Octave 3.8, jcobi/4

Percentage Accurate: 16.0% → 83.1%

Time: 16.0s

Alternatives: 11

Speedup: 53.0×

Specification

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

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_2 := t\_1 \cdot t\_1\\ \frac{\frac{t\_0 \cdot \left(\beta \cdot \alpha + t\_0\right)}{t\_2}}{t\_2 - 1} \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* i (+ (+ alpha beta) i)))
        (t_1 (+ (+ alpha beta) (* 2.0 i)))
        (t_2 (* t_1 t_1)))
   (/ (/ (* t_0 (+ (* beta alpha) t_0)) t_2) (- t_2 1.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = i * ((alpha + beta) + i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = t_1 * t_1;
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0);
}

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
    real(8) :: t_1
    real(8) :: t_2
    t_0 = i * ((alpha + beta) + i)
    t_1 = (alpha + beta) + (2.0d0 * i)
    t_2 = t_1 * t_1
    code = ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0d0)
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = i * ((alpha + beta) + i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = t_1 * t_1;
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0);
}

Python

def code(alpha, beta, i):
	t_0 = i * ((alpha + beta) + i)
	t_1 = (alpha + beta) + (2.0 * i)
	t_2 = t_1 * t_1
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0)

Julia

function code(alpha, beta, i)
	t_0 = Float64(i * Float64(Float64(alpha + beta) + i))
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_2 = Float64(t_1 * t_1)
	return Float64(Float64(Float64(t_0 * Float64(Float64(beta * alpha) + t_0)) / t_2) / Float64(t_2 - 1.0))
end

MATLAB

function tmp = code(alpha, beta, i)
	t_0 = i * ((alpha + beta) + i);
	t_1 = (alpha + beta) + (2.0 * i);
	t_2 = t_1 * t_1;
	tmp = ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0);
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, N[(N[(N[(t$95$0 * N[(N[(beta * alpha), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] / N[(t$95$2 - 1.0), $MachinePrecision]), $MachinePrecision]]]]

Initial Program: 16.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_2 := t\_1 \cdot t\_1\\ \frac{\frac{t\_0 \cdot \left(\beta \cdot \alpha + t\_0\right)}{t\_2}}{t\_2 - 1} \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* i (+ (+ alpha beta) i)))
        (t_1 (+ (+ alpha beta) (* 2.0 i)))
        (t_2 (* t_1 t_1)))
   (/ (/ (* t_0 (+ (* beta alpha) t_0)) t_2) (- t_2 1.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = i * ((alpha + beta) + i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = t_1 * t_1;
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0);
}

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
    real(8) :: t_1
    real(8) :: t_2
    t_0 = i * ((alpha + beta) + i)
    t_1 = (alpha + beta) + (2.0d0 * i)
    t_2 = t_1 * t_1
    code = ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0d0)
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = i * ((alpha + beta) + i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = t_1 * t_1;
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0);
}

Python

def code(alpha, beta, i):
	t_0 = i * ((alpha + beta) + i)
	t_1 = (alpha + beta) + (2.0 * i)
	t_2 = t_1 * t_1
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0)

Julia

function code(alpha, beta, i)
	t_0 = Float64(i * Float64(Float64(alpha + beta) + i))
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_2 = Float64(t_1 * t_1)
	return Float64(Float64(Float64(t_0 * Float64(Float64(beta * alpha) + t_0)) / t_2) / Float64(t_2 - 1.0))
end

MATLAB

function tmp = code(alpha, beta, i)
	t_0 = i * ((alpha + beta) + i);
	t_1 = (alpha + beta) + (2.0 * i);
	t_2 = t_1 * t_1;
	tmp = ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0);
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, N[(N[(N[(t$95$0 * N[(N[(beta * alpha), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] / N[(t$95$2 - 1.0), $MachinePrecision]), $MachinePrecision]]]]

Alternative 1: 83.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + i \cdot 2\\ t_1 := \alpha + t\_0\\ t_2 := i \cdot \left(i + \left(\alpha + \beta\right)\right)\\ \mathbf{if}\;i \leq 3.1 \cdot 10^{+144}:\\ \;\;\;\;\frac{\frac{\frac{\alpha \cdot \beta + t\_2}{\frac{t\_1}{\frac{t\_2}{t\_1}}}}{\alpha + \left(t\_0 + 1\right)}}{\alpha + \left(t\_0 + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ beta (* i 2.0)))
        (t_1 (+ alpha t_0))
        (t_2 (* i (+ i (+ alpha beta)))))
   (if (<= i 3.1e+144)
     (/
      (/ (/ (+ (* alpha beta) t_2) (/ t_1 (/ t_2 t_1))) (+ alpha (+ t_0 1.0)))
      (+ alpha (+ t_0 -1.0)))
     0.0625)))

C

double code(double alpha, double beta, double i) {
	double t_0 = beta + (i * 2.0);
	double t_1 = alpha + t_0;
	double t_2 = i * (i + (alpha + beta));
	double tmp;
	if (i <= 3.1e+144) {
		tmp = ((((alpha * beta) + t_2) / (t_1 / (t_2 / t_1))) / (alpha + (t_0 + 1.0))) / (alpha + (t_0 + -1.0));
	} else {
		tmp = 0.0625;
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = beta + (i * 2.0d0)
    t_1 = alpha + t_0
    t_2 = i * (i + (alpha + beta))
    if (i <= 3.1d+144) then
        tmp = ((((alpha * beta) + t_2) / (t_1 / (t_2 / t_1))) / (alpha + (t_0 + 1.0d0))) / (alpha + (t_0 + (-1.0d0)))
    else
        tmp = 0.0625d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = beta + (i * 2.0);
	double t_1 = alpha + t_0;
	double t_2 = i * (i + (alpha + beta));
	double tmp;
	if (i <= 3.1e+144) {
		tmp = ((((alpha * beta) + t_2) / (t_1 / (t_2 / t_1))) / (alpha + (t_0 + 1.0))) / (alpha + (t_0 + -1.0));
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = beta + (i * 2.0)
	t_1 = alpha + t_0
	t_2 = i * (i + (alpha + beta))
	tmp = 0
	if i <= 3.1e+144:
		tmp = ((((alpha * beta) + t_2) / (t_1 / (t_2 / t_1))) / (alpha + (t_0 + 1.0))) / (alpha + (t_0 + -1.0))
	else:
		tmp = 0.0625
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(beta + Float64(i * 2.0))
	t_1 = Float64(alpha + t_0)
	t_2 = Float64(i * Float64(i + Float64(alpha + beta)))
	tmp = 0.0
	if (i <= 3.1e+144)
		tmp = Float64(Float64(Float64(Float64(Float64(alpha * beta) + t_2) / Float64(t_1 / Float64(t_2 / t_1))) / Float64(alpha + Float64(t_0 + 1.0))) / Float64(alpha + Float64(t_0 + -1.0)));
	else
		tmp = 0.0625;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = beta + (i * 2.0);
	t_1 = alpha + t_0;
	t_2 = i * (i + (alpha + beta));
	tmp = 0.0;
	if (i <= 3.1e+144)
		tmp = ((((alpha * beta) + t_2) / (t_1 / (t_2 / t_1))) / (alpha + (t_0 + 1.0))) / (alpha + (t_0 + -1.0));
	else
		tmp = 0.0625;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(alpha + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, 3.1e+144], N[(N[(N[(N[(N[(alpha * beta), $MachinePrecision] + t$95$2), $MachinePrecision] / N[(t$95$1 / N[(t$95$2 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0625]]]]

Derivation

1. Split input into 2 regimes

2. if i < 3.1000000000000002e144

   1. Initial program 41.9%

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

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

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

   3. Simplified 83.2%

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

   6. Applied egg-rr 88.4%

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

   if 3.1000000000000002e144 < i

   1. Initial program 0.1%

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

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

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

   3. Simplified 1.7%

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

   5. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16}} \]
   6. Step-by-step derivation

   7. Simplified 89.0%

      \[\leadsto \color{blue}{0.0625} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 3.1 \cdot 10^{+144}:\\ \;\;\;\;\frac{\frac{\frac{\alpha \cdot \beta + i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\frac{\alpha + \left(\beta + i \cdot 2\right)}{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\alpha + \left(\beta + i \cdot 2\right)}}}}{\alpha + \left(\left(\beta + i \cdot 2\right) + 1\right)}}{\alpha + \left(\left(\beta + i \cdot 2\right) + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + i \cdot 2\\ t_1 := i \cdot \left(i + \left(\alpha + \beta\right)\right)\\ t_2 := \alpha + t\_0\\ \mathbf{if}\;i \leq 6.4 \cdot 10^{+144}:\\ \;\;\;\;\frac{\frac{t\_1}{t\_2}}{\alpha + \left(t\_0 + 1\right)} \cdot \frac{\frac{\alpha \cdot \beta + t\_1}{t\_2}}{\alpha + \left(t\_0 + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ beta (* i 2.0)))
        (t_1 (* i (+ i (+ alpha beta))))
        (t_2 (+ alpha t_0)))
   (if (<= i 6.4e+144)
     (*
      (/ (/ t_1 t_2) (+ alpha (+ t_0 1.0)))
      (/ (/ (+ (* alpha beta) t_1) t_2) (+ alpha (+ t_0 -1.0))))
     0.0625)))

C

double code(double alpha, double beta, double i) {
	double t_0 = beta + (i * 2.0);
	double t_1 = i * (i + (alpha + beta));
	double t_2 = alpha + t_0;
	double tmp;
	if (i <= 6.4e+144) {
		tmp = ((t_1 / t_2) / (alpha + (t_0 + 1.0))) * ((((alpha * beta) + t_1) / t_2) / (alpha + (t_0 + -1.0)));
	} else {
		tmp = 0.0625;
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = beta + (i * 2.0d0)
    t_1 = i * (i + (alpha + beta))
    t_2 = alpha + t_0
    if (i <= 6.4d+144) then
        tmp = ((t_1 / t_2) / (alpha + (t_0 + 1.0d0))) * ((((alpha * beta) + t_1) / t_2) / (alpha + (t_0 + (-1.0d0))))
    else
        tmp = 0.0625d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = beta + (i * 2.0);
	double t_1 = i * (i + (alpha + beta));
	double t_2 = alpha + t_0;
	double tmp;
	if (i <= 6.4e+144) {
		tmp = ((t_1 / t_2) / (alpha + (t_0 + 1.0))) * ((((alpha * beta) + t_1) / t_2) / (alpha + (t_0 + -1.0)));
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = beta + (i * 2.0)
	t_1 = i * (i + (alpha + beta))
	t_2 = alpha + t_0
	tmp = 0
	if i <= 6.4e+144:
		tmp = ((t_1 / t_2) / (alpha + (t_0 + 1.0))) * ((((alpha * beta) + t_1) / t_2) / (alpha + (t_0 + -1.0)))
	else:
		tmp = 0.0625
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(beta + Float64(i * 2.0))
	t_1 = Float64(i * Float64(i + Float64(alpha + beta)))
	t_2 = Float64(alpha + t_0)
	tmp = 0.0
	if (i <= 6.4e+144)
		tmp = Float64(Float64(Float64(t_1 / t_2) / Float64(alpha + Float64(t_0 + 1.0))) * Float64(Float64(Float64(Float64(alpha * beta) + t_1) / t_2) / Float64(alpha + Float64(t_0 + -1.0))));
	else
		tmp = 0.0625;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = beta + (i * 2.0);
	t_1 = i * (i + (alpha + beta));
	t_2 = alpha + t_0;
	tmp = 0.0;
	if (i <= 6.4e+144)
		tmp = ((t_1 / t_2) / (alpha + (t_0 + 1.0))) * ((((alpha * beta) + t_1) / t_2) / (alpha + (t_0 + -1.0)));
	else
		tmp = 0.0625;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(i * N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(alpha + t$95$0), $MachinePrecision]}, If[LessEqual[i, 6.4e+144], N[(N[(N[(t$95$1 / t$95$2), $MachinePrecision] / N[(alpha + N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(alpha * beta), $MachinePrecision] + t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision] / N[(alpha + N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0625]]]]

Derivation

1. Split input into 2 regimes

2. if i < 6.4000000000000002e144

   1. Initial program 41.9%

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

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

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

   3. Simplified 83.2%

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

   6. Applied egg-rr 88.3%

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

   if 6.4000000000000002e144 < i

   1. Initial program 0.1%

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

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

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

   3. Simplified 1.7%

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

   5. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16}} \]
   6. Step-by-step derivation

   7. Simplified 89.0%

      \[\leadsto \color{blue}{0.0625} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 6.4 \cdot 10^{+144}:\\ \;\;\;\;\frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\alpha + \left(\beta + i \cdot 2\right)}}{\alpha + \left(\left(\beta + i \cdot 2\right) + 1\right)} \cdot \frac{\frac{\alpha \cdot \beta + i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\alpha + \left(\beta + i \cdot 2\right)}}{\alpha + \left(\left(\beta + i \cdot 2\right) + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + i \cdot 2\\ t_1 := \alpha + t\_0\\ \mathbf{if}\;i \leq 4.9 \cdot 10^{+144}:\\ \;\;\;\;\frac{\left(\alpha \cdot \beta + i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \frac{i \cdot \left(i + \beta\right)}{t\_0 \cdot t\_0}}{t\_1 \cdot t\_1 + -1}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ beta (* i 2.0))) (t_1 (+ alpha t_0)))
   (if (<= i 4.9e+144)
     (/
      (*
       (+ (* alpha beta) (* i (+ i (+ alpha beta))))
       (/ (* i (+ i beta)) (* t_0 t_0)))
      (+ (* t_1 t_1) -1.0))
     0.0625)))

C

double code(double alpha, double beta, double i) {
	double t_0 = beta + (i * 2.0);
	double t_1 = alpha + t_0;
	double tmp;
	if (i <= 4.9e+144) {
		tmp = (((alpha * beta) + (i * (i + (alpha + beta)))) * ((i * (i + beta)) / (t_0 * t_0))) / ((t_1 * t_1) + -1.0);
	} else {
		tmp = 0.0625;
	}
	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 = beta + (i * 2.0d0)
    t_1 = alpha + t_0
    if (i <= 4.9d+144) then
        tmp = (((alpha * beta) + (i * (i + (alpha + beta)))) * ((i * (i + beta)) / (t_0 * t_0))) / ((t_1 * t_1) + (-1.0d0))
    else
        tmp = 0.0625d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = beta + (i * 2.0);
	double t_1 = alpha + t_0;
	double tmp;
	if (i <= 4.9e+144) {
		tmp = (((alpha * beta) + (i * (i + (alpha + beta)))) * ((i * (i + beta)) / (t_0 * t_0))) / ((t_1 * t_1) + -1.0);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = beta + (i * 2.0)
	t_1 = alpha + t_0
	tmp = 0
	if i <= 4.9e+144:
		tmp = (((alpha * beta) + (i * (i + (alpha + beta)))) * ((i * (i + beta)) / (t_0 * t_0))) / ((t_1 * t_1) + -1.0)
	else:
		tmp = 0.0625
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(beta + Float64(i * 2.0))
	t_1 = Float64(alpha + t_0)
	tmp = 0.0
	if (i <= 4.9e+144)
		tmp = Float64(Float64(Float64(Float64(alpha * beta) + Float64(i * Float64(i + Float64(alpha + beta)))) * Float64(Float64(i * Float64(i + beta)) / Float64(t_0 * t_0))) / Float64(Float64(t_1 * t_1) + -1.0));
	else
		tmp = 0.0625;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = beta + (i * 2.0);
	t_1 = alpha + t_0;
	tmp = 0.0;
	if (i <= 4.9e+144)
		tmp = (((alpha * beta) + (i * (i + (alpha + beta)))) * ((i * (i + beta)) / (t_0 * t_0))) / ((t_1 * t_1) + -1.0);
	else
		tmp = 0.0625;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(alpha + t$95$0), $MachinePrecision]}, If[LessEqual[i, 4.9e+144], N[(N[(N[(N[(alpha * beta), $MachinePrecision] + N[(i * N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(i * N[(i + beta), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$1 * t$95$1), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], 0.0625]]]

Derivation

1. Split input into 2 regimes

2. if i < 4.9e144

   1. Initial program 41.9%

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

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

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

   3. Simplified 83.2%

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

   5. Taylor expanded in alpha around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 75.4%

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

   7. Simplified 75.4%

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

   if 4.9e144 < i

   1. Initial program 0.1%

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

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

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

   3. Simplified 1.7%

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

   5. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16}} \]
   6. Step-by-step derivation

   7. Simplified 89.0%

      \[\leadsto \color{blue}{0.0625} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 4.9 \cdot 10^{+144}:\\ \;\;\;\;\frac{\left(\alpha \cdot \beta + i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \frac{i \cdot \left(i + \beta\right)}{\left(\beta + i \cdot 2\right) \cdot \left(\beta + i \cdot 2\right)}}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) \cdot \left(\alpha + \left(\beta + i \cdot 2\right)\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot \left(i + \beta\right)\\ t_1 := \beta + i \cdot 2\\ t_2 := \alpha + t\_1\\ \mathbf{if}\;i \leq 7.8 \cdot 10^{+144}:\\ \;\;\;\;\frac{t\_0 \cdot \frac{t\_0}{t\_1 \cdot t\_1}}{t\_2 \cdot t\_2 + -1}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* i (+ i beta))) (t_1 (+ beta (* i 2.0))) (t_2 (+ alpha t_1)))
   (if (<= i 7.8e+144)
     (/ (* t_0 (/ t_0 (* t_1 t_1))) (+ (* t_2 t_2) -1.0))
     0.0625)))

C

double code(double alpha, double beta, double i) {
	double t_0 = i * (i + beta);
	double t_1 = beta + (i * 2.0);
	double t_2 = alpha + t_1;
	double tmp;
	if (i <= 7.8e+144) {
		tmp = (t_0 * (t_0 / (t_1 * t_1))) / ((t_2 * t_2) + -1.0);
	} else {
		tmp = 0.0625;
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = i * (i + beta)
    t_1 = beta + (i * 2.0d0)
    t_2 = alpha + t_1
    if (i <= 7.8d+144) then
        tmp = (t_0 * (t_0 / (t_1 * t_1))) / ((t_2 * t_2) + (-1.0d0))
    else
        tmp = 0.0625d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = i * (i + beta);
	double t_1 = beta + (i * 2.0);
	double t_2 = alpha + t_1;
	double tmp;
	if (i <= 7.8e+144) {
		tmp = (t_0 * (t_0 / (t_1 * t_1))) / ((t_2 * t_2) + -1.0);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = i * (i + beta)
	t_1 = beta + (i * 2.0)
	t_2 = alpha + t_1
	tmp = 0
	if i <= 7.8e+144:
		tmp = (t_0 * (t_0 / (t_1 * t_1))) / ((t_2 * t_2) + -1.0)
	else:
		tmp = 0.0625
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(i * Float64(i + beta))
	t_1 = Float64(beta + Float64(i * 2.0))
	t_2 = Float64(alpha + t_1)
	tmp = 0.0
	if (i <= 7.8e+144)
		tmp = Float64(Float64(t_0 * Float64(t_0 / Float64(t_1 * t_1))) / Float64(Float64(t_2 * t_2) + -1.0));
	else
		tmp = 0.0625;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = i * (i + beta);
	t_1 = beta + (i * 2.0);
	t_2 = alpha + t_1;
	tmp = 0.0;
	if (i <= 7.8e+144)
		tmp = (t_0 * (t_0 / (t_1 * t_1))) / ((t_2 * t_2) + -1.0);
	else
		tmp = 0.0625;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(i * N[(i + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(alpha + t$95$1), $MachinePrecision]}, If[LessEqual[i, 7.8e+144], N[(N[(t$95$0 * N[(t$95$0 / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$2 * t$95$2), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], 0.0625]]]]

Derivation

1. Split input into 2 regimes

2. if i < 7.80000000000000036e144

   1. Initial program 41.9%

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

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

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

   3. Simplified 83.2%

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

   5. Taylor expanded in alpha around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 75.4%

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

   7. Simplified 75.4%

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

   8. Taylor expanded in alpha around 0

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

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

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

      2. +-lowering-+.f64 78.5%

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

   10. Simplified 78.5%

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

   if 7.80000000000000036e144 < i

   1. Initial program 0.1%

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

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

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

   3. Simplified 1.7%

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

   5. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16}} \]
   6. Step-by-step derivation

   7. Simplified 89.0%

      \[\leadsto \color{blue}{0.0625} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 7.8 \cdot 10^{+144}:\\ \;\;\;\;\frac{\left(i \cdot \left(i + \beta\right)\right) \cdot \frac{i \cdot \left(i + \beta\right)}{\left(\beta + i \cdot 2\right) \cdot \left(\beta + i \cdot 2\right)}}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) \cdot \left(\alpha + \left(\beta + i \cdot 2\right)\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot \left(i + \beta\right)\\ t_1 := \beta + i \cdot 2\\ t_2 := t\_1 \cdot t\_1\\ \mathbf{if}\;i \leq 1.9 \cdot 10^{+144}:\\ \;\;\;\;\frac{\frac{t\_0}{\frac{t\_2}{t\_0}}}{t\_2 + -1}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* i (+ i beta))) (t_1 (+ beta (* i 2.0))) (t_2 (* t_1 t_1)))
   (if (<= i 1.9e+144) (/ (/ t_0 (/ t_2 t_0)) (+ t_2 -1.0)) 0.0625)))

C

double code(double alpha, double beta, double i) {
	double t_0 = i * (i + beta);
	double t_1 = beta + (i * 2.0);
	double t_2 = t_1 * t_1;
	double tmp;
	if (i <= 1.9e+144) {
		tmp = (t_0 / (t_2 / t_0)) / (t_2 + -1.0);
	} else {
		tmp = 0.0625;
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = i * (i + beta)
    t_1 = beta + (i * 2.0d0)
    t_2 = t_1 * t_1
    if (i <= 1.9d+144) then
        tmp = (t_0 / (t_2 / t_0)) / (t_2 + (-1.0d0))
    else
        tmp = 0.0625d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = i * (i + beta);
	double t_1 = beta + (i * 2.0);
	double t_2 = t_1 * t_1;
	double tmp;
	if (i <= 1.9e+144) {
		tmp = (t_0 / (t_2 / t_0)) / (t_2 + -1.0);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = i * (i + beta)
	t_1 = beta + (i * 2.0)
	t_2 = t_1 * t_1
	tmp = 0
	if i <= 1.9e+144:
		tmp = (t_0 / (t_2 / t_0)) / (t_2 + -1.0)
	else:
		tmp = 0.0625
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(i * Float64(i + beta))
	t_1 = Float64(beta + Float64(i * 2.0))
	t_2 = Float64(t_1 * t_1)
	tmp = 0.0
	if (i <= 1.9e+144)
		tmp = Float64(Float64(t_0 / Float64(t_2 / t_0)) / Float64(t_2 + -1.0));
	else
		tmp = 0.0625;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = i * (i + beta);
	t_1 = beta + (i * 2.0);
	t_2 = t_1 * t_1;
	tmp = 0.0;
	if (i <= 1.9e+144)
		tmp = (t_0 / (t_2 / t_0)) / (t_2 + -1.0);
	else
		tmp = 0.0625;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(i * N[(i + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, If[LessEqual[i, 1.9e+144], N[(N[(t$95$0 / N[(t$95$2 / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 + -1.0), $MachinePrecision]), $MachinePrecision], 0.0625]]]]

Derivation

1. Split input into 2 regimes

2. if i < 1.90000000000000013e144

   1. Initial program 41.9%

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

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

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

   3. Simplified 83.2%

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

   5. Taylor expanded in alpha around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 75.4%

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

   7. Simplified 75.4%

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

   8. Taylor expanded in alpha around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

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

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

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

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

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

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

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

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

      10. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

      16. sub-neg N/A

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

      17. metadata-eval N/A

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

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

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

   10. Simplified 35.0%

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

       1. unswap-sqr N/A

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

       2. +-commutative N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. times-frac N/A

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

       6. pow2 N/A

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

       7. *-commutative N/A

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

       8. pow2 N/A

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

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

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

   12. Applied egg-rr 73.3%

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

       1. associate-*l/ N/A

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

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

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

   14. Applied egg-rr 73.3%

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

   if 1.90000000000000013e144 < i

   1. Initial program 0.1%

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

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

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

   3. Simplified 1.7%

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

   5. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16}} \]
   6. Step-by-step derivation

   7. Simplified 89.0%

      \[\leadsto \color{blue}{0.0625} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 1.9 \cdot 10^{+144}:\\ \;\;\;\;\frac{\frac{i \cdot \left(i + \beta\right)}{\frac{\left(\beta + i \cdot 2\right) \cdot \left(\beta + i \cdot 2\right)}{i \cdot \left(i + \beta\right)}}}{\left(\beta + i \cdot 2\right) \cdot \left(\beta + i \cdot 2\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot \left(i + \beta\right)\\ t_1 := \beta + i \cdot 2\\ t_2 := t\_1 \cdot t\_1\\ \mathbf{if}\;i \leq 6 \cdot 10^{+144}:\\ \;\;\;\;\frac{t\_0}{t\_2} \cdot \frac{t\_0}{t\_2 + -1}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* i (+ i beta))) (t_1 (+ beta (* i 2.0))) (t_2 (* t_1 t_1)))
   (if (<= i 6e+144) (* (/ t_0 t_2) (/ t_0 (+ t_2 -1.0))) 0.0625)))

C

double code(double alpha, double beta, double i) {
	double t_0 = i * (i + beta);
	double t_1 = beta + (i * 2.0);
	double t_2 = t_1 * t_1;
	double tmp;
	if (i <= 6e+144) {
		tmp = (t_0 / t_2) * (t_0 / (t_2 + -1.0));
	} else {
		tmp = 0.0625;
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = i * (i + beta)
    t_1 = beta + (i * 2.0d0)
    t_2 = t_1 * t_1
    if (i <= 6d+144) then
        tmp = (t_0 / t_2) * (t_0 / (t_2 + (-1.0d0)))
    else
        tmp = 0.0625d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = i * (i + beta);
	double t_1 = beta + (i * 2.0);
	double t_2 = t_1 * t_1;
	double tmp;
	if (i <= 6e+144) {
		tmp = (t_0 / t_2) * (t_0 / (t_2 + -1.0));
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = i * (i + beta)
	t_1 = beta + (i * 2.0)
	t_2 = t_1 * t_1
	tmp = 0
	if i <= 6e+144:
		tmp = (t_0 / t_2) * (t_0 / (t_2 + -1.0))
	else:
		tmp = 0.0625
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(i * Float64(i + beta))
	t_1 = Float64(beta + Float64(i * 2.0))
	t_2 = Float64(t_1 * t_1)
	tmp = 0.0
	if (i <= 6e+144)
		tmp = Float64(Float64(t_0 / t_2) * Float64(t_0 / Float64(t_2 + -1.0)));
	else
		tmp = 0.0625;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = i * (i + beta);
	t_1 = beta + (i * 2.0);
	t_2 = t_1 * t_1;
	tmp = 0.0;
	if (i <= 6e+144)
		tmp = (t_0 / t_2) * (t_0 / (t_2 + -1.0));
	else
		tmp = 0.0625;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(i * N[(i + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, If[LessEqual[i, 6e+144], N[(N[(t$95$0 / t$95$2), $MachinePrecision] * N[(t$95$0 / N[(t$95$2 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0625]]]]

Derivation

1. Split input into 2 regimes

2. if i < 5.9999999999999998e144

   1. Initial program 41.9%

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

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

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

   3. Simplified 83.2%

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

   5. Taylor expanded in alpha around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 75.4%

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

   7. Simplified 75.4%

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

   8. Taylor expanded in alpha around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

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

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

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

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

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

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

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

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

      10. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

      16. sub-neg N/A

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

      17. metadata-eval N/A

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

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

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

   10. Simplified 35.0%

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

       1. unswap-sqr N/A

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

       2. +-commutative N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. times-frac N/A

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

       6. pow2 N/A

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

       7. *-commutative N/A

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

       8. pow2 N/A

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

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

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

   12. Applied egg-rr 73.3%

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

   if 5.9999999999999998e144 < i

   1. Initial program 0.1%

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

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

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

   3. Simplified 1.7%

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

   5. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16}} \]
   6. Step-by-step derivation

   7. Simplified 89.0%

      \[\leadsto \color{blue}{0.0625} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 6 \cdot 10^{+144}:\\ \;\;\;\;\frac{i \cdot \left(i + \beta\right)}{\left(\beta + i \cdot 2\right) \cdot \left(\beta + i \cdot 2\right)} \cdot \frac{i \cdot \left(i + \beta\right)}{\left(\beta + i \cdot 2\right) \cdot \left(\beta + i \cdot 2\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.15 \cdot 10^{+201}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 2.15e+201)
   (+ 0.0625 (/ 0.015625 (* i i)))
   (* (/ (+ i alpha) beta) (/ i beta))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.15e+201) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else {
		tmp = ((i + alpha) / beta) * (i / 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 <= 2.15d+201) then
        tmp = 0.0625d0 + (0.015625d0 / (i * i))
    else
        tmp = ((i + alpha) / beta) * (i / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.15e+201) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else {
		tmp = ((i + alpha) / beta) * (i / beta);
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 2.15e+201:
		tmp = 0.0625 + (0.015625 / (i * i))
	else:
		tmp = ((i + alpha) / beta) * (i / beta)
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 2.15e+201)
		tmp = Float64(0.0625 + Float64(0.015625 / Float64(i * i)));
	else
		tmp = Float64(Float64(Float64(i + alpha) / beta) * Float64(i / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 2.15e+201)
		tmp = 0.0625 + (0.015625 / (i * i));
	else
		tmp = ((i + alpha) / beta) * (i / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 2.15e+201], N[(0.0625 + N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.14999999999999995e201

   1. Initial program 24.8%

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

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

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

   3. Simplified 48.3%

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

   5. Taylor expanded in alpha around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 43.7%

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

   7. Simplified 43.7%

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

   8. Taylor expanded in alpha around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

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

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

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

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

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

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

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

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

      10. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

      16. sub-neg N/A

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

      17. metadata-eval N/A

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

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

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

   10. Simplified 20.7%

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

   11. Taylor expanded in i around inf

       \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{{\beta}^{2}}{{i}^{2}}\right) - \frac{1}{256} \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]
   12. Step-by-step derivation

       1. associate--l+ N/A

          \[\leadsto \frac{1}{16} + \color{blue}{\left(\frac{1}{16} \cdot \frac{{\beta}^{2}}{{i}^{2}} - \frac{1}{256} \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}\right)} \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \color{blue}{\left(\frac{1}{16} \cdot \frac{{\beta}^{2}}{{i}^{2}} - \frac{1}{256} \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}\right)}\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \mathsf{\_.f64}\left(\left(\frac{1}{16} \cdot \frac{{\beta}^{2}}{{i}^{2}}\right), \color{blue}{\left(\frac{1}{256} \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}\right)}\right)\right) \]

       4. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \mathsf{\_.f64}\left(\left(\frac{\frac{1}{16} \cdot {\beta}^{2}}{{i}^{2}}\right), \left(\color{blue}{\frac{1}{256}} \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{16} \cdot {\beta}^{2}\right), \left({i}^{2}\right)\right), \left(\color{blue}{\frac{1}{256}} \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left({\beta}^{2} \cdot \frac{1}{16}\right), \left({i}^{2}\right)\right), \left(\frac{1}{256} \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\beta}^{2}\right), \frac{1}{16}\right), \left({i}^{2}\right)\right), \left(\frac{1}{256} \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\beta \cdot \beta\right), \frac{1}{16}\right), \left({i}^{2}\right)\right), \left(\frac{1}{256} \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \frac{1}{16}\right), \left({i}^{2}\right)\right), \left(\frac{1}{256} \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \frac{1}{16}\right), \left(i \cdot i\right)\right), \left(\frac{1}{256} \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \frac{1}{16}\right), \mathsf{*.f64}\left(i, i\right)\right), \left(\frac{1}{256} \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \frac{1}{16}\right), \mathsf{*.f64}\left(i, i\right)\right), \mathsf{*.f64}\left(\frac{1}{256}, \color{blue}{\left(\frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}\right)}\right)\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \frac{1}{16}\right), \mathsf{*.f64}\left(i, i\right)\right), \mathsf{*.f64}\left(\frac{1}{256}, \mathsf{/.f64}\left(\left(4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)\right), \color{blue}{\left({i}^{2}\right)}\right)\right)\right)\right) \]

   13. Simplified 76.6%

       \[\leadsto \color{blue}{0.0625 + \left(\frac{\left(\beta \cdot \beta\right) \cdot 0.0625}{i \cdot i} - 0.00390625 \cdot \frac{4 \cdot \left(\beta \cdot \beta + -1\right) + \left(\beta \cdot \beta\right) \cdot 20}{i \cdot i}\right)} \]

   14. Taylor expanded in beta around 0

       \[\leadsto \color{blue}{\frac{1}{16} + \frac{1}{64} \cdot \frac{1}{{i}^{2}}} \]
   15. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{1}{64} \cdot \frac{1}{{i}^{2}} + \color{blue}{\frac{1}{16}} \]

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

          \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{64} \cdot \frac{1}{{i}^{2}}\right), \color{blue}{\frac{1}{16}}\right) \]

       3. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(\left(\frac{\frac{1}{64} \cdot 1}{{i}^{2}}\right), \frac{1}{16}\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\left(\frac{\frac{1}{64}}{{i}^{2}}\right), \frac{1}{16}\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{64}, \left({i}^{2}\right)\right), \frac{1}{16}\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{64}, \left(i \cdot i\right)\right), \frac{1}{16}\right) \]

       7. *-lowering-*.f64 79.9%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{64}, \mathsf{*.f64}\left(i, i\right)\right), \frac{1}{16}\right) \]

   16. Simplified 79.9%

       \[\leadsto \color{blue}{\frac{0.015625}{i \cdot i} + 0.0625} \]

   if 2.14999999999999995e201 < beta

   1. Initial program 0.0%

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

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

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

   3. Simplified 19.5%

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

   5. Taylor expanded in beta around inf

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

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

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 26.0%

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

   7. Simplified 26.0%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

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

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

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

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

      5. +-commutative N/A

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

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

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

      7. /-lowering-/.f64 82.3%

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

   9. Applied egg-rr 82.3%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.15 \cdot 10^{+201}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.6% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.85 \cdot 10^{+201}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.85e+201)
   (+ 0.0625 (/ 0.015625 (* i i)))
   (* (/ i beta) (/ i beta))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.85e+201) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else {
		tmp = (i / beta) * (i / 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 <= 1.85d+201) then
        tmp = 0.0625d0 + (0.015625d0 / (i * i))
    else
        tmp = (i / beta) * (i / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.85e+201) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else {
		tmp = (i / beta) * (i / beta);
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.85e+201:
		tmp = 0.0625 + (0.015625 / (i * i))
	else:
		tmp = (i / beta) * (i / beta)
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.85e+201)
		tmp = Float64(0.0625 + Float64(0.015625 / Float64(i * i)));
	else
		tmp = Float64(Float64(i / beta) * Float64(i / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1.85e+201)
		tmp = 0.0625 + (0.015625 / (i * i));
	else
		tmp = (i / beta) * (i / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 1.85e+201], N[(0.0625 + N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.8499999999999999e201

   1. Initial program 24.8%

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

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

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

   3. Simplified 48.3%

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

   5. Taylor expanded in alpha around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 43.7%

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

   7. Simplified 43.7%

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

   8. Taylor expanded in alpha around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

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

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

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

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

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

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

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

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

      10. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

      16. sub-neg N/A

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

      17. metadata-eval N/A

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

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

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

   10. Simplified 20.7%

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

   11. Taylor expanded in i around inf

       \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{{\beta}^{2}}{{i}^{2}}\right) - \frac{1}{256} \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]
   12. Step-by-step derivation

       1. associate--l+ N/A

          \[\leadsto \frac{1}{16} + \color{blue}{\left(\frac{1}{16} \cdot \frac{{\beta}^{2}}{{i}^{2}} - \frac{1}{256} \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}\right)} \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \color{blue}{\left(\frac{1}{16} \cdot \frac{{\beta}^{2}}{{i}^{2}} - \frac{1}{256} \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}\right)}\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \mathsf{\_.f64}\left(\left(\frac{1}{16} \cdot \frac{{\beta}^{2}}{{i}^{2}}\right), \color{blue}{\left(\frac{1}{256} \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}\right)}\right)\right) \]

       4. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \mathsf{\_.f64}\left(\left(\frac{\frac{1}{16} \cdot {\beta}^{2}}{{i}^{2}}\right), \left(\color{blue}{\frac{1}{256}} \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{16} \cdot {\beta}^{2}\right), \left({i}^{2}\right)\right), \left(\color{blue}{\frac{1}{256}} \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left({\beta}^{2} \cdot \frac{1}{16}\right), \left({i}^{2}\right)\right), \left(\frac{1}{256} \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\beta}^{2}\right), \frac{1}{16}\right), \left({i}^{2}\right)\right), \left(\frac{1}{256} \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\beta \cdot \beta\right), \frac{1}{16}\right), \left({i}^{2}\right)\right), \left(\frac{1}{256} \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \frac{1}{16}\right), \left({i}^{2}\right)\right), \left(\frac{1}{256} \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \frac{1}{16}\right), \left(i \cdot i\right)\right), \left(\frac{1}{256} \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \frac{1}{16}\right), \mathsf{*.f64}\left(i, i\right)\right), \left(\frac{1}{256} \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \frac{1}{16}\right), \mathsf{*.f64}\left(i, i\right)\right), \mathsf{*.f64}\left(\frac{1}{256}, \color{blue}{\left(\frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}\right)}\right)\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(\frac{1}{16}, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\beta, \beta\right), \frac{1}{16}\right), \mathsf{*.f64}\left(i, i\right)\right), \mathsf{*.f64}\left(\frac{1}{256}, \mathsf{/.f64}\left(\left(4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)\right), \color{blue}{\left({i}^{2}\right)}\right)\right)\right)\right) \]

   13. Simplified 76.6%

       \[\leadsto \color{blue}{0.0625 + \left(\frac{\left(\beta \cdot \beta\right) \cdot 0.0625}{i \cdot i} - 0.00390625 \cdot \frac{4 \cdot \left(\beta \cdot \beta + -1\right) + \left(\beta \cdot \beta\right) \cdot 20}{i \cdot i}\right)} \]

   14. Taylor expanded in beta around 0

       \[\leadsto \color{blue}{\frac{1}{16} + \frac{1}{64} \cdot \frac{1}{{i}^{2}}} \]
   15. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{1}{64} \cdot \frac{1}{{i}^{2}} + \color{blue}{\frac{1}{16}} \]

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

          \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{64} \cdot \frac{1}{{i}^{2}}\right), \color{blue}{\frac{1}{16}}\right) \]

       3. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(\left(\frac{\frac{1}{64} \cdot 1}{{i}^{2}}\right), \frac{1}{16}\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\left(\frac{\frac{1}{64}}{{i}^{2}}\right), \frac{1}{16}\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{64}, \left({i}^{2}\right)\right), \frac{1}{16}\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{64}, \left(i \cdot i\right)\right), \frac{1}{16}\right) \]

       7. *-lowering-*.f64 79.9%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{64}, \mathsf{*.f64}\left(i, i\right)\right), \frac{1}{16}\right) \]

   16. Simplified 79.9%

       \[\leadsto \color{blue}{\frac{0.015625}{i \cdot i} + 0.0625} \]

   if 1.8499999999999999e201 < beta

   1. Initial program 0.0%

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

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

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

   3. Simplified 19.5%

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

   5. Taylor expanded in beta around inf

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

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

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 26.0%

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

   7. Simplified 26.0%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

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

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

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

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

      5. +-commutative N/A

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

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

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

      7. /-lowering-/.f64 82.3%

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

   9. Applied egg-rr 82.3%

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

   10. Taylor expanded in i around inf

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

       1. /-lowering-/.f64 73.9%

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

   12. Simplified 73.9%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.85 \cdot 10^{+201}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.4% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.82 \cdot 10^{+201}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.82e+201) 0.0625 (* (/ i beta) (/ i beta))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.82e+201) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (i / 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 <= 1.82d+201) then
        tmp = 0.0625d0
    else
        tmp = (i / beta) * (i / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.82e+201) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (i / beta);
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.82e+201:
		tmp = 0.0625
	else:
		tmp = (i / beta) * (i / beta)
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.82e+201)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / beta) * Float64(i / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1.82e+201)
		tmp = 0.0625;
	else
		tmp = (i / beta) * (i / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 1.82e+201], 0.0625, N[(N[(i / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.82e201

   1. Initial program 24.8%

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

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

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

   3. Simplified 48.3%

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

   5. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16}} \]
   6. Step-by-step derivation

   7. Simplified 79.5%

      \[\leadsto \color{blue}{0.0625} \]

   if 1.82e201 < beta

   1. Initial program 0.0%

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

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

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

   3. Simplified 19.5%

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

   5. Taylor expanded in beta around inf

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

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

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 26.0%

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

   7. Simplified 26.0%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

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

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

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

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

      5. +-commutative N/A

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

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

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

      7. /-lowering-/.f64 82.3%

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

   9. Applied egg-rr 82.3%

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

   10. Taylor expanded in i around inf

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

       1. /-lowering-/.f64 73.9%

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

   12. Simplified 73.9%

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

Alternative 10: 72.2% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.8 \cdot 10^{+253}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 4.8e+253) 0.0625 (* (/ i beta) (/ alpha beta))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 4.8e+253) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (alpha / 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 <= 4.8d+253) then
        tmp = 0.0625d0
    else
        tmp = (i / beta) * (alpha / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 4.8e+253) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (alpha / beta);
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 4.8e+253:
		tmp = 0.0625
	else:
		tmp = (i / beta) * (alpha / beta)
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 4.8e+253)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / beta) * Float64(alpha / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 4.8e+253)
		tmp = 0.0625;
	else
		tmp = (i / beta) * (alpha / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 4.8e+253], 0.0625, N[(N[(i / beta), $MachinePrecision] * N[(alpha / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.79999999999999982e253

   1. Initial program 23.8%

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

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

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

   3. Simplified 46.8%

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

   5. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16}} \]
   6. Step-by-step derivation

   7. Simplified 77.3%

      \[\leadsto \color{blue}{0.0625} \]

   if 4.79999999999999982e253 < beta

   1. Initial program 0.0%

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

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

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

   3. Simplified 27.3%

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

   5. Taylor expanded in beta around inf

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

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

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 38.5%

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

   7. Simplified 38.5%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

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

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

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

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

      5. +-commutative N/A

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

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

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

      7. /-lowering-/.f64 86.8%

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

   9. Applied egg-rr 86.8%

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

   10. Taylor expanded in i around 0

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

       1. /-lowering-/.f64 51.1%

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

   12. Simplified 51.1%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.8 \cdot 10^{+253}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 70.2% accurate, 53.0× speedup

Math

\[\begin{array}{l} \\ 0.0625 \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 0.0625)

C

double code(double alpha, double beta, double i) {
	return 0.0625;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    code = 0.0625d0
end function

Java

public static double code(double alpha, double beta, double i) {
	return 0.0625;
}

Python

def code(alpha, beta, i):
	return 0.0625

Julia

function code(alpha, beta, i)
	return 0.0625
end

MATLAB

function tmp = code(alpha, beta, i)
	tmp = 0.0625;
end

Wolfram

code[alpha_, beta_, i_] := 0.0625

Derivation

1. Initial program 22.8%

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

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

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

3. Simplified 45.9%

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

5. Taylor expanded in i around inf

   \[\leadsto \color{blue}{\frac{1}{16}} \]
6. Step-by-step derivation

7. Simplified 74.5%

   \[\leadsto \color{blue}{0.0625} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024161 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :precision binary64
  :pre (and (and (> alpha -1.0) (> beta -1.0)) (> i 1.0))
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i)))) (- (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i))) 1.0)))