Octave 3.8, jcobi/2

Percentage Accurate: 62.9% → 97.6%

Time: 15.7s

Alternatives: 9

Speedup: 2.1×

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: 62.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := 2 + t\_0\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_1} \leq -0.8:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(2 + \left(i \cdot 4 + \beta \cdot 2\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}}{t\_1} + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))) (t_1 (+ 2.0 t_0)))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) t_1) -0.8)
     (/ (/ (+ (- beta beta) (+ 2.0 (+ (* i 4.0) (* beta 2.0)))) alpha) 2.0)
     (/
      (+
       (/ (* (- beta alpha) (/ (+ alpha beta) (+ alpha (fma 2.0 i beta)))) t_1)
       1.0)
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = 2.0 + t_0;
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= -0.8) {
		tmp = (((beta - beta) + (2.0 + ((i * 4.0) + (beta * 2.0)))) / alpha) / 2.0;
	} else {
		tmp = ((((beta - alpha) * ((alpha + beta) / (alpha + fma(2.0, i, beta)))) / t_1) + 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(2.0 + t_0)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / t_1) <= -0.8)
		tmp = Float64(Float64(Float64(Float64(beta - beta) + Float64(2.0 + Float64(Float64(i * 4.0) + Float64(beta * 2.0)))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(beta - alpha) * Float64(Float64(alpha + beta) / Float64(alpha + fma(2.0, i, beta)))) / t_1) + 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 + t$95$0), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision], -0.8], N[(N[(N[(N[(beta - beta), $MachinePrecision] + N[(2.0 + N[(N[(i * 4.0), $MachinePrecision] + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(N[(beta - alpha), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] / N[(alpha + N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.80000000000000004

   1. Initial program 4.2%

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

   3. Taylor expanded in alpha around inf 92.5%

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

   if -0.80000000000000004 < (/.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 82.1%

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

      1. associate-/l* 100.0%

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

      2. +-commutative 100.0%

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

      3. +-commutative 100.0%

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

      4. associate-+r+ 100.0%

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

      5. fma-undefine 100.0%

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

      6. +-commutative 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. associate-*r/ 82.1%

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

      2. *-commutative 82.1%

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

      3. associate-*r/ 100.0%

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

      4. +-commutative 100.0%

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

      5. +-commutative 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.8:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(2 + \left(i \cdot 4 + \beta \cdot 2\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := 2 + t\_0\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_1} \leq -0.8:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(2 + \left(i \cdot 4 + \beta \cdot 2\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\left(\beta - \alpha\right) \cdot \frac{\beta}{\beta + 2 \cdot i}}{t\_1}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))) (t_1 (+ 2.0 t_0)))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) t_1) -0.8)
     (/ (/ (+ (- beta beta) (+ 2.0 (+ (* i 4.0) (* beta 2.0)))) alpha) 2.0)
     (/ (+ 1.0 (/ (* (- beta alpha) (/ beta (+ beta (* 2.0 i)))) t_1)) 2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = 2.0 + t_0;
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= -0.8) {
		tmp = (((beta - beta) + (2.0 + ((i * 4.0) + (beta * 2.0)))) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (((beta - alpha) * (beta / (beta + (2.0 * i)))) / t_1)) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    t_1 = 2.0d0 + t_0
    if (((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= (-0.8d0)) then
        tmp = (((beta - beta) + (2.0d0 + ((i * 4.0d0) + (beta * 2.0d0)))) / alpha) / 2.0d0
    else
        tmp = (1.0d0 + (((beta - alpha) * (beta / (beta + (2.0d0 * i)))) / t_1)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = 2.0 + t_0;
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= -0.8) {
		tmp = (((beta - beta) + (2.0 + ((i * 4.0) + (beta * 2.0)))) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (((beta - alpha) * (beta / (beta + (2.0 * i)))) / t_1)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	t_1 = 2.0 + t_0
	tmp = 0
	if ((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= -0.8:
		tmp = (((beta - beta) + (2.0 + ((i * 4.0) + (beta * 2.0)))) / alpha) / 2.0
	else:
		tmp = (1.0 + (((beta - alpha) * (beta / (beta + (2.0 * i)))) / t_1)) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(2.0 + t_0)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / t_1) <= -0.8)
		tmp = Float64(Float64(Float64(Float64(beta - beta) + Float64(2.0 + Float64(Float64(i * 4.0) + Float64(beta * 2.0)))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(beta - alpha) * Float64(beta / Float64(beta + Float64(2.0 * i)))) / t_1)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	t_1 = 2.0 + t_0;
	tmp = 0.0;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= -0.8)
		tmp = (((beta - beta) + (2.0 + ((i * 4.0) + (beta * 2.0)))) / alpha) / 2.0;
	else
		tmp = (1.0 + (((beta - alpha) * (beta / (beta + (2.0 * i)))) / t_1)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 + t$95$0), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision], -0.8], N[(N[(N[(N[(beta - beta), $MachinePrecision] + N[(2.0 + N[(N[(i * 4.0), $MachinePrecision] + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(N[(N[(beta - alpha), $MachinePrecision] * N[(beta / N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) < -0.80000000000000004

   1. Initial program 4.2%

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

   3. Taylor expanded in alpha around inf 92.5%

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

   if -0.80000000000000004 < (/.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 82.1%

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

      1. associate-/l* 100.0%

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

      2. +-commutative 100.0%

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

      3. +-commutative 100.0%

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

      4. associate-+r+ 100.0%

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

      5. fma-undefine 100.0%

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

      6. +-commutative 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. associate-*r/ 82.1%

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

      2. *-commutative 82.1%

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

      3. associate-*r/ 100.0%

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

      4. +-commutative 100.0%

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

      5. +-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in alpha around 0 99.6%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.8:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(2 + \left(i \cdot 4 + \beta \cdot 2\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\left(\beta - \alpha\right) \cdot \frac{\beta}{\beta + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.35 \cdot 10^{-176}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 3.3 \cdot 10^{-109}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 1.4 \cdot 10^{+128}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1.35e-176)
   (/ (+ 1.0 (/ (- beta alpha) (+ (+ alpha beta) 2.0))) 2.0)
   (if (<= alpha 3.3e-109)
     0.5
     (if (<= alpha 1.4e+128)
       (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
       (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.35e-176) {
		tmp = (1.0 + ((beta - alpha) / ((alpha + beta) + 2.0))) / 2.0;
	} else if (alpha <= 3.3e-109) {
		tmp = 0.5;
	} else if (alpha <= 1.4e+128) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 1.35d-176) then
        tmp = (1.0d0 + ((beta - alpha) / ((alpha + beta) + 2.0d0))) / 2.0d0
    else if (alpha <= 3.3d-109) then
        tmp = 0.5d0
    else if (alpha <= 1.4d+128) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.35e-176) {
		tmp = (1.0 + ((beta - alpha) / ((alpha + beta) + 2.0))) / 2.0;
	} else if (alpha <= 3.3e-109) {
		tmp = 0.5;
	} else if (alpha <= 1.4e+128) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 1.35e-176:
		tmp = (1.0 + ((beta - alpha) / ((alpha + beta) + 2.0))) / 2.0
	elif alpha <= 3.3e-109:
		tmp = 0.5
	elif alpha <= 1.4e+128:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 1.35e-176)
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0))) / 2.0);
	elseif (alpha <= 3.3e-109)
		tmp = 0.5;
	elseif (alpha <= 1.4e+128)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 1.35e-176)
		tmp = (1.0 + ((beta - alpha) / ((alpha + beta) + 2.0))) / 2.0;
	elseif (alpha <= 3.3e-109)
		tmp = 0.5;
	elseif (alpha <= 1.4e+128)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 1.35e-176], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[alpha, 3.3e-109], 0.5, If[LessEqual[alpha, 1.4e+128], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if alpha < 1.3499999999999999e-176

   1. Initial program 84.2%

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

   3. Taylor expanded in i around 0 93.0%

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

      1. +-commutative 93.0%

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

   5. Simplified 93.0%

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

   if 1.3499999999999999e-176 < alpha < 3.2999999999999999e-109

   1. Initial program 94.9%

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

   3. Taylor expanded in i around inf 84.6%

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

      1. associate-/l* 89.7%

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

      2. +-commutative 89.7%

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

   5. Simplified 89.7%

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

   6. Taylor expanded in i around inf 95.7%

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

   if 3.2999999999999999e-109 < alpha < 1.39999999999999991e128

   1. Initial program 66.0%

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

   3. Taylor expanded in i around 0 60.0%

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

      1. +-commutative 60.0%

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

   5. Simplified 60.0%

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

   6. Taylor expanded in alpha around 0 72.7%

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

      1. +-commutative 72.7%

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

   8. Simplified 72.7%

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

   if 1.39999999999999991e128 < alpha

   1. Initial program 1.7%

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

   3. Taylor expanded in i around 0 16.5%

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

      1. +-commutative 16.5%

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

   5. Simplified 16.5%

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

   6. Taylor expanded in alpha around inf 57.0%

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

      1. *-commutative 57.0%

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

   8. Simplified 57.0%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.35 \cdot 10^{-176}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 3.3 \cdot 10^{-109}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 1.4 \cdot 10^{+128}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{if}\;\alpha \leq 1.45 \cdot 10^{-177}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\alpha \leq 3.8 \cdot 10^{-111}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 10^{+129}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)))
   (if (<= alpha 1.45e-177)
     t_0
     (if (<= alpha 3.8e-111)
       0.5
       (if (<= alpha 1e+129) t_0 (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (1.0 + (beta / (beta + 2.0))) / 2.0;
	double tmp;
	if (alpha <= 1.45e-177) {
		tmp = t_0;
	} else if (alpha <= 3.8e-111) {
		tmp = 0.5;
	} else if (alpha <= 1e+129) {
		tmp = t_0;
	} else {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    if (alpha <= 1.45d-177) then
        tmp = t_0
    else if (alpha <= 3.8d-111) then
        tmp = 0.5d0
    else if (alpha <= 1d+129) then
        tmp = t_0
    else
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (1.0 + (beta / (beta + 2.0))) / 2.0;
	double tmp;
	if (alpha <= 1.45e-177) {
		tmp = t_0;
	} else if (alpha <= 3.8e-111) {
		tmp = 0.5;
	} else if (alpha <= 1e+129) {
		tmp = t_0;
	} else {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (1.0 + (beta / (beta + 2.0))) / 2.0
	tmp = 0
	if alpha <= 1.45e-177:
		tmp = t_0
	elif alpha <= 3.8e-111:
		tmp = 0.5
	elif alpha <= 1e+129:
		tmp = t_0
	else:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0)
	tmp = 0.0
	if (alpha <= 1.45e-177)
		tmp = t_0;
	elseif (alpha <= 3.8e-111)
		tmp = 0.5;
	elseif (alpha <= 1e+129)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (1.0 + (beta / (beta + 2.0))) / 2.0;
	tmp = 0.0;
	if (alpha <= 1.45e-177)
		tmp = t_0;
	elseif (alpha <= 3.8e-111)
		tmp = 0.5;
	elseif (alpha <= 1e+129)
		tmp = t_0;
	else
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[alpha, 1.45e-177], t$95$0, If[LessEqual[alpha, 3.8e-111], 0.5, If[LessEqual[alpha, 1e+129], t$95$0, N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 1.44999999999999999e-177 or 3.80000000000000022e-111 < alpha < 1e129

   1. Initial program 78.5%

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

   3. Taylor expanded in i around 0 82.7%

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

      1. +-commutative 82.7%

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

   5. Simplified 82.7%

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

   6. Taylor expanded in alpha around 0 86.1%

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

      1. +-commutative 86.1%

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

   8. Simplified 86.1%

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

   if 1.44999999999999999e-177 < alpha < 3.80000000000000022e-111

   1. Initial program 94.9%

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

   3. Taylor expanded in i around inf 84.6%

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

      1. associate-/l* 89.7%

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

      2. +-commutative 89.7%

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

   5. Simplified 89.7%

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

   6. Taylor expanded in i around inf 95.7%

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

   if 1e129 < alpha

   1. Initial program 1.7%

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

   3. Taylor expanded in i around 0 16.5%

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

      1. +-commutative 16.5%

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

   5. Simplified 16.5%

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

   6. Taylor expanded in alpha around inf 57.0%

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

      1. *-commutative 57.0%

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

   8. Simplified 57.0%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.45 \cdot 10^{-177}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 3.8 \cdot 10^{-111}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 10^{+129}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 10^{+128}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(2 + \left(i \cdot 4 + \beta \cdot 2\right)\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1e+128)
   (/ (+ 1.0 (/ beta (+ 2.0 (+ (+ alpha beta) (* 2.0 i))))) 2.0)
   (/ (/ (+ (- beta beta) (+ 2.0 (+ (* i 4.0) (* beta 2.0)))) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1e+128) {
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else {
		tmp = (((beta - beta) + (2.0 + ((i * 4.0) + (beta * 2.0)))) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 1d+128) then
        tmp = (1.0d0 + (beta / (2.0d0 + ((alpha + beta) + (2.0d0 * i))))) / 2.0d0
    else
        tmp = (((beta - beta) + (2.0d0 + ((i * 4.0d0) + (beta * 2.0d0)))) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1e+128) {
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else {
		tmp = (((beta - beta) + (2.0 + ((i * 4.0) + (beta * 2.0)))) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 1e+128:
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0
	else:
		tmp = (((beta - beta) + (2.0 + ((i * 4.0) + (beta * 2.0)))) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 1e+128)
		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(Float64(beta - beta) + Float64(2.0 + Float64(Float64(i * 4.0) + Float64(beta * 2.0)))) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 1e+128)
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	else
		tmp = (((beta - beta) + (2.0 + ((i * 4.0) + (beta * 2.0)))) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 1e+128], 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[(N[(beta - beta), $MachinePrecision] + N[(2.0 + N[(N[(i * 4.0), $MachinePrecision] + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.0000000000000001e128

   1. Initial program 80.0%

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

   3. Taylor expanded in beta around inf 92.4%

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

   if 1.0000000000000001e128 < alpha

   1. Initial program 1.7%

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

   3. Taylor expanded in alpha around inf 83.4%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 10^{+128}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(2 + \left(i \cdot 4 + \beta \cdot 2\right)\right)}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 4.4 \cdot 10^{+129}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 4.4e+129)
   (/ (+ 1.0 (/ beta (+ 2.0 (+ (+ alpha beta) (* 2.0 i))))) 2.0)
   (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 4.4e+129) {
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 4.4d+129) then
        tmp = (1.0d0 + (beta / (2.0d0 + ((alpha + beta) + (2.0d0 * i))))) / 2.0d0
    else
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 4.4e+129) {
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 4.4e+129:
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0
	else:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 4.4e+129)
		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(beta * 2.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 4.4e+129)
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	else
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 4.4e+129], 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[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 4.3999999999999999e129

   1. Initial program 80.0%

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

   3. Taylor expanded in beta around inf 92.4%

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

   if 4.3999999999999999e129 < alpha

   1. Initial program 1.7%

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

   3. Taylor expanded in i around 0 16.5%

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

      1. +-commutative 16.5%

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

   5. Simplified 16.5%

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

   6. Taylor expanded in alpha around inf 57.0%

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

      1. *-commutative 57.0%

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

   8. Simplified 57.0%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 4.4 \cdot 10^{+129}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.7 \cdot 10^{+40} \lor \neg \left(\beta \leq 2.4 \cdot 10^{+77}\right) \land \beta \leq 2.1 \cdot 10^{+102}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (or (<= beta 3.7e+40) (and (not (<= beta 2.4e+77)) (<= beta 2.1e+102)))
   0.5
   1.0))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if ((beta <= 3.7e+40) || (!(beta <= 2.4e+77) && (beta <= 2.1e+102))) {
		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 <= 3.7d+40) .or. (.not. (beta <= 2.4d+77)) .and. (beta <= 2.1d+102)) 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 <= 3.7e+40) || (!(beta <= 2.4e+77) && (beta <= 2.1e+102))) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if (beta <= 3.7e+40) or (not (beta <= 2.4e+77) and (beta <= 2.1e+102)):
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if ((beta <= 3.7e+40) || (!(beta <= 2.4e+77) && (beta <= 2.1e+102)))
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if ((beta <= 3.7e+40) || (~((beta <= 2.4e+77)) && (beta <= 2.1e+102)))
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[Or[LessEqual[beta, 3.7e+40], And[N[Not[LessEqual[beta, 2.4e+77]], $MachinePrecision], LessEqual[beta, 2.1e+102]]], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 3.7e40 or 2.3999999999999999e77 < beta < 2.10000000000000001e102

   1. Initial program 76.5%

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

   3. Taylor expanded in i around inf 57.1%

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

      1. associate-/l* 58.6%

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

      2. +-commutative 58.6%

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

   5. Simplified 58.6%

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

   6. Taylor expanded in i around inf 74.0%

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

   if 3.7e40 < beta < 2.3999999999999999e77 or 2.10000000000000001e102 < beta

   1. Initial program 34.7%

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

   3. Taylor expanded in beta around inf 80.6%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.7 \cdot 10^{+40} \lor \neg \left(\beta \leq 2.4 \cdot 10^{+77}\right) \land \beta \leq 2.1 \cdot 10^{+102}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 1.8 \cdot 10^{+147}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= i 1.8e+147) (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0) 0.5))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 1.8e+147) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = 0.5;
	}
	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 (i <= 1.8d+147) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 1.8e+147) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if i <= 1.8e+147:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = 0.5
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (i <= 1.8e+147)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (i <= 1.8e+147)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[i, 1.8e+147], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if i < 1.8000000000000001e147

   1. Initial program 62.1%

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

   3. Taylor expanded in i around 0 73.9%

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

      1. +-commutative 73.9%

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

   5. Simplified 73.9%

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

   6. Taylor expanded in alpha around 0 73.7%

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

      1. +-commutative 73.7%

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

   8. Simplified 73.7%

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

   if 1.8000000000000001e147 < i

   1. Initial program 73.9%

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

   3. Taylor expanded in i around inf 73.9%

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

      1. associate-/l* 81.6%

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

      2. +-commutative 81.6%

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

   5. Simplified 81.6%

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

   6. Taylor expanded in i around inf 82.9%

      \[\leadsto \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 1.8 \cdot 10^{+147}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.6% 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 65.0%

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

3. Taylor expanded in i around inf 43.5%

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

   1. associate-/l* 45.4%

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

   2. +-commutative 45.4%

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

5. Simplified 45.4%

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

6. Taylor expanded in i around inf 60.0%

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

Reproduce

herbie shell --seed 2024096 
(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))