Octave 3.8, jcobi/2

Percentage Accurate: 63.6% → 97.7%

Time: 15.1s

Alternatives: 10

Speedup: 4.8×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 63.6% 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: 97.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq -0.99999999:\\ \;\;\;\;\frac{\left(1 + 0.5 \cdot \left(\beta \cdot 2\right)\right) + i \cdot \left(2 + \frac{i \cdot -6}{\alpha}\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)}, \frac{\alpha + \beta}{\alpha + \left(\beta + 2 \cdot i\right)}, 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (if (<=
        (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0))
        -0.99999999)
     (/
      (+ (+ 1.0 (* 0.5 (* beta 2.0))) (* i (+ 2.0 (/ (* i -6.0) alpha))))
      alpha)
     (/
      (fma
       (/ (- beta alpha) (+ (+ alpha beta) (+ 2.0 (* 2.0 i))))
       (/ (+ alpha beta) (+ alpha (+ beta (* 2.0 i))))
       1.0)
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.99999999) {
		tmp = ((1.0 + (0.5 * (beta * 2.0))) + (i * (2.0 + ((i * -6.0) / alpha)))) / alpha;
	} else {
		tmp = fma(((beta - alpha) / ((alpha + beta) + (2.0 + (2.0 * i)))), ((alpha + beta) / (alpha + (beta + (2.0 * i)))), 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0)) <= -0.99999999)
		tmp = Float64(Float64(Float64(1.0 + Float64(0.5 * Float64(beta * 2.0))) + Float64(i * Float64(2.0 + Float64(Float64(i * -6.0) / alpha)))) / alpha);
	else
		tmp = Float64(fma(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + Float64(2.0 + Float64(2.0 * i)))), Float64(Float64(alpha + beta) / Float64(alpha + Float64(beta + Float64(2.0 * i)))), 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], -0.99999999], N[(N[(N[(1.0 + N[(0.5 * N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(i * N[(2.0 + N[(N[(i * -6.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] / N[(alpha + N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.99999998999999995

   1. Initial program 3.0%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1\right), \color{blue}{2}\right) \]

   3. Simplified 13.8%

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

   5. Taylor expanded in alpha around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)\right) + \frac{1}{2} \cdot \frac{{\beta}^{2} - \left(-1 \cdot \left(\left(2 + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\beta + 2 \cdot i\right)\right) + \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right) \cdot \left(\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)\right)\right)}{\alpha}\right), \color{blue}{\alpha}\right) \]

   7. Simplified 73.4%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(\left(0 \cdot \beta + 1 \cdot \left(\left(2 + \beta \cdot 2\right) + i \cdot 4\right)\right) + \frac{\left(\beta \cdot \beta - \left(-2 - \left(\beta + 2 \cdot i\right)\right) \cdot \left(\beta + 2 \cdot i\right)\right) - \left(\left(2 + \beta \cdot 2\right) + i \cdot 4\right) \cdot \left(0 \cdot \beta + 1 \cdot \left(\left(2 + \beta \cdot 2\right) + i \cdot 4\right)\right)}{\alpha}\right)}{\alpha}} \]

   8. Taylor expanded in i around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(0, \beta\right), \mathsf{*.f64}\left(1, \mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\beta, 2\right)\right), \mathsf{*.f64}\left(i, 4\right)\right)\right)\right), \mathsf{/.f64}\left(\color{blue}{\left(-12 \cdot {i}^{2}\right)}, \alpha\right)\right)\right), \alpha\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(0, \beta\right), \mathsf{*.f64}\left(1, \mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\beta, 2\right)\right), \mathsf{*.f64}\left(i, 4\right)\right)\right)\right), \mathsf{/.f64}\left(\left({i}^{2} \cdot -12\right), \alpha\right)\right)\right), \alpha\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(0, \beta\right), \mathsf{*.f64}\left(1, \mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\beta, 2\right)\right), \mathsf{*.f64}\left(i, 4\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({i}^{2}\right), -12\right), \alpha\right)\right)\right), \alpha\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(0, \beta\right), \mathsf{*.f64}\left(1, \mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\beta, 2\right)\right), \mathsf{*.f64}\left(i, 4\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(i \cdot i\right), -12\right), \alpha\right)\right)\right), \alpha\right) \]

      4. *-lowering-*.f64 78.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(0, \beta\right), \mathsf{*.f64}\left(1, \mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\beta, 2\right)\right), \mathsf{*.f64}\left(i, 4\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), -12\right), \alpha\right)\right)\right), \alpha\right) \]

   10. Simplified 78.0%

       \[\leadsto \frac{0.5 \cdot \left(\left(0 \cdot \beta + 1 \cdot \left(\left(2 + \beta \cdot 2\right) + i \cdot 4\right)\right) + \frac{\color{blue}{\left(i \cdot i\right) \cdot -12}}{\alpha}\right)}{\alpha} \]

   11. Taylor expanded in i around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot \left(2 + 2 \cdot \beta\right) + i \cdot \left(2 + -6 \cdot \frac{i}{\alpha}\right)\right)}, \alpha\right) \]
   12. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \left(2 + 2 \cdot \beta\right)\right), \left(i \cdot \left(2 + -6 \cdot \frac{i}{\alpha}\right)\right)\right), \alpha\right) \]

       2. distribute-lft-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{2} \cdot 2 + \frac{1}{2} \cdot \left(2 \cdot \beta\right)\right), \left(i \cdot \left(2 + -6 \cdot \frac{i}{\alpha}\right)\right)\right), \alpha\right) \]

       3. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(1 + \frac{1}{2} \cdot \left(2 \cdot \beta\right)\right), \left(i \cdot \left(2 + -6 \cdot \frac{i}{\alpha}\right)\right)\right), \alpha\right) \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \left(2 \cdot \beta\right)\right)\right), \left(i \cdot \left(2 + -6 \cdot \frac{i}{\alpha}\right)\right)\right), \alpha\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(2 \cdot \beta\right)\right)\right), \left(i \cdot \left(2 + -6 \cdot \frac{i}{\alpha}\right)\right)\right), \alpha\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(2, \beta\right)\right)\right), \left(i \cdot \left(2 + -6 \cdot \frac{i}{\alpha}\right)\right)\right), \alpha\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(2, \beta\right)\right)\right), \mathsf{*.f64}\left(i, \left(2 + -6 \cdot \frac{i}{\alpha}\right)\right)\right), \alpha\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(2, \beta\right)\right)\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(2, \left(-6 \cdot \frac{i}{\alpha}\right)\right)\right)\right), \alpha\right) \]

       9. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(2, \beta\right)\right)\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(2, \left(\frac{-6 \cdot i}{\alpha}\right)\right)\right)\right), \alpha\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(2, \beta\right)\right)\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\left(-6 \cdot i\right), \alpha\right)\right)\right)\right), \alpha\right) \]

       11. *-lowering-*.f64 89.7%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(2, \beta\right)\right)\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-6, i\right), \alpha\right)\right)\right)\right), \alpha\right) \]

   13. Simplified 89.7%

       \[\leadsto \frac{\color{blue}{\left(1 + 0.5 \cdot \left(2 \cdot \beta\right)\right) + i \cdot \left(2 + \frac{-6 \cdot i}{\alpha}\right)}}{\alpha} \]

   if -0.99999998999999995 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 80.9%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1\right), \color{blue}{2}\right) \]

   3. Simplified 85.6%

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

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)} + 1\right), 2\right) \]

      2. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)\right)} + 1\right), 2\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right)} + 1\right), 2\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)} + 1\right), 2\right) \]

      5. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\alpha + \beta}{\alpha + \left(\beta + 2 \cdot i\right)} \cdot \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1\right), 2\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \cdot \frac{\alpha + \beta}{\alpha + \left(\beta + 2 \cdot i\right)} + 1\right), 2\right) \]

      7. fma-define N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{fma}\left(\frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, \frac{\alpha + \beta}{\alpha + \left(\beta + 2 \cdot i\right)}, 1\right)\right), 2\right) \]

      8. fma-lowering-fma.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\left(\frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}\right), \left(\frac{\alpha + \beta}{\alpha + \left(\beta + 2 \cdot i\right)}\right), 1\right), 2\right) \]

   6. Applied egg-rr 99.6%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.99999999:\\ \;\;\;\;\frac{\left(1 + 0.5 \cdot \left(\beta \cdot 2\right)\right) + i \cdot \left(2 + \frac{i \cdot -6}{\alpha}\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)}, \frac{\alpha + \beta}{\alpha + \left(\beta + 2 \cdot i\right)}, 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{2 + t\_0} \leq -0.99999999:\\ \;\;\;\;\frac{\left(1 + 0.5 \cdot \left(\beta \cdot 2\right)\right) + i \cdot \left(2 + \frac{i \cdot -6}{\alpha}\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\left(-2 + i \cdot -2\right) - \left(\alpha + \beta\right)} \cdot \frac{\alpha + \beta}{i \cdot -2 - \left(\alpha + \beta\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (if (<=
        (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0))
        -0.99999999)
     (/
      (+ (+ 1.0 (* 0.5 (* beta 2.0))) (* i (+ 2.0 (/ (* i -6.0) alpha))))
      alpha)
     (/
      (+
       1.0
       (*
        (/ (- beta alpha) (- (+ -2.0 (* i -2.0)) (+ alpha beta)))
        (/ (+ alpha beta) (- (* i -2.0) (+ alpha beta)))))
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.99999999) {
		tmp = ((1.0 + (0.5 * (beta * 2.0))) + (i * (2.0 + ((i * -6.0) / alpha)))) / alpha;
	} else {
		tmp = (1.0 + (((beta - alpha) / ((-2.0 + (i * -2.0)) - (alpha + beta))) * ((alpha + beta) / ((i * -2.0) - (alpha + beta))))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0d0 + t_0)) <= (-0.99999999d0)) then
        tmp = ((1.0d0 + (0.5d0 * (beta * 2.0d0))) + (i * (2.0d0 + ((i * (-6.0d0)) / alpha)))) / alpha
    else
        tmp = (1.0d0 + (((beta - alpha) / (((-2.0d0) + (i * (-2.0d0))) - (alpha + beta))) * ((alpha + beta) / ((i * (-2.0d0)) - (alpha + beta))))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.99999999) {
		tmp = ((1.0 + (0.5 * (beta * 2.0))) + (i * (2.0 + ((i * -6.0) / alpha)))) / alpha;
	} else {
		tmp = (1.0 + (((beta - alpha) / ((-2.0 + (i * -2.0)) - (alpha + beta))) * ((alpha + beta) / ((i * -2.0) - (alpha + beta))))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	tmp = 0
	if ((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.99999999:
		tmp = ((1.0 + (0.5 * (beta * 2.0))) + (i * (2.0 + ((i * -6.0) / alpha)))) / alpha
	else:
		tmp = (1.0 + (((beta - alpha) / ((-2.0 + (i * -2.0)) - (alpha + beta))) * ((alpha + beta) / ((i * -2.0) - (alpha + beta))))) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0)) <= -0.99999999)
		tmp = Float64(Float64(Float64(1.0 + Float64(0.5 * Float64(beta * 2.0))) + Float64(i * Float64(2.0 + Float64(Float64(i * -6.0) / alpha)))) / alpha);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(beta - alpha) / Float64(Float64(-2.0 + Float64(i * -2.0)) - Float64(alpha + beta))) * Float64(Float64(alpha + beta) / Float64(Float64(i * -2.0) - Float64(alpha + beta))))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	tmp = 0.0;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.99999999)
		tmp = ((1.0 + (0.5 * (beta * 2.0))) + (i * (2.0 + ((i * -6.0) / alpha)))) / alpha;
	else
		tmp = (1.0 + (((beta - alpha) / ((-2.0 + (i * -2.0)) - (alpha + beta))) * ((alpha + beta) / ((i * -2.0) - (alpha + beta))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision], -0.99999999], N[(N[(N[(1.0 + N[(0.5 * N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(i * N[(2.0 + N[(N[(i * -6.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(1.0 + N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(-2.0 + N[(i * -2.0), $MachinePrecision]), $MachinePrecision] - N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] / N[(N[(i * -2.0), $MachinePrecision] - N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.99999998999999995

   1. Initial program 3.0%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1\right), \color{blue}{2}\right) \]

   3. Simplified 13.8%

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

   5. Taylor expanded in alpha around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)\right) + \frac{1}{2} \cdot \frac{{\beta}^{2} - \left(-1 \cdot \left(\left(2 + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\beta + 2 \cdot i\right)\right) + \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right) \cdot \left(\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)\right)\right)}{\alpha}\right), \color{blue}{\alpha}\right) \]

   7. Simplified 73.4%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(\left(0 \cdot \beta + 1 \cdot \left(\left(2 + \beta \cdot 2\right) + i \cdot 4\right)\right) + \frac{\left(\beta \cdot \beta - \left(-2 - \left(\beta + 2 \cdot i\right)\right) \cdot \left(\beta + 2 \cdot i\right)\right) - \left(\left(2 + \beta \cdot 2\right) + i \cdot 4\right) \cdot \left(0 \cdot \beta + 1 \cdot \left(\left(2 + \beta \cdot 2\right) + i \cdot 4\right)\right)}{\alpha}\right)}{\alpha}} \]

   8. Taylor expanded in i around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(0, \beta\right), \mathsf{*.f64}\left(1, \mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\beta, 2\right)\right), \mathsf{*.f64}\left(i, 4\right)\right)\right)\right), \mathsf{/.f64}\left(\color{blue}{\left(-12 \cdot {i}^{2}\right)}, \alpha\right)\right)\right), \alpha\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(0, \beta\right), \mathsf{*.f64}\left(1, \mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\beta, 2\right)\right), \mathsf{*.f64}\left(i, 4\right)\right)\right)\right), \mathsf{/.f64}\left(\left({i}^{2} \cdot -12\right), \alpha\right)\right)\right), \alpha\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(0, \beta\right), \mathsf{*.f64}\left(1, \mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\beta, 2\right)\right), \mathsf{*.f64}\left(i, 4\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({i}^{2}\right), -12\right), \alpha\right)\right)\right), \alpha\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(0, \beta\right), \mathsf{*.f64}\left(1, \mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\beta, 2\right)\right), \mathsf{*.f64}\left(i, 4\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(i \cdot i\right), -12\right), \alpha\right)\right)\right), \alpha\right) \]

      4. *-lowering-*.f64 78.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(0, \beta\right), \mathsf{*.f64}\left(1, \mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\beta, 2\right)\right), \mathsf{*.f64}\left(i, 4\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), -12\right), \alpha\right)\right)\right), \alpha\right) \]

   10. Simplified 78.0%

       \[\leadsto \frac{0.5 \cdot \left(\left(0 \cdot \beta + 1 \cdot \left(\left(2 + \beta \cdot 2\right) + i \cdot 4\right)\right) + \frac{\color{blue}{\left(i \cdot i\right) \cdot -12}}{\alpha}\right)}{\alpha} \]

   11. Taylor expanded in i around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot \left(2 + 2 \cdot \beta\right) + i \cdot \left(2 + -6 \cdot \frac{i}{\alpha}\right)\right)}, \alpha\right) \]
   12. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \left(2 + 2 \cdot \beta\right)\right), \left(i \cdot \left(2 + -6 \cdot \frac{i}{\alpha}\right)\right)\right), \alpha\right) \]

       2. distribute-lft-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{2} \cdot 2 + \frac{1}{2} \cdot \left(2 \cdot \beta\right)\right), \left(i \cdot \left(2 + -6 \cdot \frac{i}{\alpha}\right)\right)\right), \alpha\right) \]

       3. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(1 + \frac{1}{2} \cdot \left(2 \cdot \beta\right)\right), \left(i \cdot \left(2 + -6 \cdot \frac{i}{\alpha}\right)\right)\right), \alpha\right) \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \left(2 \cdot \beta\right)\right)\right), \left(i \cdot \left(2 + -6 \cdot \frac{i}{\alpha}\right)\right)\right), \alpha\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(2 \cdot \beta\right)\right)\right), \left(i \cdot \left(2 + -6 \cdot \frac{i}{\alpha}\right)\right)\right), \alpha\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(2, \beta\right)\right)\right), \left(i \cdot \left(2 + -6 \cdot \frac{i}{\alpha}\right)\right)\right), \alpha\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(2, \beta\right)\right)\right), \mathsf{*.f64}\left(i, \left(2 + -6 \cdot \frac{i}{\alpha}\right)\right)\right), \alpha\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(2, \beta\right)\right)\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(2, \left(-6 \cdot \frac{i}{\alpha}\right)\right)\right)\right), \alpha\right) \]

       9. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(2, \beta\right)\right)\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(2, \left(\frac{-6 \cdot i}{\alpha}\right)\right)\right)\right), \alpha\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(2, \beta\right)\right)\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\left(-6 \cdot i\right), \alpha\right)\right)\right)\right), \alpha\right) \]

       11. *-lowering-*.f64 89.7%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(2, \beta\right)\right)\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-6, i\right), \alpha\right)\right)\right)\right), \alpha\right) \]

   13. Simplified 89.7%

       \[\leadsto \frac{\color{blue}{\left(1 + 0.5 \cdot \left(2 \cdot \beta\right)\right) + i \cdot \left(2 + \frac{-6 \cdot i}{\alpha}\right)}}{\alpha} \]

   if -0.99999998999999995 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64)))

   1. Initial program 80.9%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1\right), \color{blue}{2}\right) \]

   3. Simplified 85.6%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\beta - \alpha}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)} \cdot \left(\alpha + \beta\right)\right), 1\right), 2\right) \]

      2. frac-2neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\mathsf{neg}\left(\left(\beta - \alpha\right)\right)}{\mathsf{neg}\left(\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)\right)} \cdot \left(\alpha + \beta\right)\right), 1\right), 2\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\left(\mathsf{neg}\left(\left(\beta - \alpha\right)\right)\right) \cdot \left(\alpha + \beta\right)}{\mathsf{neg}\left(\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)\right)}\right), 1\right), 2\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\left(\mathsf{neg}\left(\left(\beta - \alpha\right)\right)\right) \cdot \left(\alpha + \beta\right)}{\mathsf{neg}\left(\left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)\right)}\right), 1\right), 2\right) \]

      5. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\left(\mathsf{neg}\left(\left(\beta - \alpha\right)\right)\right) \cdot \left(\alpha + \beta\right)}{\mathsf{neg}\left(\left(\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)\right)}\right), 1\right), 2\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\left(\mathsf{neg}\left(\left(\beta - \alpha\right)\right)\right) \cdot \left(\alpha + \beta\right)}{\mathsf{neg}\left(\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)\right)}\right), 1\right), 2\right) \]

      7. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\left(\mathsf{neg}\left(\left(\beta - \alpha\right)\right)\right) \cdot \left(\alpha + \beta\right)}{\mathsf{neg}\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)\right)}\right), 1\right), 2\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\left(\mathsf{neg}\left(\left(\beta - \alpha\right)\right)\right) \cdot \left(\alpha + \beta\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right) \cdot \left(\mathsf{neg}\left(\left(\alpha + \left(\beta + 2 \cdot i\right)\right)\right)\right)}\right), 1\right), 2\right) \]

      9. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\left(\mathsf{neg}\left(\left(\beta - \alpha\right)\right)\right) \cdot \left(\alpha + \beta\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right) \cdot \left(\mathsf{neg}\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)\right)}\right), 1\right), 2\right) \]

      10. times-frac N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\mathsf{neg}\left(\left(\beta - \alpha\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \cdot \frac{\alpha + \beta}{\mathsf{neg}\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}\right), 1\right), 2\right) \]

   6. Applied egg-rr 99.5%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.99999999:\\ \;\;\;\;\frac{\left(1 + 0.5 \cdot \left(\beta \cdot 2\right)\right) + i \cdot \left(2 + \frac{i \cdot -6}{\alpha}\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\left(-2 + i \cdot -2\right) - \left(\alpha + \beta\right)} \cdot \frac{\alpha + \beta}{i \cdot -2 - \left(\alpha + \beta\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.9 \cdot 10^{+42}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + 0.5 \cdot \left(\beta \cdot 2\right)}{\alpha} + \frac{2 \cdot i}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1.9e+42)
   (/ (+ 1.0 (/ (- beta alpha) (+ 2.0 (+ (+ alpha beta) (* 2.0 i))))) 2.0)
   (+ (/ (+ 1.0 (* 0.5 (* beta 2.0))) alpha) (/ (* 2.0 i) alpha))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.9e+42) {
		tmp = (1.0 + ((beta - alpha) / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else {
		tmp = ((1.0 + (0.5 * (beta * 2.0))) / alpha) + ((2.0 * i) / alpha);
	}
	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 <= 1.9d+42) then
        tmp = (1.0d0 + ((beta - alpha) / (2.0d0 + ((alpha + beta) + (2.0d0 * i))))) / 2.0d0
    else
        tmp = ((1.0d0 + (0.5d0 * (beta * 2.0d0))) / alpha) + ((2.0d0 * i) / alpha)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.9e+42) {
		tmp = (1.0 + ((beta - alpha) / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else {
		tmp = ((1.0 + (0.5 * (beta * 2.0))) / alpha) + ((2.0 * i) / alpha);
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 1.9e+42:
		tmp = (1.0 + ((beta - alpha) / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0
	else:
		tmp = ((1.0 + (0.5 * (beta * 2.0))) / alpha) + ((2.0 * i) / alpha)
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 1.9e+42)
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / Float64(2.0 + Float64(Float64(alpha + beta) + Float64(2.0 * i))))) / 2.0);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(0.5 * Float64(beta * 2.0))) / alpha) + Float64(Float64(2.0 * i) / alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 1.9e+42)
		tmp = (1.0 + ((beta - alpha) / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	else
		tmp = ((1.0 + (0.5 * (beta * 2.0))) / alpha) + ((2.0 * i) / alpha);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 1.9e+42], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(1.0 + N[(0.5 * N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] + N[(N[(2.0 * i), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.8999999999999999e42

   1. Initial program 80.7%

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

   3. Taylor expanded in i around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(\beta - \alpha\right)}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), 2\right)\right), 1\right), 2\right) \]
   4. Step-by-step derivation

      1. --lowering--.f64 98.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), 2\right)\right), 1\right), 2\right) \]

   5. Simplified 98.1%

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

   if 1.8999999999999999e42 < alpha

   1. Initial program 13.2%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1\right), \color{blue}{2}\right) \]

   3. Simplified 23.4%

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

   5. Taylor expanded in alpha around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)\right) + \frac{1}{2} \cdot \frac{{\beta}^{2} - \left(-1 \cdot \left(\left(2 + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\beta + 2 \cdot i\right)\right) + \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right) \cdot \left(\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)\right)\right)}{\alpha}\right), \color{blue}{\alpha}\right) \]

   7. Simplified 62.4%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(\left(0 \cdot \beta + 1 \cdot \left(\left(2 + \beta \cdot 2\right) + i \cdot 4\right)\right) + \frac{\left(\beta \cdot \beta - \left(-2 - \left(\beta + 2 \cdot i\right)\right) \cdot \left(\beta + 2 \cdot i\right)\right) - \left(\left(2 + \beta \cdot 2\right) + i \cdot 4\right) \cdot \left(0 \cdot \beta + 1 \cdot \left(\left(2 + \beta \cdot 2\right) + i \cdot 4\right)\right)}{\alpha}\right)}{\alpha}} \]

   8. Taylor expanded in i around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(0, \beta\right), \mathsf{*.f64}\left(1, \mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\beta, 2\right)\right), \mathsf{*.f64}\left(i, 4\right)\right)\right)\right), \mathsf{/.f64}\left(\color{blue}{\left(-12 \cdot {i}^{2}\right)}, \alpha\right)\right)\right), \alpha\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(0, \beta\right), \mathsf{*.f64}\left(1, \mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\beta, 2\right)\right), \mathsf{*.f64}\left(i, 4\right)\right)\right)\right), \mathsf{/.f64}\left(\left({i}^{2} \cdot -12\right), \alpha\right)\right)\right), \alpha\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(0, \beta\right), \mathsf{*.f64}\left(1, \mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\beta, 2\right)\right), \mathsf{*.f64}\left(i, 4\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({i}^{2}\right), -12\right), \alpha\right)\right)\right), \alpha\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(0, \beta\right), \mathsf{*.f64}\left(1, \mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\beta, 2\right)\right), \mathsf{*.f64}\left(i, 4\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(i \cdot i\right), -12\right), \alpha\right)\right)\right), \alpha\right) \]

      4. *-lowering-*.f64 66.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(0, \beta\right), \mathsf{*.f64}\left(1, \mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\beta, 2\right)\right), \mathsf{*.f64}\left(i, 4\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), -12\right), \alpha\right)\right)\right), \alpha\right) \]

   10. Simplified 66.5%

       \[\leadsto \frac{0.5 \cdot \left(\left(0 \cdot \beta + 1 \cdot \left(\left(2 + \beta \cdot 2\right) + i \cdot 4\right)\right) + \frac{\color{blue}{\left(i \cdot i\right) \cdot -12}}{\alpha}\right)}{\alpha} \]

   11. Taylor expanded in i around 0

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

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{2 + 2 \cdot \beta}{\alpha}\right), \color{blue}{\left(2 \cdot \frac{i}{\alpha}\right)}\right) \]

       2. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(\left(\frac{\frac{1}{2} \cdot \left(2 + 2 \cdot \beta\right)}{\alpha}\right), \left(\color{blue}{2} \cdot \frac{i}{\alpha}\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(2 + 2 \cdot \beta\right)\right), \alpha\right), \left(\color{blue}{2} \cdot \frac{i}{\alpha}\right)\right) \]

       4. distribute-lft-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2} \cdot 2 + \frac{1}{2} \cdot \left(2 \cdot \beta\right)\right), \alpha\right), \left(2 \cdot \frac{i}{\alpha}\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(1 + \frac{1}{2} \cdot \left(2 \cdot \beta\right)\right), \alpha\right), \left(2 \cdot \frac{i}{\alpha}\right)\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \left(2 \cdot \beta\right)\right)\right), \alpha\right), \left(2 \cdot \frac{i}{\alpha}\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(2 \cdot \beta\right)\right)\right), \alpha\right), \left(2 \cdot \frac{i}{\alpha}\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(2, \beta\right)\right)\right), \alpha\right), \left(2 \cdot \frac{i}{\alpha}\right)\right) \]

       9. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(2, \beta\right)\right)\right), \alpha\right), \left(\frac{2 \cdot i}{\color{blue}{\alpha}}\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(2, \beta\right)\right)\right), \alpha\right), \mathsf{/.f64}\left(\left(2 \cdot i\right), \color{blue}{\alpha}\right)\right) \]

       11. *-lowering-*.f64 77.0%

           \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(2, \beta\right)\right)\right), \alpha\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, i\right), \alpha\right)\right) \]

   13. Simplified 77.0%

       \[\leadsto \color{blue}{\frac{1 + 0.5 \cdot \left(2 \cdot \beta\right)}{\alpha} + \frac{2 \cdot i}{\alpha}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.9 \cdot 10^{+42}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + 0.5 \cdot \left(\beta \cdot 2\right)}{\alpha} + \frac{2 \cdot i}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.05 \cdot 10^{+70}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + 0.5 \cdot \left(\beta \cdot 2\right)}{\alpha} + \frac{2 \cdot i}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 2.05e+70)
   (/ (+ 1.0 (/ beta (+ 2.0 (+ (+ alpha beta) (* 2.0 i))))) 2.0)
   (+ (/ (+ 1.0 (* 0.5 (* beta 2.0))) alpha) (/ (* 2.0 i) alpha))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2.05e+70) {
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else {
		tmp = ((1.0 + (0.5 * (beta * 2.0))) / alpha) + ((2.0 * i) / alpha);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 2.05d+70) then
        tmp = (1.0d0 + (beta / (2.0d0 + ((alpha + beta) + (2.0d0 * i))))) / 2.0d0
    else
        tmp = ((1.0d0 + (0.5d0 * (beta * 2.0d0))) / alpha) + ((2.0d0 * i) / alpha)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2.05e+70) {
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else {
		tmp = ((1.0 + (0.5 * (beta * 2.0))) / alpha) + ((2.0 * i) / alpha);
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 2.05e+70:
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0
	else:
		tmp = ((1.0 + (0.5 * (beta * 2.0))) / alpha) + ((2.0 * i) / alpha)
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 2.05e+70)
		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(1.0 + Float64(0.5 * Float64(beta * 2.0))) / alpha) + Float64(Float64(2.0 * i) / alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 2.05e+70)
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	else
		tmp = ((1.0 + (0.5 * (beta * 2.0))) / alpha) + ((2.0 * i) / alpha);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 2.05e+70], 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[(1.0 + N[(0.5 * N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] + N[(N[(2.0 * i), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.0500000000000001e70

   1. Initial program 78.9%

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

   3. Taylor expanded in beta around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\color{blue}{\beta}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), 2\right)\right), 1\right), 2\right) \]
   4. Step-by-step derivation

   5. Simplified 94.3%

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

   if 2.0500000000000001e70 < alpha

   1. Initial program 9.9%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1\right), \color{blue}{2}\right) \]

   3. Simplified 21.2%

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

   5. Taylor expanded in alpha around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)\right) + \frac{1}{2} \cdot \frac{{\beta}^{2} - \left(-1 \cdot \left(\left(2 + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\beta + 2 \cdot i\right)\right) + \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right) \cdot \left(\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)\right)\right)}{\alpha}\right), \color{blue}{\alpha}\right) \]

   7. Simplified 63.9%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(\left(0 \cdot \beta + 1 \cdot \left(\left(2 + \beta \cdot 2\right) + i \cdot 4\right)\right) + \frac{\left(\beta \cdot \beta - \left(-2 - \left(\beta + 2 \cdot i\right)\right) \cdot \left(\beta + 2 \cdot i\right)\right) - \left(\left(2 + \beta \cdot 2\right) + i \cdot 4\right) \cdot \left(0 \cdot \beta + 1 \cdot \left(\left(2 + \beta \cdot 2\right) + i \cdot 4\right)\right)}{\alpha}\right)}{\alpha}} \]

   8. Taylor expanded in i around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(0, \beta\right), \mathsf{*.f64}\left(1, \mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\beta, 2\right)\right), \mathsf{*.f64}\left(i, 4\right)\right)\right)\right), \mathsf{/.f64}\left(\color{blue}{\left(-12 \cdot {i}^{2}\right)}, \alpha\right)\right)\right), \alpha\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(0, \beta\right), \mathsf{*.f64}\left(1, \mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\beta, 2\right)\right), \mathsf{*.f64}\left(i, 4\right)\right)\right)\right), \mathsf{/.f64}\left(\left({i}^{2} \cdot -12\right), \alpha\right)\right)\right), \alpha\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(0, \beta\right), \mathsf{*.f64}\left(1, \mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\beta, 2\right)\right), \mathsf{*.f64}\left(i, 4\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({i}^{2}\right), -12\right), \alpha\right)\right)\right), \alpha\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(0, \beta\right), \mathsf{*.f64}\left(1, \mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\beta, 2\right)\right), \mathsf{*.f64}\left(i, 4\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(i \cdot i\right), -12\right), \alpha\right)\right)\right), \alpha\right) \]

      4. *-lowering-*.f64 68.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(0, \beta\right), \mathsf{*.f64}\left(1, \mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\beta, 2\right)\right), \mathsf{*.f64}\left(i, 4\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), -12\right), \alpha\right)\right)\right), \alpha\right) \]

   10. Simplified 68.4%

       \[\leadsto \frac{0.5 \cdot \left(\left(0 \cdot \beta + 1 \cdot \left(\left(2 + \beta \cdot 2\right) + i \cdot 4\right)\right) + \frac{\color{blue}{\left(i \cdot i\right) \cdot -12}}{\alpha}\right)}{\alpha} \]

   11. Taylor expanded in i around 0

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

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{2 + 2 \cdot \beta}{\alpha}\right), \color{blue}{\left(2 \cdot \frac{i}{\alpha}\right)}\right) \]

       2. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(\left(\frac{\frac{1}{2} \cdot \left(2 + 2 \cdot \beta\right)}{\alpha}\right), \left(\color{blue}{2} \cdot \frac{i}{\alpha}\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(2 + 2 \cdot \beta\right)\right), \alpha\right), \left(\color{blue}{2} \cdot \frac{i}{\alpha}\right)\right) \]

       4. distribute-lft-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2} \cdot 2 + \frac{1}{2} \cdot \left(2 \cdot \beta\right)\right), \alpha\right), \left(2 \cdot \frac{i}{\alpha}\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(1 + \frac{1}{2} \cdot \left(2 \cdot \beta\right)\right), \alpha\right), \left(2 \cdot \frac{i}{\alpha}\right)\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \left(2 \cdot \beta\right)\right)\right), \alpha\right), \left(2 \cdot \frac{i}{\alpha}\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(2 \cdot \beta\right)\right)\right), \alpha\right), \left(2 \cdot \frac{i}{\alpha}\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(2, \beta\right)\right)\right), \alpha\right), \left(2 \cdot \frac{i}{\alpha}\right)\right) \]

       9. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(2, \beta\right)\right)\right), \alpha\right), \left(\frac{2 \cdot i}{\color{blue}{\alpha}}\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(2, \beta\right)\right)\right), \alpha\right), \mathsf{/.f64}\left(\left(2 \cdot i\right), \color{blue}{\alpha}\right)\right) \]

       11. *-lowering-*.f64 80.0%

           \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(2, \beta\right)\right)\right), \alpha\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, i\right), \alpha\right)\right) \]

   13. Simplified 80.0%

       \[\leadsto \color{blue}{\frac{1 + 0.5 \cdot \left(2 \cdot \beta\right)}{\alpha} + \frac{2 \cdot i}{\alpha}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.05 \cdot 10^{+70}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + 0.5 \cdot \left(\beta \cdot 2\right)}{\alpha} + \frac{2 \cdot i}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 4 \cdot 10^{+50}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + 0.5 \cdot \left(\beta \cdot 2\right)}{\alpha} + \frac{2 \cdot i}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 4e+50)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (+ (/ (+ 1.0 (* 0.5 (* beta 2.0))) alpha) (/ (* 2.0 i) alpha))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 4e+50) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((1.0 + (0.5 * (beta * 2.0))) / alpha) + ((2.0 * i) / alpha);
	}
	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 <= 4d+50) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((1.0d0 + (0.5d0 * (beta * 2.0d0))) / alpha) + ((2.0d0 * i) / alpha)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 4e+50) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((1.0 + (0.5 * (beta * 2.0))) / alpha) + ((2.0 * i) / alpha);
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 4e+50:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((1.0 + (0.5 * (beta * 2.0))) / alpha) + ((2.0 * i) / alpha)
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 4e+50)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(0.5 * Float64(beta * 2.0))) / alpha) + Float64(Float64(2.0 * i) / alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 4e+50)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((1.0 + (0.5 * (beta * 2.0))) / alpha) + ((2.0 * i) / alpha);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 4e+50], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(1.0 + N[(0.5 * N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] + N[(N[(2.0 * i), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 4.0000000000000003e50

   1. Initial program 79.5%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1\right), \color{blue}{2}\right) \]

   3. Simplified 84.1%

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

   5. Taylor expanded in i around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \color{blue}{\left(\frac{\beta - \alpha}{\left(2 + \left(\alpha + \beta\right)\right) \cdot \left(\alpha + \beta\right)}\right)}\right), 1\right), 2\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{/.f64}\left(\left(\beta - \alpha\right), \left(\left(2 + \left(\alpha + \beta\right)\right) \cdot \left(\alpha + \beta\right)\right)\right)\right), 1\right), 2\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\left(2 + \left(\alpha + \beta\right)\right) \cdot \left(\alpha + \beta\right)\right)\right)\right), 1\right), 2\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\left(\alpha + \beta\right) \cdot \left(2 + \left(\alpha + \beta\right)\right)\right)\right)\right), 1\right), 2\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{*.f64}\left(\left(\alpha + \beta\right), \left(2 + \left(\alpha + \beta\right)\right)\right)\right)\right), 1\right), 2\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{*.f64}\left(\left(\beta + \alpha\right), \left(2 + \left(\alpha + \beta\right)\right)\right)\right)\right), 1\right), 2\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, \alpha\right), \left(2 + \left(\alpha + \beta\right)\right)\right)\right)\right), 1\right), 2\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(2, \left(\alpha + \beta\right)\right)\right)\right)\right), 1\right), 2\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(2, \left(\beta + \alpha\right)\right)\right)\right)\right), 1\right), 2\right) \]

      9. +-lowering-+.f64 77.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\beta, \alpha\right)\right)\right)\right)\right), 1\right), 2\right) \]

   7. Simplified 77.9%

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

   8. Taylor expanded in alpha around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{\beta}{2 + \beta}\right)}, 1\right), 2\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\beta, \left(2 + \beta\right)\right), 1\right), 2\right) \]

      2. +-lowering-+.f64 91.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\beta, \mathsf{+.f64}\left(2, \beta\right)\right), 1\right), 2\right) \]

   10. Simplified 91.0%

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

   if 4.0000000000000003e50 < alpha

   1. Initial program 12.3%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1\right), \color{blue}{2}\right) \]

   3. Simplified 22.9%

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

   5. Taylor expanded in alpha around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)\right) + \frac{1}{2} \cdot \frac{{\beta}^{2} - \left(-1 \cdot \left(\left(2 + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\beta + 2 \cdot i\right)\right) + \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right) \cdot \left(\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)\right)\right)}{\alpha}\right), \color{blue}{\alpha}\right) \]

   7. Simplified 63.1%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(\left(0 \cdot \beta + 1 \cdot \left(\left(2 + \beta \cdot 2\right) + i \cdot 4\right)\right) + \frac{\left(\beta \cdot \beta - \left(-2 - \left(\beta + 2 \cdot i\right)\right) \cdot \left(\beta + 2 \cdot i\right)\right) - \left(\left(2 + \beta \cdot 2\right) + i \cdot 4\right) \cdot \left(0 \cdot \beta + 1 \cdot \left(\left(2 + \beta \cdot 2\right) + i \cdot 4\right)\right)}{\alpha}\right)}{\alpha}} \]

   8. Taylor expanded in i around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(0, \beta\right), \mathsf{*.f64}\left(1, \mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\beta, 2\right)\right), \mathsf{*.f64}\left(i, 4\right)\right)\right)\right), \mathsf{/.f64}\left(\color{blue}{\left(-12 \cdot {i}^{2}\right)}, \alpha\right)\right)\right), \alpha\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(0, \beta\right), \mathsf{*.f64}\left(1, \mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\beta, 2\right)\right), \mathsf{*.f64}\left(i, 4\right)\right)\right)\right), \mathsf{/.f64}\left(\left({i}^{2} \cdot -12\right), \alpha\right)\right)\right), \alpha\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(0, \beta\right), \mathsf{*.f64}\left(1, \mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\beta, 2\right)\right), \mathsf{*.f64}\left(i, 4\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({i}^{2}\right), -12\right), \alpha\right)\right)\right), \alpha\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(0, \beta\right), \mathsf{*.f64}\left(1, \mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\beta, 2\right)\right), \mathsf{*.f64}\left(i, 4\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(i \cdot i\right), -12\right), \alpha\right)\right)\right), \alpha\right) \]

      4. *-lowering-*.f64 67.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(0, \beta\right), \mathsf{*.f64}\left(1, \mathsf{+.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\beta, 2\right)\right), \mathsf{*.f64}\left(i, 4\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), -12\right), \alpha\right)\right)\right), \alpha\right) \]

   10. Simplified 67.4%

       \[\leadsto \frac{0.5 \cdot \left(\left(0 \cdot \beta + 1 \cdot \left(\left(2 + \beta \cdot 2\right) + i \cdot 4\right)\right) + \frac{\color{blue}{\left(i \cdot i\right) \cdot -12}}{\alpha}\right)}{\alpha} \]

   11. Taylor expanded in i around 0

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

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{2 + 2 \cdot \beta}{\alpha}\right), \color{blue}{\left(2 \cdot \frac{i}{\alpha}\right)}\right) \]

       2. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(\left(\frac{\frac{1}{2} \cdot \left(2 + 2 \cdot \beta\right)}{\alpha}\right), \left(\color{blue}{2} \cdot \frac{i}{\alpha}\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(2 + 2 \cdot \beta\right)\right), \alpha\right), \left(\color{blue}{2} \cdot \frac{i}{\alpha}\right)\right) \]

       4. distribute-lft-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2} \cdot 2 + \frac{1}{2} \cdot \left(2 \cdot \beta\right)\right), \alpha\right), \left(2 \cdot \frac{i}{\alpha}\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(1 + \frac{1}{2} \cdot \left(2 \cdot \beta\right)\right), \alpha\right), \left(2 \cdot \frac{i}{\alpha}\right)\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \left(2 \cdot \beta\right)\right)\right), \alpha\right), \left(2 \cdot \frac{i}{\alpha}\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(2 \cdot \beta\right)\right)\right), \alpha\right), \left(2 \cdot \frac{i}{\alpha}\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(2, \beta\right)\right)\right), \alpha\right), \left(2 \cdot \frac{i}{\alpha}\right)\right) \]

       9. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(2, \beta\right)\right)\right), \alpha\right), \left(\frac{2 \cdot i}{\color{blue}{\alpha}}\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(2, \beta\right)\right)\right), \alpha\right), \mathsf{/.f64}\left(\left(2 \cdot i\right), \color{blue}{\alpha}\right)\right) \]

       11. *-lowering-*.f64 78.4%

           \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(2, \beta\right)\right)\right), \alpha\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, i\right), \alpha\right)\right) \]

   13. Simplified 78.4%

       \[\leadsto \color{blue}{\frac{1 + 0.5 \cdot \left(2 \cdot \beta\right)}{\alpha} + \frac{2 \cdot i}{\alpha}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 4 \cdot 10^{+50}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + 0.5 \cdot \left(\beta \cdot 2\right)}{\alpha} + \frac{2 \cdot i}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.15 \cdot 10^{+53}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + 0.5 \cdot \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1.15e+53)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (+ 1.0 (* 0.5 (+ (* beta 2.0) (* i 4.0)))) alpha)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.15e+53) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (1.0 + (0.5 * ((beta * 2.0) + (i * 4.0)))) / alpha;
	}
	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+53) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = (1.0d0 + (0.5d0 * ((beta * 2.0d0) + (i * 4.0d0)))) / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.15e+53) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (1.0 + (0.5 * ((beta * 2.0) + (i * 4.0)))) / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 1.15e+53:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = (1.0 + (0.5 * ((beta * 2.0) + (i * 4.0)))) / alpha
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 1.15e+53)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(0.5 * Float64(Float64(beta * 2.0) + Float64(i * 4.0)))) / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 1.15e+53)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = (1.0 + (0.5 * ((beta * 2.0) + (i * 4.0)))) / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 1.15e+53], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(0.5 * N[(N[(beta * 2.0), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.1500000000000001e53

   1. Initial program 79.5%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1\right), \color{blue}{2}\right) \]

   3. Simplified 84.1%

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

   5. Taylor expanded in i around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \color{blue}{\left(\frac{\beta - \alpha}{\left(2 + \left(\alpha + \beta\right)\right) \cdot \left(\alpha + \beta\right)}\right)}\right), 1\right), 2\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{/.f64}\left(\left(\beta - \alpha\right), \left(\left(2 + \left(\alpha + \beta\right)\right) \cdot \left(\alpha + \beta\right)\right)\right)\right), 1\right), 2\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\left(2 + \left(\alpha + \beta\right)\right) \cdot \left(\alpha + \beta\right)\right)\right)\right), 1\right), 2\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\left(\alpha + \beta\right) \cdot \left(2 + \left(\alpha + \beta\right)\right)\right)\right)\right), 1\right), 2\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{*.f64}\left(\left(\alpha + \beta\right), \left(2 + \left(\alpha + \beta\right)\right)\right)\right)\right), 1\right), 2\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{*.f64}\left(\left(\beta + \alpha\right), \left(2 + \left(\alpha + \beta\right)\right)\right)\right)\right), 1\right), 2\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, \alpha\right), \left(2 + \left(\alpha + \beta\right)\right)\right)\right)\right), 1\right), 2\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(2, \left(\alpha + \beta\right)\right)\right)\right)\right), 1\right), 2\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(2, \left(\beta + \alpha\right)\right)\right)\right)\right), 1\right), 2\right) \]

      9. +-lowering-+.f64 77.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\beta, \alpha\right)\right)\right)\right)\right), 1\right), 2\right) \]

   7. Simplified 77.9%

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

   8. Taylor expanded in alpha around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{\beta}{2 + \beta}\right)}, 1\right), 2\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\beta, \left(2 + \beta\right)\right), 1\right), 2\right) \]

      2. +-lowering-+.f64 91.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\beta, \mathsf{+.f64}\left(2, \beta\right)\right), 1\right), 2\right) \]

   10. Simplified 91.0%

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

   if 1.1500000000000001e53 < alpha

   1. Initial program 12.3%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1\right), \color{blue}{2}\right) \]

   3. Simplified 22.9%

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

   5. Taylor expanded in alpha around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)\right) + \frac{1}{2} \cdot \frac{{\beta}^{2} - \left(-1 \cdot \left(\left(2 + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\beta + 2 \cdot i\right)\right) + \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right) \cdot \left(\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)\right)\right)}{\alpha}\right), \color{blue}{\alpha}\right) \]

   7. Simplified 63.1%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(\left(0 \cdot \beta + 1 \cdot \left(\left(2 + \beta \cdot 2\right) + i \cdot 4\right)\right) + \frac{\left(\beta \cdot \beta - \left(-2 - \left(\beta + 2 \cdot i\right)\right) \cdot \left(\beta + 2 \cdot i\right)\right) - \left(\left(2 + \beta \cdot 2\right) + i \cdot 4\right) \cdot \left(0 \cdot \beta + 1 \cdot \left(\left(2 + \beta \cdot 2\right) + i \cdot 4\right)\right)}{\alpha}\right)}{\alpha}} \]

   8. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)\right), \color{blue}{\alpha}\right) \]

      3. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot 2 + \frac{1}{2} \cdot \left(2 \cdot \beta + 4 \cdot i\right)\right), \alpha\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \frac{1}{2} \cdot \left(2 \cdot \beta + 4 \cdot i\right)\right), \alpha\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \left(2 \cdot \beta + 4 \cdot i\right)\right)\right), \alpha\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(2 \cdot \beta + 4 \cdot i\right)\right)\right), \alpha\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(4 \cdot i + 2 \cdot \beta\right)\right)\right), \alpha\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\left(4 \cdot i\right), \left(2 \cdot \beta\right)\right)\right)\right), \alpha\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\left(i \cdot 4\right), \left(2 \cdot \beta\right)\right)\right)\right), \alpha\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{*.f64}\left(i, 4\right), \left(2 \cdot \beta\right)\right)\right)\right), \alpha\right) \]

      11. *-lowering-*.f64 78.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{*.f64}\left(i, 4\right), \mathsf{*.f64}\left(2, \beta\right)\right)\right)\right), \alpha\right) \]

   10. Simplified 78.4%

       \[\leadsto \color{blue}{\frac{1 + 0.5 \cdot \left(i \cdot 4 + 2 \cdot \beta\right)}{\alpha}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.15 \cdot 10^{+53}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + 0.5 \cdot \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.6 \cdot 10^{+53}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1.6e+53)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (+ beta 1.0) alpha)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.6e+53) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	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+53) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = (beta + 1.0d0) / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.6e+53) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 1.6e+53:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = (beta + 1.0) / alpha
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 1.6e+53)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(beta + 1.0) / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 1.6e+53)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = (beta + 1.0) / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 1.6e+53], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.6e53

   1. Initial program 79.5%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1\right), \color{blue}{2}\right) \]

   3. Simplified 84.1%

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

   5. Taylor expanded in i around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \color{blue}{\left(\frac{\beta - \alpha}{\left(2 + \left(\alpha + \beta\right)\right) \cdot \left(\alpha + \beta\right)}\right)}\right), 1\right), 2\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{/.f64}\left(\left(\beta - \alpha\right), \left(\left(2 + \left(\alpha + \beta\right)\right) \cdot \left(\alpha + \beta\right)\right)\right)\right), 1\right), 2\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\left(2 + \left(\alpha + \beta\right)\right) \cdot \left(\alpha + \beta\right)\right)\right)\right), 1\right), 2\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\left(\alpha + \beta\right) \cdot \left(2 + \left(\alpha + \beta\right)\right)\right)\right)\right), 1\right), 2\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{*.f64}\left(\left(\alpha + \beta\right), \left(2 + \left(\alpha + \beta\right)\right)\right)\right)\right), 1\right), 2\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{*.f64}\left(\left(\beta + \alpha\right), \left(2 + \left(\alpha + \beta\right)\right)\right)\right)\right), 1\right), 2\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, \alpha\right), \left(2 + \left(\alpha + \beta\right)\right)\right)\right)\right), 1\right), 2\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(2, \left(\alpha + \beta\right)\right)\right)\right)\right), 1\right), 2\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(2, \left(\beta + \alpha\right)\right)\right)\right)\right), 1\right), 2\right) \]

      9. +-lowering-+.f64 77.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\beta, \alpha\right)\right)\right)\right)\right), 1\right), 2\right) \]

   7. Simplified 77.9%

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

   8. Taylor expanded in alpha around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{\beta}{2 + \beta}\right)}, 1\right), 2\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\beta, \left(2 + \beta\right)\right), 1\right), 2\right) \]

      2. +-lowering-+.f64 91.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\beta, \mathsf{+.f64}\left(2, \beta\right)\right), 1\right), 2\right) \]

   10. Simplified 91.0%

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

   if 1.6e53 < alpha

   1. Initial program 12.3%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1\right), \color{blue}{2}\right) \]

   3. Simplified 22.9%

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

   5. Taylor expanded in i around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \color{blue}{\left(\frac{\beta - \alpha}{\left(2 + \left(\alpha + \beta\right)\right) \cdot \left(\alpha + \beta\right)}\right)}\right), 1\right), 2\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{/.f64}\left(\left(\beta - \alpha\right), \left(\left(2 + \left(\alpha + \beta\right)\right) \cdot \left(\alpha + \beta\right)\right)\right)\right), 1\right), 2\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\left(2 + \left(\alpha + \beta\right)\right) \cdot \left(\alpha + \beta\right)\right)\right)\right), 1\right), 2\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\left(\alpha + \beta\right) \cdot \left(2 + \left(\alpha + \beta\right)\right)\right)\right)\right), 1\right), 2\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{*.f64}\left(\left(\alpha + \beta\right), \left(2 + \left(\alpha + \beta\right)\right)\right)\right)\right), 1\right), 2\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{*.f64}\left(\left(\beta + \alpha\right), \left(2 + \left(\alpha + \beta\right)\right)\right)\right)\right), 1\right), 2\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, \alpha\right), \left(2 + \left(\alpha + \beta\right)\right)\right)\right)\right), 1\right), 2\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(2, \left(\alpha + \beta\right)\right)\right)\right)\right), 1\right), 2\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(2, \left(\beta + \alpha\right)\right)\right)\right)\right), 1\right), 2\right) \]

      9. +-lowering-+.f64 15.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\beta, \alpha\right)\right)\right)\right)\right), 1\right), 2\right) \]

   7. Simplified 15.6%

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

   8. Taylor expanded in alpha around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{2 + 2 \cdot \beta}{\alpha}\right)}, 2\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 + 2 \cdot \beta\right), \alpha\right), 2\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(2 \cdot \beta\right)\right), \alpha\right), 2\right) \]

      3. *-lowering-*.f64 54.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(2, \beta\right)\right), \alpha\right), 2\right) \]

   10. Simplified 54.6%

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

       1. div-inv N/A

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

       2. metadata-eval N/A

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

       3. associate-*l/ N/A

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

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(2 + 2 \cdot \beta\right) \cdot \frac{1}{2}\right), \color{blue}{\alpha}\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(2 + \beta \cdot 2\right) \cdot \frac{1}{2}\right), \alpha\right) \]

       6. distribute-rgt1-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\beta + 1\right) \cdot 2\right) \cdot \frac{1}{2}\right), \alpha\right) \]

       7. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot \left(2 \cdot \frac{1}{2}\right)\right), \alpha\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\beta + 1\right) \cdot 1\right), \alpha\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\beta + 1\right), 1\right), \alpha\right) \]

       10. +-lowering-+.f64 54.6%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), 1\right), \alpha\right) \]

   12. Applied egg-rr 54.6%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.6 \cdot 10^{+53}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 62000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 62000000.0) 0.5 (+ 1.0 (/ -1.0 beta))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 62000000.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0 + (-1.0 / beta);
	}
	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 <= 62000000.0d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0 + ((-1.0d0) / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 62000000.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0 + (-1.0 / beta);
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 62000000.0:
		tmp = 0.5
	else:
		tmp = 1.0 + (-1.0 / beta)
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 62000000.0)
		tmp = 0.5;
	else
		tmp = Float64(1.0 + Float64(-1.0 / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 62000000.0)
		tmp = 0.5;
	else
		tmp = 1.0 + (-1.0 / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 62000000.0], 0.5, N[(1.0 + N[(-1.0 / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.2e7

   1. Initial program 72.2%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1\right), \color{blue}{2}\right) \]

   3. Simplified 74.4%

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

   5. Taylor expanded in i around inf

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

   7. Simplified 70.8%

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

   if 6.2e7 < beta

   1. Initial program 35.0%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1\right), \color{blue}{2}\right) \]

   3. Simplified 50.9%

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

   5. Taylor expanded in i around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \color{blue}{\left(\frac{\beta - \alpha}{\left(2 + \left(\alpha + \beta\right)\right) \cdot \left(\alpha + \beta\right)}\right)}\right), 1\right), 2\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{/.f64}\left(\left(\beta - \alpha\right), \left(\left(2 + \left(\alpha + \beta\right)\right) \cdot \left(\alpha + \beta\right)\right)\right)\right), 1\right), 2\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\left(2 + \left(\alpha + \beta\right)\right) \cdot \left(\alpha + \beta\right)\right)\right)\right), 1\right), 2\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\left(\alpha + \beta\right) \cdot \left(2 + \left(\alpha + \beta\right)\right)\right)\right)\right), 1\right), 2\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{*.f64}\left(\left(\alpha + \beta\right), \left(2 + \left(\alpha + \beta\right)\right)\right)\right)\right), 1\right), 2\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{*.f64}\left(\left(\beta + \alpha\right), \left(2 + \left(\alpha + \beta\right)\right)\right)\right)\right), 1\right), 2\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, \alpha\right), \left(2 + \left(\alpha + \beta\right)\right)\right)\right)\right), 1\right), 2\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(2, \left(\alpha + \beta\right)\right)\right)\right)\right), 1\right), 2\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(2, \left(\beta + \alpha\right)\right)\right)\right)\right), 1\right), 2\right) \]

      9. +-lowering-+.f64 44.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\beta, \alpha\right)\right)\right)\right)\right), 1\right), 2\right) \]

   7. Simplified 44.8%

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

   8. Taylor expanded in alpha around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \color{blue}{\left(\beta \cdot \left(2 + \beta\right)\right)}\right)\right), 1\right), 2\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{*.f64}\left(\beta, \left(2 + \beta\right)\right)\right)\right), 1\right), 2\right) \]

      2. +-lowering-+.f64 39.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{*.f64}\left(\beta, \mathsf{+.f64}\left(2, \beta\right)\right)\right)\right), 1\right), 2\right) \]

   10. Simplified 39.8%

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

   11. Taylor expanded in beta around inf

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

       1. associate-*r/ N/A

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

       2. distribute-rgt1-in N/A

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

       3. metadata-eval N/A

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

       4. mul0-lft N/A

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

       5. metadata-eval N/A

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

       6. metadata-eval N/A

          \[\leadsto 1 + \frac{-1}{\beta} \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{\beta}\right)}\right) \]

       8. /-lowering-/.f64 72.9%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \color{blue}{\beta}\right)\right) \]

   13. Simplified 72.9%

       \[\leadsto \color{blue}{1 + \frac{-1}{\beta}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 72.3% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 26000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 (if (<= beta 26000000.0) 0.5 1.0))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 26000000.0) {
		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 <= 26000000.0d0) 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 <= 26000000.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 26000000.0:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 26000000.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 26000000.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 26000000.0], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 2.6e7

   1. Initial program 72.2%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1\right), \color{blue}{2}\right) \]

   3. Simplified 74.4%

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

   5. Taylor expanded in i around inf

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

   7. Simplified 70.8%

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

   if 2.6e7 < beta

   1. Initial program 35.0%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1\right), \color{blue}{2}\right) \]

   3. Simplified 50.9%

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

   5. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{1} \]
   6. Step-by-step derivation

   7. Simplified 72.0%

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

Alternative 10: 62.0% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[alpha_, beta_, i_] := 0.5

Derivation

1. Initial program 61.4%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1\right), \color{blue}{2}\right) \]

3. Simplified 67.6%

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

5. Taylor expanded in i around inf

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

7. Simplified 58.9%

   \[\leadsto \color{blue}{0.5} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024159 
(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))