Octave 3.8, jcobi/2

Percentage Accurate: 63.6% → 96.5%

Time: 14.9s

Alternatives: 10

Speedup: 4.8×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 63.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.5

   1. Initial program 4.5%

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

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

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

   3. Simplified 17.3%

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

   5. Taylor expanded in alpha around inf

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

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

      3. mul-1-neg N/A

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

      4. remove-double-neg N/A

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

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

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

      6. distribute-rgt1-in N/A

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

      7. metadata-eval N/A

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

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

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

      9. associate-+r+ N/A

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

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

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

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

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

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

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 87.9%

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

   7. Simplified 87.9%

      \[\leadsto \frac{\color{blue}{\frac{0 \cdot \beta + \left(\left(2 + 2 \cdot \beta\right) + i \cdot 4\right)}{\alpha}}}{2} \]
   8. Step-by-step derivation

      1. frac-2neg N/A

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

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

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

   9. Applied egg-rr 87.9%

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

   if -0.5 < (/.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 83.0%

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

      1. /-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 87.4%

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-+r+ N/A

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

      4. associate-+r+ N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. associate-/r* N/A

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

      8. div-inv N/A

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

      9. clear-num N/A

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

      10. frac-times N/A

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

      11. metadata-eval N/A

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

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

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

   6. Applied egg-rr 83.0%

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

   7. Taylor expanded in beta around inf

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

      1. /-lowering-/.f64 98.8%

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

   9. Simplified 98.8%

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

4. Final simplification 96.4%

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

Alternative 2: 89.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.2 \cdot 10^{+98}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2 - \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{-2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1.2e+98)
   (/ (+ 1.0 (/ beta (+ 2.0 (+ (+ alpha beta) (* 2.0 i))))) 2.0)
   (/ (/ (- -2.0 (+ (* beta 2.0) (* i 4.0))) alpha) -2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.2e+98) {
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else {
		tmp = ((-2.0 - ((beta * 2.0) + (i * 4.0))) / alpha) / -2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 1.2d+98) then
        tmp = (1.0d0 + (beta / (2.0d0 + ((alpha + beta) + (2.0d0 * i))))) / 2.0d0
    else
        tmp = (((-2.0d0) - ((beta * 2.0d0) + (i * 4.0d0))) / alpha) / (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.2e+98) {
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else {
		tmp = ((-2.0 - ((beta * 2.0) + (i * 4.0))) / alpha) / -2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 1.2e+98:
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0
	else:
		tmp = ((-2.0 - ((beta * 2.0) + (i * 4.0))) / alpha) / -2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 1.2e+98)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(2.0 + Float64(Float64(alpha + beta) + Float64(2.0 * i))))) / 2.0);
	else
		tmp = Float64(Float64(Float64(-2.0 - Float64(Float64(beta * 2.0) + Float64(i * 4.0))) / alpha) / -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 1.2e+98)
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	else
		tmp = ((-2.0 - ((beta * 2.0) + (i * 4.0))) / alpha) / -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 1.2e+98], N[(N[(1.0 + N[(beta / N[(2.0 + N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(-2.0 - N[(N[(beta * 2.0), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.1999999999999999e98

   1. Initial program 83.7%

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

   3. Taylor expanded in 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 95.9%

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

   if 1.1999999999999999e98 < alpha

   1. Initial program 3.5%

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

      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 19.6%

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

   5. Taylor expanded in alpha around inf

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

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

      3. mul-1-neg N/A

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

      4. remove-double-neg N/A

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

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

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

      6. distribute-rgt1-in N/A

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

      7. metadata-eval N/A

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

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

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

      9. associate-+r+ N/A

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

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

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

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

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

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

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 77.0%

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

   7. Simplified 77.0%

      \[\leadsto \frac{\color{blue}{\frac{0 \cdot \beta + \left(\left(2 + 2 \cdot \beta\right) + i \cdot 4\right)}{\alpha}}}{2} \]
   8. Step-by-step derivation

      1. frac-2neg N/A

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

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

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

   9. Applied egg-rr 77.0%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.2 \cdot 10^{+98}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2 - \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{-2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.7 \cdot 10^{+100}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\left(\alpha + \beta\right) + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2 - \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{-2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1.7e+100)
   (/ (+ 1.0 (/ beta (+ (+ alpha beta) 2.0))) 2.0)
   (/ (/ (- -2.0 (+ (* beta 2.0) (* i 4.0))) alpha) -2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.7e+100) {
		tmp = (1.0 + (beta / ((alpha + beta) + 2.0))) / 2.0;
	} else {
		tmp = ((-2.0 - ((beta * 2.0) + (i * 4.0))) / alpha) / -2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 1.7d+100) then
        tmp = (1.0d0 + (beta / ((alpha + beta) + 2.0d0))) / 2.0d0
    else
        tmp = (((-2.0d0) - ((beta * 2.0d0) + (i * 4.0d0))) / alpha) / (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.7e+100) {
		tmp = (1.0 + (beta / ((alpha + beta) + 2.0))) / 2.0;
	} else {
		tmp = ((-2.0 - ((beta * 2.0) + (i * 4.0))) / alpha) / -2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 1.7e+100:
		tmp = (1.0 + (beta / ((alpha + beta) + 2.0))) / 2.0
	else:
		tmp = ((-2.0 - ((beta * 2.0) + (i * 4.0))) / alpha) / -2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 1.7e+100)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(Float64(alpha + beta) + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(-2.0 - Float64(Float64(beta * 2.0) + Float64(i * 4.0))) / alpha) / -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 1.7e+100)
		tmp = (1.0 + (beta / ((alpha + beta) + 2.0))) / 2.0;
	else
		tmp = ((-2.0 - ((beta * 2.0) + (i * 4.0))) / alpha) / -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 1.7e+100], N[(N[(1.0 + N[(beta / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(-2.0 - N[(N[(beta * 2.0), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.69999999999999997e100

   1. Initial program 83.7%

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

   3. Taylor expanded in 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 95.9%

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

   6. Taylor expanded in i around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 91.5%

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

   8. Simplified 91.5%

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

   if 1.69999999999999997e100 < alpha

   1. Initial program 3.5%

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

      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 19.6%

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

   5. Taylor expanded in alpha around inf

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

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

      3. mul-1-neg N/A

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

      4. remove-double-neg N/A

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

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

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

      6. distribute-rgt1-in N/A

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

      7. metadata-eval N/A

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

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

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

      9. associate-+r+ N/A

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

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

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

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

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

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

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 77.0%

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

   7. Simplified 77.0%

      \[\leadsto \frac{\color{blue}{\frac{0 \cdot \beta + \left(\left(2 + 2 \cdot \beta\right) + i \cdot 4\right)}{\alpha}}}{2} \]
   8. Step-by-step derivation

      1. frac-2neg N/A

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

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

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

   9. Applied egg-rr 77.0%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.7 \cdot 10^{+100}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\left(\alpha + \beta\right) + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2 - \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{-2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.75 \cdot 10^{+103}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\left(\alpha + \beta\right) + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1.75e+103)
   (/ (+ 1.0 (/ beta (+ (+ alpha 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 <= 1.75e+103) {
		tmp = (1.0 + (beta / ((alpha + 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 <= 1.75d+103) then
        tmp = (1.0d0 + (beta / ((alpha + 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 <= 1.75e+103) {
		tmp = (1.0 + (beta / ((alpha + 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 <= 1.75e+103:
		tmp = (1.0 + (beta / ((alpha + 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 <= 1.75e+103)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(Float64(alpha + 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 <= 1.75e+103)
		tmp = (1.0 + (beta / ((alpha + 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, 1.75e+103], N[(N[(1.0 + N[(beta / N[(N[(alpha + beta), $MachinePrecision] + 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 < 1.75e103

   1. Initial program 83.7%

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

   3. Taylor expanded in 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 95.9%

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

   6. Taylor expanded in i around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 91.5%

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

   8. Simplified 91.5%

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

   if 1.75e103 < alpha

   1. Initial program 3.5%

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

      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 19.6%

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

   5. Taylor expanded in alpha around inf

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

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

      3. mul-1-neg N/A

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

      4. remove-double-neg N/A

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

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

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

      6. distribute-rgt1-in N/A

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

      7. metadata-eval N/A

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

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

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

      9. associate-+r+ N/A

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

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

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

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

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

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

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 77.0%

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

   7. Simplified 77.0%

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

   8. Taylor expanded in beta around 0

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 68.2%

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

   10. Simplified 68.2%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.75 \cdot 10^{+103}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\left(\alpha + \beta\right) + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 6.6 \cdot 10^{+100}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 6.6e+100)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (+ 2.0 (* i 4.0)) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 6.6e+100) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 6.6d+100) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((2.0d0 + (i * 4.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 6.6e+100) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 6.6e+100:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 6.6e+100)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(i * 4.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 6.6e+100)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 6.6e+100], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 6.6000000000000002e100

   1. Initial program 83.7%

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

      1. /-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 87.1%

      \[\leadsto \color{blue}{\frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)} + 1}{2}} \]
   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 88.1%

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

   7. Simplified 88.1%

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

   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. +-commutative N/A

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

      3. +-lowering-+.f64 91.3%

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

   10. Simplified 91.3%

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

   if 6.6000000000000002e100 < alpha

   1. Initial program 3.5%

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

      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 19.6%

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

   5. Taylor expanded in alpha around inf

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

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

      3. mul-1-neg N/A

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

      4. remove-double-neg N/A

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

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

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

      6. distribute-rgt1-in N/A

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

      7. metadata-eval N/A

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

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

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

      9. associate-+r+ N/A

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

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

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

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

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

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

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 77.0%

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

   7. Simplified 77.0%

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

   8. Taylor expanded in beta around 0

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 68.2%

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

   10. Simplified 68.2%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 6.6 \cdot 10^{+100}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 3.1 \cdot 10^{+102}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{\left(-2 - \beta\right) - \beta}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 3.1e+102)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (* -0.5 (/ (- (- -2.0 beta) beta) alpha))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 3.1e+102) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = -0.5 * (((-2.0 - beta) - beta) / 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 <= 3.1d+102) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = (-0.5d0) * ((((-2.0d0) - beta) - beta) / alpha)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 3.1e+102) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = -0.5 * (((-2.0 - beta) - beta) / alpha);
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 3.1e+102:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = -0.5 * (((-2.0 - beta) - beta) / alpha)
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 3.1e+102)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(-0.5 * Float64(Float64(Float64(-2.0 - beta) - beta) / alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 3.1e+102)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = -0.5 * (((-2.0 - beta) - beta) / alpha);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 3.1e+102], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(-0.5 * N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 3.09999999999999987e102

   1. Initial program 83.7%

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

      1. /-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 87.1%

      \[\leadsto \color{blue}{\frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)} + 1}{2}} \]
   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 88.1%

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

   7. Simplified 88.1%

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

   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. +-commutative N/A

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

      3. +-lowering-+.f64 91.3%

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

   10. Simplified 91.3%

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

   if 3.09999999999999987e102 < alpha

   1. Initial program 3.5%

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

      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 19.6%

      \[\leadsto \color{blue}{\frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)\right)} + 1}{2}} \]
   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 15.5%

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

   7. Simplified 15.5%

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

   8. Taylor expanded in alpha around -inf

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

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

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

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

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

      3. associate--r+ N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

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

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

      9. distribute-neg-in N/A

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

      10. metadata-eval N/A

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

      11. unsub-neg N/A

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

      12. --lowering--.f64 57.2%

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

   10. Simplified 57.2%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 3.1 \cdot 10^{+102}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{\left(-2 - \beta\right) - \beta}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 182:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{2}{\beta} + -1}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 182.0) 0.5 (+ 1.0 (/ (+ (/ 2.0 beta) -1.0) beta))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 182.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0 + (((2.0 / beta) + -1.0) / beta);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 182.0d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0 + (((2.0d0 / beta) + (-1.0d0)) / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 182.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0 + (((2.0 / beta) + -1.0) / beta);
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 182.0:
		tmp = 0.5
	else:
		tmp = 1.0 + (((2.0 / beta) + -1.0) / beta)
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 182.0)
		tmp = 0.5;
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(2.0 / beta) + -1.0) / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 182.0)
		tmp = 0.5;
	else
		tmp = 1.0 + (((2.0 / beta) + -1.0) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 182.0], 0.5, N[(1.0 + N[(N[(N[(2.0 / beta), $MachinePrecision] + -1.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 182

   1. Initial program 75.7%

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

      1. /-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 79.4%

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

   5. Taylor expanded in i around inf

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

   7. Simplified 77.5%

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

   if 182 < beta

   1. Initial program 38.5%

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

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

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

   3. Simplified 51.8%

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

   5. Taylor expanded in i around 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 75.7%

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

   7. Simplified 75.7%

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

   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. +-commutative N/A

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

      3. +-lowering-+.f64 74.2%

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

   10. Simplified 74.2%

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

   11. Taylor expanded in beta around -inf

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

       1. mul-1-neg N/A

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

       2. unsub-neg N/A

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

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

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

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

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

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

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

       6. associate-*r/ N/A

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

       7. metadata-eval N/A

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

       8. /-lowering-/.f64 73.6%

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

   13. Simplified 73.6%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 182:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{2}{\beta} + -1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 420:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 420.0) 0.5 (+ 1.0 (/ -1.0 beta))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 420.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0 + (-1.0 / beta);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 420.0d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0 + ((-1.0d0) / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 420.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0 + (-1.0 / beta);
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 420.0:
		tmp = 0.5
	else:
		tmp = 1.0 + (-1.0 / beta)
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 420.0)
		tmp = 0.5;
	else
		tmp = Float64(1.0 + Float64(-1.0 / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 420.0)
		tmp = 0.5;
	else
		tmp = 1.0 + (-1.0 / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 420.0], 0.5, N[(1.0 + N[(-1.0 / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 420

   1. Initial program 75.7%

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

      1. /-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 79.4%

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

   5. Taylor expanded in i around inf

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

   7. Simplified 77.5%

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

   if 420 < beta

   1. Initial program 38.5%

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

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

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

   3. Simplified 51.8%

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

   5. Taylor expanded in i around 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 75.7%

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

   7. Simplified 75.7%

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

   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. +-commutative N/A

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

      3. +-lowering-+.f64 74.2%

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

   10. Simplified 74.2%

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

   11. Taylor expanded in beta around inf

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

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

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

       2. /-lowering-/.f64 73.3%

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

   13. Simplified 73.3%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 420:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.0% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 420:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 (if (<= beta 420.0) 0.5 1.0))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 420.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 420.0d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 420.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 420.0:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 420.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 420.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 420.0], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 420

   1. Initial program 75.7%

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

      1. /-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 79.4%

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

   5. Taylor expanded in i around inf

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

   7. Simplified 77.5%

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

   if 420 < beta

   1. Initial program 38.5%

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

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

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

   3. Simplified 51.8%

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

   5. Taylor expanded in beta around inf

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

   7. Simplified 72.6%

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

Alternative 10: 61.8% 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 65.9%

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

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

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

3. Simplified 72.1%

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

5. Taylor expanded in i around inf

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

7. Simplified 64.6%

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

Reproduce

herbie shell --seed 2024161 
(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))