Octave 3.8, jcobi/2

Percentage Accurate: 62.9% → 97.6%

Time: 20.4s

Alternatives: 10

Speedup: 2.1×

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: 97.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + 2 \cdot i\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_1}}{2 + t\_1} \leq -0.8:\\ \;\;\;\;\frac{\frac{t\_0 + \left(t\_0 - -2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)} + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ beta (* 2.0 i))) (t_1 (+ (+ alpha beta) (* 2.0 i))))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_1) (+ 2.0 t_1)) -0.8)
     (/ (/ (+ t_0 (- t_0 -2.0)) alpha) 2.0)
     (/
      (+
       (/
        (* (- beta alpha) (/ (+ alpha beta) (fma 2.0 i (+ alpha beta))))
        (+ alpha (+ beta (fma 2.0 i 2.0))))
       1.0)
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = beta + (2.0 * i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_1) / (2.0 + t_1)) <= -0.8) {
		tmp = ((t_0 + (t_0 - -2.0)) / alpha) / 2.0;
	} else {
		tmp = ((((beta - alpha) * ((alpha + beta) / fma(2.0, i, (alpha + beta)))) / (alpha + (beta + fma(2.0, i, 2.0)))) + 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(beta + Float64(2.0 * i))
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_1) / Float64(2.0 + t_1)) <= -0.8)
		tmp = Float64(Float64(Float64(t_0 + Float64(t_0 - -2.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(beta - alpha) * Float64(Float64(alpha + beta) / fma(2.0, i, Float64(alpha + beta)))) / Float64(alpha + Float64(beta + fma(2.0, i, 2.0)))) + 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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$1), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], -0.8], N[(N[(N[(t$95$0 + N[(t$95$0 - -2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(N[(beta - alpha), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] / N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + N[(2.0 * i + 2.0), $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))) < -0.80000000000000004

   1. Initial program 4.2%

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

   3. Simplified 14.2%

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

   6. Applied egg-rr 12.0%

      \[\leadsto \frac{\color{blue}{{\left({\left(\mathsf{fma}\left(\beta - \alpha, \frac{\frac{\beta + \alpha}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}}{\left(\beta + \alpha\right) + \mathsf{fma}\left(2, i, 2\right)}, 1\right)\right)}^{3}\right)}^{0.3333333333333333}}}{2} \]

   7. Taylor expanded in alpha around -inf 92.5%

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

      1. associate-*r/ 92.5%

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

   9. Simplified 92.5%

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

   if -0.80000000000000004 < (/.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 82.1%

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

   3. Simplified 100.0%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.8:\\ \;\;\;\;\frac{\frac{\left(\beta + 2 \cdot i\right) + \left(\left(\beta + 2 \cdot i\right) - -2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_2 := \frac{\frac{t\_0}{t\_1}}{2 + t\_1}\\ t_3 := \beta + 2 \cdot i\\ \mathbf{if}\;t\_2 \leq -0.8:\\ \;\;\;\;\frac{\frac{t\_3 + \left(t\_3 - -2\right)}{\alpha}}{2}\\ \mathbf{elif}\;t\_2 \leq 0.999999999996:\\ \;\;\;\;\frac{\frac{t\_0}{\left(\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)\right) \cdot \left(\beta + \left(\alpha + 2 \cdot i\right)\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* (+ alpha beta) (- beta alpha)))
        (t_1 (+ (+ alpha beta) (* 2.0 i)))
        (t_2 (/ (/ t_0 t_1) (+ 2.0 t_1)))
        (t_3 (+ beta (* 2.0 i))))
   (if (<= t_2 -0.8)
     (/ (/ (+ t_3 (- t_3 -2.0)) alpha) 2.0)
     (if (<= t_2 0.999999999996)
       (/
        (+
         (/
          t_0
          (*
           (+ (+ alpha beta) (+ 2.0 (* 2.0 i)))
           (+ beta (+ alpha (* 2.0 i)))))
         1.0)
        2.0)
       1.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) * (beta - alpha);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = (t_0 / t_1) / (2.0 + t_1);
	double t_3 = beta + (2.0 * i);
	double tmp;
	if (t_2 <= -0.8) {
		tmp = ((t_3 + (t_3 - -2.0)) / alpha) / 2.0;
	} else if (t_2 <= 0.999999999996) {
		tmp = ((t_0 / (((alpha + beta) + (2.0 + (2.0 * i))) * (beta + (alpha + (2.0 * i))))) + 1.0) / 2.0;
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = (alpha + beta) * (beta - alpha)
    t_1 = (alpha + beta) + (2.0d0 * i)
    t_2 = (t_0 / t_1) / (2.0d0 + t_1)
    t_3 = beta + (2.0d0 * i)
    if (t_2 <= (-0.8d0)) then
        tmp = ((t_3 + (t_3 - (-2.0d0))) / alpha) / 2.0d0
    else if (t_2 <= 0.999999999996d0) then
        tmp = ((t_0 / (((alpha + beta) + (2.0d0 + (2.0d0 * i))) * (beta + (alpha + (2.0d0 * i))))) + 1.0d0) / 2.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) * (beta - alpha);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = (t_0 / t_1) / (2.0 + t_1);
	double t_3 = beta + (2.0 * i);
	double tmp;
	if (t_2 <= -0.8) {
		tmp = ((t_3 + (t_3 - -2.0)) / alpha) / 2.0;
	} else if (t_2 <= 0.999999999996) {
		tmp = ((t_0 / (((alpha + beta) + (2.0 + (2.0 * i))) * (beta + (alpha + (2.0 * i))))) + 1.0) / 2.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) * (beta - alpha)
	t_1 = (alpha + beta) + (2.0 * i)
	t_2 = (t_0 / t_1) / (2.0 + t_1)
	t_3 = beta + (2.0 * i)
	tmp = 0
	if t_2 <= -0.8:
		tmp = ((t_3 + (t_3 - -2.0)) / alpha) / 2.0
	elif t_2 <= 0.999999999996:
		tmp = ((t_0 / (((alpha + beta) + (2.0 + (2.0 * i))) * (beta + (alpha + (2.0 * i))))) + 1.0) / 2.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) * Float64(beta - alpha))
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_2 = Float64(Float64(t_0 / t_1) / Float64(2.0 + t_1))
	t_3 = Float64(beta + Float64(2.0 * i))
	tmp = 0.0
	if (t_2 <= -0.8)
		tmp = Float64(Float64(Float64(t_3 + Float64(t_3 - -2.0)) / alpha) / 2.0);
	elseif (t_2 <= 0.999999999996)
		tmp = Float64(Float64(Float64(t_0 / Float64(Float64(Float64(alpha + beta) + Float64(2.0 + Float64(2.0 * i))) * Float64(beta + Float64(alpha + Float64(2.0 * i))))) + 1.0) / 2.0);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) * (beta - alpha);
	t_1 = (alpha + beta) + (2.0 * i);
	t_2 = (t_0 / t_1) / (2.0 + t_1);
	t_3 = beta + (2.0 * i);
	tmp = 0.0;
	if (t_2 <= -0.8)
		tmp = ((t_3 + (t_3 - -2.0)) / alpha) / 2.0;
	elseif (t_2 <= 0.999999999996)
		tmp = ((t_0 / (((alpha + beta) + (2.0 + (2.0 * i))) * (beta + (alpha + (2.0 * i))))) + 1.0) / 2.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 / t$95$1), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.8], N[(N[(N[(t$95$3 + N[(t$95$3 - -2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[t$95$2, 0.999999999996], N[(N[(N[(t$95$0 / N[(N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(beta + N[(alpha + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], 1.0]]]]]]

Derivation

1. Split input into 3 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))) < -0.80000000000000004

   1. Initial program 4.2%

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

   3. Simplified 14.2%

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

   6. Applied egg-rr 12.0%

      \[\leadsto \frac{\color{blue}{{\left({\left(\mathsf{fma}\left(\beta - \alpha, \frac{\frac{\beta + \alpha}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}}{\left(\beta + \alpha\right) + \mathsf{fma}\left(2, i, 2\right)}, 1\right)\right)}^{3}\right)}^{0.3333333333333333}}}{2} \]

   7. Taylor expanded in alpha around -inf 92.5%

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

      1. associate-*r/ 92.5%

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

   9. Simplified 92.5%

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

   if -0.80000000000000004 < (/.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))) < 0.999999999995999977

   1. Initial program 100.0%

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

      1. associate-/l/ 100.0%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. associate-+l+ 100.0%

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

   3. Simplified 100.0%

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

   if 0.999999999995999977 < (/.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 38.2%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in beta around inf 97.2%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.8:\\ \;\;\;\;\frac{\frac{\left(\beta + 2 \cdot i\right) + \left(\left(\beta + 2 \cdot i\right) - -2\right)}{\alpha}}{2}\\ \mathbf{elif}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq 0.999999999996:\\ \;\;\;\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)\right) \cdot \left(\beta + \left(\alpha + 2 \cdot i\right)\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + 2 \cdot i\\ \mathbf{if}\;\alpha \leq 1.7 \cdot 10^{-183}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\\ \mathbf{elif}\;\alpha \leq 5 \cdot 10^{-109}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 3.35 \cdot 10^{+128}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0 + \left(t\_0 - -2\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ beta (* 2.0 i))))
   (if (<= alpha 1.7e-183)
     (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0)
     (if (<= alpha 5e-109)
       0.5
       (if (<= alpha 3.35e+128)
         (/ (+ (/ beta (+ beta 2.0)) 1.0) 2.0)
         (/ (/ (+ t_0 (- t_0 -2.0)) alpha) 2.0))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = beta + (2.0 * i);
	double tmp;
	if (alpha <= 1.7e-183) {
		tmp = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
	} else if (alpha <= 5e-109) {
		tmp = 0.5;
	} else if (alpha <= 3.35e+128) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	} else {
		tmp = ((t_0 + (t_0 - -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) :: t_0
    real(8) :: tmp
    t_0 = beta + (2.0d0 * i)
    if (alpha <= 1.7d-183) then
        tmp = (((beta - alpha) / ((alpha + beta) + 2.0d0)) + 1.0d0) / 2.0d0
    else if (alpha <= 5d-109) then
        tmp = 0.5d0
    else if (alpha <= 3.35d+128) then
        tmp = ((beta / (beta + 2.0d0)) + 1.0d0) / 2.0d0
    else
        tmp = ((t_0 + (t_0 - (-2.0d0))) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = beta + (2.0 * i);
	double tmp;
	if (alpha <= 1.7e-183) {
		tmp = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
	} else if (alpha <= 5e-109) {
		tmp = 0.5;
	} else if (alpha <= 3.35e+128) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	} else {
		tmp = ((t_0 + (t_0 - -2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = beta + (2.0 * i)
	tmp = 0
	if alpha <= 1.7e-183:
		tmp = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0
	elif alpha <= 5e-109:
		tmp = 0.5
	elif alpha <= 3.35e+128:
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0
	else:
		tmp = ((t_0 + (t_0 - -2.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(beta + Float64(2.0 * i))
	tmp = 0.0
	if (alpha <= 1.7e-183)
		tmp = Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) + 1.0) / 2.0);
	elseif (alpha <= 5e-109)
		tmp = 0.5;
	elseif (alpha <= 3.35e+128)
		tmp = Float64(Float64(Float64(beta / Float64(beta + 2.0)) + 1.0) / 2.0);
	else
		tmp = Float64(Float64(Float64(t_0 + Float64(t_0 - -2.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = beta + (2.0 * i);
	tmp = 0.0;
	if (alpha <= 1.7e-183)
		tmp = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
	elseif (alpha <= 5e-109)
		tmp = 0.5;
	elseif (alpha <= 3.35e+128)
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	else
		tmp = ((t_0 + (t_0 - -2.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[alpha, 1.7e-183], N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 5e-109], 0.5, If[LessEqual[alpha, 3.35e+128], N[(N[(N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(t$95$0 + N[(t$95$0 - -2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if alpha < 1.70000000000000007e-183

   1. Initial program 84.2%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in i around 0 93.0%

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

   if 1.70000000000000007e-183 < alpha < 5.0000000000000002e-109

   1. Initial program 94.9%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in i around inf 95.7%

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

   if 5.0000000000000002e-109 < alpha < 3.34999999999999996e128

   1. Initial program 66.0%

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

   3. Simplified 82.6%

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

   5. Taylor expanded in i around 0 60.0%

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

   6. Taylor expanded in alpha around 0 72.7%

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

   if 3.34999999999999996e128 < alpha

   1. Initial program 1.7%

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

   3. Simplified 22.9%

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

   6. Applied egg-rr 20.3%

      \[\leadsto \frac{\color{blue}{{\left({\left(\mathsf{fma}\left(\beta - \alpha, \frac{\frac{\beta + \alpha}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}}{\left(\beta + \alpha\right) + \mathsf{fma}\left(2, i, 2\right)}, 1\right)\right)}^{3}\right)}^{0.3333333333333333}}}{2} \]

   7. Taylor expanded in alpha around -inf 83.4%

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

      1. associate-*r/ 83.4%

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

   9. Simplified 83.4%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.7 \cdot 10^{-183}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\\ \mathbf{elif}\;\alpha \leq 5 \cdot 10^{-109}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 3.35 \cdot 10^{+128}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta + 2 \cdot i\right) + \left(\left(\beta + 2 \cdot i\right) - -2\right)}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{if}\;\alpha \leq 1.35 \cdot 10^{-176}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\alpha \leq 3.8 \cdot 10^{-111}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 2.3 \cdot 10^{+130}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{\beta}{\alpha} + \frac{1}{\alpha}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (/ (+ (/ beta (+ beta 2.0)) 1.0) 2.0)))
   (if (<= alpha 1.35e-176)
     t_0
     (if (<= alpha 3.8e-111)
       0.5
       (if (<= alpha 2.3e+130)
         t_0
         (/ (* 2.0 (+ (/ beta alpha) (/ 1.0 alpha))) 2.0))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (alpha <= 1.35e-176) {
		tmp = t_0;
	} else if (alpha <= 3.8e-111) {
		tmp = 0.5;
	} else if (alpha <= 2.3e+130) {
		tmp = t_0;
	} else {
		tmp = (2.0 * ((beta / alpha) + (1.0 / alpha))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((beta / (beta + 2.0d0)) + 1.0d0) / 2.0d0
    if (alpha <= 1.35d-176) then
        tmp = t_0
    else if (alpha <= 3.8d-111) then
        tmp = 0.5d0
    else if (alpha <= 2.3d+130) then
        tmp = t_0
    else
        tmp = (2.0d0 * ((beta / alpha) + (1.0d0 / alpha))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (alpha <= 1.35e-176) {
		tmp = t_0;
	} else if (alpha <= 3.8e-111) {
		tmp = 0.5;
	} else if (alpha <= 2.3e+130) {
		tmp = t_0;
	} else {
		tmp = (2.0 * ((beta / alpha) + (1.0 / alpha))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = ((beta / (beta + 2.0)) + 1.0) / 2.0
	tmp = 0
	if alpha <= 1.35e-176:
		tmp = t_0
	elif alpha <= 3.8e-111:
		tmp = 0.5
	elif alpha <= 2.3e+130:
		tmp = t_0
	else:
		tmp = (2.0 * ((beta / alpha) + (1.0 / alpha))) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(Float64(beta / Float64(beta + 2.0)) + 1.0) / 2.0)
	tmp = 0.0
	if (alpha <= 1.35e-176)
		tmp = t_0;
	elseif (alpha <= 3.8e-111)
		tmp = 0.5;
	elseif (alpha <= 2.3e+130)
		tmp = t_0;
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(beta / alpha) + Float64(1.0 / alpha))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	tmp = 0.0;
	if (alpha <= 1.35e-176)
		tmp = t_0;
	elseif (alpha <= 3.8e-111)
		tmp = 0.5;
	elseif (alpha <= 2.3e+130)
		tmp = t_0;
	else
		tmp = (2.0 * ((beta / alpha) + (1.0 / alpha))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[alpha, 1.35e-176], t$95$0, If[LessEqual[alpha, 3.8e-111], 0.5, If[LessEqual[alpha, 2.3e+130], t$95$0, N[(N[(2.0 * N[(N[(beta / alpha), $MachinePrecision] + N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 1.3499999999999999e-176 or 3.80000000000000022e-111 < alpha < 2.30000000000000021e130

   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. Step-by-step derivation

   3. Simplified 94.5%

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

   5. Taylor expanded in i around 0 82.7%

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

   6. Taylor expanded in alpha around 0 86.1%

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

   if 1.3499999999999999e-176 < alpha < 3.80000000000000022e-111

   1. Initial program 94.9%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in i around inf 95.7%

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

   if 2.30000000000000021e130 < alpha

   1. Initial program 1.7%

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

   3. Simplified 22.9%

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

   5. Taylor expanded in i around 0 16.5%

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

   6. Taylor expanded in alpha around inf 57.0%

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

   7. Taylor expanded in beta around 0 57.0%

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

      1. distribute-lft-out 57.0%

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

   9. Applied egg-rr 57.0%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.35 \cdot 10^{-176}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{elif}\;\alpha \leq 3.8 \cdot 10^{-111}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 2.3 \cdot 10^{+130}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{\beta}{\alpha} + \frac{1}{\alpha}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.06 \cdot 10^{-176}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\\ \mathbf{elif}\;\alpha \leq 3.8 \cdot 10^{-111}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 6.8 \cdot 10^{+128}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1.06e-176)
   (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0)
   (if (<= alpha 3.8e-111)
     0.5
     (if (<= alpha 6.8e+128)
       (/ (+ (/ beta (+ beta 2.0)) 1.0) 2.0)
       (/ (/ (+ 2.0 (* i 4.0)) alpha) 2.0)))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.06e-176) {
		tmp = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
	} else if (alpha <= 3.8e-111) {
		tmp = 0.5;
	} else if (alpha <= 6.8e+128) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	} 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 <= 1.06d-176) then
        tmp = (((beta - alpha) / ((alpha + beta) + 2.0d0)) + 1.0d0) / 2.0d0
    else if (alpha <= 3.8d-111) then
        tmp = 0.5d0
    else if (alpha <= 6.8d+128) then
        tmp = ((beta / (beta + 2.0d0)) + 1.0d0) / 2.0d0
    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 <= 1.06e-176) {
		tmp = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
	} else if (alpha <= 3.8e-111) {
		tmp = 0.5;
	} else if (alpha <= 6.8e+128) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 1.06e-176:
		tmp = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0
	elif alpha <= 3.8e-111:
		tmp = 0.5
	elif alpha <= 6.8e+128:
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0
	else:
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 1.06e-176)
		tmp = Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) + 1.0) / 2.0);
	elseif (alpha <= 3.8e-111)
		tmp = 0.5;
	elseif (alpha <= 6.8e+128)
		tmp = Float64(Float64(Float64(beta / Float64(beta + 2.0)) + 1.0) / 2.0);
	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 <= 1.06e-176)
		tmp = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
	elseif (alpha <= 3.8e-111)
		tmp = 0.5;
	elseif (alpha <= 6.8e+128)
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	else
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 1.06e-176], N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 3.8e-111], 0.5, If[LessEqual[alpha, 6.8e+128], N[(N[(N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if alpha < 1.06000000000000006e-176

   1. Initial program 84.2%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in i around 0 93.0%

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

   if 1.06000000000000006e-176 < alpha < 3.80000000000000022e-111

   1. Initial program 94.9%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in i around inf 95.7%

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

   if 3.80000000000000022e-111 < alpha < 6.7999999999999997e128

   1. Initial program 66.0%

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

   3. Simplified 82.6%

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

   5. Taylor expanded in i around 0 60.0%

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

   6. Taylor expanded in alpha around 0 72.7%

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

   if 6.7999999999999997e128 < alpha

   1. Initial program 1.7%

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

      1. associate-/l/ 0.5%

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

      2. associate-+l+ 0.5%

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

      3. +-commutative 0.5%

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

      4. associate-+l+ 0.5%

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

   3. Simplified 0.5%

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

   5. Taylor expanded in beta around 0 0.6%

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

   6. Taylor expanded in alpha around inf 64.6%

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

      1. distribute-rgt1-in 64.6%

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

      2. metadata-eval 64.6%

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

      3. mul0-lft 64.6%

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

      4. mul-1-neg 64.6%

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

      5. *-commutative 64.6%

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

   8. Simplified 64.6%

      \[\leadsto \frac{\color{blue}{\frac{0 - \left(-\left(2 + i \cdot 4\right)\right)}{\alpha}}}{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.06 \cdot 10^{-176}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\\ \mathbf{elif}\;\alpha \leq 3.8 \cdot 10^{-111}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 6.8 \cdot 10^{+128}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{if}\;\alpha \leq 2.45 \cdot 10^{-178}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\alpha \leq 3.8 \cdot 10^{-111}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 5.2 \cdot 10^{+129}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (/ (+ (/ beta (+ beta 2.0)) 1.0) 2.0)))
   (if (<= alpha 2.45e-178)
     t_0
     (if (<= alpha 3.8e-111)
       0.5
       (if (<= alpha 5.2e+129) t_0 (/ (/ (+ 2.0 (* i 4.0)) alpha) 2.0))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (alpha <= 2.45e-178) {
		tmp = t_0;
	} else if (alpha <= 3.8e-111) {
		tmp = 0.5;
	} else if (alpha <= 5.2e+129) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = ((beta / (beta + 2.0d0)) + 1.0d0) / 2.0d0
    if (alpha <= 2.45d-178) then
        tmp = t_0
    else if (alpha <= 3.8d-111) then
        tmp = 0.5d0
    else if (alpha <= 5.2d+129) then
        tmp = t_0
    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 t_0 = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (alpha <= 2.45e-178) {
		tmp = t_0;
	} else if (alpha <= 3.8e-111) {
		tmp = 0.5;
	} else if (alpha <= 5.2e+129) {
		tmp = t_0;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = ((beta / (beta + 2.0)) + 1.0) / 2.0
	tmp = 0
	if alpha <= 2.45e-178:
		tmp = t_0
	elif alpha <= 3.8e-111:
		tmp = 0.5
	elif alpha <= 5.2e+129:
		tmp = t_0
	else:
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(Float64(beta / Float64(beta + 2.0)) + 1.0) / 2.0)
	tmp = 0.0
	if (alpha <= 2.45e-178)
		tmp = t_0;
	elseif (alpha <= 3.8e-111)
		tmp = 0.5;
	elseif (alpha <= 5.2e+129)
		tmp = t_0;
	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)
	t_0 = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	tmp = 0.0;
	if (alpha <= 2.45e-178)
		tmp = t_0;
	elseif (alpha <= 3.8e-111)
		tmp = 0.5;
	elseif (alpha <= 5.2e+129)
		tmp = t_0;
	else
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[alpha, 2.45e-178], t$95$0, If[LessEqual[alpha, 3.8e-111], 0.5, If[LessEqual[alpha, 5.2e+129], t$95$0, 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 < 2.4500000000000001e-178 or 3.80000000000000022e-111 < alpha < 5.20000000000000024e129

   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. Step-by-step derivation

   3. Simplified 94.5%

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

   5. Taylor expanded in i around 0 82.7%

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

   6. Taylor expanded in alpha around 0 86.1%

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

   if 2.4500000000000001e-178 < alpha < 3.80000000000000022e-111

   1. Initial program 94.9%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in i around inf 95.7%

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

   if 5.20000000000000024e129 < alpha

   1. Initial program 1.7%

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

      1. associate-/l/ 0.5%

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

      2. associate-+l+ 0.5%

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

      3. +-commutative 0.5%

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

      4. associate-+l+ 0.5%

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

   3. Simplified 0.5%

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

   5. Taylor expanded in beta around 0 0.6%

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

   6. Taylor expanded in alpha around inf 64.6%

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

      1. distribute-rgt1-in 64.6%

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

      2. metadata-eval 64.6%

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

      3. mul0-lft 64.6%

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

      4. mul-1-neg 64.6%

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

      5. *-commutative 64.6%

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

   8. Simplified 64.6%

      \[\leadsto \frac{\color{blue}{\frac{0 - \left(-\left(2 + i \cdot 4\right)\right)}{\alpha}}}{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.45 \cdot 10^{-178}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{elif}\;\alpha \leq 3.8 \cdot 10^{-111}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 5.2 \cdot 10^{+129}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{if}\;\alpha \leq 1.28 \cdot 10^{-176}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\alpha \leq 3.8 \cdot 10^{-111}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 1.2 \cdot 10^{+128}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (/ (+ (/ beta (+ beta 2.0)) 1.0) 2.0)))
   (if (<= alpha 1.28e-176)
     t_0
     (if (<= alpha 3.8e-111)
       0.5
       (if (<= alpha 1.2e+128) t_0 (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (alpha <= 1.28e-176) {
		tmp = t_0;
	} else if (alpha <= 3.8e-111) {
		tmp = 0.5;
	} else if (alpha <= 1.2e+128) {
		tmp = t_0;
	} else {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((beta / (beta + 2.0d0)) + 1.0d0) / 2.0d0
    if (alpha <= 1.28d-176) then
        tmp = t_0
    else if (alpha <= 3.8d-111) then
        tmp = 0.5d0
    else if (alpha <= 1.2d+128) then
        tmp = t_0
    else
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (alpha <= 1.28e-176) {
		tmp = t_0;
	} else if (alpha <= 3.8e-111) {
		tmp = 0.5;
	} else if (alpha <= 1.2e+128) {
		tmp = t_0;
	} else {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = ((beta / (beta + 2.0)) + 1.0) / 2.0
	tmp = 0
	if alpha <= 1.28e-176:
		tmp = t_0
	elif alpha <= 3.8e-111:
		tmp = 0.5
	elif alpha <= 1.2e+128:
		tmp = t_0
	else:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(Float64(beta / Float64(beta + 2.0)) + 1.0) / 2.0)
	tmp = 0.0
	if (alpha <= 1.28e-176)
		tmp = t_0;
	elseif (alpha <= 3.8e-111)
		tmp = 0.5;
	elseif (alpha <= 1.2e+128)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	tmp = 0.0;
	if (alpha <= 1.28e-176)
		tmp = t_0;
	elseif (alpha <= 3.8e-111)
		tmp = 0.5;
	elseif (alpha <= 1.2e+128)
		tmp = t_0;
	else
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[alpha, 1.28e-176], t$95$0, If[LessEqual[alpha, 3.8e-111], 0.5, If[LessEqual[alpha, 1.2e+128], t$95$0, N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 1.2799999999999999e-176 or 3.80000000000000022e-111 < alpha < 1.2000000000000001e128

   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. Step-by-step derivation

   3. Simplified 94.5%

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

   5. Taylor expanded in i around 0 82.7%

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

   6. Taylor expanded in alpha around 0 86.1%

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

   if 1.2799999999999999e-176 < alpha < 3.80000000000000022e-111

   1. Initial program 94.9%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in i around inf 95.7%

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

   if 1.2000000000000001e128 < alpha

   1. Initial program 1.7%

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

   3. Simplified 22.9%

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

   5. Taylor expanded in i around 0 16.5%

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

   6. Taylor expanded in alpha around inf 57.0%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.28 \cdot 10^{-176}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{elif}\;\alpha \leq 3.8 \cdot 10^{-111}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 1.2 \cdot 10^{+128}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.4 \cdot 10^{+40}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 2.65 \cdot 10^{+76}:\\ \;\;\;\;1\\ \mathbf{elif}\;\beta \leq 6.2 \cdot 10^{+101}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 3.4e+40)
   0.5
   (if (<= beta 2.65e+76) 1.0 (if (<= beta 6.2e+101) 0.5 1.0))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3.4e+40) {
		tmp = 0.5;
	} else if (beta <= 2.65e+76) {
		tmp = 1.0;
	} else if (beta <= 6.2e+101) {
		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 <= 3.4d+40) then
        tmp = 0.5d0
    else if (beta <= 2.65d+76) then
        tmp = 1.0d0
    else if (beta <= 6.2d+101) 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 <= 3.4e+40) {
		tmp = 0.5;
	} else if (beta <= 2.65e+76) {
		tmp = 1.0;
	} else if (beta <= 6.2e+101) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 3.4e+40:
		tmp = 0.5
	elif beta <= 2.65e+76:
		tmp = 1.0
	elif beta <= 6.2e+101:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 3.4e+40)
		tmp = 0.5;
	elseif (beta <= 2.65e+76)
		tmp = 1.0;
	elseif (beta <= 6.2e+101)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 3.4e+40)
		tmp = 0.5;
	elseif (beta <= 2.65e+76)
		tmp = 1.0;
	elseif (beta <= 6.2e+101)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 3.4e+40], 0.5, If[LessEqual[beta, 2.65e+76], 1.0, If[LessEqual[beta, 6.2e+101], 0.5, 1.0]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.39999999999999989e40 or 2.65000000000000008e76 < beta < 6.19999999999999998e101

   1. Initial program 76.5%

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

   3. Simplified 78.8%

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

   5. Taylor expanded in i around inf 74.0%

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

   if 3.39999999999999989e40 < beta < 2.65000000000000008e76 or 6.19999999999999998e101 < beta

   1. Initial program 34.7%

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

   3. Simplified 87.7%

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

   5. Taylor expanded in beta around inf 80.6%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.4 \cdot 10^{+40}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 2.65 \cdot 10^{+76}:\\ \;\;\;\;1\\ \mathbf{elif}\;\beta \leq 6.2 \cdot 10^{+101}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 2.8 \cdot 10^{+146}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= i 2.8e+146) (/ (+ (/ beta (+ beta 2.0)) 1.0) 2.0) 0.5))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 2.8e+146) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (i <= 2.8d+146) then
        tmp = ((beta / (beta + 2.0d0)) + 1.0d0) / 2.0d0
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 2.8e+146) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if i <= 2.8e+146:
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0
	else:
		tmp = 0.5
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (i <= 2.8e+146)
		tmp = Float64(Float64(Float64(beta / Float64(beta + 2.0)) + 1.0) / 2.0);
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (i <= 2.8e+146)
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[i, 2.8e+146], N[(N[(N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if i < 2.8000000000000001e146

   1. Initial program 62.1%

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

   3. Simplified 78.4%

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

   5. Taylor expanded in i around 0 73.9%

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

   6. Taylor expanded in alpha around 0 73.7%

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

   if 2.8000000000000001e146 < i

   1. Initial program 73.9%

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

   3. Simplified 89.8%

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

   5. Taylor expanded in i around inf 82.9%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 2.8 \cdot 10^{+146}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.6% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[alpha_, beta_, i_] := 0.5

Derivation

1. Initial program 65.0%

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

3. Simplified 81.2%

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

5. Taylor expanded in i around inf 60.0%

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

6. Final simplification 60.0%

   \[\leadsto 0.5 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(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))