Octave 3.8, jcobi/2

Percentage Accurate: 63.3% → 97.6%

Time: 31.4s

Alternatives: 14

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: 63.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.6% accurate, 0.1× speedup

Math

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

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 3.1%

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

      1. associate-/l/ 2.4%

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

      2. *-commutative 2.4%

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

      3. times-frac 11.5%

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

      4. associate-+l+ 11.5%

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

      5. fma-def 11.5%

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

      6. +-commutative 11.5%

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

      7. fma-def 11.5%

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

   3. Simplified 11.5%

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

   4. Taylor expanded in alpha around inf 7.7%

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

   5. Taylor expanded in i around 0 94.4%

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

      1. sub-neg 94.4%

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

      2. neg-mul-1 94.4%

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

      3. associate-+r+ 94.5%

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

      4. +-commutative 94.5%

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

      5. neg-mul-1 94.5%

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

      6. distribute-lft1-in 94.5%

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

      7. metadata-eval 94.5%

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

      8. mul0-lft 94.5%

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

      9. +-lft-identity 94.5%

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

      10. mul-1-neg 94.5%

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

      11. remove-double-neg 94.5%

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

   7. Simplified 94.5%

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

   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 81.9%

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

      1. associate-/l/ 81.3%

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

      2. *-commutative 81.3%

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

      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%

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

4. Final simplification 98.6%

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

Alternative 2: 97.2% accurate, 0.2× speedup

Math

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

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)
     (/ (+ (* 4.0 (/ i alpha)) (/ (+ 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 = ((4.0 * (i / alpha)) + ((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(4.0 * Float64(i / alpha)) + Float64(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[(4.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(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 3.1%

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

      1. associate-/l/ 2.4%

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

      2. *-commutative 2.4%

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

      3. times-frac 11.5%

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

      4. associate-+l+ 11.5%

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

      5. fma-def 11.5%

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

      6. +-commutative 11.5%

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

      7. fma-def 11.5%

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

   3. Simplified 11.5%

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

   4. Taylor expanded in alpha around inf 7.7%

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

   5. Taylor expanded in i around 0 94.4%

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

      1. sub-neg 94.4%

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

      2. neg-mul-1 94.4%

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

      3. associate-+r+ 94.5%

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

      4. +-commutative 94.5%

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

      5. neg-mul-1 94.5%

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

      6. distribute-lft1-in 94.5%

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

      7. metadata-eval 94.5%

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

      8. mul0-lft 94.5%

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

      9. +-lft-identity 94.5%

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

      10. mul-1-neg 94.5%

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

      11. remove-double-neg 94.5%

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

   7. Simplified 94.5%

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

   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 81.9%

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

      1. associate-/l/ 81.3%

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

      2. *-commutative 81.3%

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

      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 99.2%

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

4. Final simplification 98.1%

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

Alternative 3: 97.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0}\\ \mathbf{if}\;t_1 \leq -0.5:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha} + \frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{elif}\;t_1 \leq 0.995:\\ \;\;\;\;\frac{t_1 + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + \left(2 + 2 \cdot i\right)}}{2}\\ \end{array} \end{array} \]

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 -0.5)
     (/ (+ (* 4.0 (/ i alpha)) (/ (+ 2.0 (* beta 2.0)) alpha)) 2.0)
     (if (<= t_1 0.995)
       (/ (+ t_1 1.0) 2.0)
       (/ (+ 1.0 (/ 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 <= -0.5) {
		tmp = ((4.0 * (i / alpha)) + ((2.0 + (beta * 2.0)) / alpha)) / 2.0;
	} else if (t_1 <= 0.995) {
		tmp = (t_1 + 1.0) / 2.0;
	} else {
		tmp = (1.0 + (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 <= (-0.5d0)) then
        tmp = ((4.0d0 * (i / alpha)) + ((2.0d0 + (beta * 2.0d0)) / alpha)) / 2.0d0
    else if (t_1 <= 0.995d0) then
        tmp = (t_1 + 1.0d0) / 2.0d0
    else
        tmp = (1.0d0 + (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 <= -0.5) {
		tmp = ((4.0 * (i / alpha)) + ((2.0 + (beta * 2.0)) / alpha)) / 2.0;
	} else if (t_1 <= 0.995) {
		tmp = (t_1 + 1.0) / 2.0;
	} else {
		tmp = (1.0 + (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 <= -0.5:
		tmp = ((4.0 * (i / alpha)) + ((2.0 + (beta * 2.0)) / alpha)) / 2.0
	elif t_1 <= 0.995:
		tmp = (t_1 + 1.0) / 2.0
	else:
		tmp = (1.0 + (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 <= -0.5)
		tmp = Float64(Float64(Float64(4.0 * Float64(i / alpha)) + Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha)) / 2.0);
	elseif (t_1 <= 0.995)
		tmp = Float64(Float64(t_1 + 1.0) / 2.0);
	else
		tmp = Float64(Float64(1.0 + 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 <= -0.5)
		tmp = ((4.0 * (i / alpha)) + ((2.0 + (beta * 2.0)) / alpha)) / 2.0;
	elseif (t_1 <= 0.995)
		tmp = (t_1 + 1.0) / 2.0;
	else
		tmp = (1.0 + (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, -0.5], N[(N[(N[(4.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[t$95$1, 0.995], N[(N[(t$95$1 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(beta / N[(beta + N[(2.0 + N[(2.0 * i), $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)) < -0.5

   1. Initial program 3.1%

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

      1. associate-/l/ 2.4%

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

      2. *-commutative 2.4%

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

      3. times-frac 11.5%

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

      4. associate-+l+ 11.5%

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

      5. fma-def 11.5%

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

      6. +-commutative 11.5%

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

      7. fma-def 11.5%

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

   3. Simplified 11.5%

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

   4. Taylor expanded in alpha around inf 7.7%

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

   5. Taylor expanded in i around 0 94.4%

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

      1. sub-neg 94.4%

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

      2. neg-mul-1 94.4%

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

      3. associate-+r+ 94.5%

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

      4. +-commutative 94.5%

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

      5. neg-mul-1 94.5%

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

      6. distribute-lft1-in 94.5%

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

      7. metadata-eval 94.5%

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

      8. mul0-lft 94.5%

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

      9. +-lft-identity 94.5%

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

      10. mul-1-neg 94.5%

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

      11. remove-double-neg 94.5%

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

   7. Simplified 94.5%

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

   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)) < 0.994999999999999996

   1. Initial program 100.0%

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

   if 0.994999999999999996 < (/.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 42.8%

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

   2. Taylor expanded in beta around inf 99.3%

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

   3. Taylor expanded in alpha around 0 99.3%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.5:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha} + \frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{elif}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq 0.995:\\ \;\;\;\;\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + \left(2 + 2 \cdot i\right)}}{2}\\ \end{array} \]

Alternative 4: 87.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.8 \cdot 10^{+22}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + \left(2 + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha} + \frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 2.8e+22)
   (/ (+ 1.0 (/ beta (+ beta (+ 2.0 (* 2.0 i))))) 2.0)
   (/ (+ (* 4.0 (/ i alpha)) (/ (+ 2.0 (* beta 2.0)) alpha)) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2.8e+22) {
		tmp = (1.0 + (beta / (beta + (2.0 + (2.0 * i))))) / 2.0;
	} else {
		tmp = ((4.0 * (i / alpha)) + ((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 <= 2.8d+22) then
        tmp = (1.0d0 + (beta / (beta + (2.0d0 + (2.0d0 * i))))) / 2.0d0
    else
        tmp = ((4.0d0 * (i / alpha)) + ((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 <= 2.8e+22) {
		tmp = (1.0 + (beta / (beta + (2.0 + (2.0 * i))))) / 2.0;
	} else {
		tmp = ((4.0 * (i / alpha)) + ((2.0 + (beta * 2.0)) / alpha)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 2.8e+22:
		tmp = (1.0 + (beta / (beta + (2.0 + (2.0 * i))))) / 2.0
	else:
		tmp = ((4.0 * (i / alpha)) + ((2.0 + (beta * 2.0)) / alpha)) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 2.8e+22)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + Float64(2.0 + Float64(2.0 * i))))) / 2.0);
	else
		tmp = Float64(Float64(Float64(4.0 * Float64(i / alpha)) + Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 2.8e+22)
		tmp = (1.0 + (beta / (beta + (2.0 + (2.0 * i))))) / 2.0;
	else
		tmp = ((4.0 * (i / alpha)) + ((2.0 + (beta * 2.0)) / alpha)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 2.8e+22], N[(N[(1.0 + N[(beta / N[(beta + N[(2.0 + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(4.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.8e22

   1. Initial program 82.6%

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

   2. Taylor expanded in beta around inf 96.8%

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

   3. Taylor expanded in alpha around 0 96.8%

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

   if 2.8e22 < alpha

   1. Initial program 14.1%

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

      1. associate-/l/ 13.3%

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

      2. *-commutative 13.3%

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

      3. times-frac 26.3%

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

      4. associate-+l+ 26.3%

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

      5. fma-def 26.3%

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

      6. +-commutative 26.3%

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

      7. fma-def 26.3%

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

   3. Simplified 26.3%

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

   4. Taylor expanded in alpha around inf 7.1%

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

   5. Taylor expanded in i around 0 79.7%

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

      1. sub-neg 79.7%

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

      2. neg-mul-1 79.7%

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

      3. associate-+r+ 79.7%

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

      4. +-commutative 79.7%

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

      5. neg-mul-1 79.7%

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

      6. distribute-lft1-in 79.7%

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

      7. metadata-eval 79.7%

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

      8. mul0-lft 79.7%

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

      9. +-lft-identity 79.7%

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

      10. mul-1-neg 79.7%

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

      11. remove-double-neg 79.7%

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

   7. Simplified 79.7%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.8 \cdot 10^{+22}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + \left(2 + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha} + \frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 5: 87.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 3.8 \cdot 10^{+18}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha} + \frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 3.8e+18)
   (/ (+ 1.0 (/ beta (+ 2.0 (+ (+ alpha beta) (* 2.0 i))))) 2.0)
   (/ (+ (* 4.0 (/ i alpha)) (/ (+ 2.0 (* beta 2.0)) alpha)) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 3.8e+18) {
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else {
		tmp = ((4.0 * (i / alpha)) + ((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 <= 3.8d+18) then
        tmp = (1.0d0 + (beta / (2.0d0 + ((alpha + beta) + (2.0d0 * i))))) / 2.0d0
    else
        tmp = ((4.0d0 * (i / alpha)) + ((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 <= 3.8e+18) {
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else {
		tmp = ((4.0 * (i / alpha)) + ((2.0 + (beta * 2.0)) / alpha)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 3.8e+18:
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0
	else:
		tmp = ((4.0 * (i / alpha)) + ((2.0 + (beta * 2.0)) / alpha)) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 3.8e+18)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(2.0 + Float64(Float64(alpha + beta) + Float64(2.0 * i))))) / 2.0);
	else
		tmp = Float64(Float64(Float64(4.0 * Float64(i / alpha)) + Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 3.8e+18)
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	else
		tmp = ((4.0 * (i / alpha)) + ((2.0 + (beta * 2.0)) / alpha)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 3.8e+18], N[(N[(1.0 + N[(beta / N[(2.0 + N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(4.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 3.8e18

   1. Initial program 82.6%

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

   2. Taylor expanded in beta around inf 96.8%

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

   if 3.8e18 < alpha

   1. Initial program 14.1%

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

      1. associate-/l/ 13.3%

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

      2. *-commutative 13.3%

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

      3. times-frac 26.3%

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

      4. associate-+l+ 26.3%

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

      5. fma-def 26.3%

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

      6. +-commutative 26.3%

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

      7. fma-def 26.3%

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

   3. Simplified 26.3%

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

   4. Taylor expanded in alpha around inf 7.1%

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

   5. Taylor expanded in i around 0 79.7%

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

      1. sub-neg 79.7%

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

      2. neg-mul-1 79.7%

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

      3. associate-+r+ 79.7%

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

      4. +-commutative 79.7%

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

      5. neg-mul-1 79.7%

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

      6. distribute-lft1-in 79.7%

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

      7. metadata-eval 79.7%

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

      8. mul0-lft 79.7%

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

      9. +-lft-identity 79.7%

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

      10. mul-1-neg 79.7%

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

      11. remove-double-neg 79.7%

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

   7. Simplified 79.7%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 3.8 \cdot 10^{+18}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha} + \frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 6: 76.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.6 \cdot 10^{+21}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 3.8 \cdot 10^{+276} \lor \neg \left(\alpha \leq 1.55 \cdot 10^{+304}\right):\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1.6e+21)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (if (or (<= alpha 3.8e+276) (not (<= alpha 1.55e+304)))
     (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)
     (/ (* 4.0 (/ i alpha)) 2.0))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.6e+21) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else if ((alpha <= 3.8e+276) || !(alpha <= 1.55e+304)) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = (4.0 * (i / alpha)) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 1.6d+21) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else if ((alpha <= 3.8d+276) .or. (.not. (alpha <= 1.55d+304))) then
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    else
        tmp = (4.0d0 * (i / alpha)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.6e+21) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else if ((alpha <= 3.8e+276) || !(alpha <= 1.55e+304)) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = (4.0 * (i / alpha)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 1.6e+21:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	elif (alpha <= 3.8e+276) or not (alpha <= 1.55e+304):
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	else:
		tmp = (4.0 * (i / alpha)) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 1.6e+21)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	elseif ((alpha <= 3.8e+276) || !(alpha <= 1.55e+304))
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(4.0 * Float64(i / alpha)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 1.6e+21)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	elseif ((alpha <= 3.8e+276) || ~((alpha <= 1.55e+304)))
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	else
		tmp = (4.0 * (i / alpha)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 1.6e+21], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[alpha, 3.8e+276], N[Not[LessEqual[alpha, 1.55e+304]], $MachinePrecision]], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(4.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 1.6e21

   1. Initial program 82.6%

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

   2. Taylor expanded in beta around inf 63.4%

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

      1. cancel-sign-sub-inv 63.4%

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

      2. mul-1-neg 63.4%

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

      3. sub-neg 63.4%

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

      4. metadata-eval 63.4%

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

   4. Simplified 63.4%

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

   5. Taylor expanded in alpha around 0 61.8%

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

   6. Taylor expanded in i around 0 90.0%

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

   if 1.6e21 < alpha < 3.79999999999999976e276 or 1.54999999999999995e304 < alpha

   1. Initial program 15.8%

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

      1. associate-/l/ 15.1%

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

      2. *-commutative 15.1%

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

      3. times-frac 29.1%

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

      4. associate-+l+ 29.1%

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

      5. fma-def 29.1%

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

      6. +-commutative 29.1%

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

      7. fma-def 29.1%

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

   3. Simplified 29.1%

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

   4. Taylor expanded in alpha around inf 7.3%

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

   5. Taylor expanded in i around 0 61.7%

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

      1. distribute-lft1-in 61.7%

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

      2. metadata-eval 61.7%

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

      3. mul0-lft 61.7%

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

      4. neg-sub0 61.7%

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

      5. mul-1-neg 61.7%

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

      6. remove-double-neg 61.7%

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

   7. Simplified 61.7%

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

   if 3.79999999999999976e276 < alpha < 1.54999999999999995e304

   1. Initial program 1.1%

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

      1. associate-/l/ 0.0%

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

      2. *-commutative 0.0%

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

      3. times-frac 5.7%

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

      4. associate-+l+ 5.7%

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

      5. fma-def 5.7%

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

      6. +-commutative 5.7%

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

      7. fma-def 5.7%

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

   3. Simplified 5.7%

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

   4. Taylor expanded in alpha around inf 5.7%

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

   5. Taylor expanded in i around inf 79.7%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.6 \cdot 10^{+21}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 3.8 \cdot 10^{+276} \lor \neg \left(\alpha \leq 1.55 \cdot 10^{+304}\right):\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha}}{2}\\ \end{array} \]

Alternative 7: 79.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.8 \cdot 10^{+22}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 2.8e+22)
   (/ (+ 1.0 (/ beta (+ beta (+ alpha 2.0)))) 2.0)
   (/ (+ (* 4.0 (/ i alpha)) (* 2.0 (/ 1.0 alpha))) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2.8e+22) {
		tmp = (1.0 + (beta / (beta + (alpha + 2.0)))) / 2.0;
	} else {
		tmp = ((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 2.8d+22) then
        tmp = (1.0d0 + (beta / (beta + (alpha + 2.0d0)))) / 2.0d0
    else
        tmp = ((4.0d0 * (i / alpha)) + (2.0d0 * (1.0d0 / alpha))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2.8e+22) {
		tmp = (1.0 + (beta / (beta + (alpha + 2.0)))) / 2.0;
	} else {
		tmp = ((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 2.8e+22:
		tmp = (1.0 + (beta / (beta + (alpha + 2.0)))) / 2.0
	else:
		tmp = ((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha))) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 2.8e+22)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + Float64(alpha + 2.0)))) / 2.0);
	else
		tmp = Float64(Float64(Float64(4.0 * Float64(i / alpha)) + Float64(2.0 * Float64(1.0 / alpha))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 2.8e+22)
		tmp = (1.0 + (beta / (beta + (alpha + 2.0)))) / 2.0;
	else
		tmp = ((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 2.8e+22], N[(N[(1.0 + N[(beta / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(4.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.8e22

   1. Initial program 82.6%

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

   2. Taylor expanded in beta around inf 96.8%

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

   3. Taylor expanded in i around 0 90.0%

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

      1. +-commutative 90.0%

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

   5. Simplified 90.0%

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

   if 2.8e22 < alpha

   1. Initial program 14.1%

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

      1. associate-/l/ 13.3%

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

      2. *-commutative 13.3%

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

      3. times-frac 26.3%

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

      4. associate-+l+ 26.3%

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

      5. fma-def 26.3%

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

      6. +-commutative 26.3%

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

      7. fma-def 26.3%

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

   3. Simplified 26.3%

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

   4. Taylor expanded in alpha around inf 7.1%

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

   5. Taylor expanded in beta around 0 66.8%

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

   6. Taylor expanded in i around 0 66.8%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.8 \cdot 10^{+22}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}}{2}\\ \end{array} \]

Alternative 8: 83.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 3.7 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + \left(2 + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 3.7e+15)
   (/ (+ 1.0 (/ beta (+ beta (+ 2.0 (* 2.0 i))))) 2.0)
   (/ (+ (* 4.0 (/ i alpha)) (* 2.0 (/ 1.0 alpha))) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 3.7e+15) {
		tmp = (1.0 + (beta / (beta + (2.0 + (2.0 * i))))) / 2.0;
	} else {
		tmp = ((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 3.7d+15) then
        tmp = (1.0d0 + (beta / (beta + (2.0d0 + (2.0d0 * i))))) / 2.0d0
    else
        tmp = ((4.0d0 * (i / alpha)) + (2.0d0 * (1.0d0 / alpha))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 3.7e+15) {
		tmp = (1.0 + (beta / (beta + (2.0 + (2.0 * i))))) / 2.0;
	} else {
		tmp = ((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 3.7e+15:
		tmp = (1.0 + (beta / (beta + (2.0 + (2.0 * i))))) / 2.0
	else:
		tmp = ((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha))) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 3.7e+15)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + Float64(2.0 + Float64(2.0 * i))))) / 2.0);
	else
		tmp = Float64(Float64(Float64(4.0 * Float64(i / alpha)) + Float64(2.0 * Float64(1.0 / alpha))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 3.7e+15)
		tmp = (1.0 + (beta / (beta + (2.0 + (2.0 * i))))) / 2.0;
	else
		tmp = ((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 3.7e+15], N[(N[(1.0 + N[(beta / N[(beta + N[(2.0 + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(4.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 3.7e15

   1. Initial program 82.6%

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

   2. Taylor expanded in beta around inf 96.8%

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

   3. Taylor expanded in alpha around 0 96.8%

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

   if 3.7e15 < alpha

   1. Initial program 14.1%

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

      1. associate-/l/ 13.3%

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

      2. *-commutative 13.3%

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

      3. times-frac 26.3%

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

      4. associate-+l+ 26.3%

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

      5. fma-def 26.3%

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

      6. +-commutative 26.3%

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

      7. fma-def 26.3%

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

   3. Simplified 26.3%

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

   4. Taylor expanded in alpha around inf 7.1%

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

   5. Taylor expanded in beta around 0 66.8%

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

   6. Taylor expanded in i around 0 66.8%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 3.7 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + \left(2 + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}}{2}\\ \end{array} \]

Alternative 9: 87.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 4.4 \cdot 10^{+21}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + \left(2 + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(2 + \beta \cdot 2\right) + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 4.4e+21)
   (/ (+ 1.0 (/ beta (+ beta (+ 2.0 (* 2.0 i))))) 2.0)
   (/ (/ (+ (+ 2.0 (* beta 2.0)) (* i 4.0)) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 4.4e+21) {
		tmp = (1.0 + (beta / (beta + (2.0 + (2.0 * i))))) / 2.0;
	} else {
		tmp = (((2.0 + (beta * 2.0)) + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 4.4d+21) then
        tmp = (1.0d0 + (beta / (beta + (2.0d0 + (2.0d0 * i))))) / 2.0d0
    else
        tmp = (((2.0d0 + (beta * 2.0d0)) + (i * 4.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 4.4e+21) {
		tmp = (1.0 + (beta / (beta + (2.0 + (2.0 * i))))) / 2.0;
	} else {
		tmp = (((2.0 + (beta * 2.0)) + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 4.4e+21:
		tmp = (1.0 + (beta / (beta + (2.0 + (2.0 * i))))) / 2.0
	else:
		tmp = (((2.0 + (beta * 2.0)) + (i * 4.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 4.4e+21)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + Float64(2.0 + Float64(2.0 * i))))) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) + Float64(i * 4.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 4.4e+21)
		tmp = (1.0 + (beta / (beta + (2.0 + (2.0 * i))))) / 2.0;
	else
		tmp = (((2.0 + (beta * 2.0)) + (i * 4.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 4.4e+21], N[(N[(1.0 + N[(beta / N[(beta + N[(2.0 + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 4.4e21

   1. Initial program 82.6%

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

   2. Taylor expanded in beta around inf 96.8%

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

   3. Taylor expanded in alpha around 0 96.8%

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

   if 4.4e21 < alpha

   1. Initial program 14.1%

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

      1. associate-/l/ 13.3%

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

      2. *-commutative 13.3%

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

      3. times-frac 26.3%

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

      4. associate-+l+ 26.3%

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

      5. fma-def 26.3%

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

      6. +-commutative 26.3%

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

      7. fma-def 26.3%

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

   3. Simplified 26.3%

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

   4. Taylor expanded in alpha around inf 7.1%

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

   5. Taylor expanded in beta around 0 7.1%

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

   6. Taylor expanded in alpha around 0 79.7%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 4.4 \cdot 10^{+21}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + \left(2 + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(2 + \beta \cdot 2\right) + i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 10: 74.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.15 \cdot 10^{+22}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 1.4 \cdot 10^{+274} \lor \neg \left(\alpha \leq 3.6 \cdot 10^{+304}\right):\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1.15e+22)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (if (or (<= alpha 1.4e+274) (not (<= alpha 3.6e+304)))
     (/ (/ 2.0 alpha) 2.0)
     (/ (* 4.0 (/ i alpha)) 2.0))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.15e+22) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else if ((alpha <= 1.4e+274) || !(alpha <= 3.6e+304)) {
		tmp = (2.0 / alpha) / 2.0;
	} else {
		tmp = (4.0 * (i / alpha)) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 1.15d+22) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else if ((alpha <= 1.4d+274) .or. (.not. (alpha <= 3.6d+304))) then
        tmp = (2.0d0 / alpha) / 2.0d0
    else
        tmp = (4.0d0 * (i / alpha)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.15e+22) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else if ((alpha <= 1.4e+274) || !(alpha <= 3.6e+304)) {
		tmp = (2.0 / alpha) / 2.0;
	} else {
		tmp = (4.0 * (i / alpha)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 1.15e+22:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	elif (alpha <= 1.4e+274) or not (alpha <= 3.6e+304):
		tmp = (2.0 / alpha) / 2.0
	else:
		tmp = (4.0 * (i / alpha)) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 1.15e+22)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	elseif ((alpha <= 1.4e+274) || !(alpha <= 3.6e+304))
		tmp = Float64(Float64(2.0 / alpha) / 2.0);
	else
		tmp = Float64(Float64(4.0 * Float64(i / alpha)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 1.15e+22)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	elseif ((alpha <= 1.4e+274) || ~((alpha <= 3.6e+304)))
		tmp = (2.0 / alpha) / 2.0;
	else
		tmp = (4.0 * (i / alpha)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 1.15e+22], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[alpha, 1.4e+274], N[Not[LessEqual[alpha, 3.6e+304]], $MachinePrecision]], N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(4.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 1.1500000000000001e22

   1. Initial program 82.6%

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

   2. Taylor expanded in beta around inf 63.4%

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

      1. cancel-sign-sub-inv 63.4%

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

      2. mul-1-neg 63.4%

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

      3. sub-neg 63.4%

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

      4. metadata-eval 63.4%

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

   4. Simplified 63.4%

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

   5. Taylor expanded in alpha around 0 61.8%

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

   6. Taylor expanded in i around 0 90.0%

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

   if 1.1500000000000001e22 < alpha < 1.40000000000000004e274 or 3.59999999999999988e304 < alpha

   1. Initial program 16.1%

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

      1. associate-/l/ 15.3%

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

      2. *-commutative 15.3%

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

      3. times-frac 29.5%

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

      4. associate-+l+ 29.5%

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

      5. fma-def 29.5%

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

      6. +-commutative 29.5%

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

      7. fma-def 29.5%

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

   3. Simplified 29.5%

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

   4. Taylor expanded in alpha around inf 7.3%

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

   5. Taylor expanded in beta around 0 63.1%

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

   6. Taylor expanded in i around 0 47.6%

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

   if 1.40000000000000004e274 < alpha < 3.59999999999999988e304

   1. Initial program 1.2%

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

      1. associate-/l/ 0.0%

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

      2. *-commutative 0.0%

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

      3. times-frac 5.9%

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

      4. associate-+l+ 5.9%

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

      5. fma-def 5.9%

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

      6. +-commutative 5.9%

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

      7. fma-def 5.9%

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

   3. Simplified 5.9%

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

   4. Taylor expanded in alpha around inf 5.9%

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

   5. Taylor expanded in i around inf 72.8%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.15 \cdot 10^{+22}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 1.4 \cdot 10^{+274} \lor \neg \left(\alpha \leq 3.6 \cdot 10^{+304}\right):\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha}}{2}\\ \end{array} \]

Alternative 11: 79.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 6.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 6.2e+14)
   (/ (+ 1.0 (/ beta (+ beta (+ alpha 2.0)))) 2.0)
   (/ (/ (+ 2.0 (* i 4.0)) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 6.2e+14) {
		tmp = (1.0 + (beta / (beta + (alpha + 2.0)))) / 2.0;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 6.2d+14) then
        tmp = (1.0d0 + (beta / (beta + (alpha + 2.0d0)))) / 2.0d0
    else
        tmp = ((2.0d0 + (i * 4.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 6.2e+14) {
		tmp = (1.0 + (beta / (beta + (alpha + 2.0)))) / 2.0;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 6.2e+14:
		tmp = (1.0 + (beta / (beta + (alpha + 2.0)))) / 2.0
	else:
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 6.2e+14)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + Float64(alpha + 2.0)))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(i * 4.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 6.2e+14)
		tmp = (1.0 + (beta / (beta + (alpha + 2.0)))) / 2.0;
	else
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 6.2e+14], N[(N[(1.0 + N[(beta / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 6.2e14

   1. Initial program 82.6%

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

   2. Taylor expanded in beta around inf 96.8%

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

   3. Taylor expanded in i around 0 90.0%

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

      1. +-commutative 90.0%

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

   5. Simplified 90.0%

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

   if 6.2e14 < alpha

   1. Initial program 14.1%

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

      1. associate-/l/ 13.3%

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

      2. *-commutative 13.3%

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

      3. times-frac 26.3%

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

      4. associate-+l+ 26.3%

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

      5. fma-def 26.3%

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

      6. +-commutative 26.3%

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

      7. fma-def 26.3%

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

   3. Simplified 26.3%

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

   4. Taylor expanded in alpha around inf 7.1%

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

   5. Taylor expanded in beta around 0 66.8%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 6.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 12: 79.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 6.5 \cdot 10^{+20}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 6.5e+20)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (+ 2.0 (* i 4.0)) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 6.5e+20) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 6.5d+20) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((2.0d0 + (i * 4.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 6.5e+20) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 6.5e+20:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 6.5e+20)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(i * 4.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 6.5e+20)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 6.5e+20], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 6.5e20

   1. Initial program 82.6%

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

   2. Taylor expanded in beta around inf 63.4%

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

      1. cancel-sign-sub-inv 63.4%

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

      2. mul-1-neg 63.4%

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

      3. sub-neg 63.4%

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

      4. metadata-eval 63.4%

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

   4. Simplified 63.4%

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

   5. Taylor expanded in alpha around 0 61.8%

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

   6. Taylor expanded in i around 0 90.0%

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

   if 6.5e20 < alpha

   1. Initial program 14.1%

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

      1. associate-/l/ 13.3%

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

      2. *-commutative 13.3%

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

      3. times-frac 26.3%

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

      4. associate-+l+ 26.3%

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

      5. fma-def 26.3%

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

      6. +-commutative 26.3%

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

      7. fma-def 26.3%

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

   3. Simplified 26.3%

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

   4. Taylor expanded in alpha around inf 7.1%

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

   5. Taylor expanded in beta around 0 66.8%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 6.5 \cdot 10^{+20}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 13: 73.4% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.8 \cdot 10^{+61}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 (if (<= beta 2.8e+61) 0.5 1.0))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.8e+61) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 2.8d+61) then
        tmp = 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.8e+61) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 2.8e+61:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 2.8e+61)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 2.8e+61)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 2.8e+61], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 2.8000000000000001e61

   1. Initial program 71.9%

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

      1. associate-/l/ 71.7%

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

      2. *-commutative 71.7%

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

      3. times-frac 73.5%

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

      4. associate-+l+ 73.5%

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

      5. fma-def 73.5%

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

      6. +-commutative 73.5%

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

      7. fma-def 73.5%

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

   3. Simplified 73.5%

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

   4. Taylor expanded in i around inf 68.0%

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

   if 2.8000000000000001e61 < beta

   1. Initial program 39.5%

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

      1. associate-/l/ 37.9%

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

      2. *-commutative 37.9%

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

      3. times-frac 89.8%

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

      4. associate-+l+ 89.8%

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

      5. fma-def 89.8%

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

      6. +-commutative 89.8%

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

      7. fma-def 89.8%

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

   3. Simplified 89.8%

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

   4. Taylor expanded in beta around inf 76.1%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.8 \cdot 10^{+61}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 14: 61.4% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[alpha_, beta_, i_] := 0.5

Derivation

1. Initial program 62.5%

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

   1. associate-/l/ 61.9%

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

   2. *-commutative 61.9%

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

   3. times-frac 78.2%

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

   4. associate-+l+ 78.2%

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

   5. fma-def 78.2%

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

   6. +-commutative 78.2%

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

   7. fma-def 78.2%

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

3. Simplified 78.2%

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

4. Taylor expanded in i around inf 56.6%

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

5. Final simplification 56.6%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023189 
(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))