Octave 3.8, jcobi/2

Percentage Accurate: 62.8% → 97.6%

Time: 13.4s

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: 62.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.6% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (if (<=
        (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0))
        -0.99999999)
     (/ (/ (+ 2.0 (+ (* i 4.0) (* beta 2.0))) alpha) 2.0)
     (/
      (+
       (*
        (/ (- beta alpha) (+ alpha (+ beta (+ 2.0 (* 2.0 i)))))
        (/ (+ alpha beta) (+ alpha (+ beta (* 2.0 i)))))
       1.0)
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.99999999) {
		tmp = ((2.0 + ((i * 4.0) + (beta * 2.0))) / alpha) / 2.0;
	} else {
		tmp = ((((beta - alpha) / (alpha + (beta + (2.0 + (2.0 * i))))) * ((alpha + beta) / (alpha + (beta + (2.0 * i))))) + 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) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0d0 + t_0)) <= (-0.99999999d0)) then
        tmp = ((2.0d0 + ((i * 4.0d0) + (beta * 2.0d0))) / alpha) / 2.0d0
    else
        tmp = ((((beta - alpha) / (alpha + (beta + (2.0d0 + (2.0d0 * i))))) * ((alpha + beta) / (alpha + (beta + (2.0d0 * i))))) + 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 tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.99999999) {
		tmp = ((2.0 + ((i * 4.0) + (beta * 2.0))) / alpha) / 2.0;
	} else {
		tmp = ((((beta - alpha) / (alpha + (beta + (2.0 + (2.0 * i))))) * ((alpha + beta) / (alpha + (beta + (2.0 * i))))) + 1.0) / 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)) <= -0.99999999:
		tmp = ((2.0 + ((i * 4.0) + (beta * 2.0))) / alpha) / 2.0
	else:
		tmp = ((((beta - alpha) / (alpha + (beta + (2.0 + (2.0 * i))))) * ((alpha + beta) / (alpha + (beta + (2.0 * i))))) + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0)) <= -0.99999999)
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(i * 4.0) + Float64(beta * 2.0))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(beta - alpha) / Float64(alpha + Float64(beta + Float64(2.0 + Float64(2.0 * i))))) * Float64(Float64(alpha + beta) / Float64(alpha + Float64(beta + Float64(2.0 * i))))) + 1.0) / 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)) <= -0.99999999)
		tmp = ((2.0 + ((i * 4.0) + (beta * 2.0))) / alpha) / 2.0;
	else
		tmp = ((((beta - alpha) / (alpha + (beta + (2.0 + (2.0 * i))))) * ((alpha + beta) / (alpha + (beta + (2.0 * i))))) + 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]}, 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], -0.99999999], N[(N[(N[(2.0 + N[(N[(i * 4.0), $MachinePrecision] + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(alpha + N[(beta + N[(2.0 + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] / N[(alpha + N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #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 2.5%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   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 16.0%

      \[\leadsto \color{blue}{\frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)} + 1}{2}} \]
   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. associate-+r+ N/A

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

      3. *-commutative N/A

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

      4. associate-+r+ N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. times-frac N/A

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

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

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

   6. Applied egg-rr 18.3%

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

   7. Taylor expanded in alpha around inf

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

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

      9. +-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. *-lowering-*.f64 88.1%

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

   9. Simplified 88.1%

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

   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 80.8%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   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 86.2%

      \[\leadsto \color{blue}{\frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)} + 1}{2}} \]
   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. associate-+r+ N/A

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

      3. *-commutative N/A

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

      4. associate-+r+ N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. times-frac N/A

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

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

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

   6. Applied egg-rr 99.8%

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

4. Final simplification 97.2%

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

Alternative 2: 88.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.3 \cdot 10^{+86}:\\ \;\;\;\;\frac{1 + \frac{1}{\frac{\beta + \left(\alpha + \left(2 + 2 \cdot i\right)\right)}{\beta - \alpha}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(i \cdot 4 + \beta \cdot 2\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1.3e+86)
   (/
    (+ 1.0 (/ 1.0 (/ (+ beta (+ alpha (+ 2.0 (* 2.0 i)))) (- beta alpha))))
    2.0)
   (/ (/ (+ 2.0 (+ (* i 4.0) (* beta 2.0))) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.3e+86) {
		tmp = (1.0 + (1.0 / ((beta + (alpha + (2.0 + (2.0 * i)))) / (beta - alpha)))) / 2.0;
	} else {
		tmp = ((2.0 + ((i * 4.0) + (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 <= 1.3d+86) then
        tmp = (1.0d0 + (1.0d0 / ((beta + (alpha + (2.0d0 + (2.0d0 * i)))) / (beta - alpha)))) / 2.0d0
    else
        tmp = ((2.0d0 + ((i * 4.0d0) + (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 <= 1.3e+86) {
		tmp = (1.0 + (1.0 / ((beta + (alpha + (2.0 + (2.0 * i)))) / (beta - alpha)))) / 2.0;
	} else {
		tmp = ((2.0 + ((i * 4.0) + (beta * 2.0))) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 1.3e+86:
		tmp = (1.0 + (1.0 / ((beta + (alpha + (2.0 + (2.0 * i)))) / (beta - alpha)))) / 2.0
	else:
		tmp = ((2.0 + ((i * 4.0) + (beta * 2.0))) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 1.3e+86)
		tmp = Float64(Float64(1.0 + Float64(1.0 / Float64(Float64(beta + Float64(alpha + Float64(2.0 + Float64(2.0 * i)))) / Float64(beta - alpha)))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(i * 4.0) + Float64(beta * 2.0))) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 1.3e+86)
		tmp = (1.0 + (1.0 / ((beta + (alpha + (2.0 + (2.0 * i)))) / (beta - alpha)))) / 2.0;
	else
		tmp = ((2.0 + ((i * 4.0) + (beta * 2.0))) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 1.3e+86], N[(N[(1.0 + N[(1.0 / N[(N[(beta + N[(alpha + N[(2.0 + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta - alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(N[(i * 4.0), $MachinePrecision] + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.2999999999999999e86

   1. Initial program 82.3%

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

   3. Taylor expanded in i around 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 98.5%

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

   5. Simplified 98.5%

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

      1. clear-num N/A

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

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

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

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

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

      4. associate-+l+ N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

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

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

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

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

      9. +-commutative N/A

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

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

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

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

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

      12. --lowering--.f64 98.5%

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

   7. Applied egg-rr 98.5%

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

   if 1.2999999999999999e86 < alpha

   1. Initial program 13.0%

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

      1. /-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 25.7%

      \[\leadsto \color{blue}{\frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)} + 1}{2}} \]
   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. associate-+r+ N/A

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

      3. *-commutative N/A

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

      4. associate-+r+ N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. times-frac N/A

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

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

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

   6. Applied egg-rr 35.7%

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

   7. Taylor expanded in alpha around inf

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

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

      9. +-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. *-lowering-*.f64 71.1%

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

   9. Simplified 71.1%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.3 \cdot 10^{+86}:\\ \;\;\;\;\frac{1 + \frac{1}{\frac{\beta + \left(\alpha + \left(2 + 2 \cdot i\right)\right)}{\beta - \alpha}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(i \cdot 4 + \beta \cdot 2\right)}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 1.9 \cdot 10^{+86}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(i \cdot 4 + \beta \cdot 2\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 2e-17)
   (/ (+ 1.0 (/ (- beta alpha) (+ (+ alpha beta) 2.0))) 2.0)
   (if (<= alpha 1.9e+86)
     0.5
     (/ (/ (+ 2.0 (+ (* i 4.0) (* beta 2.0))) alpha) 2.0))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2e-17) {
		tmp = (1.0 + ((beta - alpha) / ((alpha + beta) + 2.0))) / 2.0;
	} else if (alpha <= 1.9e+86) {
		tmp = 0.5;
	} else {
		tmp = ((2.0 + ((i * 4.0) + (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 <= 2d-17) then
        tmp = (1.0d0 + ((beta - alpha) / ((alpha + beta) + 2.0d0))) / 2.0d0
    else if (alpha <= 1.9d+86) then
        tmp = 0.5d0
    else
        tmp = ((2.0d0 + ((i * 4.0d0) + (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 <= 2e-17) {
		tmp = (1.0 + ((beta - alpha) / ((alpha + beta) + 2.0))) / 2.0;
	} else if (alpha <= 1.9e+86) {
		tmp = 0.5;
	} else {
		tmp = ((2.0 + ((i * 4.0) + (beta * 2.0))) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 2e-17:
		tmp = (1.0 + ((beta - alpha) / ((alpha + beta) + 2.0))) / 2.0
	elif alpha <= 1.9e+86:
		tmp = 0.5
	else:
		tmp = ((2.0 + ((i * 4.0) + (beta * 2.0))) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 2e-17)
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0))) / 2.0);
	elseif (alpha <= 1.9e+86)
		tmp = 0.5;
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(i * 4.0) + Float64(beta * 2.0))) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 2e-17)
		tmp = (1.0 + ((beta - alpha) / ((alpha + beta) + 2.0))) / 2.0;
	elseif (alpha <= 1.9e+86)
		tmp = 0.5;
	else
		tmp = ((2.0 + ((i * 4.0) + (beta * 2.0))) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 2e-17], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 1.9e+86], 0.5, N[(N[(N[(2.0 + N[(N[(i * 4.0), $MachinePrecision] + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 2.00000000000000014e-17

   1. Initial program 82.5%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   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 87.7%

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

   5. 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 92.0%

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

   7. Simplified 92.0%

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

   if 2.00000000000000014e-17 < alpha < 1.89999999999999989e86

   1. Initial program 80.8%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   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 85.7%

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

   5. Taylor expanded in i around inf

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

   7. Simplified 76.3%

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

   if 1.89999999999999989e86 < alpha

   1. Initial program 13.0%

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

      1. /-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 25.7%

      \[\leadsto \color{blue}{\frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)} + 1}{2}} \]
   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. associate-+r+ N/A

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

      3. *-commutative N/A

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

      4. associate-+r+ N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. times-frac N/A

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

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

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

   6. Applied egg-rr 35.7%

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

   7. Taylor expanded in alpha around inf

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

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

      9. +-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. *-lowering-*.f64 71.1%

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

   9. Simplified 71.1%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 1.9 \cdot 10^{+86}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(i \cdot 4 + \beta \cdot 2\right)}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.4 \cdot 10^{+86}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(i \cdot 4 + \beta \cdot 2\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1.4e+86)
   (/ (+ 1.0 (/ (- beta alpha) (+ 2.0 (+ (+ alpha beta) (* 2.0 i))))) 2.0)
   (/ (/ (+ 2.0 (+ (* i 4.0) (* beta 2.0))) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.4e+86) {
		tmp = (1.0 + ((beta - alpha) / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else {
		tmp = ((2.0 + ((i * 4.0) + (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 <= 1.4d+86) then
        tmp = (1.0d0 + ((beta - alpha) / (2.0d0 + ((alpha + beta) + (2.0d0 * i))))) / 2.0d0
    else
        tmp = ((2.0d0 + ((i * 4.0d0) + (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 <= 1.4e+86) {
		tmp = (1.0 + ((beta - alpha) / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else {
		tmp = ((2.0 + ((i * 4.0) + (beta * 2.0))) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 1.4e+86:
		tmp = (1.0 + ((beta - alpha) / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0
	else:
		tmp = ((2.0 + ((i * 4.0) + (beta * 2.0))) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 1.4e+86)
		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(2.0 + Float64(Float64(i * 4.0) + Float64(beta * 2.0))) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 1.4e+86)
		tmp = (1.0 + ((beta - alpha) / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	else
		tmp = ((2.0 + ((i * 4.0) + (beta * 2.0))) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 1.4e+86], 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[(2.0 + N[(N[(i * 4.0), $MachinePrecision] + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.40000000000000002e86

   1. Initial program 82.3%

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

   3. Taylor expanded in i around 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 98.5%

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

   5. Simplified 98.5%

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

   if 1.40000000000000002e86 < alpha

   1. Initial program 13.0%

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

      1. /-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 25.7%

      \[\leadsto \color{blue}{\frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)} + 1}{2}} \]
   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. associate-+r+ N/A

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

      3. *-commutative N/A

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

      4. associate-+r+ N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. times-frac N/A

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

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

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

   6. Applied egg-rr 35.7%

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

   7. Taylor expanded in alpha around inf

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

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

      9. +-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. *-lowering-*.f64 71.1%

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

   9. Simplified 71.1%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.4 \cdot 10^{+86}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(i \cdot 4 + \beta \cdot 2\right)}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 5 \cdot 10^{-17}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 4.5 \cdot 10^{+91}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 5e-17)
   (/ (+ 1.0 (/ (- beta alpha) (+ (+ alpha beta) 2.0))) 2.0)
   (if (<= alpha 4.5e+91) 0.5 (/ (/ (+ 2.0 (* i 4.0)) alpha) 2.0))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 5e-17) {
		tmp = (1.0 + ((beta - alpha) / ((alpha + beta) + 2.0))) / 2.0;
	} else if (alpha <= 4.5e+91) {
		tmp = 0.5;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 5d-17) then
        tmp = (1.0d0 + ((beta - alpha) / ((alpha + beta) + 2.0d0))) / 2.0d0
    else if (alpha <= 4.5d+91) then
        tmp = 0.5d0
    else
        tmp = ((2.0d0 + (i * 4.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 5e-17) {
		tmp = (1.0 + ((beta - alpha) / ((alpha + beta) + 2.0))) / 2.0;
	} else if (alpha <= 4.5e+91) {
		tmp = 0.5;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 5e-17:
		tmp = (1.0 + ((beta - alpha) / ((alpha + beta) + 2.0))) / 2.0
	elif alpha <= 4.5e+91:
		tmp = 0.5
	else:
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 5e-17)
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0))) / 2.0);
	elseif (alpha <= 4.5e+91)
		tmp = 0.5;
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(i * 4.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 5e-17)
		tmp = (1.0 + ((beta - alpha) / ((alpha + beta) + 2.0))) / 2.0;
	elseif (alpha <= 4.5e+91)
		tmp = 0.5;
	else
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 5e-17], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 4.5e+91], 0.5, N[(N[(N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 4.9999999999999999e-17

   1. Initial program 82.5%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   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 87.7%

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

   5. 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 92.0%

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

   7. Simplified 92.0%

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

   if 4.9999999999999999e-17 < alpha < 4.5e91

   1. Initial program 75.9%

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

      1. /-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 80.3%

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

   5. Taylor expanded in i around inf

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

   7. Simplified 72.2%

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

   if 4.5e91 < alpha

   1. Initial program 12.0%

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

      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 25.2%

      \[\leadsto \color{blue}{\frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)} + 1}{2}} \]
   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. associate-+r+ N/A

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

      3. *-commutative N/A

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

      4. associate-+r+ N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. times-frac N/A

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

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

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

   6. Applied egg-rr 35.7%

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

   7. Taylor expanded in beta around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. associate-/r* N/A

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. associate-+r+ N/A

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

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

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

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

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

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

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

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

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

      14. *-lowering-*.f64 11.9%

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

   9. Simplified 11.9%

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

   10. Taylor expanded in alpha around inf

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

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

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

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

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

       3. *-commutative N/A

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

       4. *-lowering-*.f64 61.1%

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

   12. Simplified 61.1%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 5 \cdot 10^{-17}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 4.5 \cdot 10^{+91}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 5.1 \cdot 10^{-17}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 2.3 \cdot 10^{+92}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 5.1e-17)
   (/ (+ 1.0 (/ (- beta alpha) (+ beta 2.0))) 2.0)
   (if (<= alpha 2.3e+92) 0.5 (/ (/ (+ 2.0 (* i 4.0)) alpha) 2.0))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 5.1e-17) {
		tmp = (1.0 + ((beta - alpha) / (beta + 2.0))) / 2.0;
	} else if (alpha <= 2.3e+92) {
		tmp = 0.5;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 5.1d-17) then
        tmp = (1.0d0 + ((beta - alpha) / (beta + 2.0d0))) / 2.0d0
    else if (alpha <= 2.3d+92) then
        tmp = 0.5d0
    else
        tmp = ((2.0d0 + (i * 4.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 5.1e-17) {
		tmp = (1.0 + ((beta - alpha) / (beta + 2.0))) / 2.0;
	} else if (alpha <= 2.3e+92) {
		tmp = 0.5;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 5.1e-17:
		tmp = (1.0 + ((beta - alpha) / (beta + 2.0))) / 2.0
	elif alpha <= 2.3e+92:
		tmp = 0.5
	else:
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 5.1e-17)
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / Float64(beta + 2.0))) / 2.0);
	elseif (alpha <= 2.3e+92)
		tmp = 0.5;
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(i * 4.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 5.1e-17)
		tmp = (1.0 + ((beta - alpha) / (beta + 2.0))) / 2.0;
	elseif (alpha <= 2.3e+92)
		tmp = 0.5;
	else
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 5.1e-17], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 2.3e+92], 0.5, N[(N[(N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 5.1000000000000003e-17

   1. Initial program 82.5%

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

   3. Taylor expanded in i around 0

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

      1. --lowering--.f64 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in beta around inf

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

   8. Simplified 91.3%

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

   if 5.1000000000000003e-17 < alpha < 2.29999999999999998e92

   1. Initial program 75.9%

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

      1. /-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 80.3%

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

   5. Taylor expanded in i around inf

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

   7. Simplified 72.2%

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

   if 2.29999999999999998e92 < alpha

   1. Initial program 12.0%

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

      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 25.2%

      \[\leadsto \color{blue}{\frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)} + 1}{2}} \]
   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. associate-+r+ N/A

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

      3. *-commutative N/A

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

      4. associate-+r+ N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. times-frac N/A

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

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

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

   6. Applied egg-rr 35.7%

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

   7. Taylor expanded in beta around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. associate-/r* N/A

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. associate-+r+ N/A

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

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

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

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

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

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

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

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

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

      14. *-lowering-*.f64 11.9%

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

   9. Simplified 11.9%

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

   10. Taylor expanded in alpha around inf

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

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

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

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

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

       3. *-commutative N/A

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

       4. *-lowering-*.f64 61.1%

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

   12. Simplified 61.1%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 5.1 \cdot 10^{-17}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 2.3 \cdot 10^{+92}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.9% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.4 \cdot 10^{+93}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 (if (<= beta 4.4e+93) 0.5 1.0))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 4.4e+93) {
		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 <= 4.4d+93) 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 <= 4.4e+93) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 4.4e+93:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 4.4e+93)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 4.4e+93)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 4.4e+93], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 4.40000000000000042e93

   1. Initial program 75.7%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   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 78.9%

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

   5. Taylor expanded in i around inf

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

   7. Simplified 73.7%

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

   if 4.40000000000000042e93 < beta

   1. Initial program 24.7%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   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 44.6%

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

   5. Taylor expanded in beta around inf

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

   7. Simplified 74.0%

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

Alternative 8: 61.8% 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 63.4%

   \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
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 70.6%

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

5. Taylor expanded in i around inf

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

7. Simplified 63.2%

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

Reproduce

herbie shell --seed 2024149 
(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))