Octave 3.8, jcobi/2

Percentage Accurate: 63.3% → 97.8%

Time: 15.7s

Alternatives: 6

Speedup: 4.8×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 63.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq -0.96:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(2 + \left(\beta \cdot 2 + i \cdot 4\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\alpha + \beta}{\frac{\alpha + \left(\beta + 2 \cdot i\right)}{\beta - \alpha}}}{\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)} + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0)) -0.96)
     (/ (/ (+ (- beta beta) (+ 2.0 (+ (* beta 2.0) (* i 4.0)))) alpha) 2.0)
     (/
      (+
       (/
        (/ (+ alpha beta) (/ (+ alpha (+ beta (* 2.0 i))) (- beta alpha)))
        (+ (+ alpha beta) (+ 2.0 (* 2.0 i))))
       1.0)
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.96) {
		tmp = (((beta - beta) + (2.0 + ((beta * 2.0) + (i * 4.0)))) / alpha) / 2.0;
	} else {
		tmp = ((((alpha + beta) / ((alpha + (beta + (2.0 * i))) / (beta - alpha))) / ((alpha + beta) + (2.0 + (2.0 * i)))) + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0d0 + t_0)) <= (-0.96d0)) then
        tmp = (((beta - beta) + (2.0d0 + ((beta * 2.0d0) + (i * 4.0d0)))) / alpha) / 2.0d0
    else
        tmp = ((((alpha + beta) / ((alpha + (beta + (2.0d0 * i))) / (beta - alpha))) / ((alpha + beta) + (2.0d0 + (2.0d0 * i)))) + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.96) {
		tmp = (((beta - beta) + (2.0 + ((beta * 2.0) + (i * 4.0)))) / alpha) / 2.0;
	} else {
		tmp = ((((alpha + beta) / ((alpha + (beta + (2.0 * i))) / (beta - alpha))) / ((alpha + beta) + (2.0 + (2.0 * i)))) + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	tmp = 0
	if ((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.96:
		tmp = (((beta - beta) + (2.0 + ((beta * 2.0) + (i * 4.0)))) / alpha) / 2.0
	else:
		tmp = ((((alpha + beta) / ((alpha + (beta + (2.0 * i))) / (beta - alpha))) / ((alpha + beta) + (2.0 + (2.0 * i)))) + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0)) <= -0.96)
		tmp = Float64(Float64(Float64(Float64(beta - beta) + Float64(2.0 + Float64(Float64(beta * 2.0) + Float64(i * 4.0)))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(alpha + beta) / Float64(Float64(alpha + Float64(beta + Float64(2.0 * i))) / Float64(beta - alpha))) / Float64(Float64(alpha + beta) + Float64(2.0 + Float64(2.0 * i)))) + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	tmp = 0.0;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.96)
		tmp = (((beta - beta) + (2.0 + ((beta * 2.0) + (i * 4.0)))) / alpha) / 2.0;
	else
		tmp = ((((alpha + beta) / ((alpha + (beta + (2.0 * i))) / (beta - alpha))) / ((alpha + beta) + (2.0 + (2.0 * i)))) + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision], -0.96], N[(N[(N[(N[(beta - beta), $MachinePrecision] + N[(2.0 + N[(N[(beta * 2.0), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] / N[(N[(alpha + N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta - alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 + N[(2.0 * i), $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 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2)) < -0.95999999999999996

   1. Initial program 4.1%

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

   3. Simplified 19.6%

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

   5. Taylor expanded in alpha around inf 86.6%

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

   if -0.95999999999999996 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2))

   1. Initial program 83.0%

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

   3. Simplified 100.0%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.96:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(2 + \left(\beta \cdot 2 + i \cdot 4\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\alpha + \beta}{\frac{\alpha + \left(\beta + 2 \cdot i\right)}{\beta - \alpha}}}{\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := 2 + t\_0\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_1} \leq -0.96:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(2 + \left(\beta \cdot 2 + i \cdot 4\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\frac{1}{\frac{1}{\beta - \alpha}}}{t\_1}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))) (t_1 (+ 2.0 t_0)))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) t_1) -0.96)
     (/ (/ (+ (- beta beta) (+ 2.0 (+ (* beta 2.0) (* i 4.0)))) alpha) 2.0)
     (/ (+ 1.0 (/ (/ 1.0 (/ 1.0 (- beta alpha))) t_1)) 2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = 2.0 + t_0;
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= -0.96) {
		tmp = (((beta - beta) + (2.0 + ((beta * 2.0) + (i * 4.0)))) / alpha) / 2.0;
	} else {
		tmp = (1.0 + ((1.0 / (1.0 / (beta - alpha))) / t_1)) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    t_1 = 2.0d0 + t_0
    if (((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= (-0.96d0)) then
        tmp = (((beta - beta) + (2.0d0 + ((beta * 2.0d0) + (i * 4.0d0)))) / alpha) / 2.0d0
    else
        tmp = (1.0d0 + ((1.0d0 / (1.0d0 / (beta - alpha))) / t_1)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = 2.0 + t_0;
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= -0.96) {
		tmp = (((beta - beta) + (2.0 + ((beta * 2.0) + (i * 4.0)))) / alpha) / 2.0;
	} else {
		tmp = (1.0 + ((1.0 / (1.0 / (beta - alpha))) / t_1)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	t_1 = 2.0 + t_0
	tmp = 0
	if ((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= -0.96:
		tmp = (((beta - beta) + (2.0 + ((beta * 2.0) + (i * 4.0)))) / alpha) / 2.0
	else:
		tmp = (1.0 + ((1.0 / (1.0 / (beta - alpha))) / t_1)) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(2.0 + t_0)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / t_1) <= -0.96)
		tmp = Float64(Float64(Float64(Float64(beta - beta) + Float64(2.0 + Float64(Float64(beta * 2.0) + Float64(i * 4.0)))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(1.0 / Float64(1.0 / Float64(beta - alpha))) / t_1)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	t_1 = 2.0 + t_0;
	tmp = 0.0;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= -0.96)
		tmp = (((beta - beta) + (2.0 + ((beta * 2.0) + (i * 4.0)))) / alpha) / 2.0;
	else
		tmp = (1.0 + ((1.0 / (1.0 / (beta - alpha))) / t_1)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 + t$95$0), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision], -0.96], N[(N[(N[(N[(beta - beta), $MachinePrecision] + N[(2.0 + N[(N[(beta * 2.0), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(N[(1.0 / N[(1.0 / N[(beta - alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2)) < -0.95999999999999996

   1. Initial program 4.1%

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

   3. Simplified 19.6%

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

   5. Taylor expanded in alpha around inf 86.6%

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

   if -0.95999999999999996 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2))

   1. Initial program 83.0%

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

      1. associate-/l* 100.0%

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

      2. associate-+r+ 100.0%

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

      3. clear-num 100.0%

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

      4. inv-pow 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-undefine 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in i around 0 99.3%

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

      1. unpow-1 99.3%

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

   7. Applied egg-rr 99.3%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.96:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(2 + \left(\beta \cdot 2 + i \cdot 4\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\frac{1}{\frac{1}{\beta - \alpha}}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 5 \cdot 10^{+252}:\\ \;\;\;\;\frac{1 + \frac{\frac{1}{\frac{1}{\beta - \alpha}}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + 2 \cdot \left(\beta + i\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 5e+252)
   (/
    (+
     1.0
     (/ (/ 1.0 (/ 1.0 (- beta alpha))) (+ 2.0 (+ (+ alpha beta) (* 2.0 i)))))
    2.0)
   (/ (/ (+ 2.0 (* 2.0 (+ beta i))) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 5e+252) {
		tmp = (1.0 + ((1.0 / (1.0 / (beta - alpha))) / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else {
		tmp = ((2.0 + (2.0 * (beta + i))) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 5d+252) then
        tmp = (1.0d0 + ((1.0d0 / (1.0d0 / (beta - alpha))) / (2.0d0 + ((alpha + beta) + (2.0d0 * i))))) / 2.0d0
    else
        tmp = ((2.0d0 + (2.0d0 * (beta + i))) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 5e+252) {
		tmp = (1.0 + ((1.0 / (1.0 / (beta - alpha))) / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else {
		tmp = ((2.0 + (2.0 * (beta + i))) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 5e+252:
		tmp = (1.0 + ((1.0 / (1.0 / (beta - alpha))) / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0
	else:
		tmp = ((2.0 + (2.0 * (beta + i))) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 5e+252)
		tmp = Float64(Float64(1.0 + Float64(Float64(1.0 / Float64(1.0 / Float64(beta - alpha))) / Float64(2.0 + Float64(Float64(alpha + beta) + Float64(2.0 * i))))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(2.0 * Float64(beta + i))) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 5e+252)
		tmp = (1.0 + ((1.0 / (1.0 / (beta - alpha))) / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	else
		tmp = ((2.0 + (2.0 * (beta + i))) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 5e+252], N[(N[(1.0 + N[(N[(1.0 / N[(1.0 / N[(beta - alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(2.0 * N[(beta + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 4.9999999999999997e252

   1. Initial program 70.5%

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

      1. associate-/l* 87.8%

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

      2. associate-+r+ 87.8%

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

      3. clear-num 87.8%

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

      4. inv-pow 87.8%

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

      5. +-commutative 87.8%

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

      6. fma-undefine 87.8%

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

   4. Applied egg-rr 87.8%

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

   5. Taylor expanded in i around 0 87.0%

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

      1. unpow-1 87.0%

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

   7. Applied egg-rr 87.0%

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

   if 4.9999999999999997e252 < alpha

   1. Initial program 1.2%

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

      1. associate-/l* 5.7%

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

      2. associate-+r+ 5.7%

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

      3. clear-num 6.0%

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

      4. inv-pow 6.0%

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

      5. +-commutative 6.0%

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

      6. fma-undefine 6.0%

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

   4. Applied egg-rr 6.0%

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

   5. Taylor expanded in i around 0 6.0%

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

   6. Taylor expanded in alpha around inf 59.4%

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

      1. distribute-lft-out 59.4%

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

   8. Simplified 59.4%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 5 \cdot 10^{+252}:\\ \;\;\;\;\frac{1 + \frac{\frac{1}{\frac{1}{\beta - \alpha}}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + 2 \cdot \left(\beta + i\right)}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot i \leq 10^{+49}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= (* 2.0 i) 1e+49)
   (/ (+ 1.0 (/ (- beta alpha) (+ (+ alpha beta) 2.0))) 2.0)
   0.5))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if ((2.0 * i) <= 1e+49) {
		tmp = (1.0 + ((beta - alpha) / ((alpha + beta) + 2.0))) / 2.0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if ((2.0d0 * i) <= 1d+49) then
        tmp = (1.0d0 + ((beta - alpha) / ((alpha + beta) + 2.0d0))) / 2.0d0
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if ((2.0 * i) <= 1e+49) {
		tmp = (1.0 + ((beta - alpha) / ((alpha + beta) + 2.0))) / 2.0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if (2.0 * i) <= 1e+49:
		tmp = (1.0 + ((beta - alpha) / ((alpha + beta) + 2.0))) / 2.0
	else:
		tmp = 0.5
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (Float64(2.0 * i) <= 1e+49)
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0))) / 2.0);
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if ((2.0 * i) <= 1e+49)
		tmp = (1.0 + ((beta - alpha) / ((alpha + beta) + 2.0))) / 2.0;
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[N[(2.0 * i), $MachinePrecision], 1e+49], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if (*.f64 2 i) < 9.99999999999999946e48

   1. Initial program 63.4%

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

   3. Simplified 80.8%

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

   5. Taylor expanded in i around 0 80.8%

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

      1. +-commutative 80.8%

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

   7. Simplified 80.8%

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

   if 9.99999999999999946e48 < (*.f64 2 i)

   1. Initial program 73.6%

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

   3. Simplified 89.1%

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

   5. Taylor expanded in i around inf 81.5%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot i \leq 10^{+49}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.6% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.9 \cdot 10^{+46}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 (if (<= beta 5.9e+46) 0.5 1.0))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 5.9e+46) {
		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 <= 5.9d+46) 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 <= 5.9e+46) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 5.9e+46:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 5.9e+46)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 5.9e+46)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 5.9e+46], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 5.8999999999999999e46

   1. Initial program 77.7%

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

   3. Simplified 80.6%

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

   5. Taylor expanded in i around inf 76.7%

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

   if 5.8999999999999999e46 < beta

   1. Initial program 32.5%

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

   3. Simplified 95.1%

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

   5. Taylor expanded in beta around inf 85.0%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.9 \cdot 10^{+46}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 61.8% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[alpha_, beta_, i_] := 0.5

Derivation

1. Initial program 67.3%

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

3. Simplified 84.0%

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

5. Taylor expanded in i around inf 65.1%

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

6. Final simplification 65.1%

   \[\leadsto 0.5 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024036 
(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))