Octave 3.8, jcobi/4

Percentage Accurate: 17.0% → 86.8%

Time: 6.2s

Alternatives: 14

Speedup: 115.0×

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: 17.0% 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: 86.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, i, \alpha + \beta\right)\\ t_1 := t\_0 - 1\\ t_2 := 1 + t\_0\\ t_3 := \left(\left(\alpha + \beta\right) + i\right) \cdot \frac{i}{t\_0}\\ \mathbf{if}\;\beta \leq 5.9 \cdot 10^{+162}:\\ \;\;\;\;\frac{\frac{\left(-i\right) \cdot \mathsf{fma}\left(\left(-\alpha\right) - \beta, \frac{0.25}{i}, -0.5\right)}{t\_1}}{\frac{t\_2}{t\_3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{t\_1} \cdot \frac{t\_3}{t\_2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma 2.0 i (+ alpha beta)))
        (t_1 (- t_0 1.0))
        (t_2 (+ 1.0 t_0))
        (t_3 (* (+ (+ alpha beta) i) (/ i t_0))))
   (if (<= beta 5.9e+162)
     (/ (/ (* (- i) (fma (- (- alpha) beta) (/ 0.25 i) -0.5)) t_1) (/ t_2 t_3))
     (* (/ (+ i alpha) t_1) (/ t_3 t_2)))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = fma(2.0, i, (alpha + beta));
	double t_1 = t_0 - 1.0;
	double t_2 = 1.0 + t_0;
	double t_3 = ((alpha + beta) + i) * (i / t_0);
	double tmp;
	if (beta <= 5.9e+162) {
		tmp = ((-i * fma((-alpha - beta), (0.25 / i), -0.5)) / t_1) / (t_2 / t_3);
	} else {
		tmp = ((i + alpha) / t_1) * (t_3 / t_2);
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = fma(2.0, i, Float64(alpha + beta))
	t_1 = Float64(t_0 - 1.0)
	t_2 = Float64(1.0 + t_0)
	t_3 = Float64(Float64(Float64(alpha + beta) + i) * Float64(i / t_0))
	tmp = 0.0
	if (beta <= 5.9e+162)
		tmp = Float64(Float64(Float64(Float64(-i) * fma(Float64(Float64(-alpha) - beta), Float64(0.25 / i), -0.5)) / t_1) / Float64(t_2 / t_3));
	else
		tmp = Float64(Float64(Float64(i + alpha) / t_1) * Float64(t_3 / t_2));
	end
	return tmp
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision] * N[(i / t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 5.9e+162], N[(N[(N[((-i) * N[(N[((-alpha) - beta), $MachinePrecision] * N[(0.25 / i), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$2 / t$95$3), $MachinePrecision]), $MachinePrecision], N[(N[(N[(i + alpha), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(t$95$3 / t$95$2), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.90000000000000027e162

   1. Initial program 18.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. difference-of-sqr-1 N/A

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

   4. Applied rewrites 41.3%

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

   5. Taylor expanded in i around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. distribute-rgt-out-- N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 80.8

          \[\leadsto \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{1 + \mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \frac{\left(-i\right) \cdot \mathsf{fma}\left(-1, \frac{\color{blue}{\left(\beta + \alpha\right)} \cdot 0.25}{i}, -0.5\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - 1} \]

   7. Applied rewrites 80.8%

      \[\leadsto \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{1 + \mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \frac{\color{blue}{\left(-i\right) \cdot \mathsf{fma}\left(-1, \frac{\left(\beta + \alpha\right) \cdot 0.25}{i}, -0.5\right)}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - 1} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

   9. Applied rewrites 80.8%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\left(-\alpha\right) - \beta, \frac{0.25}{i}, -0.5\right) \cdot \left(-i\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - 1}}{\frac{1 + \mathsf{fma}\left(2, i, \alpha + \beta\right)}{\frac{i}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \left(\left(\alpha + \beta\right) + i\right)}}} \]

   if 5.90000000000000027e162 < 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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. difference-of-sqr-1 N/A

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

   4. Applied rewrites 24.7%

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

   5. Taylor expanded in i around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. distribute-rgt-out-- N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 24.6

          \[\leadsto \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{1 + \mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \frac{\left(-i\right) \cdot \mathsf{fma}\left(-1, \frac{\color{blue}{\left(\beta + \alpha\right)} \cdot 0.25}{i}, -0.5\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - 1} \]

   7. Applied rewrites 24.6%

      \[\leadsto \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{1 + \mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \frac{\color{blue}{\left(-i\right) \cdot \mathsf{fma}\left(-1, \frac{\left(\beta + \alpha\right) \cdot 0.25}{i}, -0.5\right)}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - 1} \]

   8. Taylor expanded in beta around inf

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

      1. lower-+.f64 80.7

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

   10. Applied rewrites 80.7%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.9 \cdot 10^{+162}:\\ \;\;\;\;\frac{\frac{\left(-i\right) \cdot \mathsf{fma}\left(\left(-\alpha\right) - \beta, \frac{0.25}{i}, -0.5\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - 1}}{\frac{1 + \mathsf{fma}\left(2, i, \alpha + \beta\right)}{\left(\left(\alpha + \beta\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{1 + \mathsf{fma}\left(2, i, \alpha + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 73.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := 2 \cdot i + \left(\alpha + \beta\right)\\ t_1 := t\_0 \cdot t\_0\\ t_2 := \left(\left(\alpha + \beta\right) + i\right) \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha \cdot \beta + t\_2\right) \cdot t\_2}{t\_1}}{t\_1 - 1} \leq 5 \cdot 10^{-34}:\\ \;\;\;\;\frac{\left(i + \alpha\right) \cdot i}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (* 2.0 i) (+ alpha beta)))
        (t_1 (* t_0 t_0))
        (t_2 (* (+ (+ alpha beta) i) i)))
   (if (<= (/ (/ (* (+ (* alpha beta) t_2) t_2) t_1) (- t_1 1.0)) 5e-34)
     (/ (* (+ i alpha) i) (* beta beta))
     0.0625)))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = (2.0 * i) + (alpha + beta);
	double t_1 = t_0 * t_0;
	double t_2 = ((alpha + beta) + i) * i;
	double tmp;
	if ((((((alpha * beta) + t_2) * t_2) / t_1) / (t_1 - 1.0)) <= 5e-34) {
		tmp = ((i + alpha) * i) / (beta * beta);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
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 = (2.0d0 * i) + (alpha + beta)
    t_1 = t_0 * t_0
    t_2 = ((alpha + beta) + i) * i
    if ((((((alpha * beta) + t_2) * t_2) / t_1) / (t_1 - 1.0d0)) <= 5d-34) then
        tmp = ((i + alpha) * i) / (beta * beta)
    else
        tmp = 0.0625d0
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double t_0 = (2.0 * i) + (alpha + beta);
	double t_1 = t_0 * t_0;
	double t_2 = ((alpha + beta) + i) * i;
	double tmp;
	if ((((((alpha * beta) + t_2) * t_2) / t_1) / (t_1 - 1.0)) <= 5e-34) {
		tmp = ((i + alpha) * i) / (beta * beta);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	t_0 = (2.0 * i) + (alpha + beta)
	t_1 = t_0 * t_0
	t_2 = ((alpha + beta) + i) * i
	tmp = 0
	if (((((alpha * beta) + t_2) * t_2) / t_1) / (t_1 - 1.0)) <= 5e-34:
		tmp = ((i + alpha) * i) / (beta * beta)
	else:
		tmp = 0.0625
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = Float64(Float64(2.0 * i) + Float64(alpha + beta))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(Float64(Float64(alpha + beta) + i) * i)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(alpha * beta) + t_2) * t_2) / t_1) / Float64(t_1 - 1.0)) <= 5e-34)
		tmp = Float64(Float64(Float64(i + alpha) * i) / Float64(beta * beta));
	else
		tmp = 0.0625;
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	t_0 = (2.0 * i) + (alpha + beta);
	t_1 = t_0 * t_0;
	t_2 = ((alpha + beta) + i) * i;
	tmp = 0.0;
	if ((((((alpha * beta) + t_2) * t_2) / t_1) / (t_1 - 1.0)) <= 5e-34)
		tmp = ((i + alpha) * i) / (beta * beta);
	else
		tmp = 0.0625;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(2.0 * i), $MachinePrecision] + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[(alpha * beta), $MachinePrecision] + t$95$2), $MachinePrecision] * t$95$2), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 - 1.0), $MachinePrecision]), $MachinePrecision], 5e-34], N[(N[(N[(i + alpha), $MachinePrecision] * i), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision], 0.0625]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64))) < 5.0000000000000003e-34

   1. Initial program 98.9%

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

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 72.8

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

   5. Applied rewrites 72.8%

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

   7. Applied rewrites 72.8%

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

   if 5.0000000000000003e-34 < (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64)))

   1. Initial program 12.3%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 73.8%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha \cdot \beta + \left(\left(\alpha + \beta\right) + i\right) \cdot i\right) \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot i\right)}{\left(2 \cdot i + \left(\alpha + \beta\right)\right) \cdot \left(2 \cdot i + \left(\alpha + \beta\right)\right)}}{\left(2 \cdot i + \left(\alpha + \beta\right)\right) \cdot \left(2 \cdot i + \left(\alpha + \beta\right)\right) - 1} \leq 5 \cdot 10^{-34}:\\ \;\;\;\;\frac{\left(i + \alpha\right) \cdot i}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := 2 \cdot i + \left(\alpha + \beta\right)\\ t_1 := t\_0 \cdot t\_0\\ t_2 := \left(\left(\alpha + \beta\right) + i\right) \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha \cdot \beta + t\_2\right) \cdot t\_2}{t\_1}}{t\_1 - 1} \leq 5 \cdot 10^{-34}:\\ \;\;\;\;\frac{i}{\beta \cdot \beta} \cdot \left(i + \alpha\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (* 2.0 i) (+ alpha beta)))
        (t_1 (* t_0 t_0))
        (t_2 (* (+ (+ alpha beta) i) i)))
   (if (<= (/ (/ (* (+ (* alpha beta) t_2) t_2) t_1) (- t_1 1.0)) 5e-34)
     (* (/ i (* beta beta)) (+ i alpha))
     0.0625)))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = (2.0 * i) + (alpha + beta);
	double t_1 = t_0 * t_0;
	double t_2 = ((alpha + beta) + i) * i;
	double tmp;
	if ((((((alpha * beta) + t_2) * t_2) / t_1) / (t_1 - 1.0)) <= 5e-34) {
		tmp = (i / (beta * beta)) * (i + alpha);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
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 = (2.0d0 * i) + (alpha + beta)
    t_1 = t_0 * t_0
    t_2 = ((alpha + beta) + i) * i
    if ((((((alpha * beta) + t_2) * t_2) / t_1) / (t_1 - 1.0d0)) <= 5d-34) then
        tmp = (i / (beta * beta)) * (i + alpha)
    else
        tmp = 0.0625d0
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double t_0 = (2.0 * i) + (alpha + beta);
	double t_1 = t_0 * t_0;
	double t_2 = ((alpha + beta) + i) * i;
	double tmp;
	if ((((((alpha * beta) + t_2) * t_2) / t_1) / (t_1 - 1.0)) <= 5e-34) {
		tmp = (i / (beta * beta)) * (i + alpha);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	t_0 = (2.0 * i) + (alpha + beta)
	t_1 = t_0 * t_0
	t_2 = ((alpha + beta) + i) * i
	tmp = 0
	if (((((alpha * beta) + t_2) * t_2) / t_1) / (t_1 - 1.0)) <= 5e-34:
		tmp = (i / (beta * beta)) * (i + alpha)
	else:
		tmp = 0.0625
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = Float64(Float64(2.0 * i) + Float64(alpha + beta))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(Float64(Float64(alpha + beta) + i) * i)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(alpha * beta) + t_2) * t_2) / t_1) / Float64(t_1 - 1.0)) <= 5e-34)
		tmp = Float64(Float64(i / Float64(beta * beta)) * Float64(i + alpha));
	else
		tmp = 0.0625;
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	t_0 = (2.0 * i) + (alpha + beta);
	t_1 = t_0 * t_0;
	t_2 = ((alpha + beta) + i) * i;
	tmp = 0.0;
	if ((((((alpha * beta) + t_2) * t_2) / t_1) / (t_1 - 1.0)) <= 5e-34)
		tmp = (i / (beta * beta)) * (i + alpha);
	else
		tmp = 0.0625;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(2.0 * i), $MachinePrecision] + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[(alpha * beta), $MachinePrecision] + t$95$2), $MachinePrecision] * t$95$2), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 - 1.0), $MachinePrecision]), $MachinePrecision], 5e-34], N[(N[(i / N[(beta * beta), $MachinePrecision]), $MachinePrecision] * N[(i + alpha), $MachinePrecision]), $MachinePrecision], 0.0625]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64))) < 5.0000000000000003e-34

   1. Initial program 98.9%

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

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 72.8

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

   5. Applied rewrites 72.8%

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

   7. Applied rewrites 72.9%

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

   if 5.0000000000000003e-34 < (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64)))

   1. Initial program 12.3%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 73.8%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha \cdot \beta + \left(\left(\alpha + \beta\right) + i\right) \cdot i\right) \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot i\right)}{\left(2 \cdot i + \left(\alpha + \beta\right)\right) \cdot \left(2 \cdot i + \left(\alpha + \beta\right)\right)}}{\left(2 \cdot i + \left(\alpha + \beta\right)\right) \cdot \left(2 \cdot i + \left(\alpha + \beta\right)\right) - 1} \leq 5 \cdot 10^{-34}:\\ \;\;\;\;\frac{i}{\beta \cdot \beta} \cdot \left(i + \alpha\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := 2 \cdot i + \left(\alpha + \beta\right)\\ t_1 := t\_0 \cdot t\_0\\ t_2 := \left(\left(\alpha + \beta\right) + i\right) \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha \cdot \beta + t\_2\right) \cdot t\_2}{t\_1}}{t\_1 - 1} \leq 5 \cdot 10^{-34}:\\ \;\;\;\;\frac{i \cdot i}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (* 2.0 i) (+ alpha beta)))
        (t_1 (* t_0 t_0))
        (t_2 (* (+ (+ alpha beta) i) i)))
   (if (<= (/ (/ (* (+ (* alpha beta) t_2) t_2) t_1) (- t_1 1.0)) 5e-34)
     (/ (* i i) (* beta beta))
     0.0625)))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = (2.0 * i) + (alpha + beta);
	double t_1 = t_0 * t_0;
	double t_2 = ((alpha + beta) + i) * i;
	double tmp;
	if ((((((alpha * beta) + t_2) * t_2) / t_1) / (t_1 - 1.0)) <= 5e-34) {
		tmp = (i * i) / (beta * beta);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
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 = (2.0d0 * i) + (alpha + beta)
    t_1 = t_0 * t_0
    t_2 = ((alpha + beta) + i) * i
    if ((((((alpha * beta) + t_2) * t_2) / t_1) / (t_1 - 1.0d0)) <= 5d-34) then
        tmp = (i * i) / (beta * beta)
    else
        tmp = 0.0625d0
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double t_0 = (2.0 * i) + (alpha + beta);
	double t_1 = t_0 * t_0;
	double t_2 = ((alpha + beta) + i) * i;
	double tmp;
	if ((((((alpha * beta) + t_2) * t_2) / t_1) / (t_1 - 1.0)) <= 5e-34) {
		tmp = (i * i) / (beta * beta);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	t_0 = (2.0 * i) + (alpha + beta)
	t_1 = t_0 * t_0
	t_2 = ((alpha + beta) + i) * i
	tmp = 0
	if (((((alpha * beta) + t_2) * t_2) / t_1) / (t_1 - 1.0)) <= 5e-34:
		tmp = (i * i) / (beta * beta)
	else:
		tmp = 0.0625
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = Float64(Float64(2.0 * i) + Float64(alpha + beta))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(Float64(Float64(alpha + beta) + i) * i)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(alpha * beta) + t_2) * t_2) / t_1) / Float64(t_1 - 1.0)) <= 5e-34)
		tmp = Float64(Float64(i * i) / Float64(beta * beta));
	else
		tmp = 0.0625;
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	t_0 = (2.0 * i) + (alpha + beta);
	t_1 = t_0 * t_0;
	t_2 = ((alpha + beta) + i) * i;
	tmp = 0.0;
	if ((((((alpha * beta) + t_2) * t_2) / t_1) / (t_1 - 1.0)) <= 5e-34)
		tmp = (i * i) / (beta * beta);
	else
		tmp = 0.0625;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(2.0 * i), $MachinePrecision] + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[(alpha * beta), $MachinePrecision] + t$95$2), $MachinePrecision] * t$95$2), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 - 1.0), $MachinePrecision]), $MachinePrecision], 5e-34], N[(N[(i * i), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision], 0.0625]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64))) < 5.0000000000000003e-34

   1. Initial program 98.9%

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

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 72.8

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

   5. Applied rewrites 72.8%

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

   7. Applied rewrites 72.9%

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

   8. Taylor expanded in alpha around 0

      \[\leadsto \frac{{i}^{2}}{\color{blue}{{\beta}^{2}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 71.7%

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

   if 5.0000000000000003e-34 < (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64)))

   1. Initial program 12.3%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 73.8%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha \cdot \beta + \left(\left(\alpha + \beta\right) + i\right) \cdot i\right) \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot i\right)}{\left(2 \cdot i + \left(\alpha + \beta\right)\right) \cdot \left(2 \cdot i + \left(\alpha + \beta\right)\right)}}{\left(2 \cdot i + \left(\alpha + \beta\right)\right) \cdot \left(2 \cdot i + \left(\alpha + \beta\right)\right) - 1} \leq 5 \cdot 10^{-34}:\\ \;\;\;\;\frac{i \cdot i}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, i, \alpha + \beta\right)\\ t_1 := 1 + t\_0\\ t_2 := \left(\left(\alpha + \beta\right) + i\right) \cdot \frac{i}{t\_0}\\ t_3 := t\_0 - 1\\ \mathbf{if}\;\beta \leq 6.2 \cdot 10^{+162}:\\ \;\;\;\;\frac{\left(-i\right) \cdot \mathsf{fma}\left(\left(-\alpha\right) - \beta, \frac{0.25}{i}, -0.5\right)}{\frac{t\_1}{t\_2} \cdot t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{t\_3} \cdot \frac{t\_2}{t\_1}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma 2.0 i (+ alpha beta)))
        (t_1 (+ 1.0 t_0))
        (t_2 (* (+ (+ alpha beta) i) (/ i t_0)))
        (t_3 (- t_0 1.0)))
   (if (<= beta 6.2e+162)
     (/ (* (- i) (fma (- (- alpha) beta) (/ 0.25 i) -0.5)) (* (/ t_1 t_2) t_3))
     (* (/ (+ i alpha) t_3) (/ t_2 t_1)))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = fma(2.0, i, (alpha + beta));
	double t_1 = 1.0 + t_0;
	double t_2 = ((alpha + beta) + i) * (i / t_0);
	double t_3 = t_0 - 1.0;
	double tmp;
	if (beta <= 6.2e+162) {
		tmp = (-i * fma((-alpha - beta), (0.25 / i), -0.5)) / ((t_1 / t_2) * t_3);
	} else {
		tmp = ((i + alpha) / t_3) * (t_2 / t_1);
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = fma(2.0, i, Float64(alpha + beta))
	t_1 = Float64(1.0 + t_0)
	t_2 = Float64(Float64(Float64(alpha + beta) + i) * Float64(i / t_0))
	t_3 = Float64(t_0 - 1.0)
	tmp = 0.0
	if (beta <= 6.2e+162)
		tmp = Float64(Float64(Float64(-i) * fma(Float64(Float64(-alpha) - beta), Float64(0.25 / i), -0.5)) / Float64(Float64(t_1 / t_2) * t_3));
	else
		tmp = Float64(Float64(Float64(i + alpha) / t_3) * Float64(t_2 / t_1));
	end
	return tmp
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision] * N[(i / t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 - 1.0), $MachinePrecision]}, If[LessEqual[beta, 6.2e+162], N[(N[((-i) * N[(N[((-alpha) - beta), $MachinePrecision] * N[(0.25 / i), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$1 / t$95$2), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision], N[(N[(N[(i + alpha), $MachinePrecision] / t$95$3), $MachinePrecision] * N[(t$95$2 / t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.1999999999999999e162

   1. Initial program 18.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. difference-of-sqr-1 N/A

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

   4. Applied rewrites 41.3%

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

   5. Taylor expanded in i around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. distribute-rgt-out-- N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 80.8

          \[\leadsto \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{1 + \mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \frac{\left(-i\right) \cdot \mathsf{fma}\left(-1, \frac{\color{blue}{\left(\beta + \alpha\right)} \cdot 0.25}{i}, -0.5\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - 1} \]

   7. Applied rewrites 80.8%

      \[\leadsto \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{1 + \mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \frac{\color{blue}{\left(-i\right) \cdot \mathsf{fma}\left(-1, \frac{\left(\beta + \alpha\right) \cdot 0.25}{i}, -0.5\right)}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - 1} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

   9. Applied rewrites 83.3%

      \[\leadsto \color{blue}{\frac{\left(\mathsf{fma}\left(\left(-\alpha\right) - \beta, \frac{0.25}{i}, -0.5\right) \cdot \left(-i\right)\right) \cdot 1}{\left(\mathsf{fma}\left(2, i, \alpha + \beta\right) - 1\right) \cdot \frac{1 + \mathsf{fma}\left(2, i, \alpha + \beta\right)}{\frac{i}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \left(\left(\alpha + \beta\right) + i\right)}}} \]

   if 6.1999999999999999e162 < 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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. difference-of-sqr-1 N/A

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

   4. Applied rewrites 24.7%

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

   5. Taylor expanded in i around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. distribute-rgt-out-- N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 24.6

          \[\leadsto \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{1 + \mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \frac{\left(-i\right) \cdot \mathsf{fma}\left(-1, \frac{\color{blue}{\left(\beta + \alpha\right)} \cdot 0.25}{i}, -0.5\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - 1} \]

   7. Applied rewrites 24.6%

      \[\leadsto \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{1 + \mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \frac{\color{blue}{\left(-i\right) \cdot \mathsf{fma}\left(-1, \frac{\left(\beta + \alpha\right) \cdot 0.25}{i}, -0.5\right)}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - 1} \]

   8. Taylor expanded in beta around inf

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

      1. lower-+.f64 80.7

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

   10. Applied rewrites 80.7%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.2 \cdot 10^{+162}:\\ \;\;\;\;\frac{\left(-i\right) \cdot \mathsf{fma}\left(\left(-\alpha\right) - \beta, \frac{0.25}{i}, -0.5\right)}{\frac{1 + \mathsf{fma}\left(2, i, \alpha + \beta\right)}{\left(\left(\alpha + \beta\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}} \cdot \left(\mathsf{fma}\left(2, i, \alpha + \beta\right) - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{1 + \mathsf{fma}\left(2, i, \alpha + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, i, \alpha + \beta\right)\\ t_1 := t\_0 - 1\\ t_2 := \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \frac{i}{t\_0}}{1 + t\_0}\\ \mathbf{if}\;\beta \leq 5.9 \cdot 10^{+162}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, i, \left(\alpha + \beta\right) \cdot 0.25\right)}{t\_1} \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{t\_1} \cdot t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma 2.0 i (+ alpha beta)))
        (t_1 (- t_0 1.0))
        (t_2 (/ (* (+ (+ alpha beta) i) (/ i t_0)) (+ 1.0 t_0))))
   (if (<= beta 5.9e+162)
     (* (/ (fma 0.5 i (* (+ alpha beta) 0.25)) t_1) t_2)
     (* (/ (+ i alpha) t_1) t_2))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = fma(2.0, i, (alpha + beta));
	double t_1 = t_0 - 1.0;
	double t_2 = (((alpha + beta) + i) * (i / t_0)) / (1.0 + t_0);
	double tmp;
	if (beta <= 5.9e+162) {
		tmp = (fma(0.5, i, ((alpha + beta) * 0.25)) / t_1) * t_2;
	} else {
		tmp = ((i + alpha) / t_1) * t_2;
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = fma(2.0, i, Float64(alpha + beta))
	t_1 = Float64(t_0 - 1.0)
	t_2 = Float64(Float64(Float64(Float64(alpha + beta) + i) * Float64(i / t_0)) / Float64(1.0 + t_0))
	tmp = 0.0
	if (beta <= 5.9e+162)
		tmp = Float64(Float64(fma(0.5, i, Float64(Float64(alpha + beta) * 0.25)) / t_1) * t_2);
	else
		tmp = Float64(Float64(Float64(i + alpha) / t_1) * t_2);
	end
	return tmp
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision] * N[(i / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 5.9e+162], N[(N[(N[(0.5 * i + N[(N[(alpha + beta), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision], N[(N[(N[(i + alpha), $MachinePrecision] / t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.90000000000000027e162

   1. Initial program 18.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. difference-of-sqr-1 N/A

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

   4. Applied rewrites 41.3%

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

   5. Taylor expanded in i around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. distribute-rgt-out-- N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 80.8

          \[\leadsto \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{1 + \mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \frac{\left(-i\right) \cdot \mathsf{fma}\left(-1, \frac{\color{blue}{\left(\beta + \alpha\right)} \cdot 0.25}{i}, -0.5\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - 1} \]

   7. Applied rewrites 80.8%

      \[\leadsto \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{1 + \mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \frac{\color{blue}{\left(-i\right) \cdot \mathsf{fma}\left(-1, \frac{\left(\beta + \alpha\right) \cdot 0.25}{i}, -0.5\right)}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - 1} \]

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 80.8%

       \[\leadsto \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{1 + \mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \frac{\mathsf{fma}\left(0.5, \color{blue}{i}, 0.25 \cdot \left(\alpha + \beta\right)\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - 1} \]

   if 5.90000000000000027e162 < 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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. difference-of-sqr-1 N/A

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

   4. Applied rewrites 24.7%

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

   5. Taylor expanded in i around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. distribute-rgt-out-- N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 24.6

          \[\leadsto \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{1 + \mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \frac{\left(-i\right) \cdot \mathsf{fma}\left(-1, \frac{\color{blue}{\left(\beta + \alpha\right)} \cdot 0.25}{i}, -0.5\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - 1} \]

   7. Applied rewrites 24.6%

      \[\leadsto \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{1 + \mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \frac{\color{blue}{\left(-i\right) \cdot \mathsf{fma}\left(-1, \frac{\left(\beta + \alpha\right) \cdot 0.25}{i}, -0.5\right)}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - 1} \]

   8. Taylor expanded in beta around inf

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

      1. lower-+.f64 80.7

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

   10. Applied rewrites 80.7%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.9 \cdot 10^{+162}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, i, \left(\alpha + \beta\right) \cdot 0.25\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{1 + \mathsf{fma}\left(2, i, \alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{1 + \mathsf{fma}\left(2, i, \alpha + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, i, \alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 5.9 \cdot 10^{+162}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{t\_0 - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \frac{i}{t\_0}}{1 + t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma 2.0 i (+ alpha beta))))
   (if (<= beta 5.9e+162)
     0.0625
     (*
      (/ (+ i alpha) (- t_0 1.0))
      (/ (* (+ (+ alpha beta) i) (/ i t_0)) (+ 1.0 t_0))))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = fma(2.0, i, (alpha + beta));
	double tmp;
	if (beta <= 5.9e+162) {
		tmp = 0.0625;
	} else {
		tmp = ((i + alpha) / (t_0 - 1.0)) * ((((alpha + beta) + i) * (i / t_0)) / (1.0 + t_0));
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = fma(2.0, i, Float64(alpha + beta))
	tmp = 0.0
	if (beta <= 5.9e+162)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(i + alpha) / Float64(t_0 - 1.0)) * Float64(Float64(Float64(Float64(alpha + beta) + i) * Float64(i / t_0)) / Float64(1.0 + t_0)));
	end
	return tmp
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 5.9e+162], 0.0625, N[(N[(N[(i + alpha), $MachinePrecision] / N[(t$95$0 - 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision] * N[(i / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.90000000000000027e162

   1. Initial program 18.9%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 79.6%

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

   if 5.90000000000000027e162 < 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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. difference-of-sqr-1 N/A

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

   4. Applied rewrites 24.7%

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

   5. Taylor expanded in i around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. distribute-rgt-out-- N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 24.6

          \[\leadsto \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{1 + \mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \frac{\left(-i\right) \cdot \mathsf{fma}\left(-1, \frac{\color{blue}{\left(\beta + \alpha\right)} \cdot 0.25}{i}, -0.5\right)}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - 1} \]

   7. Applied rewrites 24.6%

      \[\leadsto \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{1 + \mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \frac{\color{blue}{\left(-i\right) \cdot \mathsf{fma}\left(-1, \frac{\left(\beta + \alpha\right) \cdot 0.25}{i}, -0.5\right)}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - 1} \]

   8. Taylor expanded in beta around inf

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

      1. lower-+.f64 80.7

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

   10. Applied rewrites 80.7%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.9 \cdot 10^{+162}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{1 + \mathsf{fma}\left(2, i, \alpha + \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.2 \cdot 10^{+162}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 6.2e+162) 0.0625 (/ (/ (+ i alpha) beta) (/ beta i))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 6.2e+162) {
		tmp = 0.0625;
	} else {
		tmp = ((i + alpha) / beta) / (beta / i);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
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 <= 6.2d+162) then
        tmp = 0.0625d0
    else
        tmp = ((i + alpha) / beta) / (beta / i)
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 6.2e+162) {
		tmp = 0.0625;
	} else {
		tmp = ((i + alpha) / beta) / (beta / i);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 6.2e+162:
		tmp = 0.0625
	else:
		tmp = ((i + alpha) / beta) / (beta / i)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 6.2e+162)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(i + alpha) / beta) / Float64(beta / i));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 6.2e+162)
		tmp = 0.0625;
	else
		tmp = ((i + alpha) / beta) / (beta / i);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 6.2e+162], 0.0625, N[(N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta / i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.1999999999999999e162

   1. Initial program 18.9%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 79.6%

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

   if 6.1999999999999999e162 < 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. Add Preprocessing

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 78.1

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

   5. Applied rewrites 78.1%

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

   7. Applied rewrites 78.2%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.2 \cdot 10^{+162}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.4% accurate, 3.1× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.5 \cdot 10^{+162}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta} \cdot \left(i + \alpha\right)}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 6.5e+162) 0.0625 (/ (* (/ i beta) (+ i alpha)) beta)))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 6.5e+162) {
		tmp = 0.0625;
	} else {
		tmp = ((i / beta) * (i + alpha)) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
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 <= 6.5d+162) then
        tmp = 0.0625d0
    else
        tmp = ((i / beta) * (i + alpha)) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 6.5e+162) {
		tmp = 0.0625;
	} else {
		tmp = ((i / beta) * (i + alpha)) / beta;
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 6.5e+162:
		tmp = 0.0625
	else:
		tmp = ((i / beta) * (i + alpha)) / beta
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 6.5e+162)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(i / beta) * Float64(i + alpha)) / beta);
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 6.5e+162)
		tmp = 0.0625;
	else
		tmp = ((i / beta) * (i + alpha)) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 6.5e+162], 0.0625, N[(N[(N[(i / beta), $MachinePrecision] * N[(i + alpha), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.5000000000000004e162

   1. Initial program 18.9%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 79.6%

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

   if 6.5000000000000004e162 < 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. Add Preprocessing

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 78.1

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

   5. Applied rewrites 78.1%

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

   7. Applied rewrites 78.1%

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

   9. Applied rewrites 78.1%

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

Alternative 10: 85.4% accurate, 3.1× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.2 \cdot 10^{+162}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 6.2e+162) 0.0625 (* (/ (+ i alpha) beta) (/ i beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 6.2e+162) {
		tmp = 0.0625;
	} else {
		tmp = ((i + alpha) / beta) * (i / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
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 <= 6.2d+162) then
        tmp = 0.0625d0
    else
        tmp = ((i + alpha) / beta) * (i / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 6.2e+162) {
		tmp = 0.0625;
	} else {
		tmp = ((i + alpha) / beta) * (i / beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 6.2e+162:
		tmp = 0.0625
	else:
		tmp = ((i + alpha) / beta) * (i / beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 6.2e+162)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(i + alpha) / beta) * Float64(i / beta));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 6.2e+162)
		tmp = 0.0625;
	else
		tmp = ((i + alpha) / beta) * (i / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 6.2e+162], 0.0625, N[(N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.1999999999999999e162

   1. Initial program 18.9%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 79.6%

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

   if 6.1999999999999999e162 < 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. Add Preprocessing

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 78.1

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

   5. Applied rewrites 78.1%

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

Alternative 11: 83.4% accurate, 3.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.2 \cdot 10^{+162}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 6.2e+162) 0.0625 (* (/ i beta) (/ i beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 6.2e+162) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (i / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
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 <= 6.2d+162) then
        tmp = 0.0625d0
    else
        tmp = (i / beta) * (i / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 6.2e+162) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (i / beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 6.2e+162:
		tmp = 0.0625
	else:
		tmp = (i / beta) * (i / beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 6.2e+162)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / beta) * Float64(i / beta));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 6.2e+162)
		tmp = 0.0625;
	else
		tmp = (i / beta) * (i / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 6.2e+162], 0.0625, N[(N[(i / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.1999999999999999e162

   1. Initial program 18.9%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 79.6%

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

   if 6.1999999999999999e162 < 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. Add Preprocessing

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 78.1

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

   5. Applied rewrites 78.1%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 72.9%

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

Alternative 12: 75.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.9 \cdot 10^{+205}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta} \cdot \alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 2.9e+205) 0.0625 (/ (* (/ i beta) alpha) beta)))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.9e+205) {
		tmp = 0.0625;
	} else {
		tmp = ((i / beta) * alpha) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
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 <= 2.9d+205) then
        tmp = 0.0625d0
    else
        tmp = ((i / beta) * alpha) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.9e+205) {
		tmp = 0.0625;
	} else {
		tmp = ((i / beta) * alpha) / beta;
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 2.9e+205:
		tmp = 0.0625
	else:
		tmp = ((i / beta) * alpha) / beta
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 2.9e+205)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(i / beta) * alpha) / beta);
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 2.9e+205)
		tmp = 0.0625;
	else
		tmp = ((i / beta) * alpha) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 2.9e+205], 0.0625, N[(N[(N[(i / beta), $MachinePrecision] * alpha), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.9000000000000001e205

   1. Initial program 17.8%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 77.3%

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

   if 2.9000000000000001e205 < 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. Add Preprocessing

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 88.9

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

   5. Applied rewrites 88.9%

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

   7. Applied rewrites 88.9%

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

   8. Taylor expanded in alpha around inf

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

   10. Applied rewrites 38.7%

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

   12. Applied rewrites 40.7%

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

Alternative 13: 74.6% accurate, 4.1× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.6 \cdot 10^{+231}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta \cdot \beta} \cdot \alpha\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 6.6e+231) 0.0625 (* (/ i (* beta beta)) alpha)))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 6.6e+231) {
		tmp = 0.0625;
	} else {
		tmp = (i / (beta * beta)) * alpha;
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
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 <= 6.6d+231) then
        tmp = 0.0625d0
    else
        tmp = (i / (beta * beta)) * alpha
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 6.6e+231) {
		tmp = 0.0625;
	} else {
		tmp = (i / (beta * beta)) * alpha;
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 6.6e+231:
		tmp = 0.0625
	else:
		tmp = (i / (beta * beta)) * alpha
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 6.6e+231)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / Float64(beta * beta)) * alpha);
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 6.6e+231)
		tmp = 0.0625;
	else
		tmp = (i / (beta * beta)) * alpha;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 6.6e+231], 0.0625, N[(N[(i / N[(beta * beta), $MachinePrecision]), $MachinePrecision] * alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.5999999999999994e231

   1. Initial program 17.6%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 76.3%

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

   if 6.5999999999999994e231 < 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. Add Preprocessing

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 87.6

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

   5. Applied rewrites 87.6%

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

   7. Applied rewrites 87.5%

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

   8. Taylor expanded in alpha around inf

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

   10. Applied rewrites 43.1%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.6 \cdot 10^{+231}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta \cdot \beta} \cdot \alpha\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 71.3% accurate, 115.0× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ 0.0625 \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i) :precision binary64 0.0625)

C

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

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
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

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

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	return 0.0625

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	return 0.0625
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp = code(alpha, beta, i)
	tmp = 0.0625;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := 0.0625

Derivation

1. Initial program 16.0%

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

3. Taylor expanded in i around inf

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

5. Applied rewrites 70.9%

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

Reproduce

herbie shell --seed 2024332 
(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)))