right) \]

      6. *-lowering-*.f64 89.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(2, \beta\right)\right), \mathsf{*.f64}\left(4, i\right)\right), \alpha\right), 2\right) \]

   9. Simplified 89.5%

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

   if -0.999999980000000011 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 74.3%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), 2\right)\right), 1\right), 2\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \left(\alpha + \beta\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), 2\right)\right), 1\right), 2\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}\right), \left(\alpha + \beta\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), 2\right)\right), 1\right), 2\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\beta - \alpha\right), \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right), \left(\alpha + \beta\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), 2\right)\right), 1\right), 2\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right), \left(\alpha + \beta\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), 2\right)\right), 1\right), 2\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\left(\beta + \alpha\right) + 2 \cdot i\right)\right), \left(\alpha + \beta\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), 2\right)\right), 1\right), 2\right) \]

      7. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\beta + \left(\alpha + 2 \cdot i\right)\right)\right), \left(\alpha + \beta\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), 2\right)\right), 1\right), 2\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\beta, \left(\alpha + 2 \cdot i\right)\right)\right), \left(\alpha + \beta\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), 2\right)\right), 1\right), 2\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, \left(2 \cdot i\right)\right)\right)\right), \left(\alpha + \beta\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), 2\right)\right), 1\right), 2\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, \mathsf{*.f64}\left(2, i\right)\right)\right)\right), \left(\alpha + \beta\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), 2\right)\right), 1\right), 2\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, \mathsf{*.f64}\left(2, i\right)\right)\right)\right), \left(\beta + \alpha\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), 2\right)\right), 1\right), 2\right) \]

      12. +-lowering-+.f64 99.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, \mathsf{*.f64}\left(2, i\right)\right)\right)\right), \mathsf{+.f64}\left(\beta, \alpha\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), 2\right)\right), 1\right), 2\right) \]

   4. Applied egg-rr 99.6%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2 \cdot i\right)} \cdot \frac{\beta + \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}\right), 1\right), 2\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{\frac{\beta + \left(\alpha + 2 \cdot i\right)}{\beta - \alpha}} \cdot \frac{\beta + \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}\right), 1\right), 2\right) \]

      3. frac-times N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{1 \cdot \left(\beta + \alpha\right)}{\frac{\beta + \left(\alpha + 2 \cdot i\right)}{\beta - \alpha} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)}\right), 1\right), 2\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(1 \cdot \left(\beta + \alpha\right)\right), \left(\frac{\beta + \left(\alpha + 2 \cdot i\right)}{\beta - \alpha} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)\right)\right), 1\right), 2\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(1, \left(\beta + \alpha\right)\right), \left(\frac{\beta + \left(\alpha + 2 \cdot i\right)}{\beta - \alpha} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)\right)\right), 1\right), 2\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(1, \mathsf{+.f64}\left(\beta, \alpha\right)\right), \left(\frac{\beta + \left(\alpha + 2 \cdot i\right)}{\beta - \alpha} \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)\right)\right), 1\right), 2\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(1, \mathsf{+.f64}\left(\beta, \alpha\right)\right), \mathsf{*.f64}\left(\left(\frac{\beta + \left(\alpha + 2 \cdot i\right)}{\beta - \alpha}\right), \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)\right)\right), 1\right), 2\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(1, \mathsf{+.f64}\left(\beta, \alpha\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\beta + \left(\alpha + 2 \cdot i\right)\right), \left(\beta - \alpha\right)\right), \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)\right)\right), 1\right), 2\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(1, \mathsf{+.f64}\left(\beta, \alpha\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, \left(\alpha + 2 \cdot i\right)\right), \left(\beta - \alpha\right)\right), \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)\right)\right), 1\right), 2\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(1, \mathsf{+.f64}\left(\beta, \alpha\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, \left(2 \cdot i\right)\right)\right), \left(\beta - \alpha\right)\right), \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)\right)\right), 1\right), 2\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(1, \mathsf{+.f64}\left(\beta, \alpha\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, \mathsf{*.f64}\left(2, i\right)\right)\right), \left(\beta - \alpha\right)\right), \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)\right)\right), 1\right), 2\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(1, \mathsf{+.f64}\left(\beta, \alpha\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, \mathsf{*.f64}\left(2, i\right)\right)\right), \mathsf{\_.f64}\left(\beta, \alpha\right)\right), \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)\right)\right), 1\right), 2\right) \]

      13. associate-+l+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(1, \mathsf{+.f64}\left(\beta, \alpha\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, \mathsf{*.f64}\left(2, i\right)\right)\right), \mathsf{\_.f64}\left(\beta, \alpha\right)\right), \left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right)\right)\right), 1\right), 2\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(1, \mathsf{+.f64}\left(\beta, \alpha\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, \mathsf{*.f64}\left(2, i\right)\right)\right), \mathsf{\_.f64}\left(\beta, \alpha\right)\right), \left(\left(\beta + \alpha\right) + \left(2 \cdot i + 2\right)\right)\right)\right), 1\right), 2\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(1, \mathsf{+.f64}\left(\beta, \alpha\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, \mathsf{*.f64}\left(2, i\right)\right)\right), \mathsf{\_.f64}\left(\beta, \alpha\right)\right), \left(\left(\beta + \alpha\right) + \left(2 + 2 \cdot i\right)\right)\right)\right), 1\right), 2\right) \]

      16. associate-+l+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(1, \mathsf{+.f64}\left(\beta, \alpha\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, \mathsf{*.f64}\left(2, i\right)\right)\right), \mathsf{\_.f64}\left(\beta, \alpha\right)\right), \left(\beta + \left(\alpha + \left(2 + 2 \cdot i\right)\right)\right)\right)\right), 1\right), 2\right) \]

      17. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(1, \mathsf{+.f64}\left(\beta, \alpha\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, \mathsf{*.f64}\left(2, i\right)\right)\right), \mathsf{\_.f64}\left(\beta, \alpha\right)\right), \mathsf{+.f64}\left(\beta, \left(\alpha + \left(2 + 2 \cdot i\right)\right)\right)\right)\right), 1\right), 2\right) \]

   6. Applied egg-rr 99.6%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.99999998:\\ \;\;\;\;\frac{\frac{\left(2 + \beta \cdot 2\right) + i \cdot 4}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\alpha + \beta}{\frac{\beta + \left(\alpha + 2 \cdot i\right)}{\beta - \alpha} \cdot \left(\beta + \left(\alpha + \left(2 + 2 \cdot i\right)\right)\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := 2 + t\_0\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_1} \leq -0.99999998:\\ \;\;\;\;\frac{\frac{\left(2 + \beta \cdot 2\right) + i \cdot 4}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\beta + \left(\alpha + 2 \cdot i\right)}}{t\_1}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))) (t_1 (+ 2.0 t_0)))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) t_1) -0.99999998)
     (/ (/ (+ (+ 2.0 (* beta 2.0)) (* i 4.0)) alpha) 2.0)
     (/
      (+
       1.0
       (/
        (* (+ alpha beta) (/ (- beta alpha) (+ beta (+ alpha (* 2.0 i)))))
        t_1))
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = 2.0 + t_0;
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= -0.99999998) {
		tmp = (((2.0 + (beta * 2.0)) + (i * 4.0)) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (((alpha + beta) * ((beta - alpha) / (beta + (alpha + (2.0 * i))))) / t_1)) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    t_1 = 2.0d0 + t_0
    if (((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= (-0.99999998d0)) then
        tmp = (((2.0d0 + (beta * 2.0d0)) + (i * 4.0d0)) / alpha) / 2.0d0
    else
        tmp = (1.0d0 + (((alpha + beta) * ((beta - alpha) / (beta + (alpha + (2.0d0 * i))))) / t_1)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = 2.0 + t_0;
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= -0.99999998) {
		tmp = (((2.0 + (beta * 2.0)) + (i * 4.0)) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (((alpha + beta) * ((beta - alpha) / (beta + (alpha + (2.0 * i))))) / t_1)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	t_1 = 2.0 + t_0
	tmp = 0
	if ((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= -0.99999998:
		tmp = (((2.0 + (beta * 2.0)) + (i * 4.0)) / alpha) / 2.0
	else:
		tmp = (1.0 + (((alpha + beta) * ((beta - alpha) / (beta + (alpha + (2.0 * i))))) / t_1)) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(2.0 + t_0)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / t_1) <= -0.99999998)
		tmp = Float64(Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) + Float64(i * 4.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(alpha + beta) * Float64(Float64(beta - alpha) / Float64(beta + Float64(alpha + Float64(2.0 * i))))) / t_1)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	t_1 = 2.0 + t_0;
	tmp = 0.0;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= -0.99999998)
		tmp = (((2.0 + (beta * 2.0)) + (i * 4.0)) / alpha) / 2.0;
	else
		tmp = (1.0 + (((alpha + beta) * ((beta - alpha) / (beta + (alpha + (2.0 * i))))) / t_1)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 + t$95$0), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision], -0.99999998], N[(N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(N[(N[(alpha + beta), $MachinePrecision] * N[(N[(beta - alpha), $MachinePrecision] / N[(beta + N[(alpha + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.999999980000000011

   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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1\right), \color{blue}{2}\right) \]

   3. Simplified 15.0%

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

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\left(\beta - \alpha\right) \cdot \left(\alpha + \beta\right)}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}\right), 1\right), 2\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}\right), 1\right), 2\right) \]

      3. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\left(\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)\right) \cdot \frac{1}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}\right), 1\right), 2\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)\right), \left(\frac{1}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}\right)\right), 1\right), 2\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)\right), \left(\frac{1}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}\right)\right), 1\right), 2\right) \]

      6. difference-of-squares N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\beta \cdot \beta - \alpha \cdot \alpha\right), \left(\frac{1}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}\right)\right), 1\right), 2\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\beta \cdot \beta\right), \left(\alpha \cdot \alpha\right)\right), \left(\frac{1}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}\right)\right), 1\right), 2\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \left(\alpha \cdot \alpha\right)\right), \left(\frac{1}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}\right)\right), 1\right), 2\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \mathsf{*.f64}\left(\alpha, \alpha\right)\right), \left(\frac{1}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}\right)\right), 1\right), 2\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \mathsf{*.f64}\left(\alpha, \alpha\right)\right), \mathsf{/.f64}\left(1, \left(\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)\right)\right)\right), 1\right), 2\right) \]

      11. associate-+r+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \mathsf{*.f64}\left(\alpha, \alpha\right)\right), \mathsf{/.f64}\left(1, \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)\right)\right)\right), 1\right), 2\right) \]

      12. associate-+r+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \mathsf{*.f64}\left(\alpha, \alpha\right)\right), \mathsf{/.f64}\left(1, \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)\right)\right)\right)\right), 1\right), 2\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \mathsf{*.f64}\left(\alpha, \alpha\right)\right), \mathsf{/.f64}\left(1, \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right)\right)\right)\right), 1\right), 2\right) \]

      14. associate-+l+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \mathsf{*.f64}\left(\alpha, \alpha\right)\right), \mathsf{/.f64}\left(1, \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)\right)\right)\right), 1\right), 2\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \mathsf{*.f64}\left(\alpha, \alpha\right)\right), \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right), \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)\right)\right)\right), 1\right), 2\right) \]

   6. Applied egg-rr 2.6%

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

   7. Taylor expanded in alpha around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{2 + \left(2 \cdot \beta + 4 \cdot i\right)}{\alpha}\right)}, 2\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right), \alpha\right), 2\right) \]

      2. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(2 + 2 \cdot \beta\right) + 4 \cdot i\right), \alpha\right), 2\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(2 + 2 \cdot \beta\right), \left(4 \cdot i\right)\right), \alpha\right), 2\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \left(2 \cdot \beta\right)\right), \left(4 \cdot i\right)\right), \alpha\right), 2\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(2, \beta\right)\right), \left(4 \cdot i\right)\right), \alpha\right), 2\right) \]

      6. *-lowering-*.f64 89.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(2, \beta\right)\right), \mathsf{*.f64}\left(4, i\right)\right), \alpha\right), 2\right) \]

   9. Simplified 89.5%

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

   if -0.999999980000000011 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 74.3%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), 2\right)\right), 1\right), 2\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \left(\alpha + \beta\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), 2\right)\right), 1\right), 2\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}\right), \left(\alpha + \beta\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), 2\right)\right), 1\right), 2\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\beta - \alpha\right), \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right), \left(\alpha + \beta\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), 2\right)\right), 1\right), 2\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right), \left(\alpha + \beta\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), 2\right)\right), 1\right), 2\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\left(\beta + \alpha\right) + 2 \cdot i\right)\right), \left(\alpha + \beta\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), 2\right)\right), 1\right), 2\right) \]

      7. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\beta + \left(\alpha + 2 \cdot i\right)\right)\right), \left(\alpha + \beta\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), 2\right)\right), 1\right), 2\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\beta, \left(\alpha + 2 \cdot i\right)\right)\right), \left(\alpha + \beta\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), 2\right)\right), 1\right), 2\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, \left(2 \cdot i\right)\right)\right)\right), \left(\alpha + \beta\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), 2\right)\right), 1\right), 2\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, \mathsf{*.f64}\left(2, i\right)\right)\right)\right), \left(\alpha + \beta\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), 2\right)\right), 1\right), 2\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, \mathsf{*.f64}\left(2, i\right)\right)\right)\right), \left(\beta + \alpha\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), 2\right)\right), 1\right), 2\right) \]

      12. +-lowering-+.f64 99.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, \mathsf{*.f64}\left(2, i\right)\right)\right)\right), \mathsf{+.f64}\left(\beta, \alpha\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), 2\right)\right), 1\right), 2\right) \]

   4. Applied egg-rr 99.6%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.99999998:\\ \;\;\;\;\frac{\frac{\left(2 + \beta \cdot 2\right) + i \cdot 4}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\beta + \left(\alpha + 2 \cdot i\right)}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 4.5 \cdot 10^{+81}:\\ \;\;\;\;\frac{1 + \frac{\left(\alpha + \beta\right) \cdot \frac{\beta}{\beta + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(2 + \beta \cdot 2\right) + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 4.5e+81)
   (/
    (+
     1.0
     (/
      (* (+ alpha beta) (/ beta (+ beta (* 2.0 i))))
      (+ 2.0 (+ (+ alpha beta) (* 2.0 i)))))
    2.0)
   (/ (/ (+ (+ 2.0 (* beta 2.0)) (* i 4.0)) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 4.5e+81) {
		tmp = (1.0 + (((alpha + beta) * (beta / (beta + (2.0 * i)))) / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else {
		tmp = (((2.0 + (beta * 2.0)) + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 4.5d+81) then
        tmp = (1.0d0 + (((alpha + beta) * (beta / (beta + (2.0d0 * i)))) / (2.0d0 + ((alpha + beta) + (2.0d0 * i))))) / 2.0d0
    else
        tmp = (((2.0d0 + (beta * 2.0d0)) + (i * 4.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 4.5e+81) {
		tmp = (1.0 + (((alpha + beta) * (beta / (beta + (2.0 * i)))) / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else {
		tmp = (((2.0 + (beta * 2.0)) + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 4.5e+81:
		tmp = (1.0 + (((alpha + beta) * (beta / (beta + (2.0 * i)))) / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0
	else:
		tmp = (((2.0 + (beta * 2.0)) + (i * 4.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 4.5e+81)
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(alpha + beta) * Float64(beta / Float64(beta + Float64(2.0 * i)))) / Float64(2.0 + Float64(Float64(alpha + beta) + Float64(2.0 * i))))) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) + Float64(i * 4.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 4.5e+81)
		tmp = (1.0 + (((alpha + beta) * (beta / (beta + (2.0 * i)))) / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	else
		tmp = (((2.0 + (beta * 2.0)) + (i * 4.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 4.5e+81], N[(N[(1.0 + N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta / N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 4.50000000000000017e81

   1. Initial program 72.6%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), 2\right)\right), 1\right), 2\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \left(\alpha + \beta\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), 2\right)\right), 1\right), 2\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}\right), \left(\alpha + \beta\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), 2\right)\right), 1\right), 2\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\beta - \alpha\right), \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right), \left(\alpha + \beta\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), 2\right)\right), 1\right), 2\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right), \left(\alpha + \beta\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), 2\right)\right), 1\right), 2\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\left(\beta + \alpha\right) + 2 \cdot i\right)\right), \left(\alpha + \beta\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), 2\right)\right), 1\right), 2\right) \]

      7. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\beta + \left(\alpha + 2 \cdot i\right)\right)\right), \left(\alpha + \beta\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), 2\right)\right), 1\right), 2\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\beta, \left(\alpha + 2 \cdot i\right)\right)\right), \left(\alpha + \beta\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), 2\right)\right), 1\right), 2\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, \left(2 \cdot i\right)\right)\right)\right), \left(\alpha + \beta\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), 2\right)\right), 1\right), 2\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, \mathsf{*.f64}\left(2, i\right)\right)\right)\right), \left(\alpha + \beta\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), 2\right)\right), 1\right), 2\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, \mathsf{*.f64}\left(2, i\right)\right)\right)\right), \left(\beta + \alpha\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), 2\right)\right), 1\right), 2\right) \]

      12. +-lowering-+.f64 94.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, \mathsf{*.f64}\left(2, i\right)\right)\right)\right), \mathsf{+.f64}\left(\beta, \alpha\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), 2\right)\right), 1\right), 2\right) \]

   4. Applied egg-rr 94.6%

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

   5. Taylor expanded in alpha around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(\frac{\beta}{\beta + 2 \cdot i}\right)}, \mathsf{+.f64}\left(\beta, \alpha\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), 2\right)\right), 1\right), 2\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\beta, \left(\beta + 2 \cdot i\right)\right), \mathsf{+.f64}\left(\beta, \alpha\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), 2\right)\right), 1\right), 2\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\beta, \left(2 \cdot i + \beta\right)\right), \mathsf{+.f64}\left(\beta, \alpha\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), 2\right)\right), 1\right), 2\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\beta, \mathsf{+.f64}\left(\left(2 \cdot i\right), \beta\right)\right), \mathsf{+.f64}\left(\beta, \alpha\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), 2\right)\right), 1\right), 2\right) \]

      4. *-lowering-*.f64 93.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\beta, \mathsf{+.f64}\left(\mathsf{*.f64}\left(2, i\right), \beta\right)\right), \mathsf{+.f64}\left(\beta, \alpha\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), 2\right)\right), 1\right), 2\right) \]

   7. Simplified 93.9%

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

   if 4.50000000000000017e81 < alpha

   1. Initial program 10.7%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1\right), \color{blue}{2}\right) \]

   3. Simplified 25.1%

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

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\left(\beta - \alpha\right) \cdot \left(\alpha + \beta\right)}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}\right), 1\right), 2\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}\right), 1\right), 2\right) \]

      3. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\left(\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)\right) \cdot \frac{1}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}\right), 1\right), 2\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)\right), \left(\frac{1}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}\right)\right), 1\right), 2\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)\right), \left(\frac{1}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}\right)\right), 1\right), 2\right) \]

      6. difference-of-squares N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\beta \cdot \beta - \alpha \cdot \alpha\right), \left(\frac{1}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}\right)\right), 1\right), 2\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\beta \cdot \beta\right), \left(\alpha \cdot \alpha\right)\right), \left(\frac{1}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}\right)\right), 1\right), 2\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \left(\alpha \cdot \alpha\right)\right), \left(\frac{1}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}\right)\right), 1\right), 2\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \mathsf{*.f64}\left(\alpha, \alpha\right)\right), \left(\frac{1}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}\right)\right), 1\right), 2\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \mathsf{*.f64}\left(\alpha, \alpha\right)\right), \mathsf{/.f64}\left(1, \left(\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)\right)\right)\right), 1\right), 2\right) \]

      11. associate-+r+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \mathsf{*.f64}\left(\alpha, \alpha\right)\right), \mathsf{/.f64}\left(1, \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)\right)\right)\right), 1\right), 2\right) \]

      12. associate-+r+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \mathsf{*.f64}\left(\alpha, \alpha\right)\right), \mathsf{/.f64}\left(1, \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)\right)\right)\right)\right), 1\right), 2\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \mathsf{*.f64}\left(\alpha, \alpha\right)\right), \mathsf{/.f64}\left(1, \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right)\right)\right)\right), 1\right), 2\right) \]

      14. associate-+l+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \mathsf{*.f64}\left(\alpha, \alpha\right)\right), \mathsf{/.f64}\left(1, \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)\right)\right)\right), 1\right), 2\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \mathsf{*.f64}\left(\alpha, \alpha\right)\right), \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right), \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)\right)\right)\right), 1\right), 2\right) \]

   6. Applied egg-rr 9.9%

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

   7. Taylor expanded in alpha around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{2 + \left(2 \cdot \beta + 4 \cdot i\right)}{\alpha}\right)}, 2\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right), \alpha\right), 2\right) \]

      2. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(2 + 2 \cdot \beta\right) + 4 \cdot i\right), \alpha\right), 2\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(2 + 2 \cdot \beta\right), \left(4 \cdot i\right)\right), \alpha\right), 2\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \left(2 \cdot \beta\right)\right), \left(4 \cdot i\right)\right), \alpha\right), 2\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(2, \beta\right)\right), \left(4 \cdot i\right)\right), \alpha\right), 2\right) \]

      6. *-lowering-*.f64 72.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(2, \beta\right)\right), \mathsf{*.f64}\left(4, i\right)\right), \alpha\right), 2\right) \]

   9. Simplified 72.2%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 4.5 \cdot 10^{+81}:\\ \;\;\;\;\frac{1 + \frac{\left(\alpha + \beta\right) \cdot \frac{\beta}{\beta + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(2 + \beta \cdot 2\right) + i \cdot 4}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.25 \cdot 10^{+81}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(2 + \beta \cdot 2\right) + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1.25e+81)
   (/ (+ 1.0 (/ (- beta alpha) (+ 2.0 (+ (+ alpha beta) (* 2.0 i))))) 2.0)
   (/ (/ (+ (+ 2.0 (* beta 2.0)) (* i 4.0)) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.25e+81) {
		tmp = (1.0 + ((beta - alpha) / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else {
		tmp = (((2.0 + (beta * 2.0)) + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 1.25d+81) then
        tmp = (1.0d0 + ((beta - alpha) / (2.0d0 + ((alpha + beta) + (2.0d0 * i))))) / 2.0d0
    else
        tmp = (((2.0d0 + (beta * 2.0d0)) + (i * 4.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.25e+81) {
		tmp = (1.0 + ((beta - alpha) / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else {
		tmp = (((2.0 + (beta * 2.0)) + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 1.25e+81:
		tmp = (1.0 + ((beta - alpha) / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0
	else:
		tmp = (((2.0 + (beta * 2.0)) + (i * 4.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 1.25e+81)
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / Float64(2.0 + Float64(Float64(alpha + beta) + Float64(2.0 * i))))) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) + Float64(i * 4.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 1.25e+81)
		tmp = (1.0 + ((beta - alpha) / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	else
		tmp = (((2.0 + (beta * 2.0)) + (i * 4.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 1.25e+81], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.25e81

   1. Initial program 72.6%

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

   3. Taylor expanded in i around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(\beta - \alpha\right)}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), 2\right)\right), 1\right), 2\right) \]
   4. Step-by-step derivation

      1. --lowering--.f64 93.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), 2\right)\right), 1\right), 2\right) \]

   5. Simplified 93.2%

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

   if 1.25e81 < alpha

   1. Initial program 10.7%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1\right), \color{blue}{2}\right) \]

   3. Simplified 25.1%

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

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\left(\beta - \alpha\right) \cdot \left(\alpha + \beta\right)}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}\right), 1\right), 2\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}\right), 1\right), 2\right) \]

      3. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\left(\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)\right) \cdot \frac{1}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}\right), 1\right), 2\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)\right), \left(\frac{1}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}\right)\right), 1\right), 2\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)\right), \left(\frac{1}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}\right)\right), 1\right), 2\right) \]

      6. difference-of-squares N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\beta \cdot \beta - \alpha \cdot \alpha\right), \left(\frac{1}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}\right)\right), 1\right), 2\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\beta \cdot \beta\right), \left(\alpha \cdot \alpha\right)\right), \left(\frac{1}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}\right)\right), 1\right), 2\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \left(\alpha \cdot \alpha\right)\right), \left(\frac{1}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}\right)\right), 1\right), 2\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \mathsf{*.f64}\left(\alpha, \alpha\right)\right), \left(\frac{1}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}\right)\right), 1\right), 2\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \mathsf{*.f64}\left(\alpha, \alpha\right)\right), \mathsf{/.f64}\left(1, \left(\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)\right)\right)\right), 1\right), 2\right) \]

      11. associate-+r+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \mathsf{*.f64}\left(\alpha, \alpha\right)\right), \mathsf{/.f64}\left(1, \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)\right)\right)\right), 1\right), 2\right) \]

      12. associate-+r+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \mathsf{*.f64}\left(\alpha, \alpha\right)\right), \mathsf{/.f64}\left(1, \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)\right)\right)\right)\right), 1\right), 2\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \mathsf{*.f64}\left(\alpha, \alpha\right)\right), \mathsf{/.f64}\left(1, \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right)\right)\right)\right), 1\right), 2\right) \]

      14. associate-+l+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \mathsf{*.f64}\left(\alpha, \alpha\right)\right), \mathsf{/.f64}\left(1, \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)\right)\right)\right), 1\right), 2\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \mathsf{*.f64}\left(\alpha, \alpha\right)\right), \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right), \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)\right)\right)\right), 1\right), 2\right) \]

   6. Applied egg-rr 9.9%

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

   7. Taylor expanded in alpha around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{2 + \left(2 \cdot \beta + 4 \cdot i\right)}{\alpha}\right)}, 2\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right), \alpha\right), 2\right) \]

      2. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(2 + 2 \cdot \beta\right) + 4 \cdot i\right), \alpha\right), 2\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(2 + 2 \cdot \beta\right), \left(4 \cdot i\right)\right), \alpha\right), 2\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \left(2 \cdot \beta\right)\right), \left(4 \cdot i\right)\right), \alpha\right), 2\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(2, \beta\right)\right), \left(4 \cdot i\right)\right), \alpha\right), 2\right) \]

      6. *-lowering-*.f64 72.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(2, \beta\right)\right), \mathsf{*.f64}\left(4, i\right)\right), \alpha\right), 2\right) \]

   9. Simplified 72.2%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.25 \cdot 10^{+81}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(2 + \beta \cdot 2\right) + i \cdot 4}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.06 \cdot 10^{+81}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(2 + \beta \cdot 2\right) + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1.06e+81)
   (/ (+ 1.0 (/ beta (+ 2.0 (+ (+ alpha beta) (* 2.0 i))))) 2.0)
   (/ (/ (+ (+ 2.0 (* beta 2.0)) (* i 4.0)) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.06e+81) {
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else {
		tmp = (((2.0 + (beta * 2.0)) + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 1.06d+81) then
        tmp = (1.0d0 + (beta / (2.0d0 + ((alpha + beta) + (2.0d0 * i))))) / 2.0d0
    else
        tmp = (((2.0d0 + (beta * 2.0d0)) + (i * 4.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.06e+81) {
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else {
		tmp = (((2.0 + (beta * 2.0)) + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 1.06e+81:
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0
	else:
		tmp = (((2.0 + (beta * 2.0)) + (i * 4.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 1.06e+81)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(2.0 + Float64(Float64(alpha + beta) + Float64(2.0 * i))))) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) + Float64(i * 4.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 1.06e+81)
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	else
		tmp = (((2.0 + (beta * 2.0)) + (i * 4.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 1.06e+81], N[(N[(1.0 + N[(beta / N[(2.0 + N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.05999999999999993e81

   1. Initial program 72.6%

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

   3. Taylor expanded in beta around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\color{blue}{\beta}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), 2\right)\right), 1\right), 2\right) \]
   4. Step-by-step derivation

   5. Simplified 92.9%

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

   if 1.05999999999999993e81 < alpha

   1. Initial program 10.7%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1\right), \color{blue}{2}\right) \]

   3. Simplified 25.1%

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

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\left(\beta - \alpha\right) \cdot \left(\alpha + \beta\right)}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}\right), 1\right), 2\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}\right), 1\right), 2\right) \]

      3. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\left(\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)\right) \cdot \frac{1}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}\right), 1\right), 2\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)\right), \left(\frac{1}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}\right)\right), 1\right), 2\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)\right), \left(\frac{1}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}\right)\right), 1\right), 2\right) \]

      6. difference-of-squares N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\beta \cdot \beta - \alpha \cdot \alpha\right), \left(\frac{1}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}\right)\right), 1\right), 2\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\beta \cdot \beta\right), \left(\alpha \cdot \alpha\right)\right), \left(\frac{1}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}\right)\right), 1\right), 2\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \left(\alpha \cdot \alpha\right)\right), \left(\frac{1}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}\right)\right), 1\right), 2\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \mathsf{*.f64}\left(\alpha, \alpha\right)\right), \left(\frac{1}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}\right)\right), 1\right), 2\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \mathsf{*.f64}\left(\alpha, \alpha\right)\right), \mathsf{/.f64}\left(1, \left(\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)\right)\right)\right), 1\right), 2\right) \]

      11. associate-+r+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \mathsf{*.f64}\left(\alpha, \alpha\right)\right), \mathsf{/.f64}\left(1, \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)\right)\right)\right), 1\right), 2\right) \]

      12. associate-+r+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \mathsf{*.f64}\left(\alpha, \alpha\right)\right), \mathsf{/.f64}\left(1, \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)\right)\right)\right)\right), 1\right), 2\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \mathsf{*.f64}\left(\alpha, \alpha\right)\right), \mathsf{/.f64}\left(1, \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right)\right)\right)\right), 1\right), 2\right) \]

      14. associate-+l+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \mathsf{*.f64}\left(\alpha, \alpha\right)\right), \mathsf{/.f64}\left(1, \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)\right)\right)\right), 1\right), 2\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \mathsf{*.f64}\left(\alpha, \alpha\right)\right), \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right), \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)\right)\right)\right), 1\right), 2\right) \]

   6. Applied egg-rr 9.9%

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

   7. Taylor expanded in alpha around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{2 + \left(2 \cdot \beta + 4 \cdot i\right)}{\alpha}\right)}, 2\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right), \alpha\right), 2\right) \]

      2. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(2 + 2 \cdot \beta\right) + 4 \cdot i\right), \alpha\right), 2\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(2 + 2 \cdot \beta\right), \left(4 \cdot i\right)\right), \alpha\right), 2\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \left(2 \cdot \beta\right)\right), \left(4 \cdot i\right)\right), \alpha\right), 2\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(2, \beta\right)\right), \left(4 \cdot i\right)\right), \alpha\right), 2\right) \]

      6. *-lowering-*.f64 72.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(2, \beta\right)\right), \mathsf{*.f64}\left(4, i\right)\right), \alpha\right), 2\right) \]

   9. Simplified 72.2%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.06 \cdot 10^{+81}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(2 + \beta \cdot 2\right) + i \cdot 4}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.35 \cdot 10^{+17}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(2 + \beta \cdot 2\right) + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1.35e+17)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (+ (+ 2.0 (* beta 2.0)) (* i 4.0)) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.35e+17) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (((2.0 + (beta * 2.0)) + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 1.35d+17) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = (((2.0d0 + (beta * 2.0d0)) + (i * 4.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.35e+17) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (((2.0 + (beta * 2.0)) + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 1.35e+17:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = (((2.0 + (beta * 2.0)) + (i * 4.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 1.35e+17)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) + Float64(i * 4.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 1.35e+17)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = (((2.0 + (beta * 2.0)) + (i * 4.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 1.35e+17], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.35e17

   1. Initial program 76.1%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1\right), \color{blue}{2}\right) \]

   3. Simplified 81.9%

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

   5. Taylor expanded in i around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{\beta - \alpha}{2 + \left(\alpha + \beta\right)}\right)}, 1\right), 2\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\beta - \alpha\right), \left(2 + \left(\alpha + \beta\right)\right)\right), 1\right), 2\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(2 + \left(\alpha + \beta\right)\right)\right), 1\right), 2\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(2, \left(\alpha + \beta\right)\right)\right), 1\right), 2\right) \]

      4. +-lowering-+.f64 87.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right), 1\right), 2\right) \]

   7. Simplified 87.1%

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

   8. Taylor expanded in alpha around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{\beta}{2 + \beta}\right)}, 1\right), 2\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\beta, \left(2 + \beta\right)\right), 1\right), 2\right) \]

      2. +-lowering-+.f64 87.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\beta, \mathsf{+.f64}\left(2, \beta\right)\right), 1\right), 2\right) \]

   10. Simplified 87.9%

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

   if 1.35e17 < alpha

   1. Initial program 19.5%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1\right), \color{blue}{2}\right) \]

   3. Simplified 31.7%

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

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\left(\beta - \alpha\right) \cdot \left(\alpha + \beta\right)}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}\right), 1\right), 2\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}\right), 1\right), 2\right) \]

      3. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\left(\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)\right) \cdot \frac{1}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}\right), 1\right), 2\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)\right), \left(\frac{1}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}\right)\right), 1\right), 2\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)\right), \left(\frac{1}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}\right)\right), 1\right), 2\right) \]

      6. difference-of-squares N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\beta \cdot \beta - \alpha \cdot \alpha\right), \left(\frac{1}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}\right)\right), 1\right), 2\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\beta \cdot \beta\right), \left(\alpha \cdot \alpha\right)\right), \left(\frac{1}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}\right)\right), 1\right), 2\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \left(\alpha \cdot \alpha\right)\right), \left(\frac{1}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}\right)\right), 1\right), 2\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \mathsf{*.f64}\left(\alpha, \alpha\right)\right), \left(\frac{1}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)}\right)\right), 1\right), 2\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \mathsf{*.f64}\left(\alpha, \alpha\right)\right), \mathsf{/.f64}\left(1, \left(\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)\right)\right)\right), 1\right), 2\right) \]

      11. associate-+r+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \mathsf{*.f64}\left(\alpha, \alpha\right)\right), \mathsf{/.f64}\left(1, \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)\right)\right)\right), 1\right), 2\right) \]

      12. associate-+r+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \mathsf{*.f64}\left(\alpha, \alpha\right)\right), \mathsf{/.f64}\left(1, \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)\right)\right)\right)\right), 1\right), 2\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \mathsf{*.f64}\left(\alpha, \alpha\right)\right), \mathsf{/.f64}\left(1, \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right)\right)\right)\right), 1\right), 2\right) \]

      14. associate-+l+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \mathsf{*.f64}\left(\alpha, \alpha\right)\right), \mathsf{/.f64}\left(1, \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)\right)\right)\right), 1\right), 2\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \mathsf{*.f64}\left(\alpha, \alpha\right)\right), \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right), \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)\right)\right)\right), 1\right), 2\right) \]

   6. Applied egg-rr 18.8%

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

   7. Taylor expanded in alpha around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{2 + \left(2 \cdot \beta + 4 \cdot i\right)}{\alpha}\right)}, 2\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right), \alpha\right), 2\right) \]

      2. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(2 + 2 \cdot \beta\right) + 4 \cdot i\right), \alpha\right), 2\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(2 + 2 \cdot \beta\right), \left(4 \cdot i\right)\right), \alpha\right), 2\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \left(2 \cdot \beta\right)\right), \left(4 \cdot i\right)\right), \alpha\right), 2\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(2, \beta\right)\right), \left(4 \cdot i\right)\right), \alpha\right), 2\right) \]

      6. *-lowering-*.f64 65.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(2, \beta\right)\right), \mathsf{*.f64}\left(4, i\right)\right), \alpha\right), 2\right) \]

   9. Simplified 65.8%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.35 \cdot 10^{+17}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(2 + \beta \cdot 2\right) + i \cdot 4}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 8.5 \cdot 10^{+116}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= i 8.5e+116)
   (/ (+ 1.0 (/ (- beta alpha) (+ (+ alpha beta) 2.0))) 2.0)
   0.5))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 8.5e+116) {
		tmp = (1.0 + ((beta - alpha) / ((alpha + beta) + 2.0))) / 2.0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (i <= 8.5d+116) then
        tmp = (1.0d0 + ((beta - alpha) / ((alpha + beta) + 2.0d0))) / 2.0d0
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 8.5e+116) {
		tmp = (1.0 + ((beta - alpha) / ((alpha + beta) + 2.0))) / 2.0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if i <= 8.5e+116:
		tmp = (1.0 + ((beta - alpha) / ((alpha + beta) + 2.0))) / 2.0
	else:
		tmp = 0.5
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (i <= 8.5e+116)
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0))) / 2.0);
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (i <= 8.5e+116)
		tmp = (1.0 + ((beta - alpha) / ((alpha + beta) + 2.0))) / 2.0;
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[i, 8.5e+116], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if i < 8.5000000000000002e116

   1. Initial program 52.8%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1\right), \color{blue}{2}\right) \]

   3. Simplified 56.5%

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

   5. Taylor expanded in i around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{\beta - \alpha}{2 + \left(\alpha + \beta\right)}\right)}, 1\right), 2\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\beta - \alpha\right), \left(2 + \left(\alpha + \beta\right)\right)\right), 1\right), 2\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(2 + \left(\alpha + \beta\right)\right)\right), 1\right), 2\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(2, \left(\alpha + \beta\right)\right)\right), 1\right), 2\right) \]

      4. +-lowering-+.f64 71.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right), 1\right), 2\right) \]

   7. Simplified 71.0%

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

   if 8.5000000000000002e116 < i

   1. Initial program 68.3%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1\right), \color{blue}{2}\right) \]

   3. Simplified 86.4%

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

   5. Taylor expanded in i around inf

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

   7. Simplified 85.3%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 8.5 \cdot 10^{+116}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 2.15 \cdot 10^{+117}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= i 2.15e+117) (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0) 0.5))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 2.15e+117) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (i <= 2.15d+117) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 2.15e+117) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if i <= 2.15e+117:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = 0.5
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (i <= 2.15e+117)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (i <= 2.15e+117)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[i, 2.15e+117], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if i < 2.14999999999999999e117

   1. Initial program 52.8%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1\right), \color{blue}{2}\right) \]

   3. Simplified 56.5%

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

   5. Taylor expanded in i around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{\beta - \alpha}{2 + \left(\alpha + \beta\right)}\right)}, 1\right), 2\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\beta - \alpha\right), \left(2 + \left(\alpha + \beta\right)\right)\right), 1\right), 2\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(2 + \left(\alpha + \beta\right)\right)\right), 1\right), 2\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(2, \left(\alpha + \beta\right)\right)\right), 1\right), 2\right) \]

      4. +-lowering-+.f64 71.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\alpha, \beta\right)\right)\right), 1\right), 2\right) \]

   7. Simplified 71.0%

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

   8. Taylor expanded in alpha around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{\beta}{2 + \beta}\right)}, 1\right), 2\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\beta, \left(2 + \beta\right)\right), 1\right), 2\right) \]

      2. +-lowering-+.f64 70.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\beta, \mathsf{+.f64}\left(2, \beta\right)\right), 1\right), 2\right) \]

   10. Simplified 70.3%

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

   if 2.14999999999999999e117 < i

   1. Initial program 68.3%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1\right), \color{blue}{2}\right) \]

   3. Simplified 86.4%

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

   5. Taylor expanded in i around inf

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

   7. Simplified 85.3%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 2.15 \cdot 10^{+117}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.7% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.7 \cdot 10^{+34}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 (if (<= beta 4.7e+34) 0.5 1.0))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 4.7e+34) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 4.7d+34) then
        tmp = 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 4.7e+34) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 4.7e+34:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 4.7e+34)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 4.7e+34)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 4.7e+34], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 4.70000000000000015e34

   1. Initial program 69.4%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1\right), \color{blue}{2}\right) \]

   3. Simplified 73.5%

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

   5. Taylor expanded in i around inf

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

   7. Simplified 69.3%

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

   if 4.70000000000000015e34 < beta

   1. Initial program 35.1%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1\right), \color{blue}{2}\right) \]

   3. Simplified 50.1%

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

   5. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{1} \]
   6. Step-by-step derivation

   7. Simplified 71.6%

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

Alternative 10: 61.7% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[alpha_, beta_, i_] := 0.5

Derivation

1. Initial program 57.3%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1\right), \color{blue}{2}\right) \]

3. Simplified 65.3%

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

5. Taylor expanded in i around inf

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

7. Simplified 56.8%

   \[\leadsto \color{blue}{0.5} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024158 
(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))