Octave 3.8, jcobi/2

Percentage Accurate: 64.2% → 97.6%

Time: 16.7s

Alternatives: 13

Speedup: 9.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 64.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(\beta + 1\right)\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_2 := 2 \cdot \left(-1 - \beta\right)\\ t_3 := \beta + t_0\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_1}}{2 + t_1} \leq -0.99999:\\ \;\;\;\;\frac{\mathsf{fma}\left(-12, \frac{i}{\alpha} \cdot \frac{i}{\alpha}, \left(\left(\frac{\beta}{\alpha} + \frac{\beta}{\frac{\alpha \cdot \alpha}{t_3}}\right) + \mathsf{fma}\left(\left(4 \cdot \frac{\beta}{\alpha \cdot \alpha} + \left(4 \cdot \left(\frac{t_2}{\alpha \cdot \alpha} + \frac{t_2 - \beta}{\alpha \cdot \alpha}\right) - \frac{-2 \cdot \left(\beta + \left(\beta + 2\right)\right)}{\alpha \cdot \alpha}\right)\right) + \frac{4}{\alpha}, i, \frac{\beta}{\alpha} \cdot \frac{\beta + 2}{\alpha} - \frac{t_0}{\alpha} \cdot \frac{t_3}{\alpha}\right)\right) - \frac{\beta}{\alpha}\right) + \mathsf{fma}\left(2, \frac{\beta}{\alpha}, \frac{2}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* 2.0 (+ beta 1.0)))
        (t_1 (+ (+ alpha beta) (* 2.0 i)))
        (t_2 (* 2.0 (- -1.0 beta)))
        (t_3 (+ beta t_0)))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_1) (+ 2.0 t_1)) -0.99999)
     (/
      (+
       (fma
        -12.0
        (* (/ i alpha) (/ i alpha))
        (-
         (+
          (+ (/ beta alpha) (/ beta (/ (* alpha alpha) t_3)))
          (fma
           (+
            (+
             (* 4.0 (/ beta (* alpha alpha)))
             (-
              (*
               4.0
               (+ (/ t_2 (* alpha alpha)) (/ (- t_2 beta) (* alpha alpha))))
              (/ (* -2.0 (+ beta (+ beta 2.0))) (* alpha alpha))))
            (/ 4.0 alpha))
           i
           (-
            (* (/ beta alpha) (/ (+ beta 2.0) alpha))
            (* (/ t_0 alpha) (/ t_3 alpha)))))
         (/ beta alpha)))
       (fma 2.0 (/ beta alpha) (/ 2.0 alpha)))
      2.0)
     (/
      (+
       (*
        (/ (- beta alpha) (+ (+ alpha beta) (fma 2.0 i 2.0)))
        (/ (+ alpha beta) (fma 2.0 i (+ alpha beta))))
       1.0)
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = 2.0 * (beta + 1.0);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = 2.0 * (-1.0 - beta);
	double t_3 = beta + t_0;
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_1) / (2.0 + t_1)) <= -0.99999) {
		tmp = (fma(-12.0, ((i / alpha) * (i / alpha)), ((((beta / alpha) + (beta / ((alpha * alpha) / t_3))) + fma((((4.0 * (beta / (alpha * alpha))) + ((4.0 * ((t_2 / (alpha * alpha)) + ((t_2 - beta) / (alpha * alpha)))) - ((-2.0 * (beta + (beta + 2.0))) / (alpha * alpha)))) + (4.0 / alpha)), i, (((beta / alpha) * ((beta + 2.0) / alpha)) - ((t_0 / alpha) * (t_3 / alpha))))) - (beta / alpha))) + fma(2.0, (beta / alpha), (2.0 / alpha))) / 2.0;
	} else {
		tmp = ((((beta - alpha) / ((alpha + beta) + fma(2.0, i, 2.0))) * ((alpha + beta) / fma(2.0, i, (alpha + beta)))) + 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(2.0 * Float64(beta + 1.0))
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_2 = Float64(2.0 * Float64(-1.0 - beta))
	t_3 = Float64(beta + t_0)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_1) / Float64(2.0 + t_1)) <= -0.99999)
		tmp = Float64(Float64(fma(-12.0, Float64(Float64(i / alpha) * Float64(i / alpha)), Float64(Float64(Float64(Float64(beta / alpha) + Float64(beta / Float64(Float64(alpha * alpha) / t_3))) + fma(Float64(Float64(Float64(4.0 * Float64(beta / Float64(alpha * alpha))) + Float64(Float64(4.0 * Float64(Float64(t_2 / Float64(alpha * alpha)) + Float64(Float64(t_2 - beta) / Float64(alpha * alpha)))) - Float64(Float64(-2.0 * Float64(beta + Float64(beta + 2.0))) / Float64(alpha * alpha)))) + Float64(4.0 / alpha)), i, Float64(Float64(Float64(beta / alpha) * Float64(Float64(beta + 2.0) / alpha)) - Float64(Float64(t_0 / alpha) * Float64(t_3 / alpha))))) - Float64(beta / alpha))) + fma(2.0, Float64(beta / alpha), Float64(2.0 / alpha))) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + fma(2.0, i, 2.0))) * Float64(Float64(alpha + beta) / fma(2.0, i, Float64(alpha + beta)))) + 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(2.0 * N[(beta + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(-1.0 - beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(beta + t$95$0), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], -0.99999], N[(N[(N[(-12.0 * N[(N[(i / alpha), $MachinePrecision] * N[(i / alpha), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(beta / alpha), $MachinePrecision] + N[(beta / N[(N[(alpha * alpha), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(4.0 * N[(beta / N[(alpha * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(4.0 * N[(N[(t$95$2 / N[(alpha * alpha), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 - beta), $MachinePrecision] / N[(alpha * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(-2.0 * N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(4.0 / alpha), $MachinePrecision]), $MachinePrecision] * i + N[(N[(N[(beta / alpha), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] - N[(N[(t$95$0 / alpha), $MachinePrecision] * N[(t$95$3 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(beta / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(beta / alpha), $MachinePrecision] + N[(2.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] / N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2)) < -0.999990000000000046

   1. Initial program 3.0%

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

   3. Simplified 12.7%

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

   4. Taylor expanded in alpha around inf 87.6%

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

   5. Taylor expanded in i around inf 87.6%

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

   7. Simplified 92.3%

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

   if -0.999990000000000046 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2))

   1. Initial program 82.1%

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

      1. associate-/l/ 81.6%

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

      2. *-commutative 81.6%

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

      3. times-frac 99.9%

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

      4. associate-+l+ 99.9%

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

      5. fma-def 99.9%

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

      6. +-commutative 99.9%

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

      7. fma-def 99.9%

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

   3. Simplified 99.9%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.99999:\\ \;\;\;\;\frac{\mathsf{fma}\left(-12, \frac{i}{\alpha} \cdot \frac{i}{\alpha}, \left(\left(\frac{\beta}{\alpha} + \frac{\beta}{\frac{\alpha \cdot \alpha}{\beta + 2 \cdot \left(\beta + 1\right)}}\right) + \mathsf{fma}\left(\left(4 \cdot \frac{\beta}{\alpha \cdot \alpha} + \left(4 \cdot \left(\frac{2 \cdot \left(-1 - \beta\right)}{\alpha \cdot \alpha} + \frac{2 \cdot \left(-1 - \beta\right) - \beta}{\alpha \cdot \alpha}\right) - \frac{-2 \cdot \left(\beta + \left(\beta + 2\right)\right)}{\alpha \cdot \alpha}\right)\right) + \frac{4}{\alpha}, i, \frac{\beta}{\alpha} \cdot \frac{\beta + 2}{\alpha} - \frac{2 \cdot \left(\beta + 1\right)}{\alpha} \cdot \frac{\beta + 2 \cdot \left(\beta + 1\right)}{\alpha}\right)\right) - \frac{\beta}{\alpha}\right) + \mathsf{fma}\left(2, \frac{\beta}{\alpha}, \frac{2}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1}{2}\\ \end{array} \]

Alternative 2: 97.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.999999998:\\ \;\;\;\;\frac{\frac{0}{\alpha} + \left(\frac{i}{\alpha} \cdot 4 + \left(2 \cdot \frac{\beta}{\alpha} + 2 \cdot \frac{1}{\alpha}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1}{2}\\ \end{array} \end{array} \]

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))
        -0.999999998)
     (/
      (+
       (/ 0.0 alpha)
       (+
        (* (/ i alpha) 4.0)
        (+ (* 2.0 (/ beta alpha)) (* 2.0 (/ 1.0 alpha)))))
      2.0)
     (/
      (+
       (*
        (/ (- beta alpha) (+ (+ alpha beta) (fma 2.0 i 2.0)))
        (/ (+ alpha beta) (fma 2.0 i (+ alpha beta))))
       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)) <= -0.999999998) {
		tmp = ((0.0 / alpha) + (((i / alpha) * 4.0) + ((2.0 * (beta / alpha)) + (2.0 * (1.0 / alpha))))) / 2.0;
	} else {
		tmp = ((((beta - alpha) / ((alpha + beta) + fma(2.0, i, 2.0))) * ((alpha + beta) / fma(2.0, i, (alpha + beta)))) + 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)) <= -0.999999998)
		tmp = Float64(Float64(Float64(0.0 / alpha) + Float64(Float64(Float64(i / alpha) * 4.0) + Float64(Float64(2.0 * Float64(beta / alpha)) + Float64(2.0 * Float64(1.0 / alpha))))) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + fma(2.0, i, 2.0))) * Float64(Float64(alpha + beta) / fma(2.0, i, Float64(alpha + beta)))) + 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], -0.999999998], N[(N[(N[(0.0 / alpha), $MachinePrecision] + N[(N[(N[(i / alpha), $MachinePrecision] * 4.0), $MachinePrecision] + N[(N[(2.0 * N[(beta / alpha), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] / N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2)) < -0.999999997999999946

   1. Initial program 2.0%

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

   3. Simplified 12.0%

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

   4. Taylor expanded in alpha around inf 87.5%

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

   5. Taylor expanded in alpha around inf 92.2%

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

      1. distribute-lft1-in 92.2%

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

      2. metadata-eval 92.2%

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

      3. mul0-lft 92.2%

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

   7. Simplified 92.2%

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

   8. Taylor expanded in i around 0 92.3%

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

   if -0.999999997999999946 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2))

   1. Initial program 82.0%

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

      1. associate-/l/ 81.5%

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

      2. *-commutative 81.5%

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

      3. times-frac 99.7%

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

      4. associate-+l+ 99.7%

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

      5. fma-def 99.7%

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

      6. +-commutative 99.7%

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

      7. fma-def 99.7%

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

   3. Simplified 99.7%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.999999998:\\ \;\;\;\;\frac{\frac{0}{\alpha} + \left(\frac{i}{\alpha} \cdot 4 + \left(2 \cdot \frac{\beta}{\alpha} + 2 \cdot \frac{1}{\alpha}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1}{2}\\ \end{array} \]

Alternative 3: 95.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2)) < -0.5

   1. Initial program 4.1%

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

   3. Simplified 13.7%

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

   4. Taylor expanded in alpha around inf 87.4%

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

   5. Taylor expanded in alpha around inf 91.0%

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

      1. distribute-lft1-in 91.0%

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

      2. metadata-eval 91.0%

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

      3. mul0-lft 91.0%

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

   7. Simplified 91.0%

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

   8. Taylor expanded in i around 0 91.1%

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

   if -0.5 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2))

   1. Initial program 82.1%

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

   2. Taylor expanded in beta around inf 98.9%

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

      1. div-inv 98.9%

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

      2. associate-+l+ 98.9%

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

      3. +-commutative 98.9%

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

      4. +-commutative 98.9%

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

   4. Applied egg-rr 98.9%

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

4. Final simplification 96.9%

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

Alternative 4: 86.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 6.8 \cdot 10^{+90}:\\ \;\;\;\;\frac{\beta \cdot \frac{1}{\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 6.8e+90)
   (/ (+ (* beta (/ 1.0 (+ (+ alpha beta) (+ 2.0 (* 2.0 i))))) 1.0) 2.0)
   (/ (/ (+ 2.0 (* i 4.0)) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 6.8e+90) {
		tmp = ((beta * (1.0 / ((alpha + beta) + (2.0 + (2.0 * i))))) + 1.0) / 2.0;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 6.8d+90) then
        tmp = ((beta * (1.0d0 / ((alpha + beta) + (2.0d0 + (2.0d0 * i))))) + 1.0d0) / 2.0d0
    else
        tmp = ((2.0d0 + (i * 4.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 6.8e+90) {
		tmp = ((beta * (1.0 / ((alpha + beta) + (2.0 + (2.0 * i))))) + 1.0) / 2.0;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 6.8e+90:
		tmp = ((beta * (1.0 / ((alpha + beta) + (2.0 + (2.0 * i))))) + 1.0) / 2.0
	else:
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 6.8e+90)
		tmp = Float64(Float64(Float64(beta * Float64(1.0 / Float64(Float64(alpha + beta) + Float64(2.0 + Float64(2.0 * i))))) + 1.0) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(i * 4.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 6.8e+90)
		tmp = ((beta * (1.0 / ((alpha + beta) + (2.0 + (2.0 * i))))) + 1.0) / 2.0;
	else
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 6.8e+90], N[(N[(N[(beta * N[(1.0 / N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 6.80000000000000036e90

   1. Initial program 82.6%

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

   2. Taylor expanded in beta around inf 96.7%

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

      1. div-inv 96.8%

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

      2. associate-+l+ 96.8%

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

      3. +-commutative 96.8%

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

      4. +-commutative 96.8%

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

   4. Applied egg-rr 96.8%

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

   if 6.80000000000000036e90 < alpha

   1. Initial program 6.3%

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

      1. associate-/l/ 5.4%

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

      2. *-commutative 5.4%

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

      3. times-frac 23.8%

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

      4. associate-+l+ 23.8%

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

      5. fma-def 23.8%

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

      6. +-commutative 23.8%

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

      7. fma-def 23.8%

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

   3. Simplified 23.8%

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

   4. Taylor expanded in beta around 0 5.4%

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

      1. mul-1-neg 5.4%

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

      2. unsub-neg 5.4%

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

      3. unpow2 5.4%

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

      4. associate-+r+ 5.4%

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

      5. +-commutative 5.4%

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

   6. Simplified 5.4%

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

   7. Taylor expanded in alpha around inf 64.6%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 6.8 \cdot 10^{+90}:\\ \;\;\;\;\frac{\beta \cdot \frac{1}{\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 5: 88.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 7.6 \cdot 10^{+92}:\\ \;\;\;\;\frac{\beta \cdot \frac{1}{\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0}{\alpha} + \frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 7.6e+92)
   (/ (+ (* beta (/ 1.0 (+ (+ alpha beta) (+ 2.0 (* 2.0 i))))) 1.0) 2.0)
   (/ (+ (/ 0.0 alpha) (/ (+ (* i 4.0) (+ 2.0 (* beta 2.0))) alpha)) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 7.6e+92) {
		tmp = ((beta * (1.0 / ((alpha + beta) + (2.0 + (2.0 * i))))) + 1.0) / 2.0;
	} else {
		tmp = ((0.0 / alpha) + (((i * 4.0) + (2.0 + (beta * 2.0))) / alpha)) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 7.6d+92) then
        tmp = ((beta * (1.0d0 / ((alpha + beta) + (2.0d0 + (2.0d0 * i))))) + 1.0d0) / 2.0d0
    else
        tmp = ((0.0d0 / alpha) + (((i * 4.0d0) + (2.0d0 + (beta * 2.0d0))) / alpha)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 7.6e+92) {
		tmp = ((beta * (1.0 / ((alpha + beta) + (2.0 + (2.0 * i))))) + 1.0) / 2.0;
	} else {
		tmp = ((0.0 / alpha) + (((i * 4.0) + (2.0 + (beta * 2.0))) / alpha)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 7.6e+92:
		tmp = ((beta * (1.0 / ((alpha + beta) + (2.0 + (2.0 * i))))) + 1.0) / 2.0
	else:
		tmp = ((0.0 / alpha) + (((i * 4.0) + (2.0 + (beta * 2.0))) / alpha)) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 7.6e+92)
		tmp = Float64(Float64(Float64(beta * Float64(1.0 / Float64(Float64(alpha + beta) + Float64(2.0 + Float64(2.0 * i))))) + 1.0) / 2.0);
	else
		tmp = Float64(Float64(Float64(0.0 / alpha) + Float64(Float64(Float64(i * 4.0) + Float64(2.0 + Float64(beta * 2.0))) / alpha)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 7.6e+92)
		tmp = ((beta * (1.0 / ((alpha + beta) + (2.0 + (2.0 * i))))) + 1.0) / 2.0;
	else
		tmp = ((0.0 / alpha) + (((i * 4.0) + (2.0 + (beta * 2.0))) / alpha)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 7.6e+92], N[(N[(N[(beta * N[(1.0 / N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(0.0 / alpha), $MachinePrecision] + N[(N[(N[(i * 4.0), $MachinePrecision] + N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 7.6000000000000001e92

   1. Initial program 82.6%

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

   2. Taylor expanded in beta around inf 96.7%

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

      1. div-inv 96.8%

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

      2. associate-+l+ 96.8%

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

      3. +-commutative 96.8%

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

      4. +-commutative 96.8%

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

   4. Applied egg-rr 96.8%

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

   if 7.6000000000000001e92 < alpha

   1. Initial program 6.3%

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

   3. Simplified 17.8%

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

   4. Taylor expanded in alpha around inf 76.8%

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

   5. Taylor expanded in alpha around inf 82.0%

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

      1. distribute-lft1-in 82.0%

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

      2. metadata-eval 82.0%

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

      3. mul0-lft 82.0%

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

   7. Simplified 82.0%

      \[\leadsto \frac{\color{blue}{\frac{0}{\alpha}} - -1 \cdot \frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 7.6 \cdot 10^{+92}:\\ \;\;\;\;\frac{\beta \cdot \frac{1}{\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0}{\alpha} + \frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 6: 85.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 4.8 \cdot 10^{+92}:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 4.8e+92)
   (/ (+ (/ beta (+ 2.0 (+ (+ alpha beta) (* 2.0 i)))) 1.0) 2.0)
   (/ (/ (+ 2.0 (* i 4.0)) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 4.8e+92) {
		tmp = ((beta / (2.0 + ((alpha + beta) + (2.0 * i)))) + 1.0) / 2.0;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 4.8d+92) then
        tmp = ((beta / (2.0d0 + ((alpha + beta) + (2.0d0 * i)))) + 1.0d0) / 2.0d0
    else
        tmp = ((2.0d0 + (i * 4.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 4.8e+92) {
		tmp = ((beta / (2.0 + ((alpha + beta) + (2.0 * i)))) + 1.0) / 2.0;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 4.8e+92:
		tmp = ((beta / (2.0 + ((alpha + beta) + (2.0 * i)))) + 1.0) / 2.0
	else:
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 4.8e+92)
		tmp = Float64(Float64(Float64(beta / Float64(2.0 + Float64(Float64(alpha + beta) + Float64(2.0 * i)))) + 1.0) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(i * 4.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 4.8e+92)
		tmp = ((beta / (2.0 + ((alpha + beta) + (2.0 * i)))) + 1.0) / 2.0;
	else
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 4.8e+92], N[(N[(N[(beta / N[(2.0 + N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 4.80000000000000009e92

   1. Initial program 82.6%

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

   2. Taylor expanded in beta around inf 96.7%

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

   if 4.80000000000000009e92 < alpha

   1. Initial program 6.3%

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

      1. associate-/l/ 5.4%

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

      2. *-commutative 5.4%

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

      3. times-frac 23.8%

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

      4. associate-+l+ 23.8%

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

      5. fma-def 23.8%

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

      6. +-commutative 23.8%

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

      7. fma-def 23.8%

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

   3. Simplified 23.8%

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

   4. Taylor expanded in beta around 0 5.4%

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

      1. mul-1-neg 5.4%

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

      2. unsub-neg 5.4%

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

      3. unpow2 5.4%

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

      4. associate-+r+ 5.4%

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

      5. +-commutative 5.4%

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

   6. Simplified 5.4%

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

   7. Taylor expanded in alpha around inf 64.6%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 4.8 \cdot 10^{+92}:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 7: 80.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2 \cdot 10^{+96}:\\ \;\;\;\;\frac{\beta \cdot \frac{1}{\beta + \left(\alpha + 2\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 2e+96)
   (/ (+ (* beta (/ 1.0 (+ beta (+ alpha 2.0)))) 1.0) 2.0)
   (/ (/ (+ 2.0 (* i 4.0)) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2e+96) {
		tmp = ((beta * (1.0 / (beta + (alpha + 2.0)))) + 1.0) / 2.0;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 2d+96) then
        tmp = ((beta * (1.0d0 / (beta + (alpha + 2.0d0)))) + 1.0d0) / 2.0d0
    else
        tmp = ((2.0d0 + (i * 4.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2e+96) {
		tmp = ((beta * (1.0 / (beta + (alpha + 2.0)))) + 1.0) / 2.0;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 2e+96:
		tmp = ((beta * (1.0 / (beta + (alpha + 2.0)))) + 1.0) / 2.0
	else:
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 2e+96)
		tmp = Float64(Float64(Float64(beta * Float64(1.0 / Float64(beta + Float64(alpha + 2.0)))) + 1.0) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(i * 4.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 2e+96)
		tmp = ((beta * (1.0 / (beta + (alpha + 2.0)))) + 1.0) / 2.0;
	else
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 2e+96], N[(N[(N[(beta * N[(1.0 / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.0000000000000001e96

   1. Initial program 82.6%

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

   2. Taylor expanded in beta around inf 96.7%

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

      1. div-inv 96.8%

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

      2. associate-+l+ 96.8%

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

      3. +-commutative 96.8%

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

      4. +-commutative 96.8%

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

   4. Applied egg-rr 96.8%

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

   5. Taylor expanded in i around 0 87.0%

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

   if 2.0000000000000001e96 < alpha

   1. Initial program 6.3%

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

      1. associate-/l/ 5.4%

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

      2. *-commutative 5.4%

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

      3. times-frac 23.8%

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

      4. associate-+l+ 23.8%

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

      5. fma-def 23.8%

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

      6. +-commutative 23.8%

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

      7. fma-def 23.8%

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

   3. Simplified 23.8%

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

   4. Taylor expanded in beta around 0 5.4%

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

      1. mul-1-neg 5.4%

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

      2. unsub-neg 5.4%

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

      3. unpow2 5.4%

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

      4. associate-+r+ 5.4%

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

      5. +-commutative 5.4%

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

   6. Simplified 5.4%

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

   7. Taylor expanded in alpha around inf 64.6%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2 \cdot 10^{+96}:\\ \;\;\;\;\frac{\beta \cdot \frac{1}{\beta + \left(\alpha + 2\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 8: 85.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.26 \cdot 10^{+95}:\\ \;\;\;\;\frac{\frac{\beta}{2 \cdot i + \left(\beta + 2\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1.26e+95)
   (/ (+ (/ beta (+ (* 2.0 i) (+ beta 2.0))) 1.0) 2.0)
   (/ (/ (+ 2.0 (* i 4.0)) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.26e+95) {
		tmp = ((beta / ((2.0 * i) + (beta + 2.0))) + 1.0) / 2.0;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 1.26d+95) then
        tmp = ((beta / ((2.0d0 * i) + (beta + 2.0d0))) + 1.0d0) / 2.0d0
    else
        tmp = ((2.0d0 + (i * 4.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.26e+95) {
		tmp = ((beta / ((2.0 * i) + (beta + 2.0))) + 1.0) / 2.0;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 1.26e+95:
		tmp = ((beta / ((2.0 * i) + (beta + 2.0))) + 1.0) / 2.0
	else:
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 1.26e+95)
		tmp = Float64(Float64(Float64(beta / Float64(Float64(2.0 * i) + Float64(beta + 2.0))) + 1.0) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(i * 4.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 1.26e+95)
		tmp = ((beta / ((2.0 * i) + (beta + 2.0))) + 1.0) / 2.0;
	else
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 1.26e+95], N[(N[(N[(beta / N[(N[(2.0 * i), $MachinePrecision] + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.26e95

   1. Initial program 82.6%

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

   2. Taylor expanded in beta around inf 96.7%

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

   3. Taylor expanded in alpha around 0 96.7%

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

      1. associate-+r+ 96.7%

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

   5. Simplified 96.7%

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

   if 1.26e95 < alpha

   1. Initial program 6.3%

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

      1. associate-/l/ 5.4%

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

      2. *-commutative 5.4%

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

      3. times-frac 23.8%

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

      4. associate-+l+ 23.8%

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

      5. fma-def 23.8%

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

      6. +-commutative 23.8%

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

      7. fma-def 23.8%

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

   3. Simplified 23.8%

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

   4. Taylor expanded in beta around 0 5.4%

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

      1. mul-1-neg 5.4%

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

      2. unsub-neg 5.4%

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

      3. unpow2 5.4%

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

      4. associate-+r+ 5.4%

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

      5. +-commutative 5.4%

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

   6. Simplified 5.4%

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

   7. Taylor expanded in alpha around inf 64.6%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.26 \cdot 10^{+95}:\\ \;\;\;\;\frac{\frac{\beta}{2 \cdot i + \left(\beta + 2\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 9: 80.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.25 \cdot 10^{+92}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1.25e+92)
   (/ (+ (/ beta (+ beta (+ alpha 2.0))) 1.0) 2.0)
   (/ (/ (+ 2.0 (* i 4.0)) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.25e+92) {
		tmp = ((beta / (beta + (alpha + 2.0))) + 1.0) / 2.0;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 1.25d+92) then
        tmp = ((beta / (beta + (alpha + 2.0d0))) + 1.0d0) / 2.0d0
    else
        tmp = ((2.0d0 + (i * 4.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.25e+92) {
		tmp = ((beta / (beta + (alpha + 2.0))) + 1.0) / 2.0;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 1.25e+92:
		tmp = ((beta / (beta + (alpha + 2.0))) + 1.0) / 2.0
	else:
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 1.25e+92)
		tmp = Float64(Float64(Float64(beta / Float64(beta + Float64(alpha + 2.0))) + 1.0) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(i * 4.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 1.25e+92)
		tmp = ((beta / (beta + (alpha + 2.0))) + 1.0) / 2.0;
	else
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 1.25e+92], N[(N[(N[(beta / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.25000000000000005e92

   1. Initial program 82.6%

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

   2. Taylor expanded in beta around inf 96.7%

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

   3. Taylor expanded in i around 0 87.0%

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

   if 1.25000000000000005e92 < alpha

   1. Initial program 6.3%

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

      1. associate-/l/ 5.4%

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

      2. *-commutative 5.4%

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

      3. times-frac 23.8%

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

      4. associate-+l+ 23.8%

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

      5. fma-def 23.8%

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

      6. +-commutative 23.8%

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

      7. fma-def 23.8%

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

   3. Simplified 23.8%

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

   4. Taylor expanded in beta around 0 5.4%

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

      1. mul-1-neg 5.4%

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

      2. unsub-neg 5.4%

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

      3. unpow2 5.4%

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

      4. associate-+r+ 5.4%

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

      5. +-commutative 5.4%

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

   6. Simplified 5.4%

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

   7. Taylor expanded in alpha around inf 64.6%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.25 \cdot 10^{+92}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 10: 75.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 5.6 \cdot 10^{+96}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - -2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 5.6e+96)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (- beta -2.0) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 5.6e+96) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((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 <= 5.6d+96) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((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 <= 5.6e+96) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((beta - -2.0) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 5.6e+96:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((beta - -2.0) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 5.6e+96)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta - -2.0) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 5.6e+96)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((beta - -2.0) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 5.6e+96], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta - -2.0), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 5.59999999999999999e96

   1. Initial program 82.6%

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

   2. Taylor expanded in beta around inf 96.7%

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

   3. Taylor expanded in alpha around 0 96.7%

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

      1. associate-+r+ 96.7%

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

   5. Simplified 96.7%

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

   6. Taylor expanded in i around 0 86.6%

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

   if 5.59999999999999999e96 < alpha

   1. Initial program 6.3%

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

   2. Taylor expanded in alpha around inf 17.7%

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

      1. mul-1-neg 17.7%

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

   4. Simplified 17.7%

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

   5. Taylor expanded in alpha around inf 50.9%

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

      1. associate-+r+ 50.9%

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

   7. Simplified 50.9%

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

   8. Taylor expanded in i around 0 48.1%

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

      1. metadata-eval 48.1%

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

      2. sub-neg 48.1%

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

   10. Simplified 48.1%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 5.6 \cdot 10^{+96}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - -2}{\alpha}}{2}\\ \end{array} \]

Alternative 11: 80.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.4 \cdot 10^{+92}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1.4e+92)
   (/ (+ (/ beta (+ beta 2.0)) 1.0) 2.0)
   (/ (/ (+ 2.0 (* i 4.0)) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.4e+92) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 1.4d+92) then
        tmp = ((beta / (beta + 2.0d0)) + 1.0d0) / 2.0d0
    else
        tmp = ((2.0d0 + (i * 4.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.4e+92) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 1.4e+92:
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0
	else:
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 1.4e+92)
		tmp = Float64(Float64(Float64(beta / Float64(beta + 2.0)) + 1.0) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(i * 4.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 1.4e+92)
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	else
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 1.4e+92], N[(N[(N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.4e92

   1. Initial program 82.6%

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

   2. Taylor expanded in beta around inf 96.7%

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

   3. Taylor expanded in alpha around 0 96.7%

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

      1. associate-+r+ 96.7%

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

   5. Simplified 96.7%

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

   6. Taylor expanded in i around 0 86.6%

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

   if 1.4e92 < alpha

   1. Initial program 6.3%

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

      1. associate-/l/ 5.4%

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

      2. *-commutative 5.4%

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

      3. times-frac 23.8%

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

      4. associate-+l+ 23.8%

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

      5. fma-def 23.8%

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

      6. +-commutative 23.8%

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

      7. fma-def 23.8%

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

   3. Simplified 23.8%

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

   4. Taylor expanded in beta around 0 5.4%

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

      1. mul-1-neg 5.4%

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

      2. unsub-neg 5.4%

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

      3. unpow2 5.4%

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

      4. associate-+r+ 5.4%

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

      5. +-commutative 5.4%

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

   6. Simplified 5.4%

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

   7. Taylor expanded in alpha around inf 64.6%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.4 \cdot 10^{+92}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 12: 73.3% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.2 \cdot 10^{+18}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 (if (<= beta 5.2e+18) 0.5 1.0))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 5.2e+18) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 5.2d+18) then
        tmp = 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 5.2e+18) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 5.2e+18:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 5.2e+18)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 5.2e+18)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 5.2e+18], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 5.2e18

   1. Initial program 69.3%

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

      1. associate-/l/ 69.1%

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

      2. *-commutative 69.1%

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

      3. times-frac 73.4%

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

      4. associate-+l+ 73.4%

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

      5. fma-def 73.4%

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

      6. +-commutative 73.4%

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

      7. fma-def 73.4%

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

   3. Simplified 73.4%

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

   4. Taylor expanded in i around inf 70.2%

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

   if 5.2e18 < beta

   1. Initial program 47.3%

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

      1. associate-/l/ 45.9%

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

      2. *-commutative 45.9%

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

      3. times-frac 87.6%

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

      4. associate-+l+ 87.6%

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

      5. fma-def 87.6%

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

      6. +-commutative 87.6%

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

      7. fma-def 87.6%

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

   3. Simplified 87.6%

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

   4. Taylor expanded in beta around inf 67.5%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.2 \cdot 10^{+18}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 13: 61.9% 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 62.0%

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

   1. associate-/l/ 61.4%

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

   2. *-commutative 61.4%

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

   3. times-frac 78.1%

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

   4. associate-+l+ 78.1%

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

   5. fma-def 78.1%

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

   6. +-commutative 78.1%

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

   7. fma-def 78.1%

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

3. Simplified 78.1%

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

4. Taylor expanded in i around inf 58.9%

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

5. Final simplification 58.9%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023224 
(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))