Octave 3.8, jcobi/2

Percentage Accurate: 63.7% → 97.8%

Time: 18.9s

Alternatives: 11

Speedup: 9.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 63.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.9999998:\\ \;\;\;\;\frac{\frac{2 + 2 \cdot i}{\alpha} - \left(-2 \cdot \frac{i}{\alpha} + -2 \cdot \frac{\beta}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{\mathsf{fma}\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)}, \frac{\alpha + \beta}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, 1\right)}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (if (<=
        (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0))
        -0.9999998)
     (/
      (-
       (/ (+ 2.0 (* 2.0 i)) alpha)
       (+ (* -2.0 (/ i alpha)) (* -2.0 (/ beta alpha))))
      2.0)
     (/
      (log
       (exp
        (fma
         (/ (- beta alpha) (+ (+ alpha beta) (fma 2.0 i 2.0)))
         (/ (+ alpha beta) (+ alpha (fma 2.0 i 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)) <= -0.9999998) {
		tmp = (((2.0 + (2.0 * i)) / alpha) - ((-2.0 * (i / alpha)) + (-2.0 * (beta / alpha)))) / 2.0;
	} else {
		tmp = log(exp(fma(((beta - alpha) / ((alpha + beta) + fma(2.0, i, 2.0))), ((alpha + beta) / (alpha + fma(2.0, i, 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)) <= -0.9999998)
		tmp = Float64(Float64(Float64(Float64(2.0 + Float64(2.0 * i)) / alpha) - Float64(Float64(-2.0 * Float64(i / alpha)) + Float64(-2.0 * Float64(beta / alpha)))) / 2.0);
	else
		tmp = Float64(log(exp(fma(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + fma(2.0, i, 2.0))), Float64(Float64(alpha + beta) / Float64(alpha + fma(2.0, i, 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], -0.9999998], N[(N[(N[(N[(2.0 + N[(2.0 * i), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] - N[(N[(-2.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(beta / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[Log[N[Exp[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[(alpha + N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2)) < -0.999999799999999994

   1. Initial program 3.5%

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

   3. Simplified 18.6%

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

   4. Taylor expanded in alpha around -inf 85.6%

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

   5. Taylor expanded in beta around inf 85.8%

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

   if -0.999999799999999994 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2))

   1. Initial program 83.9%

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

      1. associate-/l/ 83.4%

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

      2. *-commutative 83.4%

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

      3. times-frac 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%

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

      1. frac-times 83.4%

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

      2. *-commutative 83.4%

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

      3. fma-def 83.4%

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

      4. fma-udef 83.4%

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

      5. +-commutative 83.4%

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

      6. associate-+r+ 83.4%

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

      7. add-log-exp 83.4%

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

   5. Applied egg-rr 99.8%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.9999998:\\ \;\;\;\;\frac{\frac{2 + 2 \cdot i}{\alpha} - \left(-2 \cdot \frac{i}{\alpha} + -2 \cdot \frac{\beta}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{\mathsf{fma}\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)}, \frac{\alpha + \beta}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, 1\right)}\right)}{2}\\ \end{array} \]

Alternative 2: 97.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.9999998:\\ \;\;\;\;\frac{\frac{2 + 2 \cdot i}{\alpha} - \left(-2 \cdot \frac{i}{\alpha} + -2 \cdot \frac{\beta}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{2}\\ \end{array} \end{array} \]

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))
        -0.9999998)
     (/
      (-
       (/ (+ 2.0 (* 2.0 i)) alpha)
       (+ (* -2.0 (/ i alpha)) (* -2.0 (/ beta alpha))))
      2.0)
     (/
      (+
       1.0
       (*
        (/ (- beta alpha) (+ (+ alpha beta) (fma 2.0 i 2.0)))
        (/ (+ alpha beta) (fma 2.0 i (+ alpha beta)))))
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.9999998) {
		tmp = (((2.0 + (2.0 * i)) / alpha) - ((-2.0 * (i / alpha)) + (-2.0 * (beta / alpha)))) / 2.0;
	} else {
		tmp = (1.0 + (((beta - alpha) / ((alpha + beta) + fma(2.0, i, 2.0))) * ((alpha + beta) / fma(2.0, i, (alpha + beta))))) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0)) <= -0.9999998)
		tmp = Float64(Float64(Float64(Float64(2.0 + Float64(2.0 * i)) / alpha) - Float64(Float64(-2.0 * Float64(i / alpha)) + Float64(-2.0 * Float64(beta / alpha)))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + 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))))) / 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], -0.9999998], N[(N[(N[(N[(2.0 + N[(2.0 * i), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] - N[(N[(-2.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(beta / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + 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]), $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)) < -0.999999799999999994

   1. Initial program 3.5%

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

   3. Simplified 18.6%

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

   4. Taylor expanded in alpha around -inf 85.6%

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

   5. Taylor expanded in beta around inf 85.8%

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

   if -0.999999799999999994 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2))

   1. Initial program 83.9%

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

      1. associate-/l/ 83.4%

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

      2. *-commutative 83.4%

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

      3. times-frac 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 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.9999998:\\ \;\;\;\;\frac{\frac{2 + 2 \cdot i}{\alpha} - \left(-2 \cdot \frac{i}{\alpha} + -2 \cdot \frac{\beta}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{2}\\ \end{array} \]

Alternative 3: 96.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0}\\ \mathbf{if}\;t_1 \leq -0.9999998:\\ \;\;\;\;\frac{\frac{2 + 2 \cdot i}{\alpha} - \left(-2 \cdot \frac{i}{\alpha} + -2 \cdot \frac{\beta}{\alpha}\right)}{2}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{t_1 + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0))))
   (if (<= t_1 -0.9999998)
     (/
      (-
       (/ (+ 2.0 (* 2.0 i)) alpha)
       (+ (* -2.0 (/ i alpha)) (* -2.0 (/ beta alpha))))
      2.0)
     (if (<= t_1 2e-5)
       (/ (+ t_1 1.0) 2.0)
       (/ (+ 1.0 (/ (- beta alpha) (+ beta (+ alpha 2.0)))) 2.0)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0);
	double tmp;
	if (t_1 <= -0.9999998) {
		tmp = (((2.0 + (2.0 * i)) / alpha) - ((-2.0 * (i / alpha)) + (-2.0 * (beta / alpha)))) / 2.0;
	} else if (t_1 <= 2e-5) {
		tmp = (t_1 + 1.0) / 2.0;
	} else {
		tmp = (1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0d0 + t_0)
    if (t_1 <= (-0.9999998d0)) then
        tmp = (((2.0d0 + (2.0d0 * i)) / alpha) - (((-2.0d0) * (i / alpha)) + ((-2.0d0) * (beta / alpha)))) / 2.0d0
    else if (t_1 <= 2d-5) then
        tmp = (t_1 + 1.0d0) / 2.0d0
    else
        tmp = (1.0d0 + ((beta - alpha) / (beta + (alpha + 2.0d0)))) / 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 t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0);
	double tmp;
	if (t_1 <= -0.9999998) {
		tmp = (((2.0 + (2.0 * i)) / alpha) - ((-2.0 * (i / alpha)) + (-2.0 * (beta / alpha)))) / 2.0;
	} else if (t_1 <= 2e-5) {
		tmp = (t_1 + 1.0) / 2.0;
	} else {
		tmp = (1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)
	tmp = 0
	if t_1 <= -0.9999998:
		tmp = (((2.0 + (2.0 * i)) / alpha) - ((-2.0 * (i / alpha)) + (-2.0 * (beta / alpha)))) / 2.0
	elif t_1 <= 2e-5:
		tmp = (t_1 + 1.0) / 2.0
	else:
		tmp = (1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0))
	tmp = 0.0
	if (t_1 <= -0.9999998)
		tmp = Float64(Float64(Float64(Float64(2.0 + Float64(2.0 * i)) / alpha) - Float64(Float64(-2.0 * Float64(i / alpha)) + Float64(-2.0 * Float64(beta / alpha)))) / 2.0);
	elseif (t_1 <= 2e-5)
		tmp = Float64(Float64(t_1 + 1.0) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / Float64(beta + Float64(alpha + 2.0)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0);
	tmp = 0.0;
	if (t_1 <= -0.9999998)
		tmp = (((2.0 + (2.0 * i)) / alpha) - ((-2.0 * (i / alpha)) + (-2.0 * (beta / alpha)))) / 2.0;
	elseif (t_1 <= 2e-5)
		tmp = (t_1 + 1.0) / 2.0;
	else
		tmp = (1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))) / 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]}, Block[{t$95$1 = N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.9999998], N[(N[(N[(N[(2.0 + N[(2.0 * i), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] - N[(N[(-2.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(beta / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[t$95$1, 2e-5], N[(N[(t$95$1 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2)) < -0.999999799999999994

   1. Initial program 3.5%

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

   3. Simplified 18.6%

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

   4. Taylor expanded in alpha around -inf 85.6%

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

   5. Taylor expanded in beta around inf 85.8%

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

   if -0.999999799999999994 < (/.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)) < 2.00000000000000016e-5

   1. Initial program 99.6%

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

   if 2.00000000000000016e-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 39.7%

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

      1. associate-/l/ 37.7%

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

      2. *-commutative 37.7%

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

      3. times-frac 100.0%

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

      4. associate-+l+ 100.0%

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

      5. fma-def 100.0%

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

      6. +-commutative 100.0%

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

      7. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in i around 0 98.5%

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

      1. +-commutative 98.5%

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

   6. Simplified 98.5%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.9999998:\\ \;\;\;\;\frac{\frac{2 + 2 \cdot i}{\alpha} - \left(-2 \cdot \frac{i}{\alpha} + -2 \cdot \frac{\beta}{\alpha}\right)}{2}\\ \mathbf{elif}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}}{2}\\ \end{array} \]

Alternative 4: 82.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 5 \cdot 10^{-57}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 5.2 \cdot 10^{+24}:\\ \;\;\;\;\frac{1 - \frac{\alpha \cdot \alpha}{\left(\alpha + 2 \cdot i\right) \cdot \left(2 \cdot i + \left(\alpha + 2\right)\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 9.5 \cdot 10^{+98}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + \left(2 + i \cdot 4\right)\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 5e-57)
   (/ (+ 1.0 (/ (- beta alpha) (+ beta (+ alpha 2.0)))) 2.0)
   (if (<= alpha 5.2e+24)
     (/
      (-
       1.0
       (/ (* alpha alpha) (* (+ alpha (* 2.0 i)) (+ (* 2.0 i) (+ alpha 2.0)))))
      2.0)
     (if (<= alpha 9.5e+98)
       (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
       (/ (/ (+ beta (+ beta (+ 2.0 (* i 4.0)))) alpha) 2.0)))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 5e-57) {
		tmp = (1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))) / 2.0;
	} else if (alpha <= 5.2e+24) {
		tmp = (1.0 - ((alpha * alpha) / ((alpha + (2.0 * i)) * ((2.0 * i) + (alpha + 2.0))))) / 2.0;
	} else if (alpha <= 9.5e+98) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((beta + (beta + (2.0 + (i * 4.0)))) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 5d-57) then
        tmp = (1.0d0 + ((beta - alpha) / (beta + (alpha + 2.0d0)))) / 2.0d0
    else if (alpha <= 5.2d+24) then
        tmp = (1.0d0 - ((alpha * alpha) / ((alpha + (2.0d0 * i)) * ((2.0d0 * i) + (alpha + 2.0d0))))) / 2.0d0
    else if (alpha <= 9.5d+98) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((beta + (beta + (2.0d0 + (i * 4.0d0)))) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 5e-57) {
		tmp = (1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))) / 2.0;
	} else if (alpha <= 5.2e+24) {
		tmp = (1.0 - ((alpha * alpha) / ((alpha + (2.0 * i)) * ((2.0 * i) + (alpha + 2.0))))) / 2.0;
	} else if (alpha <= 9.5e+98) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((beta + (beta + (2.0 + (i * 4.0)))) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 5e-57:
		tmp = (1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))) / 2.0
	elif alpha <= 5.2e+24:
		tmp = (1.0 - ((alpha * alpha) / ((alpha + (2.0 * i)) * ((2.0 * i) + (alpha + 2.0))))) / 2.0
	elif alpha <= 9.5e+98:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((beta + (beta + (2.0 + (i * 4.0)))) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 5e-57)
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / Float64(beta + Float64(alpha + 2.0)))) / 2.0);
	elseif (alpha <= 5.2e+24)
		tmp = Float64(Float64(1.0 - Float64(Float64(alpha * alpha) / Float64(Float64(alpha + Float64(2.0 * i)) * Float64(Float64(2.0 * i) + Float64(alpha + 2.0))))) / 2.0);
	elseif (alpha <= 9.5e+98)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta + Float64(beta + Float64(2.0 + Float64(i * 4.0)))) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 5e-57)
		tmp = (1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))) / 2.0;
	elseif (alpha <= 5.2e+24)
		tmp = (1.0 - ((alpha * alpha) / ((alpha + (2.0 * i)) * ((2.0 * i) + (alpha + 2.0))))) / 2.0;
	elseif (alpha <= 9.5e+98)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((beta + (beta + (2.0 + (i * 4.0)))) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 5e-57], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 5.2e+24], N[(N[(1.0 - N[(N[(alpha * alpha), $MachinePrecision] / N[(N[(alpha + N[(2.0 * i), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * i), $MachinePrecision] + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 9.5e+98], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta + N[(beta + N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if alpha < 5.0000000000000002e-57

   1. Initial program 84.8%

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

      1. associate-/l/ 84.3%

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

      2. *-commutative 84.3%

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

      3. times-frac 100.0%

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

      4. associate-+l+ 100.0%

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

      5. fma-def 100.0%

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

      6. +-commutative 100.0%

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

      7. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in i around 0 94.9%

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

      1. +-commutative 94.9%

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

   6. Simplified 94.9%

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

   if 5.0000000000000002e-57 < alpha < 5.1999999999999997e24

   1. Initial program 91.6%

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

      1. associate-/l/ 91.4%

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

      2. *-commutative 91.4%

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

      3. times-frac 95.5%

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

      4. associate-+l+ 95.5%

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

      5. fma-def 95.5%

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

      6. +-commutative 95.5%

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

      7. fma-def 95.5%

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

   3. Simplified 95.5%

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

   4. Taylor expanded in beta around 0 95.4%

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

      1. associate-*r/ 95.4%

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

      2. mul-1-neg 95.4%

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

      3. unpow2 95.4%

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

      4. associate-+r+ 95.4%

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

      5. +-commutative 95.4%

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

   6. Simplified 95.4%

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

   if 5.1999999999999997e24 < alpha < 9.5000000000000001e98

   1. Initial program 30.8%

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

      1. associate-/l/ 29.4%

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

      2. *-commutative 29.4%

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

      3. times-frac 69.6%

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

      4. associate-+l+ 69.6%

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

      5. fma-def 69.6%

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

      6. +-commutative 69.6%

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

      7. fma-def 69.6%

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

   3. Simplified 69.6%

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

      1. frac-times 29.4%

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

      2. *-commutative 29.4%

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

      3. fma-def 29.4%

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

      4. fma-udef 29.4%

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

      5. +-commutative 29.4%

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

      6. associate-+r+ 29.4%

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

      7. add-log-exp 29.4%

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

   5. Applied egg-rr 70.2%

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

   6. Taylor expanded in i around 0 52.4%

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

      1. associate--l+ 52.4%

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

      2. div-sub 52.4%

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

      3. +-commutative 52.4%

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

   8. Simplified 52.4%

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

   9. Taylor expanded in alpha around 0 55.3%

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

   if 9.5000000000000001e98 < alpha

   1. Initial program 8.2%

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

   2. Taylor expanded in alpha around inf 5.2%

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

   3. Taylor expanded in alpha around -inf 77.7%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 5 \cdot 10^{-57}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 5.2 \cdot 10^{+24}:\\ \;\;\;\;\frac{1 - \frac{\alpha \cdot \alpha}{\left(\alpha + 2 \cdot i\right) \cdot \left(2 \cdot i + \left(\alpha + 2\right)\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 9.5 \cdot 10^{+98}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + \left(2 + i \cdot 4\right)\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 5: 78.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ t_1 := \frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{if}\;\alpha \leq 2.9 \cdot 10^{-53}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 1.7 \cdot 10^{+30}:\\ \;\;\;\;\frac{1 - \frac{\alpha}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 6.8 \cdot 10^{+96}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\alpha \leq 1.6 \cdot 10^{+118}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 1.35 \cdot 10^{+137}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\alpha \leq 10^{+227}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (/ (/ (+ 2.0 (* i 4.0)) alpha) 2.0))
        (t_1 (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)))
   (if (<= alpha 2.9e-53)
     (/ (+ 1.0 (/ (- beta alpha) (+ beta (+ alpha 2.0)))) 2.0)
     (if (<= alpha 1.7e+30)
       (/ (- 1.0 (/ alpha (+ 2.0 (+ (+ alpha beta) (* 2.0 i))))) 2.0)
       (if (<= alpha 6.8e+96)
         t_1
         (if (<= alpha 1.6e+118)
           t_0
           (if (<= alpha 1.35e+137)
             t_1
             (if (<= alpha 1e+227)
               (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)
               t_0))))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	double t_1 = (1.0 + (beta / (beta + 2.0))) / 2.0;
	double tmp;
	if (alpha <= 2.9e-53) {
		tmp = (1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))) / 2.0;
	} else if (alpha <= 1.7e+30) {
		tmp = (1.0 - (alpha / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else if (alpha <= 6.8e+96) {
		tmp = t_1;
	} else if (alpha <= 1.6e+118) {
		tmp = t_0;
	} else if (alpha <= 1.35e+137) {
		tmp = t_1;
	} else if (alpha <= 1e+227) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((2.0d0 + (i * 4.0d0)) / alpha) / 2.0d0
    t_1 = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    if (alpha <= 2.9d-53) then
        tmp = (1.0d0 + ((beta - alpha) / (beta + (alpha + 2.0d0)))) / 2.0d0
    else if (alpha <= 1.7d+30) then
        tmp = (1.0d0 - (alpha / (2.0d0 + ((alpha + beta) + (2.0d0 * i))))) / 2.0d0
    else if (alpha <= 6.8d+96) then
        tmp = t_1
    else if (alpha <= 1.6d+118) then
        tmp = t_0
    else if (alpha <= 1.35d+137) then
        tmp = t_1
    else if (alpha <= 1d+227) then
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	double t_1 = (1.0 + (beta / (beta + 2.0))) / 2.0;
	double tmp;
	if (alpha <= 2.9e-53) {
		tmp = (1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))) / 2.0;
	} else if (alpha <= 1.7e+30) {
		tmp = (1.0 - (alpha / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else if (alpha <= 6.8e+96) {
		tmp = t_1;
	} else if (alpha <= 1.6e+118) {
		tmp = t_0;
	} else if (alpha <= 1.35e+137) {
		tmp = t_1;
	} else if (alpha <= 1e+227) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = ((2.0 + (i * 4.0)) / alpha) / 2.0
	t_1 = (1.0 + (beta / (beta + 2.0))) / 2.0
	tmp = 0
	if alpha <= 2.9e-53:
		tmp = (1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))) / 2.0
	elif alpha <= 1.7e+30:
		tmp = (1.0 - (alpha / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0
	elif alpha <= 6.8e+96:
		tmp = t_1
	elif alpha <= 1.6e+118:
		tmp = t_0
	elif alpha <= 1.35e+137:
		tmp = t_1
	elif alpha <= 1e+227:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	else:
		tmp = t_0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(Float64(2.0 + Float64(i * 4.0)) / alpha) / 2.0)
	t_1 = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0)
	tmp = 0.0
	if (alpha <= 2.9e-53)
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / Float64(beta + Float64(alpha + 2.0)))) / 2.0);
	elseif (alpha <= 1.7e+30)
		tmp = Float64(Float64(1.0 - Float64(alpha / Float64(2.0 + Float64(Float64(alpha + beta) + Float64(2.0 * i))))) / 2.0);
	elseif (alpha <= 6.8e+96)
		tmp = t_1;
	elseif (alpha <= 1.6e+118)
		tmp = t_0;
	elseif (alpha <= 1.35e+137)
		tmp = t_1;
	elseif (alpha <= 1e+227)
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	t_1 = (1.0 + (beta / (beta + 2.0))) / 2.0;
	tmp = 0.0;
	if (alpha <= 2.9e-53)
		tmp = (1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))) / 2.0;
	elseif (alpha <= 1.7e+30)
		tmp = (1.0 - (alpha / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	elseif (alpha <= 6.8e+96)
		tmp = t_1;
	elseif (alpha <= 1.6e+118)
		tmp = t_0;
	elseif (alpha <= 1.35e+137)
		tmp = t_1;
	elseif (alpha <= 1e+227)
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[alpha, 2.9e-53], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 1.7e+30], N[(N[(1.0 - N[(alpha / N[(2.0 + N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 6.8e+96], t$95$1, If[LessEqual[alpha, 1.6e+118], t$95$0, If[LessEqual[alpha, 1.35e+137], t$95$1, If[LessEqual[alpha, 1e+227], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], t$95$0]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if alpha < 2.8999999999999998e-53

   1. Initial program 84.8%

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

      1. associate-/l/ 84.3%

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

      2. *-commutative 84.3%

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

      3. times-frac 100.0%

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

      4. associate-+l+ 100.0%

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

      5. fma-def 100.0%

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

      6. +-commutative 100.0%

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

      7. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in i around 0 94.9%

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

      1. +-commutative 94.9%

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

   6. Simplified 94.9%

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

   if 2.8999999999999998e-53 < alpha < 1.7000000000000001e30

   1. Initial program 91.3%

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

   2. Taylor expanded in alpha around inf 91.5%

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

      1. mul-1-neg 91.5%

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

   4. Simplified 91.5%

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

   if 1.7000000000000001e30 < alpha < 6.8000000000000002e96 or 1.60000000000000008e118 < alpha < 1.35000000000000009e137

   1. Initial program 33.3%

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

      1. associate-/l/ 32.1%

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

      2. *-commutative 32.1%

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

      3. times-frac 72.5%

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

      4. associate-+l+ 72.5%

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

      5. fma-def 72.5%

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

      6. +-commutative 72.5%

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

      7. fma-def 72.5%

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

   3. Simplified 72.5%

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

      1. frac-times 32.1%

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

      2. *-commutative 32.1%

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

      3. fma-def 32.1%

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

      4. fma-udef 32.1%

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

      5. +-commutative 32.1%

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

      6. associate-+r+ 32.1%

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

      7. add-log-exp 32.1%

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

   5. Applied egg-rr 72.1%

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

   6. Taylor expanded in i around 0 43.0%

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

      1. associate--l+ 43.0%

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

      2. div-sub 43.0%

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

      3. +-commutative 43.0%

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

   8. Simplified 43.0%

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

   9. Taylor expanded in alpha around 0 63.9%

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

   if 6.8000000000000002e96 < alpha < 1.60000000000000008e118 or 1.0000000000000001e227 < alpha

   1. Initial program 5.5%

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

      1. associate-/l/ 4.7%

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

      2. *-commutative 4.7%

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

      3. times-frac 16.2%

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

      4. associate-+l+ 16.2%

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

      5. fma-def 16.2%

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

      6. +-commutative 16.2%

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

      7. fma-def 16.2%

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

   3. Simplified 16.2%

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

   4. Taylor expanded in beta around 0 4.7%

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

      1. associate-*r/ 4.7%

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

      2. mul-1-neg 4.7%

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

      3. unpow2 4.7%

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

      4. associate-+r+ 4.7%

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

      5. +-commutative 4.7%

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

   6. Simplified 4.7%

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

   7. Taylor expanded in alpha around inf 89.8%

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

   if 1.35000000000000009e137 < alpha < 1.0000000000000001e227

   1. Initial program 1.2%

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

      1. associate-/l/ 0.4%

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

      2. *-commutative 0.4%

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

      3. times-frac 31.0%

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

      4. associate-+l+ 31.0%

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

      5. fma-def 31.0%

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

      6. +-commutative 31.0%

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

      7. fma-def 31.0%

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

   3. Simplified 31.0%

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

      1. frac-times 0.4%

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

      2. *-commutative 0.4%

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

      3. fma-def 0.4%

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

      4. fma-udef 0.4%

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

      5. +-commutative 0.4%

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

      6. associate-+r+ 0.4%

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

      7. add-log-exp 0.4%

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

   5. Applied egg-rr 31.0%

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

   6. Taylor expanded in i around 0 6.7%

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

      1. associate--l+ 6.7%

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

      2. div-sub 6.7%

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

      3. +-commutative 6.7%

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

   8. Simplified 6.7%

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

   9. Taylor expanded in alpha around inf 59.7%

      \[\leadsto \frac{\color{blue}{\frac{2 + 2 \cdot \beta}{\alpha}}}{2} \]
3. Recombined 5 regimes into one program.

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.9 \cdot 10^{-53}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 1.7 \cdot 10^{+30}:\\ \;\;\;\;\frac{1 - \frac{\alpha}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 6.8 \cdot 10^{+96}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 1.6 \cdot 10^{+118}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 1.35 \cdot 10^{+137}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 10^{+227}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 6: 82.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 4.5 \cdot 10^{-53}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 1.7 \cdot 10^{+30}:\\ \;\;\;\;\frac{1 - \frac{\alpha}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 8.5 \cdot 10^{+97}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + \left(2 + i \cdot 4\right)\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 4.5e-53)
   (/ (+ 1.0 (/ (- beta alpha) (+ beta (+ alpha 2.0)))) 2.0)
   (if (<= alpha 1.7e+30)
     (/ (- 1.0 (/ alpha (+ 2.0 (+ (+ alpha beta) (* 2.0 i))))) 2.0)
     (if (<= alpha 8.5e+97)
       (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
       (/ (/ (+ beta (+ beta (+ 2.0 (* i 4.0)))) alpha) 2.0)))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 4.5e-53) {
		tmp = (1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))) / 2.0;
	} else if (alpha <= 1.7e+30) {
		tmp = (1.0 - (alpha / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else if (alpha <= 8.5e+97) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((beta + (beta + (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 <= 4.5d-53) then
        tmp = (1.0d0 + ((beta - alpha) / (beta + (alpha + 2.0d0)))) / 2.0d0
    else if (alpha <= 1.7d+30) then
        tmp = (1.0d0 - (alpha / (2.0d0 + ((alpha + beta) + (2.0d0 * i))))) / 2.0d0
    else if (alpha <= 8.5d+97) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((beta + (beta + (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 <= 4.5e-53) {
		tmp = (1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))) / 2.0;
	} else if (alpha <= 1.7e+30) {
		tmp = (1.0 - (alpha / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else if (alpha <= 8.5e+97) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((beta + (beta + (2.0 + (i * 4.0)))) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 4.5e-53:
		tmp = (1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))) / 2.0
	elif alpha <= 1.7e+30:
		tmp = (1.0 - (alpha / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0
	elif alpha <= 8.5e+97:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((beta + (beta + (2.0 + (i * 4.0)))) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 4.5e-53)
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / Float64(beta + Float64(alpha + 2.0)))) / 2.0);
	elseif (alpha <= 1.7e+30)
		tmp = Float64(Float64(1.0 - Float64(alpha / Float64(2.0 + Float64(Float64(alpha + beta) + Float64(2.0 * i))))) / 2.0);
	elseif (alpha <= 8.5e+97)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta + Float64(beta + 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 <= 4.5e-53)
		tmp = (1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))) / 2.0;
	elseif (alpha <= 1.7e+30)
		tmp = (1.0 - (alpha / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	elseif (alpha <= 8.5e+97)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((beta + (beta + (2.0 + (i * 4.0)))) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 4.5e-53], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 1.7e+30], N[(N[(1.0 - N[(alpha / N[(2.0 + N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 8.5e+97], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta + N[(beta + N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if alpha < 4.49999999999999985e-53

   1. Initial program 84.8%

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

      1. associate-/l/ 84.3%

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

      2. *-commutative 84.3%

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

      3. times-frac 100.0%

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

      4. associate-+l+ 100.0%

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

      5. fma-def 100.0%

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

      6. +-commutative 100.0%

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

      7. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in i around 0 94.9%

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

      1. +-commutative 94.9%

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

   6. Simplified 94.9%

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

   if 4.49999999999999985e-53 < alpha < 1.7000000000000001e30

   1. Initial program 91.3%

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

   2. Taylor expanded in alpha around inf 91.5%

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

      1. mul-1-neg 91.5%

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

   4. Simplified 91.5%

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

   if 1.7000000000000001e30 < alpha < 8.4999999999999993e97

   1. Initial program 24.8%

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

      1. associate-/l/ 23.4%

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

      2. *-commutative 23.4%

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

      3. times-frac 67.9%

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

      4. associate-+l+ 67.9%

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

      5. fma-def 67.9%

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

      6. +-commutative 67.9%

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

      7. fma-def 67.9%

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

   3. Simplified 67.9%

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

      1. frac-times 23.4%

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

      2. *-commutative 23.4%

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

      3. fma-def 23.4%

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

      4. fma-udef 23.4%

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

      5. +-commutative 23.4%

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

      6. associate-+r+ 23.4%

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

      7. add-log-exp 23.4%

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

   5. Applied egg-rr 67.9%

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

   6. Taylor expanded in i around 0 48.1%

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

      1. associate--l+ 48.1%

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

      2. div-sub 48.1%

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

      3. +-commutative 48.1%

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

   8. Simplified 48.1%

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

   9. Taylor expanded in alpha around 0 59.9%

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

   if 8.4999999999999993e97 < alpha

   1. Initial program 8.2%

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

   2. Taylor expanded in alpha around inf 5.2%

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

   3. Taylor expanded in alpha around -inf 77.7%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 4.5 \cdot 10^{-53}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 1.7 \cdot 10^{+30}:\\ \;\;\;\;\frac{1 - \frac{\alpha}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 8.5 \cdot 10^{+97}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + \left(2 + i \cdot 4\right)\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 7: 79.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ t_1 := \frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{if}\;\alpha \leq 3.2 \cdot 10^{+97}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\alpha \leq 1.65 \cdot 10^{+118}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 4.4 \cdot 10^{+140}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\alpha \leq 4.8 \cdot 10^{+226}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (/ (/ (+ 2.0 (* i 4.0)) alpha) 2.0))
        (t_1 (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)))
   (if (<= alpha 3.2e+97)
     t_1
     (if (<= alpha 1.65e+118)
       t_0
       (if (<= alpha 4.4e+140)
         t_1
         (if (<= alpha 4.8e+226)
           (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)
           t_0))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	double t_1 = (1.0 + (beta / (beta + 2.0))) / 2.0;
	double tmp;
	if (alpha <= 3.2e+97) {
		tmp = t_1;
	} else if (alpha <= 1.65e+118) {
		tmp = t_0;
	} else if (alpha <= 4.4e+140) {
		tmp = t_1;
	} else if (alpha <= 4.8e+226) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((2.0d0 + (i * 4.0d0)) / alpha) / 2.0d0
    t_1 = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    if (alpha <= 3.2d+97) then
        tmp = t_1
    else if (alpha <= 1.65d+118) then
        tmp = t_0
    else if (alpha <= 4.4d+140) then
        tmp = t_1
    else if (alpha <= 4.8d+226) then
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	double t_1 = (1.0 + (beta / (beta + 2.0))) / 2.0;
	double tmp;
	if (alpha <= 3.2e+97) {
		tmp = t_1;
	} else if (alpha <= 1.65e+118) {
		tmp = t_0;
	} else if (alpha <= 4.4e+140) {
		tmp = t_1;
	} else if (alpha <= 4.8e+226) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = ((2.0 + (i * 4.0)) / alpha) / 2.0
	t_1 = (1.0 + (beta / (beta + 2.0))) / 2.0
	tmp = 0
	if alpha <= 3.2e+97:
		tmp = t_1
	elif alpha <= 1.65e+118:
		tmp = t_0
	elif alpha <= 4.4e+140:
		tmp = t_1
	elif alpha <= 4.8e+226:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	else:
		tmp = t_0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(Float64(2.0 + Float64(i * 4.0)) / alpha) / 2.0)
	t_1 = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0)
	tmp = 0.0
	if (alpha <= 3.2e+97)
		tmp = t_1;
	elseif (alpha <= 1.65e+118)
		tmp = t_0;
	elseif (alpha <= 4.4e+140)
		tmp = t_1;
	elseif (alpha <= 4.8e+226)
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	t_1 = (1.0 + (beta / (beta + 2.0))) / 2.0;
	tmp = 0.0;
	if (alpha <= 3.2e+97)
		tmp = t_1;
	elseif (alpha <= 1.65e+118)
		tmp = t_0;
	elseif (alpha <= 4.4e+140)
		tmp = t_1;
	elseif (alpha <= 4.8e+226)
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[alpha, 3.2e+97], t$95$1, If[LessEqual[alpha, 1.65e+118], t$95$0, If[LessEqual[alpha, 4.4e+140], t$95$1, If[LessEqual[alpha, 4.8e+226], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 3.20000000000000016e97 or 1.65e118 < alpha < 4.3999999999999997e140

   1. Initial program 81.9%

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

      1. associate-/l/ 81.4%

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

      2. *-commutative 81.4%

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

      3. times-frac 97.4%

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

      4. associate-+l+ 97.4%

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

      5. fma-def 97.4%

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

      6. +-commutative 97.4%

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

      7. fma-def 97.4%

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

   3. Simplified 97.4%

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

      1. frac-times 81.4%

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

      2. *-commutative 81.4%

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

      3. fma-def 81.4%

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

      4. fma-udef 81.4%

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

      5. +-commutative 81.4%

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

      6. associate-+r+ 81.4%

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

      7. add-log-exp 81.4%

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

   5. Applied egg-rr 97.4%

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

   6. Taylor expanded in i around 0 88.1%

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

      1. associate--l+ 88.1%

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

      2. div-sub 88.1%

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

      3. +-commutative 88.1%

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

   8. Simplified 88.1%

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

   9. Taylor expanded in alpha around 0 89.9%

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

   if 3.20000000000000016e97 < alpha < 1.65e118 or 4.8e226 < alpha

   1. Initial program 5.5%

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

      1. associate-/l/ 4.7%

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

      2. *-commutative 4.7%

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

      3. times-frac 16.2%

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

      4. associate-+l+ 16.2%

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

      5. fma-def 16.2%

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

      6. +-commutative 16.2%

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

      7. fma-def 16.2%

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

   3. Simplified 16.2%

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

   4. Taylor expanded in beta around 0 4.7%

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

      1. associate-*r/ 4.7%

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

      2. mul-1-neg 4.7%

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

      3. unpow2 4.7%

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

      4. associate-+r+ 4.7%

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

      5. +-commutative 4.7%

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

   6. Simplified 4.7%

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

   7. Taylor expanded in alpha around inf 89.8%

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

   if 4.3999999999999997e140 < alpha < 4.8e226

   1. Initial program 1.2%

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

      1. associate-/l/ 0.4%

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

      2. *-commutative 0.4%

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

      3. times-frac 31.0%

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

      4. associate-+l+ 31.0%

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

      5. fma-def 31.0%

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

      6. +-commutative 31.0%

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

      7. fma-def 31.0%

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

   3. Simplified 31.0%

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

      1. frac-times 0.4%

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

      2. *-commutative 0.4%

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

      3. fma-def 0.4%

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

      4. fma-udef 0.4%

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

      5. +-commutative 0.4%

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

      6. associate-+r+ 0.4%

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

      7. add-log-exp 0.4%

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

   5. Applied egg-rr 31.0%

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

   6. Taylor expanded in i around 0 6.7%

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

      1. associate--l+ 6.7%

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

      2. div-sub 6.7%

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

      3. +-commutative 6.7%

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

   8. Simplified 6.7%

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

   9. Taylor expanded in alpha around inf 59.7%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 3.2 \cdot 10^{+97}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 1.65 \cdot 10^{+118}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 4.4 \cdot 10^{+140}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 4.8 \cdot 10^{+226}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 8: 74.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.05 \cdot 10^{+161}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-i \cdot -4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 2.05e+161)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (- (* i -4.0)) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2.05e+161) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (-(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 <= 2.05d+161) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = (-(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 <= 2.05e+161) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (-(i * -4.0) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 2.05e+161:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = (-(i * -4.0) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 2.05e+161)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(-Float64(i * -4.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 2.05e+161)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = (-(i * -4.0) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 2.05e+161], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[((-N[(i * -4.0), $MachinePrecision]) / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.0500000000000001e161

   1. Initial program 77.7%

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

      1. associate-/l/ 77.2%

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

      2. *-commutative 77.2%

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

      3. times-frac 93.2%

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

      4. associate-+l+ 93.2%

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

      5. fma-def 93.2%

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

      6. +-commutative 93.2%

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

      7. fma-def 93.2%

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

   3. Simplified 93.2%

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

      1. frac-times 77.2%

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

      2. *-commutative 77.2%

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

      3. fma-def 77.2%

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

      4. fma-udef 77.2%

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

      5. +-commutative 77.2%

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

      6. associate-+r+ 77.2%

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

      7. add-log-exp 77.2%

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

   5. Applied egg-rr 93.2%

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

   6. Taylor expanded in i around 0 83.1%

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

      1. associate--l+ 83.1%

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

      2. div-sub 83.1%

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

      3. +-commutative 83.1%

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

   8. Simplified 83.1%

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

   9. Taylor expanded in alpha around 0 86.1%

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

   if 2.0500000000000001e161 < alpha

   1. Initial program 1.0%

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

   3. Simplified 18.5%

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

   4. Taylor expanded in alpha around -inf 85.0%

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

   5. Taylor expanded in i around inf 42.4%

      \[\leadsto \frac{-1 \cdot \frac{\color{blue}{-4 \cdot i}}{\alpha}}{2} \]
   6. Step-by-step derivation

      1. *-commutative 42.4%

         \[\leadsto \frac{-1 \cdot \frac{\color{blue}{i \cdot -4}}{\alpha}}{2} \]

   7. Simplified 42.4%

      \[\leadsto \frac{-1 \cdot \frac{\color{blue}{i \cdot -4}}{\alpha}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.05 \cdot 10^{+161}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-i \cdot -4}{\alpha}}{2}\\ \end{array} \]

Alternative 9: 78.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 10^{+98}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1e+98)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1e+98) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 1d+98) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1e+98) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 1e+98:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 1e+98)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 1e+98)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 1e+98], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 9.99999999999999998e97

   1. Initial program 83.0%

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

      1. associate-/l/ 82.5%

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

      2. *-commutative 82.5%

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

      3. times-frac 97.9%

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

      4. associate-+l+ 97.9%

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

      5. fma-def 97.9%

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

      6. +-commutative 97.9%

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

      7. fma-def 97.9%

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

   3. Simplified 97.9%

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

      1. frac-times 82.5%

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

      2. *-commutative 82.5%

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

      3. fma-def 82.5%

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

      4. fma-udef 82.5%

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

      5. +-commutative 82.5%

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

      6. associate-+r+ 82.5%

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

      7. add-log-exp 82.5%

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

   5. Applied egg-rr 98.0%

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

   6. Taylor expanded in i around 0 89.7%

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

      1. associate--l+ 89.7%

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

      2. div-sub 89.7%

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

      3. +-commutative 89.7%

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

   8. Simplified 89.7%

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

   9. Taylor expanded in alpha around 0 90.5%

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

   if 9.99999999999999998e97 < alpha

   1. Initial program 8.2%

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

      1. associate-/l/ 7.4%

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

      2. *-commutative 7.4%

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

      3. times-frac 29.0%

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

      4. associate-+l+ 29.0%

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

      5. fma-def 29.0%

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

      6. +-commutative 29.0%

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

      7. fma-def 29.0%

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

   3. Simplified 29.0%

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

      1. frac-times 7.4%

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

      2. *-commutative 7.4%

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

      3. fma-def 7.4%

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

      4. fma-udef 7.4%

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

      5. +-commutative 7.4%

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

      6. associate-+r+ 7.4%

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

      7. add-log-exp 7.4%

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

   5. Applied egg-rr 28.9%

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

   6. Taylor expanded in i around 0 9.2%

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

      1. associate--l+ 9.2%

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

      2. div-sub 9.2%

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

      3. +-commutative 9.2%

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

   8. Simplified 9.2%

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

   9. Taylor expanded in alpha around inf 48.2%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 10^{+98}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 10: 72.5% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.95 \cdot 10^{+68}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 (if (<= beta 1.95e+68) 0.5 1.0))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.95e+68) {
		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 <= 1.95d+68) 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 <= 1.95e+68) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.95e+68:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.95e+68)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1.95e+68)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 1.95e+68], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 1.95000000000000009e68

   1. Initial program 77.5%

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

      1. associate-/l/ 77.3%

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

      2. *-commutative 77.3%

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

      3. times-frac 81.4%

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

      4. associate-+l+ 81.4%

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

      5. fma-def 81.4%

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

      6. +-commutative 81.4%

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

      7. fma-def 81.4%

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

   3. Simplified 81.4%

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

   4. Taylor expanded in i around inf 76.9%

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

   if 1.95000000000000009e68 < beta

   1. Initial program 30.0%

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

      1. associate-/l/ 28.1%

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

      2. *-commutative 28.1%

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

      3. times-frac 89.2%

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

      4. associate-+l+ 89.2%

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

      5. fma-def 89.2%

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

      6. +-commutative 89.2%

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

      7. fma-def 89.2%

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

   3. Simplified 89.2%

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

   4. Taylor expanded in beta around inf 83.0%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.95 \cdot 10^{+68}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11: 62.0% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[alpha_, beta_, i_] := 0.5

Derivation

1. Initial program 66.9%

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

   1. associate-/l/ 66.4%

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

   2. *-commutative 66.4%

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

   3. times-frac 83.1%

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

   4. associate-+l+ 83.1%

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

   5. fma-def 83.1%

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

   6. +-commutative 83.1%

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

   7. fma-def 83.1%

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

3. Simplified 83.1%

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

4. Taylor expanded in i around inf 64.7%

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

5. Final simplification 64.7%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023240 
(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))