Octave 3.8, jcobi/2

Percentage Accurate: 63.7% → 97.7%

Time: 18.7s

Alternatives: 11

Speedup: 9.5×

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: 63.7% 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: 97.7% accurate, 0.1× speedup

Math

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

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)) -0.5)
     (/
      (+ (* 2.0 (/ beta alpha)) (+ (* 4.0 (/ i alpha)) (* 2.0 (/ 1.0 alpha))))
      2.0)
     (/
      (+
       1.0
       (*
        (/ (+ alpha beta) (+ (+ alpha beta) (fma 2.0 i 2.0)))
        (/ (- beta alpha) (fma 2.0 i (+ alpha beta)))))
      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)) <= -0.5) {
		tmp = ((2.0 * (beta / alpha)) + ((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha)))) / 2.0;
	} else {
		tmp = (1.0 + (((alpha + beta) / ((alpha + beta) + fma(2.0, i, 2.0))) * ((beta - alpha) / fma(2.0, i, (alpha + beta))))) / 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)) <= -0.5)
		tmp = Float64(Float64(Float64(2.0 * Float64(beta / alpha)) + Float64(Float64(4.0 * Float64(i / alpha)) + Float64(2.0 * Float64(1.0 / alpha)))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(alpha + beta) / Float64(Float64(alpha + beta) + fma(2.0, i, 2.0))) * Float64(Float64(beta - alpha) / fma(2.0, i, Float64(alpha + beta))))) / 2.0);
	end
	return 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], -0.5], N[(N[(N[(2.0 * N[(beta / alpha), $MachinePrecision]), $MachinePrecision] + N[(N[(4.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(N[(N[(alpha + beta), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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)) < -0.5

   1. Initial program 1.9%

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

   3. Simplified 8.9%

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

   4. Taylor expanded in alpha around -inf 96.4%

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

      1. associate-*r/ 96.4%

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

   6. Simplified 96.4%

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

   7. Taylor expanded in beta around 0 96.4%

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

   if -0.5 < (/.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 77.8%

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

   3. Simplified 100.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.5:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha} + \left(4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\alpha + \beta}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{2}\\ \end{array} \]

Alternative 2: 96.9% accurate, 0.6× speedup

Math

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

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)) -0.5)
     (/
      (+ (* 2.0 (/ beta alpha)) (+ (* 4.0 (/ i alpha)) (* 2.0 (/ 1.0 alpha))))
      2.0)
     (/
      (+
       1.0
       (* (/ beta (+ (* 2.0 i) (+ beta 2.0))) (/ beta (+ beta (* 2.0 i)))))
      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)) <= -0.5) {
		tmp = ((2.0 * (beta / alpha)) + ((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha)))) / 2.0;
	} else {
		tmp = (1.0 + ((beta / ((2.0 * i) + (beta + 2.0))) * (beta / (beta + (2.0 * i))))) / 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)) <= (-0.5d0)) then
        tmp = ((2.0d0 * (beta / alpha)) + ((4.0d0 * (i / alpha)) + (2.0d0 * (1.0d0 / alpha)))) / 2.0d0
    else
        tmp = (1.0d0 + ((beta / ((2.0d0 * i) + (beta + 2.0d0))) * (beta / (beta + (2.0d0 * i))))) / 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)) <= -0.5) {
		tmp = ((2.0 * (beta / alpha)) + ((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha)))) / 2.0;
	} else {
		tmp = (1.0 + ((beta / ((2.0 * i) + (beta + 2.0))) * (beta / (beta + (2.0 * i))))) / 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)) <= -0.5:
		tmp = ((2.0 * (beta / alpha)) + ((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha)))) / 2.0
	else:
		tmp = (1.0 + ((beta / ((2.0 * i) + (beta + 2.0))) * (beta / (beta + (2.0 * i))))) / 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)) <= -0.5)
		tmp = Float64(Float64(Float64(2.0 * Float64(beta / alpha)) + Float64(Float64(4.0 * Float64(i / alpha)) + Float64(2.0 * Float64(1.0 / alpha)))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(beta / Float64(Float64(2.0 * i) + Float64(beta + 2.0))) * Float64(beta / Float64(beta + Float64(2.0 * i))))) / 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)) <= -0.5)
		tmp = ((2.0 * (beta / alpha)) + ((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha)))) / 2.0;
	else
		tmp = (1.0 + ((beta / ((2.0 * i) + (beta + 2.0))) * (beta / (beta + (2.0 * i))))) / 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], -0.5], N[(N[(N[(2.0 * N[(beta / alpha), $MachinePrecision]), $MachinePrecision] + N[(N[(4.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(N[(beta / N[(N[(2.0 * i), $MachinePrecision] + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(beta / N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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)) < -0.5

   1. Initial program 1.9%

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

   3. Simplified 8.9%

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

   4. Taylor expanded in alpha around -inf 96.4%

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

      1. associate-*r/ 96.4%

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

   6. Simplified 96.4%

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

   7. Taylor expanded in beta around 0 96.4%

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

   if -0.5 < (/.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 77.8%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in alpha around 0 99.4%

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

   5. Taylor expanded in alpha around 0 99.4%

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

      1. associate-+r+ 99.4%

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

   7. Simplified 99.4%

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

4. Final simplification 98.8%

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

Alternative 3: 85.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 5 \cdot 10^{+62} \lor \neg \left(\alpha \leq 3.3 \cdot 10^{+131}\right) \land \alpha \leq 3.7 \cdot 10^{+165}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (or (<= alpha 5e+62) (and (not (<= alpha 3.3e+131)) (<= alpha 3.7e+165)))
   (/ (+ 1.0 (/ beta (+ 2.0 (+ (+ alpha beta) (* 2.0 i))))) 2.0)
   (/ (/ (+ 2.0 (* i 4.0)) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if ((alpha <= 5e+62) || (!(alpha <= 3.3e+131) && (alpha <= 3.7e+165))) {
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 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 <= 5d+62) .or. (.not. (alpha <= 3.3d+131)) .and. (alpha <= 3.7d+165)) then
        tmp = (1.0d0 + (beta / (2.0d0 + ((alpha + beta) + (2.0d0 * i))))) / 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 <= 5e+62) || (!(alpha <= 3.3e+131) && (alpha <= 3.7e+165))) {
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if (alpha <= 5e+62) or (not (alpha <= 3.3e+131) and (alpha <= 3.7e+165)):
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 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 <= 5e+62) || (!(alpha <= 3.3e+131) && (alpha <= 3.7e+165)))
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(2.0 + Float64(Float64(alpha + beta) + Float64(2.0 * i))))) / 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 <= 5e+62) || (~((alpha <= 3.3e+131)) && (alpha <= 3.7e+165)))
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	else
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[Or[LessEqual[alpha, 5e+62], And[N[Not[LessEqual[alpha, 3.3e+131]], $MachinePrecision], LessEqual[alpha, 3.7e+165]]], N[(N[(1.0 + N[(beta / 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[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 5.00000000000000029e62 or 3.2999999999999998e131 < alpha < 3.70000000000000006e165

   1. Initial program 76.8%

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

   2. Taylor expanded in beta around inf 95.4%

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

   if 5.00000000000000029e62 < alpha < 3.2999999999999998e131 or 3.70000000000000006e165 < alpha

   1. Initial program 5.5%

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

   3. Simplified 18.9%

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

   4. Taylor expanded in alpha around -inf 86.3%

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

      1. associate-*r/ 86.3%

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

   6. Simplified 86.3%

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

   7. Taylor expanded in beta around 0 71.4%

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

      1. *-commutative 71.4%

         \[\leadsto \frac{\frac{2 + \color{blue}{i \cdot 4}}{\alpha}}{2} \]

   9. Simplified 71.4%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 5 \cdot 10^{+62} \lor \neg \left(\alpha \leq 3.3 \cdot 10^{+131}\right) \land \alpha \leq 3.7 \cdot 10^{+165}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 4: 89.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 3 \cdot 10^{+61}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha} + \left(4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 3e+61)
   (/ (+ 1.0 (/ beta (+ 2.0 (+ (+ alpha beta) (* 2.0 i))))) 2.0)
   (/
    (+ (* 2.0 (/ beta alpha)) (+ (* 4.0 (/ i alpha)) (* 2.0 (/ 1.0 alpha))))
    2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 3e+61) {
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else {
		tmp = ((2.0 * (beta / alpha)) + ((4.0 * (i / alpha)) + (2.0 * (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) :: tmp
    if (alpha <= 3d+61) then
        tmp = (1.0d0 + (beta / (2.0d0 + ((alpha + beta) + (2.0d0 * i))))) / 2.0d0
    else
        tmp = ((2.0d0 * (beta / alpha)) + ((4.0d0 * (i / alpha)) + (2.0d0 * (1.0d0 / alpha)))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 3e+61) {
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else {
		tmp = ((2.0 * (beta / alpha)) + ((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha)))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 3e+61:
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0
	else:
		tmp = ((2.0 * (beta / alpha)) + ((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha)))) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 3e+61)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(2.0 + Float64(Float64(alpha + beta) + Float64(2.0 * i))))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 * Float64(beta / alpha)) + Float64(Float64(4.0 * Float64(i / alpha)) + Float64(2.0 * Float64(1.0 / alpha)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 3e+61)
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	else
		tmp = ((2.0 * (beta / alpha)) + ((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 3e+61], N[(N[(1.0 + N[(beta / 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[(beta / alpha), $MachinePrecision]), $MachinePrecision] + N[(N[(4.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 3e61

   1. Initial program 79.2%

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

   2. Taylor expanded in beta around inf 96.6%

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

   if 3e61 < alpha

   1. Initial program 9.5%

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

   3. Simplified 27.4%

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

   4. Taylor expanded in alpha around -inf 78.1%

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

      1. associate-*r/ 78.1%

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

   6. Simplified 78.1%

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

   7. Taylor expanded in beta around 0 78.2%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 3 \cdot 10^{+61}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha} + \left(4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}\right)}{2}\\ \end{array} \]

Alternative 5: 88.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 7.5 \cdot 10^{+55}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(2 \cdot i + \left(\beta + 2\right)\right) + \left(\beta + 2 \cdot i\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 7.5e+55)
   (/ (+ 1.0 (/ beta (+ 2.0 (+ (+ alpha beta) (* 2.0 i))))) 2.0)
   (/ (/ (+ (+ (* 2.0 i) (+ beta 2.0)) (+ beta (* 2.0 i))) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 7.5e+55) {
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else {
		tmp = ((((2.0 * i) + (beta + 2.0)) + (beta + (2.0 * 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 <= 7.5d+55) then
        tmp = (1.0d0 + (beta / (2.0d0 + ((alpha + beta) + (2.0d0 * i))))) / 2.0d0
    else
        tmp = ((((2.0d0 * i) + (beta + 2.0d0)) + (beta + (2.0d0 * 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 <= 7.5e+55) {
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else {
		tmp = ((((2.0 * i) + (beta + 2.0)) + (beta + (2.0 * i))) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 7.5e+55:
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0
	else:
		tmp = ((((2.0 * i) + (beta + 2.0)) + (beta + (2.0 * i))) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 7.5e+55)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(2.0 + Float64(Float64(alpha + beta) + Float64(2.0 * i))))) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(2.0 * i) + Float64(beta + 2.0)) + Float64(beta + Float64(2.0 * i))) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 7.5e+55)
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	else
		tmp = ((((2.0 * i) + (beta + 2.0)) + (beta + (2.0 * i))) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 7.5e+55], N[(N[(1.0 + N[(beta / N[(2.0 + N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(N[(2.0 * i), $MachinePrecision] + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] + N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 7.50000000000000014e55

   1. Initial program 79.2%

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

   2. Taylor expanded in beta around inf 96.6%

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

   if 7.50000000000000014e55 < alpha

   1. Initial program 9.5%

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

   3. Simplified 27.4%

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

   4. Taylor expanded in alpha around -inf 78.1%

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

      1. associate-*r/ 78.1%

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

   6. Simplified 78.1%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 7.5 \cdot 10^{+55}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(2 \cdot i + \left(\beta + 2\right)\right) + \left(\beta + 2 \cdot i\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 6: 79.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 + \frac{\beta}{\left(\alpha + \beta\right) + 2}}{2}\\ \mathbf{if}\;\alpha \leq 1.7 \cdot 10^{+30}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 2.7 \cdot 10^{+129}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 5 \cdot 10^{+165}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (/ (+ 1.0 (/ beta (+ (+ alpha beta) 2.0))) 2.0)))
   (if (<= alpha 1.7e+30)
     t_0
     (if (<= alpha 2.7e+129)
       (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)
       (if (<= alpha 5e+165) t_0 (/ (/ (+ 2.0 (* i 4.0)) alpha) 2.0))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (1.0 + (beta / ((alpha + beta) + 2.0))) / 2.0;
	double tmp;
	if (alpha <= 1.7e+30) {
		tmp = t_0;
	} else if (alpha <= 2.7e+129) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else if (alpha <= 5e+165) {
		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 = (1.0d0 + (beta / ((alpha + beta) + 2.0d0))) / 2.0d0
    if (alpha <= 1.7d+30) then
        tmp = t_0
    else if (alpha <= 2.7d+129) then
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    else if (alpha <= 5d+165) 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 = (1.0 + (beta / ((alpha + beta) + 2.0))) / 2.0;
	double tmp;
	if (alpha <= 1.7e+30) {
		tmp = t_0;
	} else if (alpha <= 2.7e+129) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else if (alpha <= 5e+165) {
		tmp = t_0;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (1.0 + (beta / ((alpha + beta) + 2.0))) / 2.0
	tmp = 0
	if alpha <= 1.7e+30:
		tmp = t_0
	elif alpha <= 2.7e+129:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	elif alpha <= 5e+165:
		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(1.0 + Float64(beta / Float64(Float64(alpha + beta) + 2.0))) / 2.0)
	tmp = 0.0
	if (alpha <= 1.7e+30)
		tmp = t_0;
	elseif (alpha <= 2.7e+129)
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	elseif (alpha <= 5e+165)
		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 = (1.0 + (beta / ((alpha + beta) + 2.0))) / 2.0;
	tmp = 0.0;
	if (alpha <= 1.7e+30)
		tmp = t_0;
	elseif (alpha <= 2.7e+129)
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	elseif (alpha <= 5e+165)
		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[(1.0 + N[(beta / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[alpha, 1.7e+30], t$95$0, If[LessEqual[alpha, 2.7e+129], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 5e+165], 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 < 1.7000000000000001e30 or 2.7000000000000001e129 < alpha < 4.9999999999999997e165

   1. Initial program 77.6%

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

   2. Taylor expanded in beta around inf 96.6%

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

   3. Taylor expanded in i around 0 90.2%

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

      1. +-commutative 90.2%

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

   5. Simplified 90.2%

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

   if 1.7000000000000001e30 < alpha < 2.7000000000000001e129

   1. Initial program 26.7%

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

   3. Simplified 31.3%

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

   4. Taylor expanded in i around 0 14.3%

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

      1. associate-+r+ 14.3%

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

   6. Simplified 14.3%

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

   7. Taylor expanded in alpha around inf 65.5%

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

      1. *-commutative 65.5%

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

   9. Simplified 65.5%

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

   if 4.9999999999999997e165 < alpha

   1. Initial program 1.3%

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

   3. Simplified 17.3%

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

   4. Taylor expanded in alpha around -inf 88.3%

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

      1. associate-*r/ 88.3%

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

   6. Simplified 88.3%

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

   7. Taylor expanded in beta around 0 70.0%

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

      1. *-commutative 70.0%

         \[\leadsto \frac{\frac{2 + \color{blue}{i \cdot 4}}{\alpha}}{2} \]

   9. Simplified 70.0%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.7 \cdot 10^{+30}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\left(\alpha + \beta\right) + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 2.7 \cdot 10^{+129}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 5 \cdot 10^{+165}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\left(\alpha + \beta\right) + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 7: 76.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.6 \cdot 10^{+30} \lor \neg \left(\alpha \leq 4.5 \cdot 10^{+131}\right) \land \alpha \leq 2.1 \cdot 10^{+189}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (or (<= alpha 1.6e+30)
         (and (not (<= alpha 4.5e+131)) (<= alpha 2.1e+189)))
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if ((alpha <= 1.6e+30) || (!(alpha <= 4.5e+131) && (alpha <= 2.1e+189))) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if ((alpha <= 1.6d+30) .or. (.not. (alpha <= 4.5d+131)) .and. (alpha <= 2.1d+189)) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if ((alpha <= 1.6e+30) || (!(alpha <= 4.5e+131) && (alpha <= 2.1e+189))) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if (alpha <= 1.6e+30) or (not (alpha <= 4.5e+131) and (alpha <= 2.1e+189)):
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if ((alpha <= 1.6e+30) || (!(alpha <= 4.5e+131) && (alpha <= 2.1e+189)))
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if ((alpha <= 1.6e+30) || (~((alpha <= 4.5e+131)) && (alpha <= 2.1e+189)))
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[Or[LessEqual[alpha, 1.6e+30], And[N[Not[LessEqual[alpha, 4.5e+131]], $MachinePrecision], LessEqual[alpha, 2.1e+189]]], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.59999999999999986e30 or 4.5000000000000002e131 < alpha < 2.09999999999999992e189

   1. Initial program 74.6%

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

   3. Simplified 96.4%

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

   4. Taylor expanded in alpha around 0 95.7%

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

   5. Taylor expanded in alpha around 0 95.7%

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

      1. associate-+r+ 95.7%

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

   7. Simplified 95.7%

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

   8. Taylor expanded in i around 0 88.0%

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

      1. +-commutative 88.0%

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

   10. Simplified 88.0%

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

   if 1.59999999999999986e30 < alpha < 4.5000000000000002e131 or 2.09999999999999992e189 < alpha

   1. Initial program 11.4%

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

   3. Simplified 19.4%

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

   4. Taylor expanded in i around 0 12.7%

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

      1. associate-+r+ 12.7%

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

   6. Simplified 12.7%

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

   7. Taylor expanded in alpha around inf 66.9%

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

      1. *-commutative 66.9%

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

   9. Simplified 66.9%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.6 \cdot 10^{+30} \lor \neg \left(\alpha \leq 4.5 \cdot 10^{+131}\right) \land \alpha \leq 2.1 \cdot 10^{+189}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 8: 79.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{if}\;\alpha \leq 9 \cdot 10^{+29}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 1.15 \cdot 10^{+129}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 3.7 \cdot 10^{+165}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)))
   (if (<= alpha 9e+29)
     t_0
     (if (<= alpha 1.15e+129)
       (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)
       (if (<= alpha 3.7e+165) t_0 (/ (/ (+ 2.0 (* i 4.0)) alpha) 2.0))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (1.0 + (beta / (beta + 2.0))) / 2.0;
	double tmp;
	if (alpha <= 9e+29) {
		tmp = t_0;
	} else if (alpha <= 1.15e+129) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else if (alpha <= 3.7e+165) {
		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 = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    if (alpha <= 9d+29) then
        tmp = t_0
    else if (alpha <= 1.15d+129) then
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    else if (alpha <= 3.7d+165) 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 = (1.0 + (beta / (beta + 2.0))) / 2.0;
	double tmp;
	if (alpha <= 9e+29) {
		tmp = t_0;
	} else if (alpha <= 1.15e+129) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else if (alpha <= 3.7e+165) {
		tmp = t_0;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (1.0 + (beta / (beta + 2.0))) / 2.0
	tmp = 0
	if alpha <= 9e+29:
		tmp = t_0
	elif alpha <= 1.15e+129:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	elif alpha <= 3.7e+165:
		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(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0)
	tmp = 0.0
	if (alpha <= 9e+29)
		tmp = t_0;
	elseif (alpha <= 1.15e+129)
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	elseif (alpha <= 3.7e+165)
		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 = (1.0 + (beta / (beta + 2.0))) / 2.0;
	tmp = 0.0;
	if (alpha <= 9e+29)
		tmp = t_0;
	elseif (alpha <= 1.15e+129)
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	elseif (alpha <= 3.7e+165)
		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[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[alpha, 9e+29], t$95$0, If[LessEqual[alpha, 1.15e+129], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 3.7e+165], 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 < 9.0000000000000005e29 or 1.14999999999999995e129 < alpha < 3.70000000000000006e165

   1. Initial program 77.6%

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

   3. Simplified 98.7%

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

   4. Taylor expanded in alpha around 0 97.9%

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

   5. Taylor expanded in alpha around 0 98.0%

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

      1. associate-+r+ 98.0%

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

   7. Simplified 98.0%

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

   8. Taylor expanded in i around 0 90.0%

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

      1. +-commutative 90.0%

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

   10. Simplified 90.0%

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

   if 9.0000000000000005e29 < alpha < 1.14999999999999995e129

   1. Initial program 26.7%

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

   3. Simplified 31.3%

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

   4. Taylor expanded in i around 0 14.3%

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

      1. associate-+r+ 14.3%

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

   6. Simplified 14.3%

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

   7. Taylor expanded in alpha around inf 65.5%

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

      1. *-commutative 65.5%

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

   9. Simplified 65.5%

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

   if 3.70000000000000006e165 < alpha

   1. Initial program 1.3%

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

   3. Simplified 17.3%

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

   4. Taylor expanded in alpha around -inf 88.3%

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

      1. associate-*r/ 88.3%

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

   6. Simplified 88.3%

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

   7. Taylor expanded in beta around 0 70.0%

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

      1. *-commutative 70.0%

         \[\leadsto \frac{\frac{2 + \color{blue}{i \cdot 4}}{\alpha}}{2} \]

   9. Simplified 70.0%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 9 \cdot 10^{+29}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 1.15 \cdot 10^{+129}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 3.7 \cdot 10^{+165}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 9: 75.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 3.8 \cdot 10^{+106}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= i 3.8e+106) (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0) 0.5))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 3.8e+106) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (i <= 3.8d+106) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 3.8e+106) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if i <= 3.8e+106:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = 0.5
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (i <= 3.8e+106)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (i <= 3.8e+106)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[i, 3.8e+106], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if i < 3.7999999999999998e106

   1. Initial program 55.7%

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

   3. Simplified 74.8%

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

   4. Taylor expanded in alpha around 0 74.0%

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

   5. Taylor expanded in alpha around 0 74.0%

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

      1. associate-+r+ 74.0%

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

   7. Simplified 74.0%

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

   8. Taylor expanded in i around 0 73.3%

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

      1. +-commutative 73.3%

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

   10. Simplified 73.3%

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

   if 3.7999999999999998e106 < i

   1. Initial program 74.3%

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

   3. Simplified 92.8%

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

   4. Taylor expanded in i around inf 86.4%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 3.8 \cdot 10^{+106}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 10: 72.4% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 800000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 (if (<= beta 800000000.0) 0.5 1.0))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 800000000.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 800000000.0d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 800000000.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 800000000.0:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 800000000.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 800000000.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 800000000.0], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 8e8

   1. Initial program 75.4%

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

   3. Simplified 77.2%

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

   4. Taylor expanded in i around inf 76.2%

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

   if 8e8 < beta

   1. Initial program 31.6%

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

   3. Simplified 87.3%

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

   4. Taylor expanded in beta around inf 68.7%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 800000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11: 62.0% 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 61.5%

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

3. Simplified 80.4%

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

4. Taylor expanded in i around inf 62.4%

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

5. Final simplification 62.4%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023277 
(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))