Octave 3.8, jcobi/2

Percentage Accurate: 62.9% → 96.1%

Time: 12.0s

Alternatives: 8

Speedup: 4.8×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.9% 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: 96.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := 2 + t\_0\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_1} \leq -0.5:\\ \;\;\;\;\frac{\frac{\beta \cdot 0 + \left(\left(2 + \beta \cdot 2\right) + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{t\_1} + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))) (t_1 (+ 2.0 t_0)))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) t_1) -0.5)
     (/ (/ (+ (* beta 0.0) (+ (+ 2.0 (* beta 2.0)) (* i 4.0))) alpha) 2.0)
     (/ (+ (/ beta t_1) 1.0) 2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = 2.0 + t_0;
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= -0.5) {
		tmp = (((beta * 0.0) + ((2.0 + (beta * 2.0)) + (i * 4.0))) / alpha) / 2.0;
	} else {
		tmp = ((beta / t_1) + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    t_1 = 2.0d0 + t_0
    if (((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= (-0.5d0)) then
        tmp = (((beta * 0.0d0) + ((2.0d0 + (beta * 2.0d0)) + (i * 4.0d0))) / alpha) / 2.0d0
    else
        tmp = ((beta / t_1) + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = 2.0 + t_0;
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= -0.5) {
		tmp = (((beta * 0.0) + ((2.0 + (beta * 2.0)) + (i * 4.0))) / alpha) / 2.0;
	} else {
		tmp = ((beta / t_1) + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	t_1 = 2.0 + t_0
	tmp = 0
	if ((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= -0.5:
		tmp = (((beta * 0.0) + ((2.0 + (beta * 2.0)) + (i * 4.0))) / alpha) / 2.0
	else:
		tmp = ((beta / t_1) + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(2.0 + t_0)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / t_1) <= -0.5)
		tmp = Float64(Float64(Float64(Float64(beta * 0.0) + Float64(Float64(2.0 + Float64(beta * 2.0)) + Float64(i * 4.0))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta / t_1) + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	t_1 = 2.0 + t_0;
	tmp = 0.0;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= -0.5)
		tmp = (((beta * 0.0) + ((2.0 + (beta * 2.0)) + (i * 4.0))) / alpha) / 2.0;
	else
		tmp = ((beta / t_1) + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 + t$95$0), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision], -0.5], N[(N[(N[(N[(beta * 0.0), $MachinePrecision] + N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta / t$95$1), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.5

   1. Initial program 5.1%

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

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

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

   3. Simplified 8.9%

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

   5. Taylor expanded in alpha around inf

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

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

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. remove-double-neg N/A

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

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

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

      6. distribute-rgt1-in N/A

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

      7. metadata-eval N/A

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

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

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

      9. associate-+r+ N/A

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

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

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

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

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

      12. *-commutative N/A

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

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

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 96.3%

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

   7. Simplified 96.3%

      \[\leadsto \frac{\color{blue}{\frac{0 \cdot \beta + \left(\left(2 + \beta \cdot 2\right) + i \cdot 4\right)}{\alpha}}}{2} \]

   if -0.5 < (/.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 82.8%

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

   3. Taylor expanded in beta around inf

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

   5. Simplified 97.7%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.5:\\ \;\;\;\;\frac{\frac{\beta \cdot 0 + \left(\left(2 + \beta \cdot 2\right) + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.52 \cdot 10^{+109}:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\alpha}}{\frac{2}{\beta \cdot 2 + \left(2 + i \cdot 4\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1.52e+109)
   (/ (+ (/ beta (+ 2.0 (+ (+ alpha beta) (* 2.0 i)))) 1.0) 2.0)
   (/ (/ 1.0 alpha) (/ 2.0 (+ (* beta 2.0) (+ 2.0 (* i 4.0)))))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.52e+109) {
		tmp = ((beta / (2.0 + ((alpha + beta) + (2.0 * i)))) + 1.0) / 2.0;
	} else {
		tmp = (1.0 / alpha) / (2.0 / ((beta * 2.0) + (2.0 + (i * 4.0))));
	}
	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 <= 1.52d+109) then
        tmp = ((beta / (2.0d0 + ((alpha + beta) + (2.0d0 * i)))) + 1.0d0) / 2.0d0
    else
        tmp = (1.0d0 / alpha) / (2.0d0 / ((beta * 2.0d0) + (2.0d0 + (i * 4.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.52e+109) {
		tmp = ((beta / (2.0 + ((alpha + beta) + (2.0 * i)))) + 1.0) / 2.0;
	} else {
		tmp = (1.0 / alpha) / (2.0 / ((beta * 2.0) + (2.0 + (i * 4.0))));
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 1.52e+109:
		tmp = ((beta / (2.0 + ((alpha + beta) + (2.0 * i)))) + 1.0) / 2.0
	else:
		tmp = (1.0 / alpha) / (2.0 / ((beta * 2.0) + (2.0 + (i * 4.0))))
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 1.52e+109)
		tmp = Float64(Float64(Float64(beta / Float64(2.0 + Float64(Float64(alpha + beta) + Float64(2.0 * i)))) + 1.0) / 2.0);
	else
		tmp = Float64(Float64(1.0 / alpha) / Float64(2.0 / Float64(Float64(beta * 2.0) + Float64(2.0 + Float64(i * 4.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 1.52e+109)
		tmp = ((beta / (2.0 + ((alpha + beta) + (2.0 * i)))) + 1.0) / 2.0;
	else
		tmp = (1.0 / alpha) / (2.0 / ((beta * 2.0) + (2.0 + (i * 4.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 1.52e+109], N[(N[(N[(beta / N[(2.0 + N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 / alpha), $MachinePrecision] / N[(2.0 / N[(N[(beta * 2.0), $MachinePrecision] + N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.52000000000000003e109

   1. Initial program 78.5%

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

   3. Taylor expanded in beta around inf

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

   5. Simplified 92.5%

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

   if 1.52000000000000003e109 < alpha

   1. Initial program 19.5%

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

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

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

   3. Simplified 25.0%

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

   5. Taylor expanded in alpha around inf

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

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

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. remove-double-neg N/A

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

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

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

      6. distribute-rgt1-in N/A

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

      7. metadata-eval N/A

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

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

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

      9. associate-+r+ N/A

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

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

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

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

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

      12. *-commutative N/A

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

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

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 75.4%

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

   7. Simplified 75.4%

      \[\leadsto \frac{\color{blue}{\frac{0 \cdot \beta + \left(\left(2 + \beta \cdot 2\right) + i \cdot 4\right)}{\alpha}}}{2} \]
   8. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{2}{\frac{0 \cdot \beta + \left(\left(2 + \beta \cdot 2\right) + i \cdot 4\right)}{\alpha}}}} \]

      2. div-inv N/A

         \[\leadsto \frac{1}{\frac{2}{\left(0 \cdot \beta + \left(\left(2 + \beta \cdot 2\right) + i \cdot 4\right)\right) \cdot \color{blue}{\frac{1}{\alpha}}}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{1}{\frac{\frac{2}{0 \cdot \beta + \left(\left(2 + \beta \cdot 2\right) + i \cdot 4\right)}}{\color{blue}{\frac{1}{\alpha}}}} \]

      4. clear-num N/A

         \[\leadsto \frac{\frac{1}{\alpha}}{\color{blue}{\frac{2}{0 \cdot \beta + \left(\left(2 + \beta \cdot 2\right) + i \cdot 4\right)}}} \]

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

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

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

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

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

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

      8. mul0-lft N/A

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

      9. +-lft-identity N/A

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

      10. +-commutative N/A

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

      11. associate-+l+ N/A

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

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

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

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

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

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

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

      15. *-lowering-*.f64 75.1%

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

   9. Applied egg-rr 75.1%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.52 \cdot 10^{+109}:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\alpha}}{\frac{2}{\beta \cdot 2 + \left(2 + i \cdot 4\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.6 \cdot 10^{+143}:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot i + 1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 2.6e+143)
   (/ (+ (/ beta (+ 2.0 (+ (+ alpha beta) (* 2.0 i)))) 1.0) 2.0)
   (/ (+ (* 2.0 i) 1.0) alpha)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2.6e+143) {
		tmp = ((beta / (2.0 + ((alpha + beta) + (2.0 * i)))) + 1.0) / 2.0;
	} else {
		tmp = ((2.0 * i) + 1.0) / alpha;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 2.6d+143) then
        tmp = ((beta / (2.0d0 + ((alpha + beta) + (2.0d0 * i)))) + 1.0d0) / 2.0d0
    else
        tmp = ((2.0d0 * i) + 1.0d0) / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2.6e+143) {
		tmp = ((beta / (2.0 + ((alpha + beta) + (2.0 * i)))) + 1.0) / 2.0;
	} else {
		tmp = ((2.0 * i) + 1.0) / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 2.6e+143:
		tmp = ((beta / (2.0 + ((alpha + beta) + (2.0 * i)))) + 1.0) / 2.0
	else:
		tmp = ((2.0 * i) + 1.0) / alpha
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 2.6e+143)
		tmp = Float64(Float64(Float64(beta / Float64(2.0 + Float64(Float64(alpha + beta) + Float64(2.0 * i)))) + 1.0) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 * i) + 1.0) / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 2.6e+143)
		tmp = ((beta / (2.0 + ((alpha + beta) + (2.0 * i)))) + 1.0) / 2.0;
	else
		tmp = ((2.0 * i) + 1.0) / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 2.6e+143], N[(N[(N[(beta / N[(2.0 + N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 * i), $MachinePrecision] + 1.0), $MachinePrecision] / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.5999999999999999e143

   1. Initial program 77.4%

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

   3. Taylor expanded in beta around inf

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

   5. Simplified 90.0%

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

   if 2.5999999999999999e143 < alpha

   1. Initial program 4.7%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   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 12.1%

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

   5. Taylor expanded in alpha around inf

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

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

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. remove-double-neg N/A

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

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

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

      6. distribute-rgt1-in N/A

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

      7. metadata-eval N/A

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

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

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

      9. associate-+r+ N/A

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

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

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

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

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

      12. *-commutative N/A

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

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

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 85.4%

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

   7. Simplified 85.4%

      \[\leadsto \frac{\color{blue}{\frac{0 \cdot \beta + \left(\left(2 + \beta \cdot 2\right) + i \cdot 4\right)}{\alpha}}}{2} \]
   8. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{2}{\frac{0 \cdot \beta + \left(\left(2 + \beta \cdot 2\right) + i \cdot 4\right)}{\alpha}}}} \]

      2. div-inv N/A

         \[\leadsto \frac{1}{\frac{2}{\left(0 \cdot \beta + \left(\left(2 + \beta \cdot 2\right) + i \cdot 4\right)\right) \cdot \color{blue}{\frac{1}{\alpha}}}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{1}{\frac{\frac{2}{0 \cdot \beta + \left(\left(2 + \beta \cdot 2\right) + i \cdot 4\right)}}{\color{blue}{\frac{1}{\alpha}}}} \]

      4. clear-num N/A

         \[\leadsto \frac{\frac{1}{\alpha}}{\color{blue}{\frac{2}{0 \cdot \beta + \left(\left(2 + \beta \cdot 2\right) + i \cdot 4\right)}}} \]

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

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

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

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

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

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

      8. mul0-lft N/A

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

      9. +-lft-identity N/A

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

      10. +-commutative N/A

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

      11. associate-+l+ N/A

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

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

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

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

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

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

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

      15. *-lowering-*.f64 85.0%

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

   9. Applied egg-rr 85.0%

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

   10. Taylor expanded in beta around 0

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

       1. associate-*r/ N/A

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

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

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

       3. distribute-rgt-in N/A

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

       4. metadata-eval N/A

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

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

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. metadata-eval N/A

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

       9. *-lowering-*.f64 75.7%

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

   12. Simplified 75.7%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.6 \cdot 10^{+143}:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot i + 1}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.2 \cdot 10^{+79}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot i + 1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 2.2e+79)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (+ (* 2.0 i) 1.0) alpha)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2.2e+79) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 * i) + 1.0) / alpha;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 2.2d+79) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((2.0d0 * i) + 1.0d0) / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2.2e+79) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 * i) + 1.0) / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 2.2e+79:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((2.0 * i) + 1.0) / alpha
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 2.2e+79)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 * i) + 1.0) / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 2.2e+79)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((2.0 * i) + 1.0) / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 2.2e+79], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 * i), $MachinePrecision] + 1.0), $MachinePrecision] / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.1999999999999999e79

   1. Initial program 80.6%

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

      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 85.0%

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

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

      5. +-lowering-+.f64 86.7%

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

   7. Simplified 86.7%

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

   8. Taylor expanded in alpha around 0

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

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

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 88.5%

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

   10. Simplified 88.5%

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

   if 2.1999999999999999e79 < alpha

   1. Initial program 23.4%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   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 30.0%

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

   5. Taylor expanded in alpha around inf

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

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

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. remove-double-neg N/A

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

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

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

      6. distribute-rgt1-in N/A

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

      7. metadata-eval N/A

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

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

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

      9. associate-+r+ N/A

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

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

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

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

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

      12. *-commutative N/A

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

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

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 70.0%

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

   7. Simplified 70.0%

      \[\leadsto \frac{\color{blue}{\frac{0 \cdot \beta + \left(\left(2 + \beta \cdot 2\right) + i \cdot 4\right)}{\alpha}}}{2} \]
   8. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{2}{\frac{0 \cdot \beta + \left(\left(2 + \beta \cdot 2\right) + i \cdot 4\right)}{\alpha}}}} \]

      2. div-inv N/A

         \[\leadsto \frac{1}{\frac{2}{\left(0 \cdot \beta + \left(\left(2 + \beta \cdot 2\right) + i \cdot 4\right)\right) \cdot \color{blue}{\frac{1}{\alpha}}}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{1}{\frac{\frac{2}{0 \cdot \beta + \left(\left(2 + \beta \cdot 2\right) + i \cdot 4\right)}}{\color{blue}{\frac{1}{\alpha}}}} \]

      4. clear-num N/A

         \[\leadsto \frac{\frac{1}{\alpha}}{\color{blue}{\frac{2}{0 \cdot \beta + \left(\left(2 + \beta \cdot 2\right) + i \cdot 4\right)}}} \]

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

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

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

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

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

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

      8. mul0-lft N/A

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

      9. +-lft-identity N/A

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

      10. +-commutative N/A

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

      11. associate-+l+ N/A

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

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

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

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

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

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

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

      15. *-lowering-*.f64 69.7%

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

   9. Applied egg-rr 69.7%

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

   10. Taylor expanded in beta around 0

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

       1. associate-*r/ N/A

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

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

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

       3. distribute-rgt-in N/A

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

       4. metadata-eval N/A

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

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

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. metadata-eval N/A

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

       9. *-lowering-*.f64 62.4%

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

   12. Simplified 62.4%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.2 \cdot 10^{+79}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot i + 1}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3800000000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 3800000000000.0) 0.5 (- 1.0 (/ 1.0 beta))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3800000000000.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0 - (1.0 / beta);
	}
	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 <= 3800000000000.0d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0 - (1.0d0 / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3800000000000.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0 - (1.0 / beta);
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 3800000000000.0:
		tmp = 0.5
	else:
		tmp = 1.0 - (1.0 / beta)
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 3800000000000.0)
		tmp = 0.5;
	else
		tmp = Float64(1.0 - Float64(1.0 / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 3800000000000.0)
		tmp = 0.5;
	else
		tmp = 1.0 - (1.0 / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 3800000000000.0], 0.5, N[(1.0 - N[(1.0 / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.8e12

   1. Initial program 78.9%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   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 79.3%

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

   5. Taylor expanded in i around inf

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

   7. Simplified 77.0%

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

   if 3.8e12 < beta

   1. Initial program 44.5%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   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 59.9%

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

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

      5. +-lowering-+.f64 74.6%

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

   7. Simplified 74.6%

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

   8. Taylor expanded in alpha around 0

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

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

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 72.9%

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

   10. Simplified 72.9%

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

   11. Taylor expanded in beta around inf

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

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

          \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{1}{\beta}\right)}\right) \]

       2. /-lowering-/.f64 72.9%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\beta}\right)\right) \]

   13. Simplified 72.9%

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

Alternative 6: 72.6% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 29000000000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 29000000000000.0) 0.5 1.0))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 29000000000000.0) {
		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 <= 29000000000000.0d0) 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 <= 29000000000000.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 29000000000000.0:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 29000000000000.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 29000000000000.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 29000000000000.0], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 2.9e13

   1. Initial program 78.9%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   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 79.3%

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

   5. Taylor expanded in i around inf

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

   7. Simplified 77.0%

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

   if 2.9e13 < beta

   1. Initial program 44.5%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   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 59.9%

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

   5. Taylor expanded in beta around inf

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

   7. Simplified 72.6%

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

Alternative 7: 61.2% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[alpha_, beta_, i_] := 0.5

Derivation

1. Initial program 68.6%

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

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

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

3. Simplified 73.4%

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

5. Taylor expanded in i around inf

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

7. Simplified 64.2%

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

Alternative 8: 3.5% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 0.0)

C

double code(double alpha, double beta, double i) {
	return 0.0;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    code = 0.0d0
end function

Java

public static double code(double alpha, double beta, double i) {
	return 0.0;
}

Python

def code(alpha, beta, i):
	return 0.0

Julia

function code(alpha, beta, i)
	return 0.0
end

MATLAB

function tmp = code(alpha, beta, i)
	tmp = 0.0;
end

Wolfram

code[alpha_, beta_, i_] := 0.0

Derivation

1. Initial program 68.6%

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

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

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

3. Simplified 73.4%

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

5. Taylor expanded in alpha around inf

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

7. Simplified 3.5%

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

   1. metadata-eval N/A

      \[\leadsto \frac{0}{2} \]

   2. metadata-eval 3.5%

      \[\leadsto 0 \]

9. Applied egg-rr 3.5%

   \[\leadsto \color{blue}{0} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024164 
(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))