Octave 3.8, jcobi/2

Percentage Accurate: 63.2% → 97.9%

Time: 17.0s

Alternatives: 8

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: 63.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := 2 + t\_0\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_1} \leq -0.99999999:\\ \;\;\;\;\frac{\frac{\left(2 + \beta \cdot 2\right) + i \cdot 4}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\alpha + \left(\beta + 2 \cdot i\right)}}{t\_1} + 1}{2}\\ \end{array} \end{array} \]

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) -0.99999999)
     (/ (/ (+ (+ 2.0 (* beta 2.0)) (* i 4.0)) alpha) 2.0)
     (/
      (+
       (/
        (* (+ alpha beta) (/ (- beta alpha) (+ alpha (+ beta (* 2.0 i)))))
        t_1)
       1.0)
      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) <= -0.99999999) {
		tmp = (((2.0 + (beta * 2.0)) + (i * 4.0)) / alpha) / 2.0;
	} else {
		tmp = ((((alpha + beta) * ((beta - alpha) / (alpha + (beta + (2.0 * i))))) / t_1) + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: 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) <= (-0.99999999d0)) then
        tmp = (((2.0d0 + (beta * 2.0d0)) + (i * 4.0d0)) / alpha) / 2.0d0
    else
        tmp = ((((alpha + beta) * ((beta - alpha) / (alpha + (beta + (2.0d0 * i))))) / t_1) + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = 2.0 + t_0;
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= -0.99999999) {
		tmp = (((2.0 + (beta * 2.0)) + (i * 4.0)) / alpha) / 2.0;
	} else {
		tmp = ((((alpha + beta) * ((beta - alpha) / (alpha + (beta + (2.0 * i))))) / t_1) + 1.0) / 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) <= -0.99999999:
		tmp = (((2.0 + (beta * 2.0)) + (i * 4.0)) / alpha) / 2.0
	else:
		tmp = ((((alpha + beta) * ((beta - alpha) / (alpha + (beta + (2.0 * i))))) / t_1) + 1.0) / 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) <= -0.99999999)
		tmp = Float64(Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) + Float64(i * 4.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(Float64(beta - alpha) / Float64(alpha + Float64(beta + Float64(2.0 * i))))) / t_1) + 1.0) / 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) <= -0.99999999)
		tmp = (((2.0 + (beta * 2.0)) + (i * 4.0)) / alpha) / 2.0;
	else
		tmp = ((((alpha + beta) * ((beta - alpha) / (alpha + (beta + (2.0 * i))))) / t_1) + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, 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], -0.99999999], 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[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(N[(beta - alpha), $MachinePrecision] / N[(alpha + N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.99999998999999995

   1. Initial program 2.7%

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

   3. Taylor expanded in alpha around inf

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

      1. distribute-rgt1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(-1 + 1\right) \cdot \beta - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha}\right), 2\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{0 \cdot \beta - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha}\right), 2\right) \]

      3. mul0-lft N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{0 - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha}\right), 2\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(-1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)\right)}{\alpha}\right), 2\right) \]

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

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

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

      8. associate-+r+ N/A

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

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

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

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

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

      11. *-commutative N/A

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

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

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 94.7%

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

   5. Simplified 94.7%

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

   if -0.99999998999999995 < (/.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 80.7%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Add Preprocessing
   3. 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. associate-+l+ N/A

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

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

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

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

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

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

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

      10. +-lowering-+.f64 99.4%

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

   4. Applied egg-rr 99.4%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.99999999:\\ \;\;\;\;\frac{\frac{\left(2 + \beta \cdot 2\right) + i \cdot 4}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\alpha + \left(\beta + 2 \cdot i\right)}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 8 \cdot 10^{+108}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(2 + \beta \cdot 2\right) + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 8e+108)
   (/ (+ 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 <= 8e+108) {
		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 <= 8d+108) 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 <= 8e+108) {
		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 <= 8e+108:
		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 <= 8e+108)
		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 <= 8e+108)
		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, 8e+108], 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 < 8.0000000000000003e108

   1. Initial program 78.4%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. 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 93.6%

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

   5. Simplified 93.6%

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

   if 8.0000000000000003e108 < alpha

   1. Initial program 7.5%

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

   3. Taylor expanded in alpha around inf

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

      1. distribute-rgt1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(-1 + 1\right) \cdot \beta - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha}\right), 2\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{0 \cdot \beta - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha}\right), 2\right) \]

      3. mul0-lft N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{0 - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha}\right), 2\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(-1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)\right)}{\alpha}\right), 2\right) \]

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

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

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

      8. associate-+r+ N/A

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

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

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

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

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

      11. *-commutative N/A

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

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

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 82.0%

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

   5. Simplified 82.0%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 8 \cdot 10^{+108}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(2 + \beta \cdot 2\right) + i \cdot 4}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.35 \cdot 10^{+107}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(2 + \beta \cdot 2\right) + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1.35e+107)
   (/ (+ 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+107) {
		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+107) 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+107) {
		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+107:
		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+107)
		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+107)
		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+107], 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 < 1.3500000000000001e107

   1. Initial program 78.4%

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

   3. Taylor expanded in i around 0

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

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

      5. +-lowering-+.f64 80.7%

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

   5. Simplified 80.7%

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

   6. Taylor expanded in alpha around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{\beta}{2 + \beta}\right)}, 1\right), 2\right) \]
   7. 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 84.1%

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

   8. Simplified 84.1%

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

   if 1.3500000000000001e107 < alpha

   1. Initial program 7.5%

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

   3. Taylor expanded in alpha around inf

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

      1. distribute-rgt1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(-1 + 1\right) \cdot \beta - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha}\right), 2\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{0 \cdot \beta - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha}\right), 2\right) \]

      3. mul0-lft N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{0 - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha}\right), 2\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(-1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)\right)}{\alpha}\right), 2\right) \]

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

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

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

      8. associate-+r+ N/A

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

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

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

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

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

      11. *-commutative N/A

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

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

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 82.0%

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

   5. Simplified 82.0%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.35 \cdot 10^{+107}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(2 + \beta \cdot 2\right) + i \cdot 4}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.7 \cdot 10^{+108}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta + i \cdot 4\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1.7e+108)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (+ 2.0 (+ beta (* i 4.0))) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.7e+108) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (beta + (i * 4.0))) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 1.7d+108) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((2.0d0 + (beta + (i * 4.0d0))) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.7e+108) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (beta + (i * 4.0))) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 1.7e+108:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((2.0 + (beta + (i * 4.0))) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 1.7e+108)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(beta + Float64(i * 4.0))) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 1.7e+108)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((2.0 + (beta + (i * 4.0))) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 1.7e+108], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(beta + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.69999999999999998e108

   1. Initial program 78.4%

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

   3. Taylor expanded in i around 0

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

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

      5. +-lowering-+.f64 80.7%

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

   5. Simplified 80.7%

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

   6. Taylor expanded in alpha around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{\beta}{2 + \beta}\right)}, 1\right), 2\right) \]
   7. 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 84.1%

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

   8. Simplified 84.1%

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

   if 1.69999999999999998e108 < alpha

   1. Initial program 7.5%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. difference-of-squares N/A

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

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

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

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

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

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

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

      10. associate-+l+ N/A

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

      11. associate-+l+ N/A

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

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

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

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

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

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

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

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

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

      16. associate-+l+ N/A

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

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

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

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

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

      19. *-lowering-*.f64 7.3%

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

   4. Applied egg-rr 7.3%

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

   5. Taylor expanded in beta around 0

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

      1. associate-*r/ N/A

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

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

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

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

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

      4. unpow2 N/A

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

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

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

      6. associate-+r+ N/A

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

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

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

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

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

      9. *-lowering-*.f64 7.4%

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

   7. Simplified 7.4%

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

   8. Taylor expanded in alpha around inf

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

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

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

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

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

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

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

      4. *-lowering-*.f64 73.3%

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

   10. Simplified 73.3%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.7 \cdot 10^{+108}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta + i \cdot 4\right)}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 7.6 \cdot 10^{+107}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 7.6e+107)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 7.6e+107) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 7.6d+107) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 7.6e+107) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 7.6e+107:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 7.6e+107)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 7.6e+107)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 7.6e+107], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 7.5999999999999996e107

   1. Initial program 78.4%

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

   3. Taylor expanded in i around 0

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

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

      5. +-lowering-+.f64 80.7%

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

   5. Simplified 80.7%

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

   6. Taylor expanded in alpha around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{\beta}{2 + \beta}\right)}, 1\right), 2\right) \]
   7. 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 84.1%

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

   8. Simplified 84.1%

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

   if 7.5999999999999996e107 < alpha

   1. Initial program 7.5%

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

   3. Taylor expanded in i around 0

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

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

      5. +-lowering-+.f64 11.0%

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

   5. Simplified 11.0%

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

   6. Taylor expanded in alpha around inf

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

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

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

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

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

      3. *-lowering-*.f64 49.8%

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

   8. Simplified 49.8%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 7.6 \cdot 10^{+107}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 3.5 \cdot 10^{+90}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= i 3.5e+90) (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0) 0.5))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 3.5e+90) {
		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 <= 3.5d+90) 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 <= 3.5e+90) {
		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 <= 3.5e+90:
		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 <= 3.5e+90)
		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 <= 3.5e+90)
		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, 3.5e+90], 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 < 3.4999999999999998e90

   1. Initial program 61.2%

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

   3. Taylor expanded in i around 0

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

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

      5. +-lowering-+.f64 72.7%

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

   5. Simplified 72.7%

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

   6. Taylor expanded in alpha around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{\beta}{2 + \beta}\right)}, 1\right), 2\right) \]
   7. 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 72.7%

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

   8. Simplified 72.7%

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

   if 3.4999999999999998e90 < i

   1. Initial program 69.0%

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

   3. Taylor expanded in i around inf

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

   5. Simplified 81.1%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 3.5 \cdot 10^{+90}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.5% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+120}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 (if (<= beta 2e+120) 0.5 1.0))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2e+120) {
		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 <= 2d+120) 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 <= 2e+120) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 2e+120:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 2e+120)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 2e+120)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 2e+120], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 2e120

   1. Initial program 75.0%

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

   3. Taylor expanded in i around inf

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

   5. Simplified 70.4%

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

   if 2e120 < beta

   1. Initial program 18.8%

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

   3. Taylor expanded in beta around inf

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

   5. Simplified 79.5%

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

Alternative 8: 61.6% 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 64.3%

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

3. Taylor expanded in i around inf

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

5. Simplified 63.6%

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

Reproduce

herbie shell --seed 2024191 
(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))