Octave 3.8, jcobi/2

Percentage Accurate: 63.6% → 97.5%

Time: 14.8s

Alternatives: 13

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: 97.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot i + \left(\alpha + \beta\right)\\ \mathbf{if}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\alpha + \beta\right)}{t\_0}}{2 + t\_0} \leq -1:\\ \;\;\;\;\frac{\beta}{\alpha} + \frac{0.5 \cdot \left(2 + i \cdot 4\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)}, \frac{\alpha + \beta}{\beta + \left(\alpha + 2 \cdot i\right)}, 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

   1. Initial program 1.6%

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

      1. /-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 10.6%

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

      1. associate-+r+ N/A

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

      2. *-commutative N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. associate-/r* N/A

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

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

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

   6. Applied egg-rr 14.1%

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

   7. Taylor expanded in alpha around inf

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

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

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

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

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. mul-1-neg N/A

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

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

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

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

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

      9. +-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. *-lowering-*.f64 92.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(0, \beta\right), \mathsf{neg.f64}\left(\mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(i, 4\right), \mathsf{*.f64}\left(2, \beta\right)\right)\right)\right)\right), \alpha\right), 2\right) \]

   9. Simplified 92.5%

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

   10. Taylor expanded in beta around 0

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

       1. +-commutative N/A

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

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

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

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

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

       4. associate-*r/ N/A

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

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

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

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

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

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

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 92.5%

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

   12. Simplified 92.5%

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

   if -1 < (/.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 78.0%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   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 83.7%

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

      1. associate-*r/ N/A

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

      2. associate-+r+ N/A

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

      3. *-commutative N/A

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

      4. associate-+r+ N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. times-frac N/A

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

      8. fma-define N/A

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

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

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

   6. Applied egg-rr 99.9%

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

4. Final simplification 98.0%

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

Alternative 2: 97.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot i + \left(\alpha + \beta\right)\\ \mathbf{if}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\alpha + \beta\right)}{t\_0}}{2 + t\_0} \leq -1:\\ \;\;\;\;\frac{\beta}{\alpha} + \frac{0.5 \cdot \left(2 + i \cdot 4\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\alpha + \beta, \frac{\frac{\beta - \alpha}{\beta + \left(\alpha + 2 \cdot i\right)}}{\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)}, 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

   1. Initial program 1.6%

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

      1. /-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 10.6%

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

      1. associate-+r+ N/A

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

      2. *-commutative N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. associate-/r* N/A

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

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

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

   6. Applied egg-rr 14.1%

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

   7. Taylor expanded in alpha around inf

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

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

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

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

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. mul-1-neg N/A

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

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

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

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

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

      9. +-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. *-lowering-*.f64 92.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(0, \beta\right), \mathsf{neg.f64}\left(\mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(i, 4\right), \mathsf{*.f64}\left(2, \beta\right)\right)\right)\right)\right), \alpha\right), 2\right) \]

   9. Simplified 92.5%

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

   10. Taylor expanded in beta around 0

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

       1. +-commutative N/A

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

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

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

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

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

       4. associate-*r/ N/A

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

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

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

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

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

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

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 92.5%

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

   12. Simplified 92.5%

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

   if -1 < (/.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 78.0%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   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 83.7%

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

      1. associate-*r/ N/A

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

      2. associate-+r+ N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. associate-/l* N/A

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

      9. associate-/l* N/A

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

      10. fma-define N/A

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

   6. Applied egg-rr 99.9%

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

4. Final simplification 98.0%

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

Alternative 3: 97.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot i + \left(\alpha + \beta\right)\\ t_1 := 2 + t\_0\\ \mathbf{if}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\alpha + \beta\right)}{t\_0}}{t\_1} \leq -1:\\ \;\;\;\;\frac{\beta}{\alpha} + \frac{0.5 \cdot \left(2 + i \cdot 4\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\beta + \left(\alpha + 2 \cdot i\right)}}{t\_1}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (* 2.0 i) (+ alpha beta))) (t_1 (+ 2.0 t_0)))
   (if (<= (/ (/ (* (- beta alpha) (+ alpha beta)) t_0) t_1) -1.0)
     (+ (/ beta alpha) (/ (* 0.5 (+ 2.0 (* i 4.0))) alpha))
     (/
      (+
       1.0
       (/
        (* (+ alpha beta) (/ (- beta alpha) (+ beta (+ alpha (* 2.0 i)))))
        t_1))
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (2.0 * i) + (alpha + beta);
	double t_1 = 2.0 + t_0;
	double tmp;
	if (((((beta - alpha) * (alpha + beta)) / t_0) / t_1) <= -1.0) {
		tmp = (beta / alpha) + ((0.5 * (2.0 + (i * 4.0))) / alpha);
	} else {
		tmp = (1.0 + (((alpha + beta) * ((beta - alpha) / (beta + (alpha + (2.0 * i))))) / t_1)) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (2.0d0 * i) + (alpha + beta)
    t_1 = 2.0d0 + t_0
    if (((((beta - alpha) * (alpha + beta)) / t_0) / t_1) <= (-1.0d0)) then
        tmp = (beta / alpha) + ((0.5d0 * (2.0d0 + (i * 4.0d0))) / alpha)
    else
        tmp = (1.0d0 + (((alpha + beta) * ((beta - alpha) / (beta + (alpha + (2.0d0 * i))))) / t_1)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (2.0 * i) + (alpha + beta);
	double t_1 = 2.0 + t_0;
	double tmp;
	if (((((beta - alpha) * (alpha + beta)) / t_0) / t_1) <= -1.0) {
		tmp = (beta / alpha) + ((0.5 * (2.0 + (i * 4.0))) / alpha);
	} else {
		tmp = (1.0 + (((alpha + beta) * ((beta - alpha) / (beta + (alpha + (2.0 * i))))) / t_1)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (2.0 * i) + (alpha + beta)
	t_1 = 2.0 + t_0
	tmp = 0
	if ((((beta - alpha) * (alpha + beta)) / t_0) / t_1) <= -1.0:
		tmp = (beta / alpha) + ((0.5 * (2.0 + (i * 4.0))) / alpha)
	else:
		tmp = (1.0 + (((alpha + beta) * ((beta - alpha) / (beta + (alpha + (2.0 * i))))) / t_1)) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(2.0 * i) + Float64(alpha + beta))
	t_1 = Float64(2.0 + t_0)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(beta - alpha) * Float64(alpha + beta)) / t_0) / t_1) <= -1.0)
		tmp = Float64(Float64(beta / alpha) + Float64(Float64(0.5 * Float64(2.0 + Float64(i * 4.0))) / alpha));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(alpha + beta) * Float64(Float64(beta - alpha) / Float64(beta + Float64(alpha + Float64(2.0 * i))))) / t_1)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (2.0 * i) + (alpha + beta);
	t_1 = 2.0 + t_0;
	tmp = 0.0;
	if (((((beta - alpha) * (alpha + beta)) / t_0) / t_1) <= -1.0)
		tmp = (beta / alpha) + ((0.5 * (2.0 + (i * 4.0))) / alpha);
	else
		tmp = (1.0 + (((alpha + beta) * ((beta - alpha) / (beta + (alpha + (2.0 * i))))) / t_1)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

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

   1. Initial program 1.6%

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

      1. /-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 10.6%

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

      1. associate-+r+ N/A

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

      2. *-commutative N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. associate-/r* N/A

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

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

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

   6. Applied egg-rr 14.1%

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

   7. Taylor expanded in alpha around inf

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

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

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

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

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. mul-1-neg N/A

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

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

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

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

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

      9. +-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. *-lowering-*.f64 92.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(0, \beta\right), \mathsf{neg.f64}\left(\mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(i, 4\right), \mathsf{*.f64}\left(2, \beta\right)\right)\right)\right)\right), \alpha\right), 2\right) \]

   9. Simplified 92.5%

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

   10. Taylor expanded in beta around 0

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

       1. +-commutative N/A

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

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

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

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

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

       4. associate-*r/ N/A

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

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

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

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

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

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

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 92.5%

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

   12. Simplified 92.5%

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

   if -1 < (/.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 78.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

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

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

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

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

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

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

      11. +-commutative N/A

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

      12. +-lowering-+.f64 99.9%

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

   4. Applied egg-rr 99.9%

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

4. Final simplification 98.0%

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

Alternative 4: 90.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 8.6 \cdot 10^{+109}:\\ \;\;\;\;\frac{1 + \left(\beta - \alpha\right) \cdot \frac{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)}}{\beta + \left(\alpha + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha} + \frac{0.5 \cdot \left(2 + i \cdot 4\right)}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 8.6e+109)
   (/
    (+
     1.0
     (*
      (- beta alpha)
      (/
       (/ (+ alpha beta) (+ (+ alpha beta) (+ 2.0 (* 2.0 i))))
       (+ beta (+ alpha (* 2.0 i))))))
    2.0)
   (+ (/ beta alpha) (/ (* 0.5 (+ 2.0 (* i 4.0))) alpha))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 8.6e+109) {
		tmp = (1.0 + ((beta - alpha) * (((alpha + beta) / ((alpha + beta) + (2.0 + (2.0 * i)))) / (beta + (alpha + (2.0 * i)))))) / 2.0;
	} else {
		tmp = (beta / alpha) + ((0.5 * (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 <= 8.6d+109) then
        tmp = (1.0d0 + ((beta - alpha) * (((alpha + beta) / ((alpha + beta) + (2.0d0 + (2.0d0 * i)))) / (beta + (alpha + (2.0d0 * i)))))) / 2.0d0
    else
        tmp = (beta / alpha) + ((0.5d0 * (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 <= 8.6e+109) {
		tmp = (1.0 + ((beta - alpha) * (((alpha + beta) / ((alpha + beta) + (2.0 + (2.0 * i)))) / (beta + (alpha + (2.0 * i)))))) / 2.0;
	} else {
		tmp = (beta / alpha) + ((0.5 * (2.0 + (i * 4.0))) / alpha);
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 8.6e+109:
		tmp = (1.0 + ((beta - alpha) * (((alpha + beta) / ((alpha + beta) + (2.0 + (2.0 * i)))) / (beta + (alpha + (2.0 * i)))))) / 2.0
	else:
		tmp = (beta / alpha) + ((0.5 * (2.0 + (i * 4.0))) / alpha)
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 8.6e+109)
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) * Float64(Float64(Float64(alpha + beta) / Float64(Float64(alpha + beta) + Float64(2.0 + Float64(2.0 * i)))) / Float64(beta + Float64(alpha + Float64(2.0 * i)))))) / 2.0);
	else
		tmp = Float64(Float64(beta / alpha) + Float64(Float64(0.5 * Float64(2.0 + Float64(i * 4.0))) / alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 8.6e+109)
		tmp = (1.0 + ((beta - alpha) * (((alpha + beta) / ((alpha + beta) + (2.0 + (2.0 * i)))) / (beta + (alpha + (2.0 * i)))))) / 2.0;
	else
		tmp = (beta / alpha) + ((0.5 * (2.0 + (i * 4.0))) / alpha);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 8.6e+109], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] * N[(N[(N[(alpha + beta), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + N[(alpha + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(beta / alpha), $MachinePrecision] + N[(N[(0.5 * N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 8.6000000000000001e109

   1. Initial program 78.2%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   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 82.8%

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

      1. associate-+r+ N/A

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

      2. *-commutative N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. associate-/r* N/A

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

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

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

   6. Applied egg-rr 96.4%

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

   if 8.6000000000000001e109 < alpha

   1. Initial program 4.4%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   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 16.1%

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

      1. associate-+r+ N/A

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

      2. *-commutative N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. associate-/r* N/A

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

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

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

   6. Applied egg-rr 27.4%

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

   7. Taylor expanded in alpha around inf

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

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

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

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

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. mul-1-neg N/A

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

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

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

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

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

      9. +-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. *-lowering-*.f64 79.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(0, \beta\right), \mathsf{neg.f64}\left(\mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(i, 4\right), \mathsf{*.f64}\left(2, \beta\right)\right)\right)\right)\right), \alpha\right), 2\right) \]

   9. Simplified 79.4%

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

   10. Taylor expanded in beta around 0

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

       1. +-commutative N/A

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

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

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

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

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

       4. associate-*r/ N/A

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

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

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

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

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

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

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 79.4%

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

   12. Simplified 79.4%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 8.6 \cdot 10^{+109}:\\ \;\;\;\;\frac{1 + \left(\beta - \alpha\right) \cdot \frac{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)}}{\beta + \left(\alpha + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha} + \frac{0.5 \cdot \left(2 + i \cdot 4\right)}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 4 \cdot 10^{+111}:\\ \;\;\;\;\frac{1 + \left(\beta - \alpha\right) \cdot \frac{\frac{\beta}{2 \cdot i + \left(\beta + 2\right)}}{\beta + 2 \cdot i}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha} + \frac{0.5 \cdot \left(2 + i \cdot 4\right)}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 4e+111)
   (/
    (+
     1.0
     (*
      (- beta alpha)
      (/ (/ beta (+ (* 2.0 i) (+ beta 2.0))) (+ beta (* 2.0 i)))))
    2.0)
   (+ (/ beta alpha) (/ (* 0.5 (+ 2.0 (* i 4.0))) alpha))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 4e+111) {
		tmp = (1.0 + ((beta - alpha) * ((beta / ((2.0 * i) + (beta + 2.0))) / (beta + (2.0 * i))))) / 2.0;
	} else {
		tmp = (beta / alpha) + ((0.5 * (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 <= 4d+111) then
        tmp = (1.0d0 + ((beta - alpha) * ((beta / ((2.0d0 * i) + (beta + 2.0d0))) / (beta + (2.0d0 * i))))) / 2.0d0
    else
        tmp = (beta / alpha) + ((0.5d0 * (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 <= 4e+111) {
		tmp = (1.0 + ((beta - alpha) * ((beta / ((2.0 * i) + (beta + 2.0))) / (beta + (2.0 * i))))) / 2.0;
	} else {
		tmp = (beta / alpha) + ((0.5 * (2.0 + (i * 4.0))) / alpha);
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 4e+111:
		tmp = (1.0 + ((beta - alpha) * ((beta / ((2.0 * i) + (beta + 2.0))) / (beta + (2.0 * i))))) / 2.0
	else:
		tmp = (beta / alpha) + ((0.5 * (2.0 + (i * 4.0))) / alpha)
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 4e+111)
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) * Float64(Float64(beta / Float64(Float64(2.0 * i) + Float64(beta + 2.0))) / Float64(beta + Float64(2.0 * i))))) / 2.0);
	else
		tmp = Float64(Float64(beta / alpha) + Float64(Float64(0.5 * Float64(2.0 + Float64(i * 4.0))) / alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 4e+111)
		tmp = (1.0 + ((beta - alpha) * ((beta / ((2.0 * i) + (beta + 2.0))) / (beta + (2.0 * i))))) / 2.0;
	else
		tmp = (beta / alpha) + ((0.5 * (2.0 + (i * 4.0))) / alpha);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 4e+111], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] * N[(N[(beta / N[(N[(2.0 * i), $MachinePrecision] + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(beta / alpha), $MachinePrecision] + N[(N[(0.5 * N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 3.99999999999999983e111

   1. Initial program 78.2%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   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 82.8%

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

      1. associate-+r+ N/A

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

      2. *-commutative N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. associate-/r* N/A

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

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

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

   6. Applied egg-rr 96.4%

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

   7. Taylor expanded in alpha around 0

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

      1. associate-/r* N/A

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

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

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

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

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

      4. associate-+r+ N/A

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

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

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

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

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

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

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

      8. +-commutative N/A

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

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

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

      10. *-lowering-*.f64 95.4%

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

   9. Simplified 95.4%

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

   if 3.99999999999999983e111 < alpha

   1. Initial program 4.4%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   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 16.1%

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

      1. associate-+r+ N/A

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

      2. *-commutative N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. associate-/r* N/A

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

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

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

   6. Applied egg-rr 27.4%

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

   7. Taylor expanded in alpha around inf

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

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

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

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

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. mul-1-neg N/A

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

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

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

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

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

      9. +-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. *-lowering-*.f64 79.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(0, \beta\right), \mathsf{neg.f64}\left(\mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(i, 4\right), \mathsf{*.f64}\left(2, \beta\right)\right)\right)\right)\right), \alpha\right), 2\right) \]

   9. Simplified 79.4%

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

   10. Taylor expanded in beta around 0

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

       1. +-commutative N/A

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

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

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

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

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

       4. associate-*r/ N/A

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

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

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

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

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

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

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 79.4%

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

   12. Simplified 79.4%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 4 \cdot 10^{+111}:\\ \;\;\;\;\frac{1 + \left(\beta - \alpha\right) \cdot \frac{\frac{\beta}{2 \cdot i + \left(\beta + 2\right)}}{\beta + 2 \cdot i}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha} + \frac{0.5 \cdot \left(2 + i \cdot 4\right)}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.16 \cdot 10^{+110}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{2 + \left(2 \cdot i + \left(\alpha + \beta\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha} + \frac{0.5 \cdot \left(2 + i \cdot 4\right)}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1.16e+110)
   (/ (+ 1.0 (/ (- beta alpha) (+ 2.0 (+ (* 2.0 i) (+ alpha beta))))) 2.0)
   (+ (/ beta alpha) (/ (* 0.5 (+ 2.0 (* i 4.0))) alpha))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.16e+110) {
		tmp = (1.0 + ((beta - alpha) / (2.0 + ((2.0 * i) + (alpha + beta))))) / 2.0;
	} else {
		tmp = (beta / alpha) + ((0.5 * (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 <= 1.16d+110) then
        tmp = (1.0d0 + ((beta - alpha) / (2.0d0 + ((2.0d0 * i) + (alpha + beta))))) / 2.0d0
    else
        tmp = (beta / alpha) + ((0.5d0 * (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 <= 1.16e+110) {
		tmp = (1.0 + ((beta - alpha) / (2.0 + ((2.0 * i) + (alpha + beta))))) / 2.0;
	} else {
		tmp = (beta / alpha) + ((0.5 * (2.0 + (i * 4.0))) / alpha);
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 1.16e+110:
		tmp = (1.0 + ((beta - alpha) / (2.0 + ((2.0 * i) + (alpha + beta))))) / 2.0
	else:
		tmp = (beta / alpha) + ((0.5 * (2.0 + (i * 4.0))) / alpha)
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 1.16e+110)
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / Float64(2.0 + Float64(Float64(2.0 * i) + Float64(alpha + beta))))) / 2.0);
	else
		tmp = Float64(Float64(beta / alpha) + Float64(Float64(0.5 * Float64(2.0 + Float64(i * 4.0))) / alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 1.16e+110)
		tmp = (1.0 + ((beta - alpha) / (2.0 + ((2.0 * i) + (alpha + beta))))) / 2.0;
	else
		tmp = (beta / alpha) + ((0.5 * (2.0 + (i * 4.0))) / alpha);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 1.16e+110], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + N[(N[(2.0 * i), $MachinePrecision] + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(beta / alpha), $MachinePrecision] + N[(N[(0.5 * N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.16e110

   1. Initial program 78.2%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. 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 95.4%

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

   5. Simplified 95.4%

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

   if 1.16e110 < alpha

   1. Initial program 4.4%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   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 16.1%

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

      1. associate-+r+ N/A

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

      2. *-commutative N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. associate-/r* N/A

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

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

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

   6. Applied egg-rr 27.4%

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

   7. Taylor expanded in alpha around inf

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

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

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

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

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. mul-1-neg N/A

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

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

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

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

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

      9. +-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. *-lowering-*.f64 79.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(0, \beta\right), \mathsf{neg.f64}\left(\mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(i, 4\right), \mathsf{*.f64}\left(2, \beta\right)\right)\right)\right)\right), \alpha\right), 2\right) \]

   9. Simplified 79.4%

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

   10. Taylor expanded in beta around 0

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

       1. +-commutative N/A

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

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

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

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

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

       4. associate-*r/ N/A

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

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

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

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

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

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

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 79.4%

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

   12. Simplified 79.4%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.16 \cdot 10^{+110}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{2 + \left(2 \cdot i + \left(\alpha + \beta\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha} + \frac{0.5 \cdot \left(2 + i \cdot 4\right)}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.8 \cdot 10^{+137}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(2 \cdot i + \left(\alpha + \beta\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha} + \frac{0.5 \cdot \left(2 + i \cdot 4\right)}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 2.8e+137)
   (/ (+ 1.0 (/ beta (+ 2.0 (+ (* 2.0 i) (+ alpha beta))))) 2.0)
   (+ (/ beta alpha) (/ (* 0.5 (+ 2.0 (* i 4.0))) alpha))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2.8e+137) {
		tmp = (1.0 + (beta / (2.0 + ((2.0 * i) + (alpha + beta))))) / 2.0;
	} else {
		tmp = (beta / alpha) + ((0.5 * (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 <= 2.8d+137) then
        tmp = (1.0d0 + (beta / (2.0d0 + ((2.0d0 * i) + (alpha + beta))))) / 2.0d0
    else
        tmp = (beta / alpha) + ((0.5d0 * (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 <= 2.8e+137) {
		tmp = (1.0 + (beta / (2.0 + ((2.0 * i) + (alpha + beta))))) / 2.0;
	} else {
		tmp = (beta / alpha) + ((0.5 * (2.0 + (i * 4.0))) / alpha);
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 2.8e+137:
		tmp = (1.0 + (beta / (2.0 + ((2.0 * i) + (alpha + beta))))) / 2.0
	else:
		tmp = (beta / alpha) + ((0.5 * (2.0 + (i * 4.0))) / alpha)
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 2.8e+137)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(2.0 + Float64(Float64(2.0 * i) + Float64(alpha + beta))))) / 2.0);
	else
		tmp = Float64(Float64(beta / alpha) + Float64(Float64(0.5 * Float64(2.0 + Float64(i * 4.0))) / alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 2.8e+137)
		tmp = (1.0 + (beta / (2.0 + ((2.0 * i) + (alpha + beta))))) / 2.0;
	else
		tmp = (beta / alpha) + ((0.5 * (2.0 + (i * 4.0))) / alpha);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 2.8e+137], N[(N[(1.0 + N[(beta / N[(2.0 + N[(N[(2.0 * i), $MachinePrecision] + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(beta / alpha), $MachinePrecision] + N[(N[(0.5 * N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.80000000000000001e137

   1. Initial program 76.3%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. 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 93.6%

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

   if 2.80000000000000001e137 < alpha

   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 15.1%

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

      1. associate-+r+ N/A

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

      2. *-commutative N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. associate-/r* N/A

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

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

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

   6. Applied egg-rr 25.1%

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

   7. Taylor expanded in alpha around inf

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

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

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

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

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. mul-1-neg N/A

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

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

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

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

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

      9. +-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. *-lowering-*.f64 82.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(0, \beta\right), \mathsf{neg.f64}\left(\mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(i, 4\right), \mathsf{*.f64}\left(2, \beta\right)\right)\right)\right)\right), \alpha\right), 2\right) \]

   9. Simplified 82.0%

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

   10. Taylor expanded in beta around 0

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

       1. +-commutative N/A

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

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

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

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

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

       4. associate-*r/ N/A

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

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

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

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

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

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

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 82.0%

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

   12. Simplified 82.0%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.8 \cdot 10^{+137}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(2 \cdot i + \left(\alpha + \beta\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha} + \frac{0.5 \cdot \left(2 + i \cdot 4\right)}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.8 \cdot 10^{+137}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha} + \frac{0.5 \cdot \left(2 + i \cdot 4\right)}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 2.8e+137)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (+ (/ beta alpha) (/ (* 0.5 (+ 2.0 (* i 4.0))) alpha))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2.8e+137) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (beta / alpha) + ((0.5 * (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 <= 2.8d+137) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = (beta / alpha) + ((0.5d0 * (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 <= 2.8e+137) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (beta / alpha) + ((0.5 * (2.0 + (i * 4.0))) / alpha);
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 2.8e+137:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = (beta / alpha) + ((0.5 * (2.0 + (i * 4.0))) / alpha)
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 2.8e+137)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(beta / alpha) + Float64(Float64(0.5 * Float64(2.0 + Float64(i * 4.0))) / alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 2.8e+137)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = (beta / alpha) + ((0.5 * (2.0 + (i * 4.0))) / alpha);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 2.8e+137], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(beta / alpha), $MachinePrecision] + N[(N[(0.5 * N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.80000000000000001e137

   1. Initial program 76.3%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   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 81.1%

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

   5. Taylor expanded in i around 0

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

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

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

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

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

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

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

      4. +-lowering-+.f64 80.7%

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

   7. Simplified 80.7%

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

   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 84.7%

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

   10. Simplified 84.7%

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

   if 2.80000000000000001e137 < alpha

   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 15.1%

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

      1. associate-+r+ N/A

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

      2. *-commutative N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. associate-/r* N/A

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

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

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

   6. Applied egg-rr 25.1%

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

   7. Taylor expanded in alpha around inf

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

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

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

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

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. mul-1-neg N/A

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

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

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

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

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

      9. +-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. *-lowering-*.f64 82.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(0, \beta\right), \mathsf{neg.f64}\left(\mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(i, 4\right), \mathsf{*.f64}\left(2, \beta\right)\right)\right)\right)\right), \alpha\right), 2\right) \]

   9. Simplified 82.0%

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

   10. Taylor expanded in beta around 0

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

       1. +-commutative N/A

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

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

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

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

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

       4. associate-*r/ N/A

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

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

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

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

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

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

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 82.0%

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

   12. Simplified 82.0%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.8 \cdot 10^{+137}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha} + \frac{0.5 \cdot \left(2 + i \cdot 4\right)}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.8 \cdot 10^{+137}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 2.8e+137)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (+ 2.0 (* i 4.0)) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2.8e+137) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 2.8d+137) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((2.0d0 + (i * 4.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2.8e+137) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 2.8e+137:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 2.8e+137)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(i * 4.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 2.8e+137)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 2.8e+137], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.80000000000000001e137

   1. Initial program 76.3%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   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 81.1%

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

   5. Taylor expanded in i around 0

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

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

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

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

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

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

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

      4. +-lowering-+.f64 80.7%

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

   7. Simplified 80.7%

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

   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 84.7%

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

   10. Simplified 84.7%

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

   if 2.80000000000000001e137 < alpha

   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 15.1%

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

      1. associate-+r+ N/A

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

      2. *-commutative N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. associate-/r* N/A

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

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

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

   6. Applied egg-rr 25.1%

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

   7. Taylor expanded in alpha around inf

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

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

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

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

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. mul-1-neg N/A

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

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

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

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

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

      9. +-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. *-lowering-*.f64 82.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(0, \beta\right), \mathsf{neg.f64}\left(\mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(i, 4\right), \mathsf{*.f64}\left(2, \beta\right)\right)\right)\right)\right), \alpha\right), 2\right) \]

   9. Simplified 82.0%

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

   10. Taylor expanded in beta around 0

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

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

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

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

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

       3. *-commutative N/A

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

       4. *-lowering-*.f64 65.1%

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

   12. Simplified 65.1%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.8 \cdot 10^{+137}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.12 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1.12e+154)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.12e+154) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 1.12d+154) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.12e+154) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 1.12e+154:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 1.12e+154)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 1.12e+154)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 1.12e+154], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.11999999999999994e154

   1. Initial program 75.0%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   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 79.6%

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

   5. Taylor expanded in i around 0

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

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

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

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

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

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

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

      4. +-lowering-+.f64 78.7%

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

   7. Simplified 78.7%

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

   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 83.2%

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

   10. Simplified 83.2%

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

   if 1.11999999999999994e154 < alpha

   1. Initial program 1.3%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   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 14.5%

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

   5. Taylor expanded in i around 0

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

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

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

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

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

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

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

      4. +-lowering-+.f64 14.9%

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

   7. Simplified 14.9%

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

   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 58.3%

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

   10. Simplified 58.3%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.12 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.65 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1.65e+154) (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0) (/ 1.0 alpha)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.65e+154) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = 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 <= 1.65d+154) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = 1.0d0 / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.65e+154) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 1.65e+154:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = 1.0 / alpha
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 1.65e+154)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(1.0 / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 1.65e+154)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = 1.0 / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 1.65e+154], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(1.0 / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.65e154

   1. Initial program 75.0%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   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 79.6%

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

   5. Taylor expanded in i around 0

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

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

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

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

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

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

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

      4. +-lowering-+.f64 78.7%

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

   7. Simplified 78.7%

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

   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 83.2%

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

   10. Simplified 83.2%

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

   if 1.65e154 < alpha

   1. Initial program 1.3%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   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 14.5%

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

   5. Taylor expanded in i around 0

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

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

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

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

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

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

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

      4. +-lowering-+.f64 14.9%

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

   7. Simplified 14.9%

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

   8. Taylor expanded in beta around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 5.0%

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

   10. Simplified 5.0%

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

   11. Taylor expanded in alpha around inf

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

       1. /-lowering-/.f64 41.7%

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

   13. Simplified 41.7%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.65 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 72.2% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.45 \cdot 10^{+97}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 (if (<= beta 1.45e+97) 0.5 1.0))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.45e+97) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 1.45d+97) then
        tmp = 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.45e+97) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.45e+97:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.45e+97)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1.45e+97)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 1.45e+97], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 1.44999999999999994e97

   1. Initial program 70.0%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   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 72.8%

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

   5. Taylor expanded in i around inf

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

   7. Simplified 69.2%

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

   if 1.44999999999999994e97 < beta

   1. Initial program 21.6%

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

      1. /-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 39.9%

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

   5. Taylor expanded in beta around inf

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

   7. Simplified 77.1%

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

Alternative 13: 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 58.3%

   \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
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 64.8%

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

5. Taylor expanded in i around inf

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

7. Simplified 59.3%

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

Reproduce

herbie shell --seed 2024154 
(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))