Octave 3.8, jcobi/2

Percentage Accurate: 63.8% → 97.8%

Time: 21.7s

Alternatives: 9

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: 63.8% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.9999999:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (if (<=
        (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0))
        -0.9999999)
     (/ (/ (+ (* i 4.0) (+ 2.0 (* beta 2.0))) alpha) 2.0)
     (/
      (+
       (*
        (/ (- beta alpha) (+ (+ alpha beta) (fma 2.0 i 2.0)))
        (/ (+ alpha beta) (fma 2.0 i (+ alpha beta))))
       1.0)
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.9999999) {
		tmp = (((i * 4.0) + (2.0 + (beta * 2.0))) / alpha) / 2.0;
	} else {
		tmp = ((((beta - alpha) / ((alpha + beta) + fma(2.0, i, 2.0))) * ((alpha + beta) / fma(2.0, i, (alpha + beta)))) + 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(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)) <= -0.9999999)
		tmp = Float64(Float64(Float64(Float64(i * 4.0) + Float64(2.0 + Float64(beta * 2.0))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + fma(2.0, i, 2.0))) * Float64(Float64(alpha + beta) / fma(2.0, i, Float64(alpha + beta)))) + 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(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], -0.9999999], N[(N[(N[(N[(i * 4.0), $MachinePrecision] + N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] / N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2)) < -0.999999900000000053

   1. Initial program 2.3%

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

      1. associate-/l/ 1.3%

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

      2. *-commutative 1.3%

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

      3. times-frac 11.3%

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

      4. associate-+l+ 11.3%

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

      5. fma-def 11.3%

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

      6. +-commutative 11.3%

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

      7. fma-def 11.3%

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

   3. Simplified 11.3%

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

      1. clear-num 11.3%

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

      2. fma-udef 11.3%

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

      3. +-commutative 11.3%

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

      4. frac-times 11.4%

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

      5. *-un-lft-identity 11.4%

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

      6. +-commutative 11.4%

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

      7. +-commutative 11.4%

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

      8. +-commutative 11.4%

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

      9. fma-udef 11.4%

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

      10. +-commutative 11.4%

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

   5. Applied egg-rr 11.4%

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

   6. Taylor expanded in alpha around -inf 82.1%

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

   8. Simplified 82.1%

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

   9. Taylor expanded in alpha around inf 94.3%

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

   if -0.999999900000000053 < (/.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 80.8%

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

      1. associate-/l/ 80.2%

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

      2. *-commutative 80.2%

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

      3. times-frac 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 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.9999999:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1}{2}\\ \end{array} \]

Alternative 2: 97.3% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 4.8%

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

      1. associate-/l/ 3.8%

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

      2. *-commutative 3.8%

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

      3. times-frac 13.4%

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

      4. associate-+l+ 13.4%

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

      5. fma-def 13.4%

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

      6. +-commutative 13.4%

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

      7. fma-def 13.4%

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

   3. Simplified 13.4%

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

      1. clear-num 13.4%

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

      2. fma-udef 13.4%

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

      3. +-commutative 13.4%

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

      4. frac-times 13.5%

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

      5. *-un-lft-identity 13.5%

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

      6. +-commutative 13.5%

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

      7. +-commutative 13.5%

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

      8. +-commutative 13.5%

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

      9. fma-udef 13.5%

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

      10. +-commutative 13.5%

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

   5. Applied egg-rr 13.5%

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

   6. Taylor expanded in alpha around -inf 81.9%

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

   8. Simplified 81.9%

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

   9. Taylor expanded in alpha around inf 92.6%

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

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

   1. Initial program 80.9%

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

      1. associate-/l/ 80.3%

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

      2. *-commutative 80.3%

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

      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 alpha around 0 98.8%

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

4. Final simplification 97.4%

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

Alternative 3: 97.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0}\\ \mathbf{if}\;t_1 \leq -0.9999999:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \mathbf{elif}\;t_1 \leq 0.999999999999998:\\ \;\;\;\;\frac{t_1 + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\alpha + \beta}{\frac{\beta + 2 \cdot i}{\frac{\beta}{\beta + \left(2 + 2 \cdot i\right)}}}}{2}\\ \end{array} \end{array} \]

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 -0.9999999)
     (/ (/ (+ (* i 4.0) (+ 2.0 (* beta 2.0))) alpha) 2.0)
     (if (<= t_1 0.999999999999998)
       (/ (+ t_1 1.0) 2.0)
       (/
        (+
         1.0
         (/
          (+ alpha beta)
          (/ (+ beta (* 2.0 i)) (/ beta (+ beta (+ 2.0 (* 2.0 i)))))))
        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 <= -0.9999999) {
		tmp = (((i * 4.0) + (2.0 + (beta * 2.0))) / alpha) / 2.0;
	} else if (t_1 <= 0.999999999999998) {
		tmp = (t_1 + 1.0) / 2.0;
	} else {
		tmp = (1.0 + ((alpha + beta) / ((beta + (2.0 * i)) / (beta / (beta + (2.0 + (2.0 * i))))))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: 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 <= (-0.9999999d0)) then
        tmp = (((i * 4.0d0) + (2.0d0 + (beta * 2.0d0))) / alpha) / 2.0d0
    else if (t_1 <= 0.999999999999998d0) then
        tmp = (t_1 + 1.0d0) / 2.0d0
    else
        tmp = (1.0d0 + ((alpha + beta) / ((beta + (2.0d0 * i)) / (beta / (beta + (2.0d0 + (2.0d0 * i))))))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0);
	double tmp;
	if (t_1 <= -0.9999999) {
		tmp = (((i * 4.0) + (2.0 + (beta * 2.0))) / alpha) / 2.0;
	} else if (t_1 <= 0.999999999999998) {
		tmp = (t_1 + 1.0) / 2.0;
	} else {
		tmp = (1.0 + ((alpha + beta) / ((beta + (2.0 * i)) / (beta / (beta + (2.0 + (2.0 * i))))))) / 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 <= -0.9999999:
		tmp = (((i * 4.0) + (2.0 + (beta * 2.0))) / alpha) / 2.0
	elif t_1 <= 0.999999999999998:
		tmp = (t_1 + 1.0) / 2.0
	else:
		tmp = (1.0 + ((alpha + beta) / ((beta + (2.0 * i)) / (beta / (beta + (2.0 + (2.0 * i))))))) / 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 <= -0.9999999)
		tmp = Float64(Float64(Float64(Float64(i * 4.0) + Float64(2.0 + Float64(beta * 2.0))) / alpha) / 2.0);
	elseif (t_1 <= 0.999999999999998)
		tmp = Float64(Float64(t_1 + 1.0) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(alpha + beta) / Float64(Float64(beta + Float64(2.0 * i)) / Float64(beta / Float64(beta + Float64(2.0 + Float64(2.0 * i))))))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	t_1 = (((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0);
	tmp = 0.0;
	if (t_1 <= -0.9999999)
		tmp = (((i * 4.0) + (2.0 + (beta * 2.0))) / alpha) / 2.0;
	elseif (t_1 <= 0.999999999999998)
		tmp = (t_1 + 1.0) / 2.0;
	else
		tmp = (1.0 + ((alpha + beta) / ((beta + (2.0 * i)) / (beta / (beta + (2.0 + (2.0 * i))))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, 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, -0.9999999], N[(N[(N[(N[(i * 4.0), $MachinePrecision] + N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[t$95$1, 0.999999999999998], N[(N[(t$95$1 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(N[(alpha + beta), $MachinePrecision] / N[(N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision] / N[(beta / N[(beta + N[(2.0 + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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)) < -0.999999900000000053

   1. Initial program 2.3%

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

      1. associate-/l/ 1.3%

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

      2. *-commutative 1.3%

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

      3. times-frac 11.3%

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

      4. associate-+l+ 11.3%

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

      5. fma-def 11.3%

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

      6. +-commutative 11.3%

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

      7. fma-def 11.3%

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

   3. Simplified 11.3%

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

      1. clear-num 11.3%

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

      2. fma-udef 11.3%

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

      3. +-commutative 11.3%

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

      4. frac-times 11.4%

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

      5. *-un-lft-identity 11.4%

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

      6. +-commutative 11.4%

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

      7. +-commutative 11.4%

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

      8. +-commutative 11.4%

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

      9. fma-udef 11.4%

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

      10. +-commutative 11.4%

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

   5. Applied egg-rr 11.4%

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

   6. Taylor expanded in alpha around -inf 82.1%

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

   8. Simplified 82.1%

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

   9. Taylor expanded in alpha around inf 94.3%

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

   if -0.999999900000000053 < (/.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)) < 0.999999999999998

   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 0.999999999999998 < (/.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 36.0%

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

      1. associate-/l/ 33.9%

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

      2. *-commutative 33.9%

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

      3. times-frac 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. Step-by-step derivation

      1. clear-num 99.9%

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

      2. fma-udef 99.9%

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

      3. +-commutative 99.9%

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

      4. frac-times 100.0%

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

      5. *-un-lft-identity 100.0%

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

      6. +-commutative 100.0%

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

      7. +-commutative 100.0%

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

      8. +-commutative 100.0%

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

      9. fma-udef 100.0%

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

      10. +-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in alpha around 0 56.5%

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

      1. associate-/l* 98.5%

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

      2. +-commutative 98.5%

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

   8. Simplified 98.5%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.9999999:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \mathbf{elif}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq 0.999999999999998:\\ \;\;\;\;\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\alpha + \beta}{\frac{\beta + 2 \cdot i}{\frac{\beta}{\beta + \left(2 + 2 \cdot i\right)}}}}{2}\\ \end{array} \]

Alternative 4: 96.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.9999999:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\alpha + \beta}{\frac{t_1}{\frac{\beta - \alpha}{\alpha + \beta}} + \left(\frac{4 \cdot \left(i \cdot i\right)}{\beta - \alpha} - i \cdot \left(-2 \cdot \left(\frac{t_1}{\beta - \alpha} + \frac{\alpha + \beta}{\beta - \alpha}\right)\right)\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))) (t_1 (+ beta (+ alpha 2.0))))
   (if (<=
        (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0))
        -0.9999999)
     (/ (/ (+ (* i 4.0) (+ 2.0 (* beta 2.0))) alpha) 2.0)
     (/
      (+
       1.0
       (/
        (+ alpha beta)
        (+
         (/ t_1 (/ (- beta alpha) (+ alpha beta)))
         (-
          (/ (* 4.0 (* i i)) (- beta alpha))
          (*
           i
           (*
            -2.0
            (+ (/ t_1 (- beta alpha)) (/ (+ alpha beta) (- beta alpha)))))))))
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = beta + (alpha + 2.0);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.9999999) {
		tmp = (((i * 4.0) + (2.0 + (beta * 2.0))) / alpha) / 2.0;
	} else {
		tmp = (1.0 + ((alpha + beta) / ((t_1 / ((beta - alpha) / (alpha + beta))) + (((4.0 * (i * i)) / (beta - alpha)) - (i * (-2.0 * ((t_1 / (beta - alpha)) + ((alpha + beta) / (beta - alpha))))))))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    t_1 = beta + (alpha + 2.0d0)
    if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0d0 + t_0)) <= (-0.9999999d0)) then
        tmp = (((i * 4.0d0) + (2.0d0 + (beta * 2.0d0))) / alpha) / 2.0d0
    else
        tmp = (1.0d0 + ((alpha + beta) / ((t_1 / ((beta - alpha) / (alpha + beta))) + (((4.0d0 * (i * i)) / (beta - alpha)) - (i * ((-2.0d0) * ((t_1 / (beta - alpha)) + ((alpha + beta) / (beta - alpha))))))))) / 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 = beta + (alpha + 2.0);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.9999999) {
		tmp = (((i * 4.0) + (2.0 + (beta * 2.0))) / alpha) / 2.0;
	} else {
		tmp = (1.0 + ((alpha + beta) / ((t_1 / ((beta - alpha) / (alpha + beta))) + (((4.0 * (i * i)) / (beta - alpha)) - (i * (-2.0 * ((t_1 / (beta - alpha)) + ((alpha + beta) / (beta - alpha))))))))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	t_1 = beta + (alpha + 2.0)
	tmp = 0
	if ((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.9999999:
		tmp = (((i * 4.0) + (2.0 + (beta * 2.0))) / alpha) / 2.0
	else:
		tmp = (1.0 + ((alpha + beta) / ((t_1 / ((beta - alpha) / (alpha + beta))) + (((4.0 * (i * i)) / (beta - alpha)) - (i * (-2.0 * ((t_1 / (beta - alpha)) + ((alpha + beta) / (beta - alpha))))))))) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(beta + Float64(alpha + 2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0)) <= -0.9999999)
		tmp = Float64(Float64(Float64(Float64(i * 4.0) + Float64(2.0 + Float64(beta * 2.0))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(alpha + beta) / Float64(Float64(t_1 / Float64(Float64(beta - alpha) / Float64(alpha + beta))) + Float64(Float64(Float64(4.0 * Float64(i * i)) / Float64(beta - alpha)) - Float64(i * Float64(-2.0 * Float64(Float64(t_1 / Float64(beta - alpha)) + Float64(Float64(alpha + beta) / Float64(beta - alpha))))))))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	t_1 = beta + (alpha + 2.0);
	tmp = 0.0;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.9999999)
		tmp = (((i * 4.0) + (2.0 + (beta * 2.0))) / alpha) / 2.0;
	else
		tmp = (1.0 + ((alpha + beta) / ((t_1 / ((beta - alpha) / (alpha + beta))) + (((4.0 * (i * i)) / (beta - alpha)) - (i * (-2.0 * ((t_1 / (beta - alpha)) + ((alpha + beta) / (beta - alpha))))))))) / 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[(beta + N[(alpha + 2.0), $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], -0.9999999], N[(N[(N[(N[(i * 4.0), $MachinePrecision] + N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(N[(alpha + beta), $MachinePrecision] / N[(N[(t$95$1 / N[(N[(beta - alpha), $MachinePrecision] / N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(4.0 * N[(i * i), $MachinePrecision]), $MachinePrecision] / N[(beta - alpha), $MachinePrecision]), $MachinePrecision] - N[(i * N[(-2.0 * N[(N[(t$95$1 / N[(beta - alpha), $MachinePrecision]), $MachinePrecision] + N[(N[(alpha + beta), $MachinePrecision] / N[(beta - alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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)) < -0.999999900000000053

   1. Initial program 2.3%

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

      1. associate-/l/ 1.3%

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

      2. *-commutative 1.3%

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

      3. times-frac 11.3%

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

      4. associate-+l+ 11.3%

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

      5. fma-def 11.3%

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

      6. +-commutative 11.3%

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

      7. fma-def 11.3%

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

   3. Simplified 11.3%

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

      1. clear-num 11.3%

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

      2. fma-udef 11.3%

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

      3. +-commutative 11.3%

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

      4. frac-times 11.4%

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

      5. *-un-lft-identity 11.4%

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

      6. +-commutative 11.4%

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

      7. +-commutative 11.4%

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

      8. +-commutative 11.4%

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

      9. fma-udef 11.4%

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

      10. +-commutative 11.4%

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

   5. Applied egg-rr 11.4%

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

   6. Taylor expanded in alpha around -inf 82.1%

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

   8. Simplified 82.1%

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

   9. Taylor expanded in alpha around inf 94.3%

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

   if -0.999999900000000053 < (/.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 80.8%

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

      1. associate-/l/ 80.2%

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

      2. *-commutative 80.2%

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

      3. times-frac 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. clear-num 99.7%

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

      2. fma-udef 99.7%

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

      3. +-commutative 99.7%

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

      4. frac-times 99.7%

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

      5. *-un-lft-identity 99.7%

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

      6. +-commutative 99.7%

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

      7. +-commutative 99.7%

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

      8. +-commutative 99.7%

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

      9. fma-udef 99.7%

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

      10. +-commutative 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in i around -inf 86.9%

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

      1. associate-/l* 97.9%

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

      2. +-commutative 97.9%

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

      3. +-commutative 97.9%

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

      4. +-commutative 97.9%

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

      5. mul-1-neg 97.9%

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

      6. unsub-neg 97.9%

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

      7. associate-*r/ 97.9%

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

      8. unpow2 97.9%

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

   8. Simplified 97.9%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.9999999:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\alpha + \beta}{\frac{\beta + \left(\alpha + 2\right)}{\frac{\beta - \alpha}{\alpha + \beta}} + \left(\frac{4 \cdot \left(i \cdot i\right)}{\beta - \alpha} - i \cdot \left(-2 \cdot \left(\frac{\beta + \left(\alpha + 2\right)}{\beta - \alpha} + \frac{\alpha + \beta}{\beta - \alpha}\right)\right)\right)}}{2}\\ \end{array} \]

Alternative 5: 89.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 4.9 \cdot 10^{+114}:\\ \;\;\;\;\frac{1 + \frac{\alpha + \beta}{\frac{\beta + 2 \cdot i}{\frac{\beta}{\beta + \left(2 + 2 \cdot i\right)}}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 4.9e+114)
   (/
    (+
     1.0
     (/
      (+ alpha beta)
      (/ (+ beta (* 2.0 i)) (/ beta (+ beta (+ 2.0 (* 2.0 i)))))))
    2.0)
   (/ (/ (+ (* i 4.0) (+ 2.0 (* beta 2.0))) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 4.9e+114) {
		tmp = (1.0 + ((alpha + beta) / ((beta + (2.0 * i)) / (beta / (beta + (2.0 + (2.0 * i))))))) / 2.0;
	} else {
		tmp = (((i * 4.0) + (2.0 + (beta * 2.0))) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 4.9d+114) then
        tmp = (1.0d0 + ((alpha + beta) / ((beta + (2.0d0 * i)) / (beta / (beta + (2.0d0 + (2.0d0 * i))))))) / 2.0d0
    else
        tmp = (((i * 4.0d0) + (2.0d0 + (beta * 2.0d0))) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 4.9e+114) {
		tmp = (1.0 + ((alpha + beta) / ((beta + (2.0 * i)) / (beta / (beta + (2.0 + (2.0 * i))))))) / 2.0;
	} else {
		tmp = (((i * 4.0) + (2.0 + (beta * 2.0))) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 4.9e+114:
		tmp = (1.0 + ((alpha + beta) / ((beta + (2.0 * i)) / (beta / (beta + (2.0 + (2.0 * i))))))) / 2.0
	else:
		tmp = (((i * 4.0) + (2.0 + (beta * 2.0))) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 4.9e+114)
		tmp = Float64(Float64(1.0 + Float64(Float64(alpha + beta) / Float64(Float64(beta + Float64(2.0 * i)) / Float64(beta / Float64(beta + Float64(2.0 + Float64(2.0 * i))))))) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(i * 4.0) + Float64(2.0 + Float64(beta * 2.0))) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 4.9e+114)
		tmp = (1.0 + ((alpha + beta) / ((beta + (2.0 * i)) / (beta / (beta + (2.0 + (2.0 * i))))))) / 2.0;
	else
		tmp = (((i * 4.0) + (2.0 + (beta * 2.0))) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 4.9e+114], N[(N[(1.0 + N[(N[(alpha + beta), $MachinePrecision] / N[(N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision] / N[(beta / N[(beta + N[(2.0 + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(i * 4.0), $MachinePrecision] + N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 4.9000000000000001e114

   1. Initial program 79.7%

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

      1. associate-/l/ 79.2%

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

      2. *-commutative 79.2%

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

      3. times-frac 94.9%

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

      4. associate-+l+ 94.9%

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

      5. fma-def 94.9%

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

      6. +-commutative 94.9%

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

      7. fma-def 94.9%

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

   3. Simplified 94.9%

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

      1. clear-num 94.8%

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

      2. fma-udef 94.8%

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

      3. +-commutative 94.8%

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

      4. frac-times 94.8%

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

      5. *-un-lft-identity 94.8%

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

      6. +-commutative 94.8%

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

      7. +-commutative 94.8%

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

      8. +-commutative 94.8%

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

      9. fma-udef 94.8%

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

      10. +-commutative 94.8%

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

   5. Applied egg-rr 94.8%

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

   6. Taylor expanded in alpha around 0 82.3%

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

      1. associate-/l* 92.6%

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

      2. +-commutative 92.6%

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

   8. Simplified 92.6%

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

   if 4.9000000000000001e114 < alpha

   1. Initial program 8.9%

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

      1. associate-/l/ 7.6%

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

      2. *-commutative 7.6%

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

      3. times-frac 30.9%

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

      4. associate-+l+ 30.9%

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

      5. fma-def 30.9%

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

      6. +-commutative 30.9%

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

      7. fma-def 30.9%

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

   3. Simplified 30.9%

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

      1. clear-num 30.9%

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

      2. fma-udef 30.9%

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

      3. +-commutative 30.9%

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

      4. frac-times 31.0%

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

      5. *-un-lft-identity 31.0%

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

      6. +-commutative 31.0%

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

      7. +-commutative 31.0%

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

      8. +-commutative 31.0%

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

      9. fma-udef 31.0%

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

      10. +-commutative 31.0%

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

   5. Applied egg-rr 31.0%

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

   6. Taylor expanded in alpha around -inf 62.1%

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

   8. Simplified 62.3%

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

   9. Taylor expanded in alpha around inf 75.5%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 4.9 \cdot 10^{+114}:\\ \;\;\;\;\frac{1 + \frac{\alpha + \beta}{\frac{\beta + 2 \cdot i}{\frac{\beta}{\beta + \left(2 + 2 \cdot i\right)}}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 6: 78.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{if}\;\alpha \leq -8.5 \cdot 10^{-108}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq -5.8 \cdot 10^{-160}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 10^{+18}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (/ (+ 1.0 (/ (- beta alpha) (+ beta (+ alpha 2.0)))) 2.0)))
   (if (<= alpha -8.5e-108)
     t_0
     (if (<= alpha -5.8e-160)
       0.5
       (if (<= alpha 1e+18) t_0 (/ (/ (+ 2.0 (* i 4.0)) alpha) 2.0))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))) / 2.0;
	double tmp;
	if (alpha <= -8.5e-108) {
		tmp = t_0;
	} else if (alpha <= -5.8e-160) {
		tmp = 0.5;
	} else if (alpha <= 1e+18) {
		tmp = t_0;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 + ((beta - alpha) / (beta + (alpha + 2.0d0)))) / 2.0d0
    if (alpha <= (-8.5d-108)) then
        tmp = t_0
    else if (alpha <= (-5.8d-160)) then
        tmp = 0.5d0
    else if (alpha <= 1d+18) then
        tmp = t_0
    else
        tmp = ((2.0d0 + (i * 4.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))) / 2.0;
	double tmp;
	if (alpha <= -8.5e-108) {
		tmp = t_0;
	} else if (alpha <= -5.8e-160) {
		tmp = 0.5;
	} else if (alpha <= 1e+18) {
		tmp = t_0;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))) / 2.0
	tmp = 0
	if alpha <= -8.5e-108:
		tmp = t_0
	elif alpha <= -5.8e-160:
		tmp = 0.5
	elif alpha <= 1e+18:
		tmp = t_0
	else:
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / Float64(beta + Float64(alpha + 2.0)))) / 2.0)
	tmp = 0.0
	if (alpha <= -8.5e-108)
		tmp = t_0;
	elseif (alpha <= -5.8e-160)
		tmp = 0.5;
	elseif (alpha <= 1e+18)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(i * 4.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))) / 2.0;
	tmp = 0.0;
	if (alpha <= -8.5e-108)
		tmp = t_0;
	elseif (alpha <= -5.8e-160)
		tmp = 0.5;
	elseif (alpha <= 1e+18)
		tmp = t_0;
	else
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[alpha, -8.5e-108], t$95$0, If[LessEqual[alpha, -5.8e-160], 0.5, If[LessEqual[alpha, 1e+18], t$95$0, N[(N[(N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if alpha < -8.49999999999999986e-108 or -5.7999999999999998e-160 < alpha < 1e18

   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 99.8%

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

      4. associate-+l+ 99.8%

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

      5. fma-def 99.8%

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

      6. +-commutative 99.8%

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

      7. fma-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in i around 0 90.7%

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

      1. +-commutative 90.7%

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

   6. Simplified 90.7%

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

   if -8.49999999999999986e-108 < alpha < -5.7999999999999998e-160

   1. Initial program 93.5%

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

   2. Taylor expanded in i around inf 87.7%

      \[\leadsto \frac{\frac{\color{blue}{0.5 \cdot \frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   3. Step-by-step derivation

      1. associate-/l* 94.2%

         \[\leadsto \frac{\frac{0.5 \cdot \color{blue}{\frac{\beta + \alpha}{\frac{i}{\beta - \alpha}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      2. +-commutative 94.2%

         \[\leadsto \frac{\frac{0.5 \cdot \frac{\color{blue}{\alpha + \beta}}{\frac{i}{\beta - \alpha}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   4. Simplified 94.2%

      \[\leadsto \frac{\frac{\color{blue}{0.5 \cdot \frac{\alpha + \beta}{\frac{i}{\beta - \alpha}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   5. Taylor expanded in i around inf 100.0%

      \[\leadsto \color{blue}{0.5} \]

   if 1e18 < alpha

   1. Initial program 22.9%

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

      1. associate-/l/ 21.8%

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

      2. *-commutative 21.8%

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

      3. times-frac 41.3%

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

      4. associate-+l+ 41.3%

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

      5. fma-def 41.3%

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

      6. +-commutative 41.3%

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

      7. fma-def 41.3%

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

   3. Simplified 41.3%

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

      1. clear-num 41.3%

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

      2. fma-udef 41.3%

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

      3. +-commutative 41.3%

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

      4. frac-times 41.3%

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

      5. *-un-lft-identity 41.3%

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

      6. +-commutative 41.3%

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

      7. +-commutative 41.3%

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

      8. +-commutative 41.3%

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

      9. fma-udef 41.3%

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

      10. +-commutative 41.3%

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

   5. Applied egg-rr 41.3%

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

   6. Taylor expanded in beta around 0 18.4%

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

      1. mul-1-neg 18.4%

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

      2. unsub-neg 18.4%

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

      3. unpow2 18.4%

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

      4. associate-+r+ 18.4%

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

      5. +-commutative 18.4%

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

   8. Simplified 18.4%

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

   9. Taylor expanded in alpha around inf 54.6%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq -8.5 \cdot 10^{-108}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{elif}\;\alpha \leq -5.8 \cdot 10^{-160}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 10^{+18}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 7: 82.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{if}\;\alpha \leq -1 \cdot 10^{-107}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq -1 \cdot 10^{-159}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 1.7 \cdot 10^{+18}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (/ (+ 1.0 (/ (- beta alpha) (+ beta (+ alpha 2.0)))) 2.0)))
   (if (<= alpha -1e-107)
     t_0
     (if (<= alpha -1e-159)
       0.5
       (if (<= alpha 1.7e+18)
         t_0
         (/ (/ (+ (* i 4.0) (+ 2.0 (* beta 2.0))) alpha) 2.0))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))) / 2.0;
	double tmp;
	if (alpha <= -1e-107) {
		tmp = t_0;
	} else if (alpha <= -1e-159) {
		tmp = 0.5;
	} else if (alpha <= 1.7e+18) {
		tmp = t_0;
	} else {
		tmp = (((i * 4.0) + (2.0 + (beta * 2.0))) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 + ((beta - alpha) / (beta + (alpha + 2.0d0)))) / 2.0d0
    if (alpha <= (-1d-107)) then
        tmp = t_0
    else if (alpha <= (-1d-159)) then
        tmp = 0.5d0
    else if (alpha <= 1.7d+18) then
        tmp = t_0
    else
        tmp = (((i * 4.0d0) + (2.0d0 + (beta * 2.0d0))) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))) / 2.0;
	double tmp;
	if (alpha <= -1e-107) {
		tmp = t_0;
	} else if (alpha <= -1e-159) {
		tmp = 0.5;
	} else if (alpha <= 1.7e+18) {
		tmp = t_0;
	} else {
		tmp = (((i * 4.0) + (2.0 + (beta * 2.0))) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))) / 2.0
	tmp = 0
	if alpha <= -1e-107:
		tmp = t_0
	elif alpha <= -1e-159:
		tmp = 0.5
	elif alpha <= 1.7e+18:
		tmp = t_0
	else:
		tmp = (((i * 4.0) + (2.0 + (beta * 2.0))) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / Float64(beta + Float64(alpha + 2.0)))) / 2.0)
	tmp = 0.0
	if (alpha <= -1e-107)
		tmp = t_0;
	elseif (alpha <= -1e-159)
		tmp = 0.5;
	elseif (alpha <= 1.7e+18)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(Float64(i * 4.0) + Float64(2.0 + Float64(beta * 2.0))) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (1.0 + ((beta - alpha) / (beta + (alpha + 2.0)))) / 2.0;
	tmp = 0.0;
	if (alpha <= -1e-107)
		tmp = t_0;
	elseif (alpha <= -1e-159)
		tmp = 0.5;
	elseif (alpha <= 1.7e+18)
		tmp = t_0;
	else
		tmp = (((i * 4.0) + (2.0 + (beta * 2.0))) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = 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, -1e-107], t$95$0, If[LessEqual[alpha, -1e-159], 0.5, If[LessEqual[alpha, 1.7e+18], t$95$0, N[(N[(N[(N[(i * 4.0), $MachinePrecision] + N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if alpha < -1e-107 or -9.99999999999999989e-160 < alpha < 1.7e18

   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 99.8%

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

      4. associate-+l+ 99.8%

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

      5. fma-def 99.8%

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

      6. +-commutative 99.8%

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

      7. fma-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in i around 0 90.7%

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

      1. +-commutative 90.7%

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

   6. Simplified 90.7%

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

   if -1e-107 < alpha < -9.99999999999999989e-160

   1. Initial program 93.5%

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

   2. Taylor expanded in i around inf 87.7%

      \[\leadsto \frac{\frac{\color{blue}{0.5 \cdot \frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   3. Step-by-step derivation

      1. associate-/l* 94.2%

         \[\leadsto \frac{\frac{0.5 \cdot \color{blue}{\frac{\beta + \alpha}{\frac{i}{\beta - \alpha}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      2. +-commutative 94.2%

         \[\leadsto \frac{\frac{0.5 \cdot \frac{\color{blue}{\alpha + \beta}}{\frac{i}{\beta - \alpha}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   4. Simplified 94.2%

      \[\leadsto \frac{\frac{\color{blue}{0.5 \cdot \frac{\alpha + \beta}{\frac{i}{\beta - \alpha}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   5. Taylor expanded in i around inf 100.0%

      \[\leadsto \color{blue}{0.5} \]

   if 1.7e18 < alpha

   1. Initial program 22.9%

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

      1. associate-/l/ 21.8%

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

      2. *-commutative 21.8%

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

      3. times-frac 41.3%

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

      4. associate-+l+ 41.3%

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

      5. fma-def 41.3%

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

      6. +-commutative 41.3%

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

      7. fma-def 41.3%

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

   3. Simplified 41.3%

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

      1. clear-num 41.3%

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

      2. fma-udef 41.3%

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

      3. +-commutative 41.3%

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

      4. frac-times 41.3%

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

      5. *-un-lft-identity 41.3%

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

      6. +-commutative 41.3%

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

      7. +-commutative 41.3%

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

      8. +-commutative 41.3%

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

      9. fma-udef 41.3%

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

      10. +-commutative 41.3%

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

   5. Applied egg-rr 41.3%

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

   6. Taylor expanded in alpha around -inf 54.9%

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

   8. Simplified 55.4%

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

   9. Taylor expanded in alpha around inf 64.9%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq -1 \cdot 10^{-107}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{elif}\;\alpha \leq -1 \cdot 10^{-159}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 1.7 \cdot 10^{+18}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 8: 72.9% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4 \cdot 10^{+36}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 (if (<= beta 4e+36) 0.5 1.0))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 4e+36) {
		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 <= 4d+36) 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 <= 4e+36) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 4e+36:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 4e+36)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 4e+36)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 4e+36], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 4.00000000000000017e36

   1. Initial program 75.4%

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

   2. Taylor expanded in i around inf 63.3%

      \[\leadsto \frac{\frac{\color{blue}{0.5 \cdot \frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   3. Step-by-step derivation

      1. associate-/l* 63.9%

         \[\leadsto \frac{\frac{0.5 \cdot \color{blue}{\frac{\beta + \alpha}{\frac{i}{\beta - \alpha}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      2. +-commutative 63.9%

         \[\leadsto \frac{\frac{0.5 \cdot \frac{\color{blue}{\alpha + \beta}}{\frac{i}{\beta - \alpha}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   4. Simplified 63.9%

      \[\leadsto \frac{\frac{\color{blue}{0.5 \cdot \frac{\alpha + \beta}{\frac{i}{\beta - \alpha}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   5. Taylor expanded in i around inf 73.1%

      \[\leadsto \color{blue}{0.5} \]

   if 4.00000000000000017e36 < beta

   1. Initial program 37.4%

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

      1. associate-/l/ 35.6%

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

      2. *-commutative 35.6%

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

      3. times-frac 87.8%

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

      4. associate-+l+ 87.8%

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

      5. fma-def 87.8%

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

      6. +-commutative 87.8%

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

      7. fma-def 87.8%

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

   3. Simplified 87.8%

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

   4. Taylor expanded in beta around inf 69.6%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4 \cdot 10^{+36}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9: 62.3% 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 63.7%

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

2. Taylor expanded in i around inf 47.4%

   \[\leadsto \frac{\frac{\color{blue}{0.5 \cdot \frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
3. Step-by-step derivation

   1. associate-/l* 51.5%

      \[\leadsto \frac{\frac{0.5 \cdot \color{blue}{\frac{\beta + \alpha}{\frac{i}{\beta - \alpha}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   2. +-commutative 51.5%

      \[\leadsto \frac{\frac{0.5 \cdot \frac{\color{blue}{\alpha + \beta}}{\frac{i}{\beta - \alpha}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

4. Simplified 51.5%

   \[\leadsto \frac{\frac{\color{blue}{0.5 \cdot \frac{\alpha + \beta}{\frac{i}{\beta - \alpha}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

5. Taylor expanded in i around inf 61.1%

   \[\leadsto \color{blue}{0.5} \]

6. Final simplification 61.1%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023216 
(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))