Octave 3.8, jcobi/4

Percentage Accurate: 16.7% → 86.3%

Time: 14.9s

Alternatives: 10

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: 16.7% 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: 86.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := i + \left(\beta + \alpha\right)\\ t_1 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ t_2 := \alpha + \mathsf{fma}\left(i, 2, \beta\right)\\ \mathbf{if}\;\beta \leq 3 \cdot 10^{+127}:\\ \;\;\;\;0.25 \cdot \frac{i}{t\_2 \cdot \frac{t\_2}{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{t\_0}{t\_2}}{t\_1 + 1} \cdot \frac{i + \alpha}{-1 + t\_1}\\ \end{array} \end{array} \]

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 (+ i (+ beta alpha)))
        (t_1 (fma i 2.0 (+ beta alpha)))
        (t_2 (+ alpha (fma i 2.0 beta))))
   (if (<= beta 3e+127)
     (* 0.25 (/ i (* t_2 (/ t_2 t_0))))
     (* (/ (* i (/ t_0 t_2)) (+ t_1 1.0)) (/ (+ i alpha) (+ -1.0 t_1))))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = i + (beta + alpha);
	double t_1 = fma(i, 2.0, (beta + alpha));
	double t_2 = alpha + fma(i, 2.0, beta);
	double tmp;
	if (beta <= 3e+127) {
		tmp = 0.25 * (i / (t_2 * (t_2 / t_0)));
	} else {
		tmp = ((i * (t_0 / t_2)) / (t_1 + 1.0)) * ((i + alpha) / (-1.0 + t_1));
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = Float64(i + Float64(beta + alpha))
	t_1 = fma(i, 2.0, Float64(beta + alpha))
	t_2 = Float64(alpha + fma(i, 2.0, beta))
	tmp = 0.0
	if (beta <= 3e+127)
		tmp = Float64(0.25 * Float64(i / Float64(t_2 * Float64(t_2 / t_0))));
	else
		tmp = Float64(Float64(Float64(i * Float64(t_0 / t_2)) / Float64(t_1 + 1.0)) * Float64(Float64(i + alpha) / Float64(-1.0 + 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[(i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(alpha + N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 3e+127], N[(0.25 * N[(i / N[(t$95$2 * N[(t$95$2 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(i * N[(t$95$0 / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(i + alpha), $MachinePrecision] / N[(-1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.0000000000000002e127

   1. Initial program 16.6%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   2. Add Preprocessing
   3. 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. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

   4. Applied rewrites 39.1%

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

   5. Taylor expanded in i around inf

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

   7. Applied rewrites 32.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-+.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. clear-num N/A

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

      13. frac-times N/A

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

      14. *-commutative N/A

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

      15. *-lft-identity N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 N/A

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

   9. Applied rewrites 83.1%

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

   if 3.0000000000000002e127 < beta

   1. Initial program 2.1%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   2. 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 28.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 28.2

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

      7. lift-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. associate-+r+ N/A

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

      11. +-commutative N/A

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

      12. lift-fma.f64 N/A

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

      13. lower-+.f64 28.2

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

   6. Applied rewrites 28.2%

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

   7. Taylor expanded in beta around inf

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

      1. lower-+.f64 76.0

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

   9. Applied rewrites 76.0%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3 \cdot 10^{+127}:\\ \;\;\;\;0.25 \cdot \frac{i}{\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right) \cdot \frac{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}{i + \left(\beta + \alpha\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{i + \left(\beta + \alpha\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1} \cdot \frac{i + \alpha}{-1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := i + \left(\beta + \alpha\right)\\ t_1 := \alpha + \mathsf{fma}\left(i, 2, \beta\right)\\ \mathbf{if}\;\beta \leq 6.6 \cdot 10^{+157}:\\ \;\;\;\;0.25 \cdot \frac{i}{t\_1 \cdot \frac{t\_1}{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{t\_0}{t\_1}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ i (+ beta alpha))) (t_1 (+ alpha (fma i 2.0 beta))))
   (if (<= beta 6.6e+157)
     (* 0.25 (/ i (* t_1 (/ t_1 t_0))))
     (*
      (/ (* i (/ t_0 t_1)) (+ (fma i 2.0 (+ beta alpha)) 1.0))
      (/ (+ i alpha) beta)))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = i + (beta + alpha);
	double t_1 = alpha + fma(i, 2.0, beta);
	double tmp;
	if (beta <= 6.6e+157) {
		tmp = 0.25 * (i / (t_1 * (t_1 / t_0)));
	} else {
		tmp = ((i * (t_0 / t_1)) / (fma(i, 2.0, (beta + alpha)) + 1.0)) * ((i + alpha) / beta);
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = Float64(i + Float64(beta + alpha))
	t_1 = Float64(alpha + fma(i, 2.0, beta))
	tmp = 0.0
	if (beta <= 6.6e+157)
		tmp = Float64(0.25 * Float64(i / Float64(t_1 * Float64(t_1 / t_0))));
	else
		tmp = Float64(Float64(Float64(i * Float64(t_0 / t_1)) / Float64(fma(i, 2.0, Float64(beta + alpha)) + 1.0)) * Float64(Float64(i + alpha) / beta));
	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[(i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(alpha + N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 6.6e+157], N[(0.25 * N[(i / N[(t$95$1 * N[(t$95$1 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(i * N[(t$95$0 / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(N[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.6000000000000003e157

   1. Initial program 16.3%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   2. Add Preprocessing
   3. 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. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

   4. Applied rewrites 39.1%

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

   5. Taylor expanded in i around inf

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

   7. Applied rewrites 31.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-+.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. clear-num N/A

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

      13. frac-times N/A

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

      14. *-commutative N/A

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

      15. *-lft-identity N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 N/A

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

   9. Applied rewrites 81.8%

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

   if 6.6000000000000003e157 < 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 22.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 22.3

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

      7. lift-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. associate-+r+ N/A

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

      11. +-commutative N/A

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

      12. lift-fma.f64 N/A

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

      13. lower-+.f64 22.3

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

   6. Applied rewrites 22.3%

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

   7. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 79.9

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

   9. Applied rewrites 79.9%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.6 \cdot 10^{+157}:\\ \;\;\;\;0.25 \cdot \frac{i}{\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right) \cdot \frac{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}{i + \left(\beta + \alpha\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{i + \left(\beta + \alpha\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \alpha + \mathsf{fma}\left(i, 2, \beta\right)\\ \mathbf{if}\;\beta \leq 7.4 \cdot 10^{+157}:\\ \;\;\;\;0.25 \cdot \frac{i}{t\_0 \cdot \frac{t\_0}{i + \left(\beta + \alpha\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ alpha (fma i 2.0 beta))))
   (if (<= beta 7.4e+157)
     (* 0.25 (/ i (* t_0 (/ t_0 (+ i (+ beta alpha))))))
     (/ (/ (+ i alpha) beta) (/ beta i)))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = alpha + fma(i, 2.0, beta);
	double tmp;
	if (beta <= 7.4e+157) {
		tmp = 0.25 * (i / (t_0 * (t_0 / (i + (beta + alpha)))));
	} else {
		tmp = ((i + alpha) / beta) / (beta / i);
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = Float64(alpha + fma(i, 2.0, beta))
	tmp = 0.0
	if (beta <= 7.4e+157)
		tmp = Float64(0.25 * Float64(i / Float64(t_0 * Float64(t_0 / Float64(i + Float64(beta + alpha))))));
	else
		tmp = Float64(Float64(Float64(i + alpha) / beta) / Float64(beta / i));
	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[(alpha + N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 7.4e+157], N[(0.25 * N[(i / N[(t$95$0 * N[(t$95$0 / N[(i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta / i), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.3999999999999997e157

   1. Initial program 16.3%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   2. Add Preprocessing
   3. 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. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

   4. Applied rewrites 39.1%

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

   5. Taylor expanded in i around inf

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

   7. Applied rewrites 31.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-+.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. clear-num N/A

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

      13. frac-times N/A

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

      14. *-commutative N/A

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

      15. *-lft-identity N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 N/A

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

   9. Applied rewrites 81.8%

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

   if 7.3999999999999997e157 < 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. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 17.5

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

   5. Applied rewrites 17.5%

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

   7. Applied rewrites 79.7%

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

   9. Applied rewrites 79.6%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.4 \cdot 10^{+157}:\\ \;\;\;\;0.25 \cdot \frac{i}{\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right) \cdot \frac{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}{i + \left(\beta + \alpha\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.6 \cdot 10^{+157}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 6.6e+157) 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 <= 6.6e+157) {
		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 <= 6.6d+157) 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 <= 6.6e+157) {
		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 <= 6.6e+157:
		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 <= 6.6e+157)
		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 <= 6.6e+157)
		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, 6.6e+157], 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 < 6.6000000000000003e157

   1. Initial program 16.3%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 81.3%

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

   if 6.6000000000000003e157 < 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. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 17.5

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

   5. Applied rewrites 17.5%

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

   7. Applied rewrites 79.7%

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

   9. Applied rewrites 79.6%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.6 \cdot 10^{+157}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.6 \cdot 10^{+157}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(i + \alpha\right) \cdot \frac{i}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 6.6e+157) 0.0625 (/ (* (+ i alpha) (/ i beta)) beta)))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 6.6e+157) {
		tmp = 0.0625;
	} else {
		tmp = ((i + alpha) * (i / beta)) / 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 <= 6.6d+157) then
        tmp = 0.0625d0
    else
        tmp = ((i + alpha) * (i / beta)) / 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 <= 6.6e+157) {
		tmp = 0.0625;
	} else {
		tmp = ((i + alpha) * (i / beta)) / beta;
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 6.6e+157:
		tmp = 0.0625
	else:
		tmp = ((i + alpha) * (i / beta)) / beta
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 6.6e+157)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(i + alpha) * Float64(i / beta)) / 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 <= 6.6e+157)
		tmp = 0.0625;
	else
		tmp = ((i + alpha) * (i / beta)) / 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, 6.6e+157], 0.0625, N[(N[(N[(i + alpha), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.6000000000000003e157

   1. Initial program 16.3%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 81.3%

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

   if 6.6000000000000003e157 < 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. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 17.5

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

   5. Applied rewrites 17.5%

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

   7. Applied rewrites 79.7%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.6 \cdot 10^{+157}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(i + \alpha\right) \cdot \frac{i}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.6 \cdot 10^{+157}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 6.6e+157) 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 <= 6.6e+157) {
		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 <= 6.6d+157) 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 <= 6.6e+157) {
		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 <= 6.6e+157:
		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 <= 6.6e+157)
		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 <= 6.6e+157)
		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, 6.6e+157], 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 < 6.6000000000000003e157

   1. Initial program 16.3%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 81.3%

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

   if 6.6000000000000003e157 < 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. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 17.5

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

   5. Applied rewrites 17.5%

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

   7. Applied rewrites 79.5%

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

Alternative 7: 83.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.4 \cdot 10^{+157}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{i}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 7.4e+157) 0.0625 (/ (* i (/ i beta)) beta)))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 7.4e+157) {
		tmp = 0.0625;
	} else {
		tmp = (i * (i / beta)) / 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 <= 7.4d+157) then
        tmp = 0.0625d0
    else
        tmp = (i * (i / beta)) / 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 <= 7.4e+157) {
		tmp = 0.0625;
	} else {
		tmp = (i * (i / beta)) / beta;
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 7.4e+157:
		tmp = 0.0625
	else:
		tmp = (i * (i / beta)) / beta
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 7.4e+157)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i * Float64(i / beta)) / 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 <= 7.4e+157)
		tmp = 0.0625;
	else
		tmp = (i * (i / beta)) / 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, 7.4e+157], 0.0625, N[(N[(i * N[(i / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.3999999999999997e157

   1. Initial program 16.3%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 81.3%

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

   if 7.3999999999999997e157 < 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. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 17.5

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

   5. Applied rewrites 17.5%

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

   7. Applied rewrites 79.7%

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

   8. Taylor expanded in i around inf

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

   10. Applied rewrites 43.1%

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

   12. Applied rewrites 75.2%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.4 \cdot 10^{+157}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{i}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.0% accurate, 3.7× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.1 \cdot 10^{+265}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;i \cdot \frac{i + \alpha}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.1e+265) 0.0625 (* i (/ (+ i alpha) (* beta beta)))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.1e+265) {
		tmp = 0.0625;
	} else {
		tmp = i * ((i + alpha) / (beta * 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 <= 1.1d+265) then
        tmp = 0.0625d0
    else
        tmp = i * ((i + alpha) / (beta * 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 <= 1.1e+265) {
		tmp = 0.0625;
	} else {
		tmp = i * ((i + alpha) / (beta * beta));
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.1e+265:
		tmp = 0.0625
	else:
		tmp = i * ((i + alpha) / (beta * beta))
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.1e+265)
		tmp = 0.0625;
	else
		tmp = Float64(i * Float64(Float64(i + alpha) / Float64(beta * 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 <= 1.1e+265)
		tmp = 0.0625;
	else
		tmp = i * ((i + alpha) / (beta * 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, 1.1e+265], 0.0625, N[(i * N[(N[(i + alpha), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.0999999999999999e265

   1. Initial program 14.5%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 75.5%

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

   if 1.0999999999999999e265 < 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. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 25.2

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

   5. Applied rewrites 25.2%

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

   7. Applied rewrites 27.9%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.1 \cdot 10^{+265}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;i \cdot \frac{i + \alpha}{\beta \cdot \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.9% accurate, 4.1× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.1 \cdot 10^{+265}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \alpha}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.1e+265) 0.0625 (/ (* i alpha) (* beta beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.1e+265) {
		tmp = 0.0625;
	} else {
		tmp = (i * alpha) / (beta * 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 <= 1.1d+265) then
        tmp = 0.0625d0
    else
        tmp = (i * alpha) / (beta * 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 <= 1.1e+265) {
		tmp = 0.0625;
	} else {
		tmp = (i * alpha) / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.1e+265:
		tmp = 0.0625
	else:
		tmp = (i * alpha) / (beta * beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.1e+265)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i * alpha) / Float64(beta * 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 <= 1.1e+265)
		tmp = 0.0625;
	else
		tmp = (i * alpha) / (beta * 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, 1.1e+265], 0.0625, N[(N[(i * alpha), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.0999999999999999e265

   1. Initial program 14.5%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 75.5%

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

   if 1.0999999999999999e265 < 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. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 25.2

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

   5. Applied rewrites 25.2%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 27.6%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.1 \cdot 10^{+265}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \alpha}{\beta \cdot \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.3% 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 13.8%

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

3. Taylor expanded in i around inf

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

5. Applied rewrites 71.9%

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

Reproduce

herbie shell --seed 2024223 
(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)))