Octave 3.8, jcobi/2

Percentage Accurate: 62.5% → 97.9%

Time: 28.8s

Alternatives: 12

Speedup: 4.8×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.5% 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: 97.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq -0.999995:\\ \;\;\;\;\frac{\frac{\beta \cdot \left(\left(2 + -2 \cdot \frac{\beta}{\alpha}\right) + \left(\left(4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}\right) - 4 \cdot \frac{2 + i \cdot 4}{\alpha}\right)\right) - \left(\left(4 \cdot \frac{1}{\alpha} + i \cdot \left(\left(\frac{i}{\alpha} \cdot 12 + \frac{1}{\alpha} \cdot 16\right) + \left(4 \cdot \frac{-1}{\alpha} - 4\right)\right)\right) - 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)}, \frac{\beta - \alpha}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, 1\right)}{2}\\ \end{array} \end{array} \]

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)) -0.999995)
     (/
      (/
       (-
        (*
         beta
         (+
          (+ 2.0 (* -2.0 (/ beta alpha)))
          (-
           (+ (* 4.0 (/ i alpha)) (* 2.0 (/ 1.0 alpha)))
           (* 4.0 (/ (+ 2.0 (* i 4.0)) alpha)))))
        (-
         (+
          (* 4.0 (/ 1.0 alpha))
          (*
           i
           (+
            (+ (* (/ i alpha) 12.0) (* (/ 1.0 alpha) 16.0))
            (- (* 4.0 (/ -1.0 alpha)) 4.0))))
         2.0))
       alpha)
      2.0)
     (/
      (fma
       (/ (+ alpha beta) (+ (+ alpha beta) (fma 2.0 i 2.0)))
       (/ (- beta alpha) (+ beta (fma 2.0 i alpha)))
       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)) <= -0.999995) {
		tmp = (((beta * ((2.0 + (-2.0 * (beta / alpha))) + (((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha))) - (4.0 * ((2.0 + (i * 4.0)) / alpha))))) - (((4.0 * (1.0 / alpha)) + (i * ((((i / alpha) * 12.0) + ((1.0 / alpha) * 16.0)) + ((4.0 * (-1.0 / alpha)) - 4.0)))) - 2.0)) / alpha) / 2.0;
	} else {
		tmp = fma(((alpha + beta) / ((alpha + beta) + fma(2.0, i, 2.0))), ((beta - alpha) / (beta + fma(2.0, i, alpha))), 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)) <= -0.999995)
		tmp = Float64(Float64(Float64(Float64(beta * Float64(Float64(2.0 + Float64(-2.0 * Float64(beta / alpha))) + Float64(Float64(Float64(4.0 * Float64(i / alpha)) + Float64(2.0 * Float64(1.0 / alpha))) - Float64(4.0 * Float64(Float64(2.0 + Float64(i * 4.0)) / alpha))))) - Float64(Float64(Float64(4.0 * Float64(1.0 / alpha)) + Float64(i * Float64(Float64(Float64(Float64(i / alpha) * 12.0) + Float64(Float64(1.0 / alpha) * 16.0)) + Float64(Float64(4.0 * Float64(-1.0 / alpha)) - 4.0)))) - 2.0)) / alpha) / 2.0);
	else
		tmp = Float64(fma(Float64(Float64(alpha + beta) / Float64(Float64(alpha + beta) + fma(2.0, i, 2.0))), Float64(Float64(beta - alpha) / Float64(beta + fma(2.0, i, alpha))), 1.0) / 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], -0.999995], N[(N[(N[(N[(beta * N[(N[(2.0 + N[(-2.0 * N[(beta / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(4.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(4.0 * N[(N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(4.0 * N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision] + N[(i * N[(N[(N[(N[(i / alpha), $MachinePrecision] * 12.0), $MachinePrecision] + N[(N[(1.0 / alpha), $MachinePrecision] * 16.0), $MachinePrecision]), $MachinePrecision] + N[(N[(4.0 * N[(-1.0 / alpha), $MachinePrecision]), $MachinePrecision] - 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(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[(beta + N[(2.0 * i + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.99999499999999997

   1. Initial program 4.3%

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

   3. Simplified 16.1%

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

   5. Taylor expanded in alpha around inf 81.0%

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

   6. Taylor expanded in beta around 0 87.9%

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

   7. Taylor expanded in i around 0 91.4%

      \[\leadsto \frac{\frac{\beta \cdot \left(\left(2 + -2 \cdot \frac{\beta}{\alpha}\right) - \left(-1 \cdot \left(4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}\right) + 4 \cdot \frac{2 + 4 \cdot i}{\alpha}\right)\right) - \color{blue}{\left(\left(4 \cdot \frac{1}{\alpha} + i \cdot \left(\left(12 \cdot \frac{i}{\alpha} + 16 \cdot \frac{1}{\alpha}\right) - \left(4 + 4 \cdot \frac{1}{\alpha}\right)\right)\right) - 2\right)}}{\alpha}}{2} \]

   if -0.99999499999999997 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 77.3%

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

   3. Simplified 99.8%

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

   6. Applied egg-rr 99.8%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.999995:\\ \;\;\;\;\frac{\frac{\beta \cdot \left(\left(2 + -2 \cdot \frac{\beta}{\alpha}\right) + \left(\left(4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}\right) - 4 \cdot \frac{2 + i \cdot 4}{\alpha}\right)\right) - \left(\left(4 \cdot \frac{1}{\alpha} + i \cdot \left(\left(\frac{i}{\alpha} \cdot 12 + \frac{1}{\alpha} \cdot 16\right) + \left(4 \cdot \frac{-1}{\alpha} - 4\right)\right)\right) - 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)}, \frac{\beta - \alpha}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq -0.999995:\\ \;\;\;\;\frac{\frac{\beta \cdot \left(\left(2 + -2 \cdot \frac{\beta}{\alpha}\right) + \left(\left(4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}\right) - 4 \cdot \frac{2 + i \cdot 4}{\alpha}\right)\right) - \left(\left(4 \cdot \frac{1}{\alpha} + i \cdot \left(\left(\frac{i}{\alpha} \cdot 12 + \frac{1}{\alpha} \cdot 16\right) + \left(4 \cdot \frac{-1}{\alpha} - 4\right)\right)\right) - 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)} + 1}{2}\\ \end{array} \end{array} \]

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)) -0.999995)
     (/
      (/
       (-
        (*
         beta
         (+
          (+ 2.0 (* -2.0 (/ beta alpha)))
          (-
           (+ (* 4.0 (/ i alpha)) (* 2.0 (/ 1.0 alpha)))
           (* 4.0 (/ (+ 2.0 (* i 4.0)) alpha)))))
        (-
         (+
          (* 4.0 (/ 1.0 alpha))
          (*
           i
           (+
            (+ (* (/ i alpha) 12.0) (* (/ 1.0 alpha) 16.0))
            (- (* 4.0 (/ -1.0 alpha)) 4.0))))
         2.0))
       alpha)
      2.0)
     (/
      (+
       (/
        (* (- beta alpha) (/ (+ alpha beta) (fma 2.0 i (+ alpha beta))))
        (+ alpha (+ beta (fma 2.0 i 2.0))))
       1.0)
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.999995) {
		tmp = (((beta * ((2.0 + (-2.0 * (beta / alpha))) + (((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha))) - (4.0 * ((2.0 + (i * 4.0)) / alpha))))) - (((4.0 * (1.0 / alpha)) + (i * ((((i / alpha) * 12.0) + ((1.0 / alpha) * 16.0)) + ((4.0 * (-1.0 / alpha)) - 4.0)))) - 2.0)) / alpha) / 2.0;
	} else {
		tmp = ((((beta - alpha) * ((alpha + beta) / fma(2.0, i, (alpha + beta)))) / (alpha + (beta + fma(2.0, i, 2.0)))) + 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(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)) <= -0.999995)
		tmp = Float64(Float64(Float64(Float64(beta * Float64(Float64(2.0 + Float64(-2.0 * Float64(beta / alpha))) + Float64(Float64(Float64(4.0 * Float64(i / alpha)) + Float64(2.0 * Float64(1.0 / alpha))) - Float64(4.0 * Float64(Float64(2.0 + Float64(i * 4.0)) / alpha))))) - Float64(Float64(Float64(4.0 * Float64(1.0 / alpha)) + Float64(i * Float64(Float64(Float64(Float64(i / alpha) * 12.0) + Float64(Float64(1.0 / alpha) * 16.0)) + Float64(Float64(4.0 * Float64(-1.0 / alpha)) - 4.0)))) - 2.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(beta - alpha) * Float64(Float64(alpha + beta) / fma(2.0, i, Float64(alpha + beta)))) / Float64(alpha + Float64(beta + fma(2.0, i, 2.0)))) + 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(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], -0.999995], N[(N[(N[(N[(beta * N[(N[(2.0 + N[(-2.0 * N[(beta / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(4.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(4.0 * N[(N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(4.0 * N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision] + N[(i * N[(N[(N[(N[(i / alpha), $MachinePrecision] * 12.0), $MachinePrecision] + N[(N[(1.0 / alpha), $MachinePrecision] * 16.0), $MachinePrecision]), $MachinePrecision] + N[(N[(4.0 * N[(-1.0 / alpha), $MachinePrecision]), $MachinePrecision] - 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(N[(beta - alpha), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] / N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.99999499999999997

   1. Initial program 4.3%

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

   3. Simplified 16.1%

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

   5. Taylor expanded in alpha around inf 81.0%

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

   6. Taylor expanded in beta around 0 87.9%

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

   7. Taylor expanded in i around 0 91.4%

      \[\leadsto \frac{\frac{\beta \cdot \left(\left(2 + -2 \cdot \frac{\beta}{\alpha}\right) - \left(-1 \cdot \left(4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}\right) + 4 \cdot \frac{2 + 4 \cdot i}{\alpha}\right)\right) - \color{blue}{\left(\left(4 \cdot \frac{1}{\alpha} + i \cdot \left(\left(12 \cdot \frac{i}{\alpha} + 16 \cdot \frac{1}{\alpha}\right) - \left(4 + 4 \cdot \frac{1}{\alpha}\right)\right)\right) - 2\right)}}{\alpha}}{2} \]

   if -0.99999499999999997 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 77.3%

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

   3. Simplified 99.8%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.999995:\\ \;\;\;\;\frac{\frac{\beta \cdot \left(\left(2 + -2 \cdot \frac{\beta}{\alpha}\right) + \left(\left(4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}\right) - 4 \cdot \frac{2 + i \cdot 4}{\alpha}\right)\right) - \left(\left(4 \cdot \frac{1}{\alpha} + i \cdot \left(\left(\frac{i}{\alpha} \cdot 12 + \frac{1}{\alpha} \cdot 16\right) + \left(4 \cdot \frac{-1}{\alpha} - 4\right)\right)\right) - 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq -0.5:\\ \;\;\;\;\frac{\frac{\beta \cdot \left(\left(2 + -2 \cdot \frac{\beta}{\alpha}\right) + \left(\left(4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}\right) - 4 \cdot \frac{2 + i \cdot 4}{\alpha}\right)\right) - \left(\left(4 \cdot \frac{1}{\alpha} + i \cdot \left(\left(\frac{i}{\alpha} \cdot 12 + \frac{1}{\alpha} \cdot 16\right) + \left(4 \cdot \frac{-1}{\alpha} - 4\right)\right)\right) - 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta - \alpha\right) \cdot \frac{\beta}{\beta + 2 \cdot i}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)} + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0)) -0.5)
     (/
      (/
       (-
        (*
         beta
         (+
          (+ 2.0 (* -2.0 (/ beta alpha)))
          (-
           (+ (* 4.0 (/ i alpha)) (* 2.0 (/ 1.0 alpha)))
           (* 4.0 (/ (+ 2.0 (* i 4.0)) alpha)))))
        (-
         (+
          (* 4.0 (/ 1.0 alpha))
          (*
           i
           (+
            (+ (* (/ i alpha) 12.0) (* (/ 1.0 alpha) 16.0))
            (- (* 4.0 (/ -1.0 alpha)) 4.0))))
         2.0))
       alpha)
      2.0)
     (/
      (+
       (/
        (* (- beta alpha) (/ beta (+ beta (* 2.0 i))))
        (+ alpha (+ beta (fma 2.0 i 2.0))))
       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)) <= -0.5) {
		tmp = (((beta * ((2.0 + (-2.0 * (beta / alpha))) + (((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha))) - (4.0 * ((2.0 + (i * 4.0)) / alpha))))) - (((4.0 * (1.0 / alpha)) + (i * ((((i / alpha) * 12.0) + ((1.0 / alpha) * 16.0)) + ((4.0 * (-1.0 / alpha)) - 4.0)))) - 2.0)) / alpha) / 2.0;
	} else {
		tmp = ((((beta - alpha) * (beta / (beta + (2.0 * i)))) / (alpha + (beta + fma(2.0, i, 2.0)))) + 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)) <= -0.5)
		tmp = Float64(Float64(Float64(Float64(beta * Float64(Float64(2.0 + Float64(-2.0 * Float64(beta / alpha))) + Float64(Float64(Float64(4.0 * Float64(i / alpha)) + Float64(2.0 * Float64(1.0 / alpha))) - Float64(4.0 * Float64(Float64(2.0 + Float64(i * 4.0)) / alpha))))) - Float64(Float64(Float64(4.0 * Float64(1.0 / alpha)) + Float64(i * Float64(Float64(Float64(Float64(i / alpha) * 12.0) + Float64(Float64(1.0 / alpha) * 16.0)) + Float64(Float64(4.0 * Float64(-1.0 / alpha)) - 4.0)))) - 2.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(beta - alpha) * Float64(beta / Float64(beta + Float64(2.0 * i)))) / Float64(alpha + Float64(beta + fma(2.0, i, 2.0)))) + 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(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[(N[(beta * N[(N[(2.0 + N[(-2.0 * N[(beta / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(4.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(4.0 * N[(N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(4.0 * N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision] + N[(i * N[(N[(N[(N[(i / alpha), $MachinePrecision] * 12.0), $MachinePrecision] + N[(N[(1.0 / alpha), $MachinePrecision] * 16.0), $MachinePrecision]), $MachinePrecision] + N[(N[(4.0 * N[(-1.0 / alpha), $MachinePrecision]), $MachinePrecision] - 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(N[(beta - alpha), $MachinePrecision] * N[(beta / N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(beta + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 8.4%

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

   3. Simplified 19.7%

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

   5. Taylor expanded in alpha around inf 79.3%

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

   6. Taylor expanded in beta around 0 85.8%

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

   7. Taylor expanded in i around 0 89.2%

      \[\leadsto \frac{\frac{\beta \cdot \left(\left(2 + -2 \cdot \frac{\beta}{\alpha}\right) - \left(-1 \cdot \left(4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}\right) + 4 \cdot \frac{2 + 4 \cdot i}{\alpha}\right)\right) - \color{blue}{\left(\left(4 \cdot \frac{1}{\alpha} + i \cdot \left(\left(12 \cdot \frac{i}{\alpha} + 16 \cdot \frac{1}{\alpha}\right) - \left(4 + 4 \cdot \frac{1}{\alpha}\right)\right)\right) - 2\right)}}{\alpha}}{2} \]

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

   1. Initial program 77.1%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in alpha around 0 100.0%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.5:\\ \;\;\;\;\frac{\frac{\beta \cdot \left(\left(2 + -2 \cdot \frac{\beta}{\alpha}\right) + \left(\left(4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}\right) - 4 \cdot \frac{2 + i \cdot 4}{\alpha}\right)\right) - \left(\left(4 \cdot \frac{1}{\alpha} + i \cdot \left(\left(\frac{i}{\alpha} \cdot 12 + \frac{1}{\alpha} \cdot 16\right) + \left(4 \cdot \frac{-1}{\alpha} - 4\right)\right)\right) - 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta - \alpha\right) \cdot \frac{\beta}{\beta + 2 \cdot i}}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_2 := \frac{\frac{t\_0}{t\_1}}{2 + t\_1}\\ \mathbf{if}\;t\_2 \leq -0.999995:\\ \;\;\;\;\frac{\frac{\beta \cdot \left(\left(2 + -2 \cdot \frac{\beta}{\alpha}\right) + \left(\left(4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}\right) - 4 \cdot \frac{2 + i \cdot 4}{\alpha}\right)\right) - \left(\left(4 \cdot \frac{1}{\alpha} + i \cdot \left(\left(\frac{i}{\alpha} \cdot 12 + \frac{1}{\alpha} \cdot 16\right) + \left(4 \cdot \frac{-1}{\alpha} - 4\right)\right)\right) - 2\right)}{\alpha}}{2}\\ \mathbf{elif}\;t\_2 \leq 0.999999999999:\\ \;\;\;\;\frac{\frac{t\_0}{\left(\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)\right) \cdot \left(\beta + \left(\alpha + 2 \cdot i\right)\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)} + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* (+ alpha beta) (- beta alpha)))
        (t_1 (+ (+ alpha beta) (* 2.0 i)))
        (t_2 (/ (/ t_0 t_1) (+ 2.0 t_1))))
   (if (<= t_2 -0.999995)
     (/
      (/
       (-
        (*
         beta
         (+
          (+ 2.0 (* -2.0 (/ beta alpha)))
          (-
           (+ (* 4.0 (/ i alpha)) (* 2.0 (/ 1.0 alpha)))
           (* 4.0 (/ (+ 2.0 (* i 4.0)) alpha)))))
        (-
         (+
          (* 4.0 (/ 1.0 alpha))
          (*
           i
           (+
            (+ (* (/ i alpha) 12.0) (* (/ 1.0 alpha) 16.0))
            (- (* 4.0 (/ -1.0 alpha)) 4.0))))
         2.0))
       alpha)
      2.0)
     (if (<= t_2 0.999999999999)
       (/
        (+
         (/
          t_0
          (*
           (+ (+ alpha beta) (+ 2.0 (* 2.0 i)))
           (+ beta (+ alpha (* 2.0 i)))))
         1.0)
        2.0)
       (/ (+ (/ (- beta alpha) (+ beta (+ alpha 2.0))) 1.0) 2.0)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) * (beta - alpha);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = (t_0 / t_1) / (2.0 + t_1);
	double tmp;
	if (t_2 <= -0.999995) {
		tmp = (((beta * ((2.0 + (-2.0 * (beta / alpha))) + (((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha))) - (4.0 * ((2.0 + (i * 4.0)) / alpha))))) - (((4.0 * (1.0 / alpha)) + (i * ((((i / alpha) * 12.0) + ((1.0 / alpha) * 16.0)) + ((4.0 * (-1.0 / alpha)) - 4.0)))) - 2.0)) / alpha) / 2.0;
	} else if (t_2 <= 0.999999999999) {
		tmp = ((t_0 / (((alpha + beta) + (2.0 + (2.0 * i))) * (beta + (alpha + (2.0 * i))))) + 1.0) / 2.0;
	} else {
		tmp = (((beta - alpha) / (beta + (alpha + 2.0))) + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (alpha + beta) * (beta - alpha)
    t_1 = (alpha + beta) + (2.0d0 * i)
    t_2 = (t_0 / t_1) / (2.0d0 + t_1)
    if (t_2 <= (-0.999995d0)) then
        tmp = (((beta * ((2.0d0 + ((-2.0d0) * (beta / alpha))) + (((4.0d0 * (i / alpha)) + (2.0d0 * (1.0d0 / alpha))) - (4.0d0 * ((2.0d0 + (i * 4.0d0)) / alpha))))) - (((4.0d0 * (1.0d0 / alpha)) + (i * ((((i / alpha) * 12.0d0) + ((1.0d0 / alpha) * 16.0d0)) + ((4.0d0 * ((-1.0d0) / alpha)) - 4.0d0)))) - 2.0d0)) / alpha) / 2.0d0
    else if (t_2 <= 0.999999999999d0) then
        tmp = ((t_0 / (((alpha + beta) + (2.0d0 + (2.0d0 * i))) * (beta + (alpha + (2.0d0 * i))))) + 1.0d0) / 2.0d0
    else
        tmp = (((beta - alpha) / (beta + (alpha + 2.0d0))) + 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) * (beta - alpha);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = (t_0 / t_1) / (2.0 + t_1);
	double tmp;
	if (t_2 <= -0.999995) {
		tmp = (((beta * ((2.0 + (-2.0 * (beta / alpha))) + (((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha))) - (4.0 * ((2.0 + (i * 4.0)) / alpha))))) - (((4.0 * (1.0 / alpha)) + (i * ((((i / alpha) * 12.0) + ((1.0 / alpha) * 16.0)) + ((4.0 * (-1.0 / alpha)) - 4.0)))) - 2.0)) / alpha) / 2.0;
	} else if (t_2 <= 0.999999999999) {
		tmp = ((t_0 / (((alpha + beta) + (2.0 + (2.0 * i))) * (beta + (alpha + (2.0 * i))))) + 1.0) / 2.0;
	} else {
		tmp = (((beta - alpha) / (beta + (alpha + 2.0))) + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) * (beta - alpha)
	t_1 = (alpha + beta) + (2.0 * i)
	t_2 = (t_0 / t_1) / (2.0 + t_1)
	tmp = 0
	if t_2 <= -0.999995:
		tmp = (((beta * ((2.0 + (-2.0 * (beta / alpha))) + (((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha))) - (4.0 * ((2.0 + (i * 4.0)) / alpha))))) - (((4.0 * (1.0 / alpha)) + (i * ((((i / alpha) * 12.0) + ((1.0 / alpha) * 16.0)) + ((4.0 * (-1.0 / alpha)) - 4.0)))) - 2.0)) / alpha) / 2.0
	elif t_2 <= 0.999999999999:
		tmp = ((t_0 / (((alpha + beta) + (2.0 + (2.0 * i))) * (beta + (alpha + (2.0 * i))))) + 1.0) / 2.0
	else:
		tmp = (((beta - alpha) / (beta + (alpha + 2.0))) + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) * Float64(beta - alpha))
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_2 = Float64(Float64(t_0 / t_1) / Float64(2.0 + t_1))
	tmp = 0.0
	if (t_2 <= -0.999995)
		tmp = Float64(Float64(Float64(Float64(beta * Float64(Float64(2.0 + Float64(-2.0 * Float64(beta / alpha))) + Float64(Float64(Float64(4.0 * Float64(i / alpha)) + Float64(2.0 * Float64(1.0 / alpha))) - Float64(4.0 * Float64(Float64(2.0 + Float64(i * 4.0)) / alpha))))) - Float64(Float64(Float64(4.0 * Float64(1.0 / alpha)) + Float64(i * Float64(Float64(Float64(Float64(i / alpha) * 12.0) + Float64(Float64(1.0 / alpha) * 16.0)) + Float64(Float64(4.0 * Float64(-1.0 / alpha)) - 4.0)))) - 2.0)) / alpha) / 2.0);
	elseif (t_2 <= 0.999999999999)
		tmp = Float64(Float64(Float64(t_0 / Float64(Float64(Float64(alpha + beta) + Float64(2.0 + Float64(2.0 * i))) * Float64(beta + Float64(alpha + Float64(2.0 * i))))) + 1.0) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(beta - alpha) / Float64(beta + Float64(alpha + 2.0))) + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) * (beta - alpha);
	t_1 = (alpha + beta) + (2.0 * i);
	t_2 = (t_0 / t_1) / (2.0 + t_1);
	tmp = 0.0;
	if (t_2 <= -0.999995)
		tmp = (((beta * ((2.0 + (-2.0 * (beta / alpha))) + (((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha))) - (4.0 * ((2.0 + (i * 4.0)) / alpha))))) - (((4.0 * (1.0 / alpha)) + (i * ((((i / alpha) * 12.0) + ((1.0 / alpha) * 16.0)) + ((4.0 * (-1.0 / alpha)) - 4.0)))) - 2.0)) / alpha) / 2.0;
	elseif (t_2 <= 0.999999999999)
		tmp = ((t_0 / (((alpha + beta) + (2.0 + (2.0 * i))) * (beta + (alpha + (2.0 * i))))) + 1.0) / 2.0;
	else
		tmp = (((beta - alpha) / (beta + (alpha + 2.0))) + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 / t$95$1), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.999995], N[(N[(N[(N[(beta * N[(N[(2.0 + N[(-2.0 * N[(beta / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(4.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(4.0 * N[(N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(4.0 * N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision] + N[(i * N[(N[(N[(N[(i / alpha), $MachinePrecision] * 12.0), $MachinePrecision] + N[(N[(1.0 / alpha), $MachinePrecision] * 16.0), $MachinePrecision]), $MachinePrecision] + N[(N[(4.0 * N[(-1.0 / alpha), $MachinePrecision]), $MachinePrecision] - 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[t$95$2, 0.999999999999], N[(N[(N[(t$95$0 / N[(N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(beta + N[(alpha + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.99999499999999997

   1. Initial program 4.3%

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

   3. Simplified 16.1%

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

   5. Taylor expanded in alpha around inf 81.0%

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

   6. Taylor expanded in beta around 0 87.9%

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

   7. Taylor expanded in i around 0 91.4%

      \[\leadsto \frac{\frac{\beta \cdot \left(\left(2 + -2 \cdot \frac{\beta}{\alpha}\right) - \left(-1 \cdot \left(4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}\right) + 4 \cdot \frac{2 + 4 \cdot i}{\alpha}\right)\right) - \color{blue}{\left(\left(4 \cdot \frac{1}{\alpha} + i \cdot \left(\left(12 \cdot \frac{i}{\alpha} + 16 \cdot \frac{1}{\alpha}\right) - \left(4 + 4 \cdot \frac{1}{\alpha}\right)\right)\right) - 2\right)}}{\alpha}}{2} \]

   if -0.99999499999999997 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < 0.999999999999000022

   1. Initial program 99.7%

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

      1. associate-/l/ 99.7%

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

      2. associate-+l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-+l+ 99.7%

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

   3. Simplified 99.7%

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

   if 0.999999999999000022 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 30.4%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in i around 0 88.0%

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

      1. associate-+r+ 88.0%

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

      2. +-commutative 88.0%

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

   7. Simplified 88.0%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.999995:\\ \;\;\;\;\frac{\frac{\beta \cdot \left(\left(2 + -2 \cdot \frac{\beta}{\alpha}\right) + \left(\left(4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}\right) - 4 \cdot \frac{2 + i \cdot 4}{\alpha}\right)\right) - \left(\left(4 \cdot \frac{1}{\alpha} + i \cdot \left(\left(\frac{i}{\alpha} \cdot 12 + \frac{1}{\alpha} \cdot 16\right) + \left(4 \cdot \frac{-1}{\alpha} - 4\right)\right)\right) - 2\right)}{\alpha}}{2}\\ \mathbf{elif}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq 0.999999999999:\\ \;\;\;\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)\right) \cdot \left(\beta + \left(\alpha + 2 \cdot i\right)\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_2 := \frac{\frac{t\_0}{t\_1}}{2 + t\_1}\\ \mathbf{if}\;t\_2 \leq -0.999995:\\ \;\;\;\;\frac{\left(4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}\right) + 2 \cdot \frac{\beta}{\alpha}}{2}\\ \mathbf{elif}\;t\_2 \leq 0.999999999999:\\ \;\;\;\;\frac{\frac{t\_0}{\left(\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)\right) \cdot \left(\beta + \left(\alpha + 2 \cdot i\right)\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)} + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* (+ alpha beta) (- beta alpha)))
        (t_1 (+ (+ alpha beta) (* 2.0 i)))
        (t_2 (/ (/ t_0 t_1) (+ 2.0 t_1))))
   (if (<= t_2 -0.999995)
     (/
      (+ (+ (* 4.0 (/ i alpha)) (* 2.0 (/ 1.0 alpha))) (* 2.0 (/ beta alpha)))
      2.0)
     (if (<= t_2 0.999999999999)
       (/
        (+
         (/
          t_0
          (*
           (+ (+ alpha beta) (+ 2.0 (* 2.0 i)))
           (+ beta (+ alpha (* 2.0 i)))))
         1.0)
        2.0)
       (/ (+ (/ (- beta alpha) (+ beta (+ alpha 2.0))) 1.0) 2.0)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) * (beta - alpha);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = (t_0 / t_1) / (2.0 + t_1);
	double tmp;
	if (t_2 <= -0.999995) {
		tmp = (((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha))) + (2.0 * (beta / alpha))) / 2.0;
	} else if (t_2 <= 0.999999999999) {
		tmp = ((t_0 / (((alpha + beta) + (2.0 + (2.0 * i))) * (beta + (alpha + (2.0 * i))))) + 1.0) / 2.0;
	} else {
		tmp = (((beta - alpha) / (beta + (alpha + 2.0))) + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (alpha + beta) * (beta - alpha)
    t_1 = (alpha + beta) + (2.0d0 * i)
    t_2 = (t_0 / t_1) / (2.0d0 + t_1)
    if (t_2 <= (-0.999995d0)) then
        tmp = (((4.0d0 * (i / alpha)) + (2.0d0 * (1.0d0 / alpha))) + (2.0d0 * (beta / alpha))) / 2.0d0
    else if (t_2 <= 0.999999999999d0) then
        tmp = ((t_0 / (((alpha + beta) + (2.0d0 + (2.0d0 * i))) * (beta + (alpha + (2.0d0 * i))))) + 1.0d0) / 2.0d0
    else
        tmp = (((beta - alpha) / (beta + (alpha + 2.0d0))) + 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) * (beta - alpha);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = (t_0 / t_1) / (2.0 + t_1);
	double tmp;
	if (t_2 <= -0.999995) {
		tmp = (((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha))) + (2.0 * (beta / alpha))) / 2.0;
	} else if (t_2 <= 0.999999999999) {
		tmp = ((t_0 / (((alpha + beta) + (2.0 + (2.0 * i))) * (beta + (alpha + (2.0 * i))))) + 1.0) / 2.0;
	} else {
		tmp = (((beta - alpha) / (beta + (alpha + 2.0))) + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) * (beta - alpha)
	t_1 = (alpha + beta) + (2.0 * i)
	t_2 = (t_0 / t_1) / (2.0 + t_1)
	tmp = 0
	if t_2 <= -0.999995:
		tmp = (((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha))) + (2.0 * (beta / alpha))) / 2.0
	elif t_2 <= 0.999999999999:
		tmp = ((t_0 / (((alpha + beta) + (2.0 + (2.0 * i))) * (beta + (alpha + (2.0 * i))))) + 1.0) / 2.0
	else:
		tmp = (((beta - alpha) / (beta + (alpha + 2.0))) + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) * Float64(beta - alpha))
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_2 = Float64(Float64(t_0 / t_1) / Float64(2.0 + t_1))
	tmp = 0.0
	if (t_2 <= -0.999995)
		tmp = Float64(Float64(Float64(Float64(4.0 * Float64(i / alpha)) + Float64(2.0 * Float64(1.0 / alpha))) + Float64(2.0 * Float64(beta / alpha))) / 2.0);
	elseif (t_2 <= 0.999999999999)
		tmp = Float64(Float64(Float64(t_0 / Float64(Float64(Float64(alpha + beta) + Float64(2.0 + Float64(2.0 * i))) * Float64(beta + Float64(alpha + Float64(2.0 * i))))) + 1.0) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(beta - alpha) / Float64(beta + Float64(alpha + 2.0))) + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) * (beta - alpha);
	t_1 = (alpha + beta) + (2.0 * i);
	t_2 = (t_0 / t_1) / (2.0 + t_1);
	tmp = 0.0;
	if (t_2 <= -0.999995)
		tmp = (((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha))) + (2.0 * (beta / alpha))) / 2.0;
	elseif (t_2 <= 0.999999999999)
		tmp = ((t_0 / (((alpha + beta) + (2.0 + (2.0 * i))) * (beta + (alpha + (2.0 * i))))) + 1.0) / 2.0;
	else
		tmp = (((beta - alpha) / (beta + (alpha + 2.0))) + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 / t$95$1), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.999995], N[(N[(N[(N[(4.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(beta / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[t$95$2, 0.999999999999], N[(N[(N[(t$95$0 / N[(N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(beta + N[(alpha + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.99999499999999997

   1. Initial program 4.3%

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

   3. Simplified 16.1%

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

   5. Taylor expanded in alpha around inf 90.4%

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

   6. Taylor expanded in beta around 0 90.5%

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

   if -0.99999499999999997 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < 0.999999999999000022

   1. Initial program 99.7%

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

      1. associate-/l/ 99.7%

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

      2. associate-+l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-+l+ 99.7%

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

   3. Simplified 99.7%

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

   if 0.999999999999000022 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 30.4%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in i around 0 88.0%

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

      1. associate-+r+ 88.0%

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

      2. +-commutative 88.0%

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

   7. Simplified 88.0%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.999995:\\ \;\;\;\;\frac{\left(4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}\right) + 2 \cdot \frac{\beta}{\alpha}}{2}\\ \mathbf{elif}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq 0.999999999999:\\ \;\;\;\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)\right) \cdot \left(\beta + \left(\alpha + 2 \cdot i\right)\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 15000000000:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{elif}\;\alpha \leq 1.35 \cdot 10^{+143} \lor \neg \left(\alpha \leq 8 \cdot 10^{+200}\right):\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 15000000000.0)
   (/ (+ (/ beta (+ beta 2.0)) 1.0) 2.0)
   (if (or (<= alpha 1.35e+143) (not (<= alpha 8e+200)))
     (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)
     (/ (* 4.0 (/ i alpha)) 2.0))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 15000000000.0) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	} else if ((alpha <= 1.35e+143) || !(alpha <= 8e+200)) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = (4.0 * (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 <= 15000000000.0d0) then
        tmp = ((beta / (beta + 2.0d0)) + 1.0d0) / 2.0d0
    else if ((alpha <= 1.35d+143) .or. (.not. (alpha <= 8d+200))) then
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    else
        tmp = (4.0d0 * (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 <= 15000000000.0) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	} else if ((alpha <= 1.35e+143) || !(alpha <= 8e+200)) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = (4.0 * (i / alpha)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 15000000000.0:
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0
	elif (alpha <= 1.35e+143) or not (alpha <= 8e+200):
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	else:
		tmp = (4.0 * (i / alpha)) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 15000000000.0)
		tmp = Float64(Float64(Float64(beta / Float64(beta + 2.0)) + 1.0) / 2.0);
	elseif ((alpha <= 1.35e+143) || !(alpha <= 8e+200))
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(4.0 * Float64(i / alpha)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 15000000000.0)
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	elseif ((alpha <= 1.35e+143) || ~((alpha <= 8e+200)))
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	else
		tmp = (4.0 * (i / alpha)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 15000000000.0], N[(N[(N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[alpha, 1.35e+143], N[Not[LessEqual[alpha, 8e+200]], $MachinePrecision]], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(4.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 1.5e10

   1. Initial program 79.3%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in i around 0 91.5%

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

      1. associate-+r+ 91.5%

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

      2. +-commutative 91.5%

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

   7. Simplified 91.5%

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

   8. Taylor expanded in alpha around 0 91.3%

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

      1. +-commutative 91.3%

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

   10. Simplified 91.3%

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

   if 1.5e10 < alpha < 1.3500000000000001e143 or 7.9999999999999998e200 < alpha

   1. Initial program 22.6%

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

   3. Simplified 37.5%

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

   5. Taylor expanded in alpha around inf 69.6%

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

   6. Taylor expanded in i around 0 56.8%

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

      1. distribute-rgt1-in 56.8%

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

      2. metadata-eval 56.8%

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

      3. mul0-lft 56.8%

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

      4. neg-sub0 56.8%

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

      5. mul-1-neg 56.8%

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

      6. remove-double-neg 56.8%

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

   8. Simplified 56.8%

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

   if 1.3500000000000001e143 < alpha < 7.9999999999999998e200

   1. Initial program 1.6%

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

   3. Simplified 36.2%

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

   5. Taylor expanded in alpha around inf 69.3%

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

   6. Taylor expanded in i around inf 56.6%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 15000000000:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{elif}\;\alpha \leq 1.35 \cdot 10^{+143} \lor \neg \left(\alpha \leq 8 \cdot 10^{+200}\right):\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 15000000000:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)} + 1}{2}\\ \mathbf{elif}\;\alpha \leq 2.8 \cdot 10^{+142} \lor \neg \left(\alpha \leq 8 \cdot 10^{+200}\right):\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 15000000000.0)
   (/ (+ (/ (- beta alpha) (+ beta (+ alpha 2.0))) 1.0) 2.0)
   (if (or (<= alpha 2.8e+142) (not (<= alpha 8e+200)))
     (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)
     (/ (* 4.0 (/ i alpha)) 2.0))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 15000000000.0) {
		tmp = (((beta - alpha) / (beta + (alpha + 2.0))) + 1.0) / 2.0;
	} else if ((alpha <= 2.8e+142) || !(alpha <= 8e+200)) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = (4.0 * (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 <= 15000000000.0d0) then
        tmp = (((beta - alpha) / (beta + (alpha + 2.0d0))) + 1.0d0) / 2.0d0
    else if ((alpha <= 2.8d+142) .or. (.not. (alpha <= 8d+200))) then
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    else
        tmp = (4.0d0 * (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 <= 15000000000.0) {
		tmp = (((beta - alpha) / (beta + (alpha + 2.0))) + 1.0) / 2.0;
	} else if ((alpha <= 2.8e+142) || !(alpha <= 8e+200)) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = (4.0 * (i / alpha)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 15000000000.0:
		tmp = (((beta - alpha) / (beta + (alpha + 2.0))) + 1.0) / 2.0
	elif (alpha <= 2.8e+142) or not (alpha <= 8e+200):
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	else:
		tmp = (4.0 * (i / alpha)) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 15000000000.0)
		tmp = Float64(Float64(Float64(Float64(beta - alpha) / Float64(beta + Float64(alpha + 2.0))) + 1.0) / 2.0);
	elseif ((alpha <= 2.8e+142) || !(alpha <= 8e+200))
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(4.0 * Float64(i / alpha)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 15000000000.0)
		tmp = (((beta - alpha) / (beta + (alpha + 2.0))) + 1.0) / 2.0;
	elseif ((alpha <= 2.8e+142) || ~((alpha <= 8e+200)))
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	else
		tmp = (4.0 * (i / alpha)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 15000000000.0], N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[alpha, 2.8e+142], N[Not[LessEqual[alpha, 8e+200]], $MachinePrecision]], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(4.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 1.5e10

   1. Initial program 79.3%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in i around 0 91.5%

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

      1. associate-+r+ 91.5%

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

      2. +-commutative 91.5%

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

   7. Simplified 91.5%

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

   if 1.5e10 < alpha < 2.8e142 or 7.9999999999999998e200 < alpha

   1. Initial program 22.6%

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

   3. Simplified 37.5%

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

   5. Taylor expanded in alpha around inf 69.6%

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

   6. Taylor expanded in i around 0 56.8%

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

      1. distribute-rgt1-in 56.8%

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

      2. metadata-eval 56.8%

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

      3. mul0-lft 56.8%

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

      4. neg-sub0 56.8%

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

      5. mul-1-neg 56.8%

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

      6. remove-double-neg 56.8%

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

   8. Simplified 56.8%

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

   if 2.8e142 < alpha < 7.9999999999999998e200

   1. Initial program 1.6%

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

   3. Simplified 36.2%

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

   5. Taylor expanded in alpha around inf 69.3%

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

   6. Taylor expanded in i around inf 56.6%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 15000000000:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)} + 1}{2}\\ \mathbf{elif}\;\alpha \leq 2.8 \cdot 10^{+142} \lor \neg \left(\alpha \leq 8 \cdot 10^{+200}\right):\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 15000000000:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}\right) + 2 \cdot \frac{\beta}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 15000000000.0)
   (/ (+ (/ (- beta alpha) (+ beta (+ alpha 2.0))) 1.0) 2.0)
   (/
    (+ (+ (* 4.0 (/ i alpha)) (* 2.0 (/ 1.0 alpha))) (* 2.0 (/ beta alpha)))
    2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 15000000000.0) {
		tmp = (((beta - alpha) / (beta + (alpha + 2.0))) + 1.0) / 2.0;
	} else {
		tmp = (((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha))) + (2.0 * (beta / 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 <= 15000000000.0d0) then
        tmp = (((beta - alpha) / (beta + (alpha + 2.0d0))) + 1.0d0) / 2.0d0
    else
        tmp = (((4.0d0 * (i / alpha)) + (2.0d0 * (1.0d0 / alpha))) + (2.0d0 * (beta / alpha))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 15000000000.0) {
		tmp = (((beta - alpha) / (beta + (alpha + 2.0))) + 1.0) / 2.0;
	} else {
		tmp = (((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha))) + (2.0 * (beta / alpha))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 15000000000.0:
		tmp = (((beta - alpha) / (beta + (alpha + 2.0))) + 1.0) / 2.0
	else:
		tmp = (((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha))) + (2.0 * (beta / alpha))) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 15000000000.0)
		tmp = Float64(Float64(Float64(Float64(beta - alpha) / Float64(beta + Float64(alpha + 2.0))) + 1.0) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(4.0 * Float64(i / alpha)) + Float64(2.0 * Float64(1.0 / alpha))) + Float64(2.0 * Float64(beta / alpha))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 15000000000.0)
		tmp = (((beta - alpha) / (beta + (alpha + 2.0))) + 1.0) / 2.0;
	else
		tmp = (((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha))) + (2.0 * (beta / alpha))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 15000000000.0], N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(4.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(beta / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.5e10

   1. Initial program 79.3%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in i around 0 91.5%

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

      1. associate-+r+ 91.5%

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

      2. +-commutative 91.5%

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

   7. Simplified 91.5%

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

   if 1.5e10 < alpha

   1. Initial program 18.6%

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

   3. Simplified 37.3%

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

   5. Taylor expanded in alpha around inf 69.6%

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

   6. Taylor expanded in beta around 0 69.6%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 15000000000:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}\right) + 2 \cdot \frac{\beta}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 15000000000:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(2 + i \cdot 4\right) + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 15000000000.0)
   (/ (+ (/ (- beta alpha) (+ beta (+ alpha 2.0))) 1.0) 2.0)
   (/ (/ (+ (+ 2.0 (* i 4.0)) (* beta 2.0)) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 15000000000.0) {
		tmp = (((beta - alpha) / (beta + (alpha + 2.0))) + 1.0) / 2.0;
	} else {
		tmp = (((2.0 + (i * 4.0)) + (beta * 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 15000000000.0d0) then
        tmp = (((beta - alpha) / (beta + (alpha + 2.0d0))) + 1.0d0) / 2.0d0
    else
        tmp = (((2.0d0 + (i * 4.0d0)) + (beta * 2.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 15000000000.0) {
		tmp = (((beta - alpha) / (beta + (alpha + 2.0))) + 1.0) / 2.0;
	} else {
		tmp = (((2.0 + (i * 4.0)) + (beta * 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 15000000000.0:
		tmp = (((beta - alpha) / (beta + (alpha + 2.0))) + 1.0) / 2.0
	else:
		tmp = (((2.0 + (i * 4.0)) + (beta * 2.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 15000000000.0)
		tmp = Float64(Float64(Float64(Float64(beta - alpha) / Float64(beta + Float64(alpha + 2.0))) + 1.0) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(2.0 + Float64(i * 4.0)) + Float64(beta * 2.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 15000000000.0)
		tmp = (((beta - alpha) / (beta + (alpha + 2.0))) + 1.0) / 2.0;
	else
		tmp = (((2.0 + (i * 4.0)) + (beta * 2.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 15000000000.0], N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.5e10

   1. Initial program 79.3%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in i around 0 91.5%

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

      1. associate-+r+ 91.5%

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

      2. +-commutative 91.5%

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

   7. Simplified 91.5%

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

   if 1.5e10 < alpha

   1. Initial program 18.6%

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

   3. Simplified 37.3%

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

   5. Taylor expanded in alpha around inf 61.6%

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

   6. Taylor expanded in beta around 0 66.8%

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

   7. Taylor expanded in alpha around inf 69.6%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 15000000000:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(2 + i \cdot 4\right) + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 2.05 \cdot 10^{+114}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= i 2.05e+114) (/ (+ (/ beta (+ beta 2.0)) 1.0) 2.0) 0.5))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 2.05e+114) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (i <= 2.05d+114) then
        tmp = ((beta / (beta + 2.0d0)) + 1.0d0) / 2.0d0
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 2.05e+114) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if i <= 2.05e+114:
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0
	else:
		tmp = 0.5
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (i <= 2.05e+114)
		tmp = Float64(Float64(Float64(beta / Float64(beta + 2.0)) + 1.0) / 2.0);
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (i <= 2.05e+114)
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[i, 2.05e+114], N[(N[(N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if i < 2.05e114

   1. Initial program 57.6%

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

   3. Simplified 74.4%

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

   5. Taylor expanded in i around 0 72.4%

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

      1. associate-+r+ 72.4%

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

      2. +-commutative 72.4%

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

   7. Simplified 72.4%

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

   8. Taylor expanded in alpha around 0 70.7%

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

      1. +-commutative 70.7%

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

   10. Simplified 70.7%

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

   if 2.05e114 < i

   1. Initial program 67.6%

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

   3. Simplified 95.1%

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

   5. Taylor expanded in i around inf 83.5%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 2.05 \cdot 10^{+114}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 71.4% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.15 \cdot 10^{+96}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 (if (<= beta 2.15e+96) 0.5 1.0))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.15e+96) {
		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 <= 2.15d+96) 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 <= 2.15e+96) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 2.15e+96:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 2.15e+96)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 2.15e+96)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 2.15e+96], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 2.15000000000000001e96

   1. Initial program 73.0%

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

   3. Simplified 76.5%

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

   5. Taylor expanded in i around inf 71.1%

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

   if 2.15000000000000001e96 < beta

   1. Initial program 28.8%

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

   3. Simplified 92.1%

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

   5. Taylor expanded in beta around inf 75.2%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.15 \cdot 10^{+96}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 60.9% 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 60.8%

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

3. Simplified 80.9%

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

5. Taylor expanded in i around inf 59.5%

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

6. Final simplification 59.5%

   \[\leadsto 0.5 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024115 
(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))