Octave 3.8, jcobi/4

Percentage Accurate: 15.7% → 81.6%

Time: 15.2s

Alternatives: 15

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: 15.7% 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: 81.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \alpha + \beta\right)\\ t_1 := i + \left(\alpha + \beta\right)\\ t_2 := \beta + \mathsf{fma}\left(i, 2, \alpha\right)\\ \mathbf{if}\;i \leq 2.15 \cdot 10^{+95}:\\ \;\;\;\;\frac{\frac{i \cdot t\_1}{t\_0}}{t\_0 + 1} \cdot \frac{\frac{\mathsf{fma}\left(i, t\_1, \alpha \cdot \beta\right)}{t\_0}}{t\_0 + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{t\_2} \cdot \left(\frac{t\_1}{t\_2} \cdot 0.25\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma i 2.0 (+ alpha beta)))
        (t_1 (+ i (+ alpha beta)))
        (t_2 (+ beta (fma i 2.0 alpha))))
   (if (<= i 2.15e+95)
     (*
      (/ (/ (* i t_1) t_0) (+ t_0 1.0))
      (/ (/ (fma i t_1 (* alpha beta)) t_0) (+ t_0 -1.0)))
     (* (/ i t_2) (* (/ t_1 t_2) 0.25)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = fma(i, 2.0, (alpha + beta));
	double t_1 = i + (alpha + beta);
	double t_2 = beta + fma(i, 2.0, alpha);
	double tmp;
	if (i <= 2.15e+95) {
		tmp = (((i * t_1) / t_0) / (t_0 + 1.0)) * ((fma(i, t_1, (alpha * beta)) / t_0) / (t_0 + -1.0));
	} else {
		tmp = (i / t_2) * ((t_1 / t_2) * 0.25);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = fma(i, 2.0, Float64(alpha + beta))
	t_1 = Float64(i + Float64(alpha + beta))
	t_2 = Float64(beta + fma(i, 2.0, alpha))
	tmp = 0.0
	if (i <= 2.15e+95)
		tmp = Float64(Float64(Float64(Float64(i * t_1) / t_0) / Float64(t_0 + 1.0)) * Float64(Float64(fma(i, t_1, Float64(alpha * beta)) / t_0) / Float64(t_0 + -1.0)));
	else
		tmp = Float64(Float64(i / t_2) * Float64(Float64(t_1 / t_2) * 0.25));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(i * 2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(beta + N[(i * 2.0 + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, 2.15e+95], N[(N[(N[(N[(i * t$95$1), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(i * t$95$1 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i / t$95$2), $MachinePrecision] * N[(N[(t$95$1 / t$95$2), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if i < 2.15e95

   1. Initial program 64.0%

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

   4. Applied rewrites 91.2%

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

   if 2.15e95 < i

   1. Initial program 0.6%

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

   4. Applied rewrites 19.6%

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

   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 19.6%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

   9. Applied rewrites 82.2%

      \[\leadsto \color{blue}{\frac{i}{\beta + \mathsf{fma}\left(i, 2, \alpha\right)} \cdot \left(\frac{i + \left(\alpha + \beta\right)}{\beta + \mathsf{fma}\left(i, 2, \alpha\right)} \cdot 0.25\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 72.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + i \cdot 2\\ t_1 := t\_0 \cdot t\_0\\ t_2 := i \cdot \left(i + \left(\alpha + \beta\right)\right)\\ \mathbf{if}\;\frac{\frac{t\_2 \cdot \left(t\_2 + \alpha \cdot \beta\right)}{t\_1}}{-1 + t\_1} \leq 0.004:\\ \;\;\;\;\frac{i \cdot i}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* i 2.0)))
        (t_1 (* t_0 t_0))
        (t_2 (* i (+ i (+ alpha beta)))))
   (if (<= (/ (/ (* t_2 (+ t_2 (* alpha beta))) t_1) (+ -1.0 t_1)) 0.004)
     (/ (* i i) (* beta beta))
     (+ 0.0625 (/ 0.015625 (* i i))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (i * 2.0);
	double t_1 = t_0 * t_0;
	double t_2 = i * (i + (alpha + beta));
	double tmp;
	if ((((t_2 * (t_2 + (alpha * beta))) / t_1) / (-1.0 + t_1)) <= 0.004) {
		tmp = (i * i) / (beta * beta);
	} else {
		tmp = 0.0625 + (0.015625 / (i * i));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (alpha + beta) + (i * 2.0d0)
    t_1 = t_0 * t_0
    t_2 = i * (i + (alpha + beta))
    if ((((t_2 * (t_2 + (alpha * beta))) / t_1) / ((-1.0d0) + t_1)) <= 0.004d0) then
        tmp = (i * i) / (beta * beta)
    else
        tmp = 0.0625d0 + (0.015625d0 / (i * i))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (i * 2.0);
	double t_1 = t_0 * t_0;
	double t_2 = i * (i + (alpha + beta));
	double tmp;
	if ((((t_2 * (t_2 + (alpha * beta))) / t_1) / (-1.0 + t_1)) <= 0.004) {
		tmp = (i * i) / (beta * beta);
	} else {
		tmp = 0.0625 + (0.015625 / (i * i));
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (i * 2.0)
	t_1 = t_0 * t_0
	t_2 = i * (i + (alpha + beta))
	tmp = 0
	if (((t_2 * (t_2 + (alpha * beta))) / t_1) / (-1.0 + t_1)) <= 0.004:
		tmp = (i * i) / (beta * beta)
	else:
		tmp = 0.0625 + (0.015625 / (i * i))
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(i * 2.0))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(i * Float64(i + Float64(alpha + beta)))
	tmp = 0.0
	if (Float64(Float64(Float64(t_2 * Float64(t_2 + Float64(alpha * beta))) / t_1) / Float64(-1.0 + t_1)) <= 0.004)
		tmp = Float64(Float64(i * i) / Float64(beta * beta));
	else
		tmp = Float64(0.0625 + Float64(0.015625 / Float64(i * i)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (i * 2.0);
	t_1 = t_0 * t_0;
	t_2 = i * (i + (alpha + beta));
	tmp = 0.0;
	if ((((t_2 * (t_2 + (alpha * beta))) / t_1) / (-1.0 + t_1)) <= 0.004)
		tmp = (i * i) / (beta * beta);
	else
		tmp = 0.0625 + (0.015625 / (i * i));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$2 * N[(t$95$2 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(-1.0 + t$95$1), $MachinePrecision]), $MachinePrecision], 0.004], N[(N[(i * i), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(0.0625 + N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64))) < 0.0040000000000000001

   1. Initial program 98.2%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   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 46.5

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

   5. Applied rewrites 46.5%

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

   6. Taylor expanded in i around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{i}^{2}}{{\beta}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{i \cdot i}}{{\beta}^{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{i \cdot i}}{{\beta}^{2}} \]

      4. unpow2 N/A

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

      5. lower-*.f64 46.5

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

   8. Applied rewrites 46.5%

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

   if 0.0040000000000000001 < (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64)))

   1. Initial program 15.9%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 35.9

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

   5. Applied rewrites 35.9%

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

   6. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 30.9

         \[\leadsto \frac{\left(i \cdot i\right) \cdot 0.25}{\color{blue}{\left(i \cdot i\right)} \cdot 4 - 1} \]

   8. Applied rewrites 30.9%

      \[\leadsto \frac{\left(i \cdot i\right) \cdot 0.25}{\color{blue}{\left(i \cdot i\right) \cdot 4} - 1} \]

   9. Taylor expanded in i around inf

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

       1. lower-+.f64 N/A

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

       2. associate-*r/ N/A

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

       3. metadata-eval N/A

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

       4. lower-/.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 77.0

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

   11. Applied rewrites 77.0%

       \[\leadsto \color{blue}{0.0625 + \frac{0.015625}{i \cdot i}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \left(i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}{-1 + \left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right)} \leq 0.004:\\ \;\;\;\;\frac{i \cdot i}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i + \left(\alpha + \beta\right)\\ t_1 := \beta + \mathsf{fma}\left(i, 2, \alpha\right)\\ t_2 := \beta + \left(i + \left(i + \alpha\right)\right)\\ \mathbf{if}\;i \leq 3.5 \cdot 10^{+88}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(t\_2, t\_2, -1\right)}{\mathsf{fma}\left(i, t\_0, \alpha \cdot \beta\right)} \cdot \frac{t\_2 \cdot t\_2}{i \cdot t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{t\_1} \cdot \left(\frac{t\_0}{t\_1} \cdot 0.25\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ i (+ alpha beta)))
        (t_1 (+ beta (fma i 2.0 alpha)))
        (t_2 (+ beta (+ i (+ i alpha)))))
   (if (<= i 3.5e+88)
     (/
      1.0
      (*
       (/ (fma t_2 t_2 -1.0) (fma i t_0 (* alpha beta)))
       (/ (* t_2 t_2) (* i t_0))))
     (* (/ i t_1) (* (/ t_0 t_1) 0.25)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = i + (alpha + beta);
	double t_1 = beta + fma(i, 2.0, alpha);
	double t_2 = beta + (i + (i + alpha));
	double tmp;
	if (i <= 3.5e+88) {
		tmp = 1.0 / ((fma(t_2, t_2, -1.0) / fma(i, t_0, (alpha * beta))) * ((t_2 * t_2) / (i * t_0)));
	} else {
		tmp = (i / t_1) * ((t_0 / t_1) * 0.25);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(i + Float64(alpha + beta))
	t_1 = Float64(beta + fma(i, 2.0, alpha))
	t_2 = Float64(beta + Float64(i + Float64(i + alpha)))
	tmp = 0.0
	if (i <= 3.5e+88)
		tmp = Float64(1.0 / Float64(Float64(fma(t_2, t_2, -1.0) / fma(i, t_0, Float64(alpha * beta))) * Float64(Float64(t_2 * t_2) / Float64(i * t_0))));
	else
		tmp = Float64(Float64(i / t_1) * Float64(Float64(t_0 / t_1) * 0.25));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(beta + N[(i * 2.0 + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(beta + N[(i + N[(i + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, 3.5e+88], N[(1.0 / N[(N[(N[(t$95$2 * t$95$2 + -1.0), $MachinePrecision] / N[(i * t$95$0 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$2 * t$95$2), $MachinePrecision] / N[(i * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i / t$95$1), $MachinePrecision] * N[(N[(t$95$0 / t$95$1), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if i < 3.4999999999999998e88

   1. Initial program 66.7%

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

   4. Applied rewrites 89.7%

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

   6. Applied rewrites 89.7%

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

   if 3.4999999999999998e88 < i

   1. Initial program 0.6%

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

   4. Applied rewrites 19.8%

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

   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 19.8%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

   9. Applied rewrites 81.5%

      \[\leadsto \color{blue}{\frac{i}{\beta + \mathsf{fma}\left(i, 2, \alpha\right)} \cdot \left(\frac{i + \left(\alpha + \beta\right)}{\beta + \mathsf{fma}\left(i, 2, \alpha\right)} \cdot 0.25\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 79.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \alpha + \beta\right)\\ t_1 := i + \left(\alpha + \beta\right)\\ t_2 := \beta + \mathsf{fma}\left(i, 2, \alpha\right)\\ \mathbf{if}\;i \leq 3.5 \cdot 10^{+88}:\\ \;\;\;\;\frac{\mathsf{fma}\left(i, t\_1, \alpha \cdot \beta\right)}{\mathsf{fma}\left(t\_0, t\_0, -1\right)} \cdot \frac{i \cdot t\_1}{\mathsf{fma}\left(i, t\_1 \cdot 4, \left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{t\_2} \cdot \left(\frac{t\_1}{t\_2} \cdot 0.25\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma i 2.0 (+ alpha beta)))
        (t_1 (+ i (+ alpha beta)))
        (t_2 (+ beta (fma i 2.0 alpha))))
   (if (<= i 3.5e+88)
     (*
      (/ (fma i t_1 (* alpha beta)) (fma t_0 t_0 -1.0))
      (/ (* i t_1) (fma i (* t_1 4.0) (* (+ alpha beta) (+ alpha beta)))))
     (* (/ i t_2) (* (/ t_1 t_2) 0.25)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = fma(i, 2.0, (alpha + beta));
	double t_1 = i + (alpha + beta);
	double t_2 = beta + fma(i, 2.0, alpha);
	double tmp;
	if (i <= 3.5e+88) {
		tmp = (fma(i, t_1, (alpha * beta)) / fma(t_0, t_0, -1.0)) * ((i * t_1) / fma(i, (t_1 * 4.0), ((alpha + beta) * (alpha + beta))));
	} else {
		tmp = (i / t_2) * ((t_1 / t_2) * 0.25);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = fma(i, 2.0, Float64(alpha + beta))
	t_1 = Float64(i + Float64(alpha + beta))
	t_2 = Float64(beta + fma(i, 2.0, alpha))
	tmp = 0.0
	if (i <= 3.5e+88)
		tmp = Float64(Float64(fma(i, t_1, Float64(alpha * beta)) / fma(t_0, t_0, -1.0)) * Float64(Float64(i * t_1) / fma(i, Float64(t_1 * 4.0), Float64(Float64(alpha + beta) * Float64(alpha + beta)))));
	else
		tmp = Float64(Float64(i / t_2) * Float64(Float64(t_1 / t_2) * 0.25));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(i * 2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(beta + N[(i * 2.0 + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, 3.5e+88], N[(N[(N[(i * t$95$1 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(i * t$95$1), $MachinePrecision] / N[(i * N[(t$95$1 * 4.0), $MachinePrecision] + N[(N[(alpha + beta), $MachinePrecision] * N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i / t$95$2), $MachinePrecision] * N[(N[(t$95$1 / t$95$2), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if i < 3.4999999999999998e88

   1. Initial program 66.7%

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

   4. Applied rewrites 89.7%

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

   5. Taylor expanded in i around 0

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

      1. lower-fma.f64 N/A

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

      2. distribute-lft-out N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 89.8

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

   7. Applied rewrites 89.8%

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

   if 3.4999999999999998e88 < i

   1. Initial program 0.6%

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

   4. Applied rewrites 19.8%

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

   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 19.8%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

   9. Applied rewrites 81.5%

      \[\leadsto \color{blue}{\frac{i}{\beta + \mathsf{fma}\left(i, 2, \alpha\right)} \cdot \left(\frac{i + \left(\alpha + \beta\right)}{\beta + \mathsf{fma}\left(i, 2, \alpha\right)} \cdot 0.25\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 3.5 \cdot 10^{+88}:\\ \;\;\;\;\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\mathsf{fma}\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right), \mathsf{fma}\left(i, 2, \alpha + \beta\right), -1\right)} \cdot \frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\mathsf{fma}\left(i, \left(i + \left(\alpha + \beta\right)\right) \cdot 4, \left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta + \mathsf{fma}\left(i, 2, \alpha\right)} \cdot \left(\frac{i + \left(\alpha + \beta\right)}{\beta + \mathsf{fma}\left(i, 2, \alpha\right)} \cdot 0.25\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \alpha + \beta\right)\\ t_1 := \beta + \mathsf{fma}\left(i, 2, \alpha\right)\\ t_2 := i + \left(\alpha + \beta\right)\\ \mathbf{if}\;i \leq 3.5 \cdot 10^{+88}:\\ \;\;\;\;\frac{\mathsf{fma}\left(i, t\_2, \alpha \cdot \beta\right)}{\mathsf{fma}\left(t\_0, t\_0, -1\right)} \cdot \frac{i \cdot t\_2}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{t\_1} \cdot \left(\frac{t\_2}{t\_1} \cdot 0.25\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma i 2.0 (+ alpha beta)))
        (t_1 (+ beta (fma i 2.0 alpha)))
        (t_2 (+ i (+ alpha beta))))
   (if (<= i 3.5e+88)
     (*
      (/ (fma i t_2 (* alpha beta)) (fma t_0 t_0 -1.0))
      (/ (* i t_2) (* t_0 t_0)))
     (* (/ i t_1) (* (/ t_2 t_1) 0.25)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = fma(i, 2.0, (alpha + beta));
	double t_1 = beta + fma(i, 2.0, alpha);
	double t_2 = i + (alpha + beta);
	double tmp;
	if (i <= 3.5e+88) {
		tmp = (fma(i, t_2, (alpha * beta)) / fma(t_0, t_0, -1.0)) * ((i * t_2) / (t_0 * t_0));
	} else {
		tmp = (i / t_1) * ((t_2 / t_1) * 0.25);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = fma(i, 2.0, Float64(alpha + beta))
	t_1 = Float64(beta + fma(i, 2.0, alpha))
	t_2 = Float64(i + Float64(alpha + beta))
	tmp = 0.0
	if (i <= 3.5e+88)
		tmp = Float64(Float64(fma(i, t_2, Float64(alpha * beta)) / fma(t_0, t_0, -1.0)) * Float64(Float64(i * t_2) / Float64(t_0 * t_0)));
	else
		tmp = Float64(Float64(i / t_1) * Float64(Float64(t_2 / t_1) * 0.25));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(i * 2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(beta + N[(i * 2.0 + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, 3.5e+88], N[(N[(N[(i * t$95$2 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(i * t$95$2), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i / t$95$1), $MachinePrecision] * N[(N[(t$95$2 / t$95$1), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if i < 3.4999999999999998e88

   1. Initial program 66.7%

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

   4. Applied rewrites 89.7%

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

   if 3.4999999999999998e88 < i

   1. Initial program 0.6%

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

   4. Applied rewrites 19.8%

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

   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 19.8%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

   9. Applied rewrites 81.5%

      \[\leadsto \color{blue}{\frac{i}{\beta + \mathsf{fma}\left(i, 2, \alpha\right)} \cdot \left(\frac{i + \left(\alpha + \beta\right)}{\beta + \mathsf{fma}\left(i, 2, \alpha\right)} \cdot 0.25\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 75.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot \left(i + \beta\right)\\ t_1 := \beta + \mathsf{fma}\left(i, 2, \alpha\right)\\ \mathbf{if}\;i \leq 3.5 \cdot 10^{+88}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(i, 2, \beta\right), \mathsf{fma}\left(i, 2, \beta\right), -1\right)}{t\_0} \cdot \frac{\mathsf{fma}\left(i, 2, \beta\right) \cdot \mathsf{fma}\left(i, 2, \beta\right)}{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{t\_1} \cdot \left(\frac{i + \left(\alpha + \beta\right)}{t\_1} \cdot 0.25\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* i (+ i beta))) (t_1 (+ beta (fma i 2.0 alpha))))
   (if (<= i 3.5e+88)
     (/
      1.0
      (*
       (/ (fma (fma i 2.0 beta) (fma i 2.0 beta) -1.0) t_0)
       (/ (* (fma i 2.0 beta) (fma i 2.0 beta)) t_0)))
     (* (/ i t_1) (* (/ (+ i (+ alpha beta)) t_1) 0.25)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = i * (i + beta);
	double t_1 = beta + fma(i, 2.0, alpha);
	double tmp;
	if (i <= 3.5e+88) {
		tmp = 1.0 / ((fma(fma(i, 2.0, beta), fma(i, 2.0, beta), -1.0) / t_0) * ((fma(i, 2.0, beta) * fma(i, 2.0, beta)) / t_0));
	} else {
		tmp = (i / t_1) * (((i + (alpha + beta)) / t_1) * 0.25);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(i * Float64(i + beta))
	t_1 = Float64(beta + fma(i, 2.0, alpha))
	tmp = 0.0
	if (i <= 3.5e+88)
		tmp = Float64(1.0 / Float64(Float64(fma(fma(i, 2.0, beta), fma(i, 2.0, beta), -1.0) / t_0) * Float64(Float64(fma(i, 2.0, beta) * fma(i, 2.0, beta)) / t_0)));
	else
		tmp = Float64(Float64(i / t_1) * Float64(Float64(Float64(i + Float64(alpha + beta)) / t_1) * 0.25));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(i * N[(i + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(beta + N[(i * 2.0 + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, 3.5e+88], N[(1.0 / N[(N[(N[(N[(i * 2.0 + beta), $MachinePrecision] * N[(i * 2.0 + beta), $MachinePrecision] + -1.0), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(N[(i * 2.0 + beta), $MachinePrecision] * N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i / t$95$1), $MachinePrecision] * N[(N[(N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if i < 3.4999999999999998e88

   1. Initial program 66.7%

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

   4. Applied rewrites 89.7%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. sub-neg N/A

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

      6. unpow2 N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 76.7

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

   7. Applied rewrites 76.7%

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

   8. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 70.9

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

   10. Applied rewrites 70.9%

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

       1. lift-+.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-fma.f64 N/A

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

       4. lift-fma.f64 N/A

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

       5. lift-fma.f64 N/A

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

       6. lift-+.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. lift-+.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. lift-+.f64 N/A

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

       12. lift-*.f64 N/A

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

   12. Applied rewrites 70.9%

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

   if 3.4999999999999998e88 < i

   1. Initial program 0.6%

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

   4. Applied rewrites 19.8%

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

   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 19.8%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

   9. Applied rewrites 81.5%

      \[\leadsto \color{blue}{\frac{i}{\beta + \mathsf{fma}\left(i, 2, \alpha\right)} \cdot \left(\frac{i + \left(\alpha + \beta\right)}{\beta + \mathsf{fma}\left(i, 2, \alpha\right)} \cdot 0.25\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 75.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + \mathsf{fma}\left(i, 2, \alpha\right)\\ t_1 := i \cdot \left(i + \beta\right)\\ \mathbf{if}\;i \leq 3.5 \cdot 10^{+88}:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(\mathsf{fma}\left(i, 2, \beta\right), \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \frac{t\_1}{\mathsf{fma}\left(i, 4 \cdot \left(i + \beta\right), \beta \cdot \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{t\_0} \cdot \left(\frac{i + \left(\alpha + \beta\right)}{t\_0} \cdot 0.25\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ beta (fma i 2.0 alpha))) (t_1 (* i (+ i beta))))
   (if (<= i 3.5e+88)
     (*
      (/ t_1 (fma (fma i 2.0 beta) (fma i 2.0 beta) -1.0))
      (/ t_1 (fma i (* 4.0 (+ i beta)) (* beta beta))))
     (* (/ i t_0) (* (/ (+ i (+ alpha beta)) t_0) 0.25)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = beta + fma(i, 2.0, alpha);
	double t_1 = i * (i + beta);
	double tmp;
	if (i <= 3.5e+88) {
		tmp = (t_1 / fma(fma(i, 2.0, beta), fma(i, 2.0, beta), -1.0)) * (t_1 / fma(i, (4.0 * (i + beta)), (beta * beta)));
	} else {
		tmp = (i / t_0) * (((i + (alpha + beta)) / t_0) * 0.25);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(beta + fma(i, 2.0, alpha))
	t_1 = Float64(i * Float64(i + beta))
	tmp = 0.0
	if (i <= 3.5e+88)
		tmp = Float64(Float64(t_1 / fma(fma(i, 2.0, beta), fma(i, 2.0, beta), -1.0)) * Float64(t_1 / fma(i, Float64(4.0 * Float64(i + beta)), Float64(beta * beta))));
	else
		tmp = Float64(Float64(i / t_0) * Float64(Float64(Float64(i + Float64(alpha + beta)) / t_0) * 0.25));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(beta + N[(i * 2.0 + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(i * N[(i + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, 3.5e+88], N[(N[(t$95$1 / N[(N[(i * 2.0 + beta), $MachinePrecision] * N[(i * 2.0 + beta), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / N[(i * N[(4.0 * N[(i + beta), $MachinePrecision]), $MachinePrecision] + N[(beta * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i / t$95$0), $MachinePrecision] * N[(N[(N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if i < 3.4999999999999998e88

   1. Initial program 66.7%

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

   4. Applied rewrites 89.7%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. sub-neg N/A

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

      6. unpow2 N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 76.7

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

   7. Applied rewrites 76.7%

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

   8. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 70.9

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

   10. Applied rewrites 70.9%

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

   11. Taylor expanded in i around 0

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

       1. lower-fma.f64 N/A

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

       2. distribute-lft-out N/A

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

       3. lower-*.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-+.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 71.0

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

   13. Applied rewrites 71.0%

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

   if 3.4999999999999998e88 < i

   1. Initial program 0.6%

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

   4. Applied rewrites 19.8%

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

   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 19.8%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

   9. Applied rewrites 81.5%

      \[\leadsto \color{blue}{\frac{i}{\beta + \mathsf{fma}\left(i, 2, \alpha\right)} \cdot \left(\frac{i + \left(\alpha + \beta\right)}{\beta + \mathsf{fma}\left(i, 2, \alpha\right)} \cdot 0.25\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 3.5 \cdot 10^{+88}:\\ \;\;\;\;\frac{i \cdot \left(i + \beta\right)}{\mathsf{fma}\left(\mathsf{fma}\left(i, 2, \beta\right), \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \frac{i \cdot \left(i + \beta\right)}{\mathsf{fma}\left(i, 4 \cdot \left(i + \beta\right), \beta \cdot \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta + \mathsf{fma}\left(i, 2, \alpha\right)} \cdot \left(\frac{i + \left(\alpha + \beta\right)}{\beta + \mathsf{fma}\left(i, 2, \alpha\right)} \cdot 0.25\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.32 \cdot 10^{+233}:\\ \;\;\;\;\frac{i}{\beta + \mathsf{fma}\left(i, 2, \alpha\right)} \cdot \frac{0.25 \cdot \left(i + \beta\right)}{\mathsf{fma}\left(i, 2, \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\frac{\beta}{i}}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.32e+233)
   (*
    (/ i (+ beta (fma i 2.0 alpha)))
    (/ (* 0.25 (+ i beta)) (fma i 2.0 beta)))
   (/ (/ (+ i alpha) (/ beta i)) beta)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.32e+233) {
		tmp = (i / (beta + fma(i, 2.0, alpha))) * ((0.25 * (i + beta)) / fma(i, 2.0, beta));
	} else {
		tmp = ((i + alpha) / (beta / i)) / beta;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.32e+233)
		tmp = Float64(Float64(i / Float64(beta + fma(i, 2.0, alpha))) * Float64(Float64(0.25 * Float64(i + beta)) / fma(i, 2.0, beta)));
	else
		tmp = Float64(Float64(Float64(i + alpha) / Float64(beta / i)) / beta);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 1.32e+233], N[(N[(i / N[(beta + N[(i * 2.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(0.25 * N[(i + beta), $MachinePrecision]), $MachinePrecision] / N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(i + alpha), $MachinePrecision] / N[(beta / i), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.32e233

   1. Initial program 20.0%

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

   4. Applied rewrites 41.5%

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

   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 34.7%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

   9. Applied rewrites 79.5%

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

   10. Taylor expanded in alpha around 0

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-+.f64 N/A

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

       6. +-commutative N/A

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

       7. *-commutative N/A

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

       8. lower-fma.f64 79.2

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

   12. Applied rewrites 79.2%

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

   if 1.32e233 < 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 21.0

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

   5. Applied rewrites 21.0%

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

      1. lift-+.f64 N/A

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

      2. times-frac N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 78.7

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 78.7

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

   7. Applied rewrites 78.7%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 78.9

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

   9. Applied rewrites 78.9%

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

Alternative 9: 77.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.32 \cdot 10^{+233}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\frac{\beta}{i}}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.32e+233)
   (+ 0.0625 (/ 0.015625 (* i i)))
   (/ (/ (+ i alpha) (/ beta i)) beta)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.32e+233) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else {
		tmp = ((i + alpha) / (beta / i)) / beta;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 1.32d+233) then
        tmp = 0.0625d0 + (0.015625d0 / (i * i))
    else
        tmp = ((i + alpha) / (beta / i)) / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.32e+233) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else {
		tmp = ((i + alpha) / (beta / i)) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.32e+233:
		tmp = 0.0625 + (0.015625 / (i * i))
	else:
		tmp = ((i + alpha) / (beta / i)) / beta
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.32e+233)
		tmp = Float64(0.0625 + Float64(0.015625 / Float64(i * i)));
	else
		tmp = Float64(Float64(Float64(i + alpha) / Float64(beta / i)) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1.32e+233)
		tmp = 0.0625 + (0.015625 / (i * i));
	else
		tmp = ((i + alpha) / (beta / i)) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 1.32e+233], N[(0.0625 + N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(i + alpha), $MachinePrecision] / N[(beta / i), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.32e233

   1. Initial program 20.0%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 36.5

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

   5. Applied rewrites 36.5%

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

   6. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 32.7

         \[\leadsto \frac{\left(i \cdot i\right) \cdot 0.25}{\color{blue}{\left(i \cdot i\right)} \cdot 4 - 1} \]

   8. Applied rewrites 32.7%

      \[\leadsto \frac{\left(i \cdot i\right) \cdot 0.25}{\color{blue}{\left(i \cdot i\right) \cdot 4} - 1} \]

   9. Taylor expanded in i around inf

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

       1. lower-+.f64 N/A

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

       2. associate-*r/ N/A

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

       3. metadata-eval N/A

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

       4. lower-/.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 79.2

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

   11. Applied rewrites 79.2%

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

   if 1.32e233 < 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 21.0

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

   5. Applied rewrites 21.0%

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

      1. lift-+.f64 N/A

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

      2. times-frac N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 78.7

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 78.7

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

   7. Applied rewrites 78.7%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 78.9

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

   9. Applied rewrites 78.9%

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

Alternative 10: 77.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.32 \cdot 10^{+233}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(i + \alpha\right) \cdot \left(i \cdot \frac{1}{\beta}\right)}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.32e+233)
   (+ 0.0625 (/ 0.015625 (* i i)))
   (/ (* (+ i alpha) (* i (/ 1.0 beta))) beta)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.32e+233) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else {
		tmp = ((i + alpha) * (i * (1.0 / beta))) / beta;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 1.32d+233) then
        tmp = 0.0625d0 + (0.015625d0 / (i * i))
    else
        tmp = ((i + alpha) * (i * (1.0d0 / beta))) / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.32e+233) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else {
		tmp = ((i + alpha) * (i * (1.0 / beta))) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.32e+233:
		tmp = 0.0625 + (0.015625 / (i * i))
	else:
		tmp = ((i + alpha) * (i * (1.0 / beta))) / beta
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.32e+233)
		tmp = Float64(0.0625 + Float64(0.015625 / Float64(i * i)));
	else
		tmp = Float64(Float64(Float64(i + alpha) * Float64(i * Float64(1.0 / beta))) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1.32e+233)
		tmp = 0.0625 + (0.015625 / (i * i));
	else
		tmp = ((i + alpha) * (i * (1.0 / beta))) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 1.32e+233], N[(0.0625 + N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(i + alpha), $MachinePrecision] * N[(i * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.32e233

   1. Initial program 20.0%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 36.5

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

   5. Applied rewrites 36.5%

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

   6. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 32.7

         \[\leadsto \frac{\left(i \cdot i\right) \cdot 0.25}{\color{blue}{\left(i \cdot i\right)} \cdot 4 - 1} \]

   8. Applied rewrites 32.7%

      \[\leadsto \frac{\left(i \cdot i\right) \cdot 0.25}{\color{blue}{\left(i \cdot i\right) \cdot 4} - 1} \]

   9. Taylor expanded in i around inf

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

       1. lower-+.f64 N/A

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

       2. associate-*r/ N/A

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

       3. metadata-eval N/A

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

       4. lower-/.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 79.2

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

   11. Applied rewrites 79.2%

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

   if 1.32e233 < 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 21.0

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

   5. Applied rewrites 21.0%

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

      1. lift-+.f64 N/A

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

      2. times-frac N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 78.7

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 78.7

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

   7. Applied rewrites 78.7%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 78.8

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

   9. Applied rewrites 78.8%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.32 \cdot 10^{+233}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(i + \alpha\right) \cdot \left(i \cdot \frac{1}{\beta}\right)}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 77.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.32 \cdot 10^{+233}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(i + \alpha\right) \cdot \frac{i}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.32e+233)
   (+ 0.0625 (/ 0.015625 (* i i)))
   (/ (* (+ i alpha) (/ i beta)) beta)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.32e+233) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else {
		tmp = ((i + alpha) * (i / beta)) / beta;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 1.32d+233) then
        tmp = 0.0625d0 + (0.015625d0 / (i * i))
    else
        tmp = ((i + alpha) * (i / beta)) / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.32e+233) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else {
		tmp = ((i + alpha) * (i / beta)) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.32e+233:
		tmp = 0.0625 + (0.015625 / (i * i))
	else:
		tmp = ((i + alpha) * (i / beta)) / beta
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.32e+233)
		tmp = Float64(0.0625 + Float64(0.015625 / Float64(i * i)));
	else
		tmp = Float64(Float64(Float64(i + alpha) * Float64(i / beta)) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1.32e+233)
		tmp = 0.0625 + (0.015625 / (i * i));
	else
		tmp = ((i + alpha) * (i / beta)) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 1.32e+233], N[(0.0625 + N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(i + alpha), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.32e233

   1. Initial program 20.0%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 36.5

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

   5. Applied rewrites 36.5%

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

   6. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 32.7

         \[\leadsto \frac{\left(i \cdot i\right) \cdot 0.25}{\color{blue}{\left(i \cdot i\right)} \cdot 4 - 1} \]

   8. Applied rewrites 32.7%

      \[\leadsto \frac{\left(i \cdot i\right) \cdot 0.25}{\color{blue}{\left(i \cdot i\right) \cdot 4} - 1} \]

   9. Taylor expanded in i around inf

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

       1. lower-+.f64 N/A

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

       2. associate-*r/ N/A

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

       3. metadata-eval N/A

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

       4. lower-/.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 79.2

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

   11. Applied rewrites 79.2%

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

   if 1.32e233 < 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 21.0

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

   5. Applied rewrites 21.0%

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

      1. lift-+.f64 N/A

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

      2. times-frac N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 78.7

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 78.7

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

   7. Applied rewrites 78.7%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.32 \cdot 10^{+233}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(i + \alpha\right) \cdot \frac{i}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 77.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.32 \cdot 10^{+233}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.32e+233)
   (+ 0.0625 (/ 0.015625 (* i i)))
   (* (/ i beta) (/ (+ i alpha) beta))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.32e+233) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else {
		tmp = (i / beta) * ((i + alpha) / beta);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 1.32d+233) then
        tmp = 0.0625d0 + (0.015625d0 / (i * i))
    else
        tmp = (i / beta) * ((i + alpha) / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.32e+233) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else {
		tmp = (i / beta) * ((i + alpha) / beta);
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.32e+233:
		tmp = 0.0625 + (0.015625 / (i * i))
	else:
		tmp = (i / beta) * ((i + alpha) / beta)
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.32e+233)
		tmp = Float64(0.0625 + Float64(0.015625 / Float64(i * i)));
	else
		tmp = Float64(Float64(i / beta) * Float64(Float64(i + alpha) / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1.32e+233)
		tmp = 0.0625 + (0.015625 / (i * i));
	else
		tmp = (i / beta) * ((i + alpha) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 1.32e+233], N[(0.0625 + N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i / beta), $MachinePrecision] * N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.32e233

   1. Initial program 20.0%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 36.5

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

   5. Applied rewrites 36.5%

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

   6. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 32.7

         \[\leadsto \frac{\left(i \cdot i\right) \cdot 0.25}{\color{blue}{\left(i \cdot i\right)} \cdot 4 - 1} \]

   8. Applied rewrites 32.7%

      \[\leadsto \frac{\left(i \cdot i\right) \cdot 0.25}{\color{blue}{\left(i \cdot i\right) \cdot 4} - 1} \]

   9. Taylor expanded in i around inf

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

       1. lower-+.f64 N/A

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

       2. associate-*r/ N/A

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

       3. metadata-eval N/A

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

       4. lower-/.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 79.2

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

   11. Applied rewrites 79.2%

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

   if 1.32e233 < 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 21.0

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

   5. Applied rewrites 21.0%

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

      1. lift-+.f64 N/A

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

      2. times-frac N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 78.7

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

   7. Applied rewrites 78.7%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.32 \cdot 10^{+233}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 73.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.56 \cdot 10^{+233}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i \cdot i}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.56e+233)
   (+ 0.0625 (/ 0.015625 (* i i)))
   (/ (/ (* i i) beta) beta)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.56e+233) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else {
		tmp = ((i * i) / beta) / beta;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 1.56d+233) then
        tmp = 0.0625d0 + (0.015625d0 / (i * i))
    else
        tmp = ((i * i) / beta) / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.56e+233) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else {
		tmp = ((i * i) / beta) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.56e+233:
		tmp = 0.0625 + (0.015625 / (i * i))
	else:
		tmp = ((i * i) / beta) / beta
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.56e+233)
		tmp = Float64(0.0625 + Float64(0.015625 / Float64(i * i)));
	else
		tmp = Float64(Float64(Float64(i * i) / beta) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1.56e+233)
		tmp = 0.0625 + (0.015625 / (i * i));
	else
		tmp = ((i * i) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 1.56e+233], N[(0.0625 + N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(i * i), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.56e233

   1. Initial program 20.0%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 36.5

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

   5. Applied rewrites 36.5%

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

   6. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 32.7

         \[\leadsto \frac{\left(i \cdot i\right) \cdot 0.25}{\color{blue}{\left(i \cdot i\right)} \cdot 4 - 1} \]

   8. Applied rewrites 32.7%

      \[\leadsto \frac{\left(i \cdot i\right) \cdot 0.25}{\color{blue}{\left(i \cdot i\right) \cdot 4} - 1} \]

   9. Taylor expanded in i around inf

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

       1. lower-+.f64 N/A

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

       2. associate-*r/ N/A

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

       3. metadata-eval N/A

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

       4. lower-/.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 79.2

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

   11. Applied rewrites 79.2%

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

   if 1.56e233 < 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 21.0

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

   5. Applied rewrites 21.0%

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

      1. lift-+.f64 N/A

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

      2. times-frac N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 78.7

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 78.7

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

   7. Applied rewrites 78.7%

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

   8. Taylor expanded in i around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{{i}^{2}}{\beta}}}{\beta} \]

      2. unpow2 N/A

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

      3. lower-*.f64 36.6

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

   10. Applied rewrites 36.6%

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

Alternative 14: 71.1% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ 0.0625 + \frac{0.015625}{i \cdot i} \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 (+ 0.0625 (/ 0.015625 (* i i))))

C

double code(double alpha, double beta, double i) {
	return 0.0625 + (0.015625 / (i * i));
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    code = 0.0625d0 + (0.015625d0 / (i * i))
end function

Java

public static double code(double alpha, double beta, double i) {
	return 0.0625 + (0.015625 / (i * i));
}

Python

def code(alpha, beta, i):
	return 0.0625 + (0.015625 / (i * i))

Julia

function code(alpha, beta, i)
	return Float64(0.0625 + Float64(0.015625 / Float64(i * i)))
end

MATLAB

function tmp = code(alpha, beta, i)
	tmp = 0.0625 + (0.015625 / (i * i));
end

Wolfram

code[alpha_, beta_, i_] := N[(0.0625 + N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.4%

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

3. Taylor expanded in i around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 35.3

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

5. Applied rewrites 35.3%

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

6. Taylor expanded in i around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 30.2

      \[\leadsto \frac{\left(i \cdot i\right) \cdot 0.25}{\color{blue}{\left(i \cdot i\right)} \cdot 4 - 1} \]

8. Applied rewrites 30.2%

   \[\leadsto \frac{\left(i \cdot i\right) \cdot 0.25}{\color{blue}{\left(i \cdot i\right) \cdot 4} - 1} \]

9. Taylor expanded in i around inf

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

    1. lower-+.f64 N/A

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

    2. associate-*r/ N/A

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

    3. metadata-eval N/A

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

    4. lower-/.f64 N/A

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

    5. unpow2 N/A

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

    6. lower-*.f64 74.9

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

11. Applied rewrites 74.9%

    \[\leadsto \color{blue}{0.0625 + \frac{0.015625}{i \cdot i}} \]
12. Add Preprocessing

Alternative 15: 70.9% accurate, 115.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[alpha_, beta_, i_] := 0.0625

Derivation

1. Initial program 18.4%

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

3. Taylor expanded in i around inf

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

5. Applied rewrites 74.8%

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

Reproduce

herbie shell --seed 2024216 
(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)))