Octave 3.8, jcobi/4

Percentage Accurate: 16.6% → 83.7%

Time: 5.4s

Alternatives: 9

Speedup: 75.4×

Specification

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

Math

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

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 16.6% accurate, 1.0× speedup

Math

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

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 83.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(2, i, \beta + \alpha\right)\\ t_1 := \left(\beta + \alpha\right) + i\\ \mathbf{if}\;i \leq 2.1 \cdot 10^{+118}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_1, i, \beta \cdot \alpha\right)}{t\_0}}{t\_0 - 1} \cdot \frac{t\_1 \cdot \frac{i}{t\_0}}{t\_0 - -1}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma 2.0 i (+ beta alpha))) (t_1 (+ (+ beta alpha) i)))
   (if (<= i 2.1e+118)
     (*
      (/ (/ (fma t_1 i (* beta alpha)) t_0) (- t_0 1.0))
      (/ (* t_1 (/ i t_0)) (- t_0 -1.0)))
     0.0625)))

C

double code(double alpha, double beta, double i) {
	double t_0 = fma(2.0, i, (beta + alpha));
	double t_1 = (beta + alpha) + i;
	double tmp;
	if (i <= 2.1e+118) {
		tmp = ((fma(t_1, i, (beta * alpha)) / t_0) / (t_0 - 1.0)) * ((t_1 * (i / t_0)) / (t_0 - -1.0));
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = fma(2.0, i, Float64(beta + alpha))
	t_1 = Float64(Float64(beta + alpha) + i)
	tmp = 0.0
	if (i <= 2.1e+118)
		tmp = Float64(Float64(Float64(fma(t_1, i, Float64(beta * alpha)) / t_0) / Float64(t_0 - 1.0)) * Float64(Float64(t_1 * Float64(i / t_0)) / Float64(t_0 - -1.0)));
	else
		tmp = 0.0625;
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(2.0 * i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(beta + alpha), $MachinePrecision] + i), $MachinePrecision]}, If[LessEqual[i, 2.1e+118], N[(N[(N[(N[(t$95$1 * i + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 - 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 * N[(i / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0625]]]

Derivation

1. Split input into 2 regimes

2. if i < 2.1e118

   1. Initial program 16.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

   3. Applied rewrites 43.0%

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

   if 2.1e118 < i

   1. Initial program 16.6%

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

   2. Taylor expanded in i around inf

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

   4. Applied rewrites 70.4%

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

Alternative 2: 83.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := t\_0 \cdot t\_0\\ t_2 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ t_3 := \left(\beta + \alpha\right) + i\\ t_4 := \mathsf{fma}\left(2, i, \beta + \alpha\right)\\ \mathbf{if}\;\frac{\frac{t\_2 \cdot \left(\beta \cdot \alpha + t\_2\right)}{t\_1}}{t\_1 - 1} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_3, i, \beta \cdot \alpha\right)}{\mathsf{fma}\left(t\_4, t\_4, -1\right)} \cdot \frac{\frac{t\_3}{t\_4} \cdot i}{t\_4}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{i}{\mathsf{fma}\left(\mathsf{fma}\left(\beta + \alpha, 2, i\right), 0.0625, -0.125 \cdot \left(\beta + \alpha\right)\right)}}\\ \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (* t_0 t_0))
        (t_2 (* i (+ (+ alpha beta) i)))
        (t_3 (+ (+ beta alpha) i))
        (t_4 (fma 2.0 i (+ beta alpha))))
   (if (<= (/ (/ (* t_2 (+ (* beta alpha) t_2)) t_1) (- t_1 1.0)) INFINITY)
     (*
      (/ (fma t_3 i (* beta alpha)) (fma t_4 t_4 -1.0))
      (/ (* (/ t_3 t_4) i) t_4))
     (/
      1.0
      (/
       i
       (fma (fma (+ beta alpha) 2.0 i) 0.0625 (* -0.125 (+ beta alpha))))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = t_0 * t_0;
	double t_2 = i * ((alpha + beta) + i);
	double t_3 = (beta + alpha) + i;
	double t_4 = fma(2.0, i, (beta + alpha));
	double tmp;
	if ((((t_2 * ((beta * alpha) + t_2)) / t_1) / (t_1 - 1.0)) <= ((double) INFINITY)) {
		tmp = (fma(t_3, i, (beta * alpha)) / fma(t_4, t_4, -1.0)) * (((t_3 / t_4) * i) / t_4);
	} else {
		tmp = 1.0 / (i / fma(fma((beta + alpha), 2.0, i), 0.0625, (-0.125 * (beta + alpha))));
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(i * Float64(Float64(alpha + beta) + i))
	t_3 = Float64(Float64(beta + alpha) + i)
	t_4 = fma(2.0, i, Float64(beta + alpha))
	tmp = 0.0
	if (Float64(Float64(Float64(t_2 * Float64(Float64(beta * alpha) + t_2)) / t_1) / Float64(t_1 - 1.0)) <= Inf)
		tmp = Float64(Float64(fma(t_3, i, Float64(beta * alpha)) / fma(t_4, t_4, -1.0)) * Float64(Float64(Float64(t_3 / t_4) * i) / t_4));
	else
		tmp = Float64(1.0 / Float64(i / fma(fma(Float64(beta + alpha), 2.0, i), 0.0625, Float64(-0.125 * Float64(beta + alpha)))));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(beta + alpha), $MachinePrecision] + i), $MachinePrecision]}, Block[{t$95$4 = N[(2.0 * i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$2 * N[(N[(beta * alpha), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 - 1.0), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(t$95$3 * i + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] / N[(t$95$4 * t$95$4 + -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$3 / t$95$4), $MachinePrecision] * i), $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(i / N[(N[(N[(beta + alpha), $MachinePrecision] * 2.0 + i), $MachinePrecision] * 0.0625 + N[(-0.125 * N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $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))) < +inf.0

   1. Initial program 16.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

   3. Applied rewrites 37.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. associate-*r/ N/A

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

      13. lower-/.f64 N/A

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

   5. Applied rewrites 37.9%

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

   if +inf.0 < (/.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 16.6%

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

   2. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i}} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \color{blue}{\frac{1}{8} \cdot \frac{\alpha + \beta}{i}} \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \color{blue}{\frac{1}{8}} \cdot \frac{\alpha + \beta}{i} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \color{blue}{\frac{\alpha + \beta}{i}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{\color{blue}{i}} \]

      9. lower-+.f64 77.0

         \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   4. Applied rewrites 77.0%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]

   5. Taylor expanded in i around 0

      \[\leadsto \frac{\left(\frac{1}{16} \cdot i + \frac{1}{16} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right)\right) - \frac{1}{8} \cdot \left(\alpha + \beta\right)}{\color{blue}{i}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{16} \cdot i + \frac{1}{16} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right)\right) - \frac{1}{8} \cdot \left(\alpha + \beta\right)}{i} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{16} \cdot i + \frac{1}{16} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right)\right) - \frac{1}{8} \cdot \left(\alpha + \beta\right)}{i} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{16}, i, \frac{1}{16} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right)\right) - \frac{1}{8} \cdot \left(\alpha + \beta\right)}{i} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{16}, i, \frac{1}{16} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right)\right) - \frac{1}{8} \cdot \left(\alpha + \beta\right)}{i} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{16}, i, \frac{1}{16} \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) - \frac{1}{8} \cdot \left(\alpha + \beta\right)}{i} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{16}, i, \frac{1}{16} \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) - \frac{1}{8} \cdot \left(\alpha + \beta\right)}{i} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{16}, i, \frac{1}{16} \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) - \frac{1}{8} \cdot \left(\alpha + \beta\right)}{i} \]

      8. lower-+.f64 77.0

         \[\leadsto \frac{\mathsf{fma}\left(0.0625, i, 0.0625 \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) - 0.125 \cdot \left(\alpha + \beta\right)}{i} \]

   7. Applied rewrites 77.0%

      \[\leadsto \frac{\mathsf{fma}\left(0.0625, i, 0.0625 \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) - 0.125 \cdot \left(\alpha + \beta\right)}{\color{blue}{i}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{16}, i, \frac{1}{16} \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) - \frac{1}{8} \cdot \left(\alpha + \beta\right)}{i} \]

      2. div-flip N/A

         \[\leadsto \frac{1}{\frac{i}{\color{blue}{\mathsf{fma}\left(\frac{1}{16}, i, \frac{1}{16} \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) - \frac{1}{8} \cdot \left(\alpha + \beta\right)}}} \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \frac{1}{\frac{i}{\color{blue}{\mathsf{fma}\left(\frac{1}{16}, i, \frac{1}{16} \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) - \frac{1}{8} \cdot \left(\alpha + \beta\right)}}} \]

      4. lower-unsound-/.f64 77.0

         \[\leadsto \frac{1}{\frac{i}{\mathsf{fma}\left(0.0625, i, 0.0625 \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) - \color{blue}{0.125 \cdot \left(\alpha + \beta\right)}}} \]

      5. lift--.f64 N/A

         \[\leadsto \frac{1}{\frac{i}{\mathsf{fma}\left(\frac{1}{16}, i, \frac{1}{16} \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) - \frac{1}{8} \cdot \color{blue}{\left(\alpha + \beta\right)}}} \]

      6. sub-flip N/A

         \[\leadsto \frac{1}{\frac{i}{\mathsf{fma}\left(\frac{1}{16}, i, \frac{1}{16} \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) + \left(\mathsf{neg}\left(\frac{1}{8} \cdot \left(\alpha + \beta\right)\right)\right)}} \]

      7. lift-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{i}{\left(\frac{1}{16} \cdot i + \frac{1}{16} \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) + \left(\mathsf{neg}\left(\frac{1}{8} \cdot \left(\alpha + \beta\right)\right)\right)}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{i}{\left(\frac{1}{16} \cdot i + \frac{1}{16} \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) + \left(\mathsf{neg}\left(\frac{1}{8} \cdot \left(\alpha + \beta\right)\right)\right)}} \]

      9. distribute-lft-out N/A

         \[\leadsto \frac{1}{\frac{i}{\frac{1}{16} \cdot \left(i + \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) + \left(\mathsf{neg}\left(\frac{1}{8} \cdot \left(\alpha + \beta\right)\right)\right)}} \]

      10. *-commutative N/A

          \[\leadsto \frac{1}{\frac{i}{\left(i + \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) \cdot \frac{1}{16} + \left(\mathsf{neg}\left(\frac{1}{8} \cdot \left(\alpha + \beta\right)\right)\right)}} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{i}{\mathsf{fma}\left(i + \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right), \frac{1}{16}, \mathsf{neg}\left(\frac{1}{8} \cdot \left(\alpha + \beta\right)\right)\right)}} \]

   9. Applied rewrites 77.0%

      \[\leadsto \frac{1}{\frac{i}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\beta + \alpha, 2, i\right), 0.0625, -0.125 \cdot \left(\beta + \alpha\right)\right)}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 83.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := t\_0 \cdot t\_0\\ t_2 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ t_3 := \left(\beta + \alpha\right) + i\\ t_4 := \mathsf{fma}\left(2, i, \beta + \alpha\right)\\ \mathbf{if}\;\frac{\frac{t\_2 \cdot \left(\beta \cdot \alpha + t\_2\right)}{t\_1}}{t\_1 - 1} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_3, i, \beta \cdot \alpha\right)}{\mathsf{fma}\left(t\_4, t\_4, -1\right)} \cdot \frac{t\_3 \cdot i}{t\_4 \cdot t\_4}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{i}{\mathsf{fma}\left(\mathsf{fma}\left(\beta + \alpha, 2, i\right), 0.0625, -0.125 \cdot \left(\beta + \alpha\right)\right)}}\\ \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (* t_0 t_0))
        (t_2 (* i (+ (+ alpha beta) i)))
        (t_3 (+ (+ beta alpha) i))
        (t_4 (fma 2.0 i (+ beta alpha))))
   (if (<= (/ (/ (* t_2 (+ (* beta alpha) t_2)) t_1) (- t_1 1.0)) INFINITY)
     (*
      (/ (fma t_3 i (* beta alpha)) (fma t_4 t_4 -1.0))
      (/ (* t_3 i) (* t_4 t_4)))
     (/
      1.0
      (/
       i
       (fma (fma (+ beta alpha) 2.0 i) 0.0625 (* -0.125 (+ beta alpha))))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = t_0 * t_0;
	double t_2 = i * ((alpha + beta) + i);
	double t_3 = (beta + alpha) + i;
	double t_4 = fma(2.0, i, (beta + alpha));
	double tmp;
	if ((((t_2 * ((beta * alpha) + t_2)) / t_1) / (t_1 - 1.0)) <= ((double) INFINITY)) {
		tmp = (fma(t_3, i, (beta * alpha)) / fma(t_4, t_4, -1.0)) * ((t_3 * i) / (t_4 * t_4));
	} else {
		tmp = 1.0 / (i / fma(fma((beta + alpha), 2.0, i), 0.0625, (-0.125 * (beta + alpha))));
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(i * Float64(Float64(alpha + beta) + i))
	t_3 = Float64(Float64(beta + alpha) + i)
	t_4 = fma(2.0, i, Float64(beta + alpha))
	tmp = 0.0
	if (Float64(Float64(Float64(t_2 * Float64(Float64(beta * alpha) + t_2)) / t_1) / Float64(t_1 - 1.0)) <= Inf)
		tmp = Float64(Float64(fma(t_3, i, Float64(beta * alpha)) / fma(t_4, t_4, -1.0)) * Float64(Float64(t_3 * i) / Float64(t_4 * t_4)));
	else
		tmp = Float64(1.0 / Float64(i / fma(fma(Float64(beta + alpha), 2.0, i), 0.0625, Float64(-0.125 * Float64(beta + alpha)))));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(beta + alpha), $MachinePrecision] + i), $MachinePrecision]}, Block[{t$95$4 = N[(2.0 * i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$2 * N[(N[(beta * alpha), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 - 1.0), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(t$95$3 * i + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] / N[(t$95$4 * t$95$4 + -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$3 * i), $MachinePrecision] / N[(t$95$4 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(i / N[(N[(N[(beta + alpha), $MachinePrecision] * 2.0 + i), $MachinePrecision] * 0.0625 + N[(-0.125 * N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $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))) < +inf.0

   1. Initial program 16.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

   3. Applied rewrites 37.8%

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

   if +inf.0 < (/.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 16.6%

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

   2. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i}} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \color{blue}{\frac{1}{8} \cdot \frac{\alpha + \beta}{i}} \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \color{blue}{\frac{1}{8}} \cdot \frac{\alpha + \beta}{i} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \color{blue}{\frac{\alpha + \beta}{i}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{\color{blue}{i}} \]

      9. lower-+.f64 77.0

         \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   4. Applied rewrites 77.0%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]

   5. Taylor expanded in i around 0

      \[\leadsto \frac{\left(\frac{1}{16} \cdot i + \frac{1}{16} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right)\right) - \frac{1}{8} \cdot \left(\alpha + \beta\right)}{\color{blue}{i}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{16} \cdot i + \frac{1}{16} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right)\right) - \frac{1}{8} \cdot \left(\alpha + \beta\right)}{i} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{16} \cdot i + \frac{1}{16} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right)\right) - \frac{1}{8} \cdot \left(\alpha + \beta\right)}{i} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{16}, i, \frac{1}{16} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right)\right) - \frac{1}{8} \cdot \left(\alpha + \beta\right)}{i} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{16}, i, \frac{1}{16} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right)\right) - \frac{1}{8} \cdot \left(\alpha + \beta\right)}{i} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{16}, i, \frac{1}{16} \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) - \frac{1}{8} \cdot \left(\alpha + \beta\right)}{i} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{16}, i, \frac{1}{16} \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) - \frac{1}{8} \cdot \left(\alpha + \beta\right)}{i} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{16}, i, \frac{1}{16} \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) - \frac{1}{8} \cdot \left(\alpha + \beta\right)}{i} \]

      8. lower-+.f64 77.0

         \[\leadsto \frac{\mathsf{fma}\left(0.0625, i, 0.0625 \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) - 0.125 \cdot \left(\alpha + \beta\right)}{i} \]

   7. Applied rewrites 77.0%

      \[\leadsto \frac{\mathsf{fma}\left(0.0625, i, 0.0625 \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) - 0.125 \cdot \left(\alpha + \beta\right)}{\color{blue}{i}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{16}, i, \frac{1}{16} \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) - \frac{1}{8} \cdot \left(\alpha + \beta\right)}{i} \]

      2. div-flip N/A

         \[\leadsto \frac{1}{\frac{i}{\color{blue}{\mathsf{fma}\left(\frac{1}{16}, i, \frac{1}{16} \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) - \frac{1}{8} \cdot \left(\alpha + \beta\right)}}} \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \frac{1}{\frac{i}{\color{blue}{\mathsf{fma}\left(\frac{1}{16}, i, \frac{1}{16} \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) - \frac{1}{8} \cdot \left(\alpha + \beta\right)}}} \]

      4. lower-unsound-/.f64 77.0

         \[\leadsto \frac{1}{\frac{i}{\mathsf{fma}\left(0.0625, i, 0.0625 \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) - \color{blue}{0.125 \cdot \left(\alpha + \beta\right)}}} \]

      5. lift--.f64 N/A

         \[\leadsto \frac{1}{\frac{i}{\mathsf{fma}\left(\frac{1}{16}, i, \frac{1}{16} \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) - \frac{1}{8} \cdot \color{blue}{\left(\alpha + \beta\right)}}} \]

      6. sub-flip N/A

         \[\leadsto \frac{1}{\frac{i}{\mathsf{fma}\left(\frac{1}{16}, i, \frac{1}{16} \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) + \left(\mathsf{neg}\left(\frac{1}{8} \cdot \left(\alpha + \beta\right)\right)\right)}} \]

      7. lift-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{i}{\left(\frac{1}{16} \cdot i + \frac{1}{16} \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) + \left(\mathsf{neg}\left(\frac{1}{8} \cdot \left(\alpha + \beta\right)\right)\right)}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{i}{\left(\frac{1}{16} \cdot i + \frac{1}{16} \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) + \left(\mathsf{neg}\left(\frac{1}{8} \cdot \left(\alpha + \beta\right)\right)\right)}} \]

      9. distribute-lft-out N/A

         \[\leadsto \frac{1}{\frac{i}{\frac{1}{16} \cdot \left(i + \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) + \left(\mathsf{neg}\left(\frac{1}{8} \cdot \left(\alpha + \beta\right)\right)\right)}} \]

      10. *-commutative N/A

          \[\leadsto \frac{1}{\frac{i}{\left(i + \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) \cdot \frac{1}{16} + \left(\mathsf{neg}\left(\frac{1}{8} \cdot \left(\alpha + \beta\right)\right)\right)}} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{i}{\mathsf{fma}\left(i + \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right), \frac{1}{16}, \mathsf{neg}\left(\frac{1}{8} \cdot \left(\alpha + \beta\right)\right)\right)}} \]

   9. Applied rewrites 77.0%

      \[\leadsto \frac{1}{\frac{i}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\beta + \alpha, 2, i\right), 0.0625, -0.125 \cdot \left(\beta + \alpha\right)\right)}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 82.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\alpha, \beta\right) + \mathsf{max}\left(\alpha, \beta\right)\\ t_1 := i \cdot \left(t\_0 + i\right)\\ t_2 := t\_0 + 2 \cdot i\\ t_3 := t\_2 \cdot t\_2\\ t_4 := \mathsf{max}\left(\alpha, \beta\right) + \mathsf{min}\left(\alpha, \beta\right)\\ t_5 := \mathsf{fma}\left(2, i, t\_4\right)\\ \mathbf{if}\;\frac{\frac{t\_1 \cdot \left(\mathsf{max}\left(\alpha, \beta\right) \cdot \mathsf{min}\left(\alpha, \beta\right) + t\_1\right)}{t\_3}}{t\_3 - 1} \leq \infty:\\ \;\;\;\;\frac{i \cdot \left(\mathsf{max}\left(\alpha, \beta\right) + i\right)}{{\left(\mathsf{max}\left(\alpha, \beta\right) + 2 \cdot i\right)}^{2} - 1} \cdot \frac{\left(t\_4 + i\right) \cdot i}{t\_5 \cdot t\_5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{i}{\mathsf{fma}\left(\mathsf{fma}\left(t\_4, 2, i\right), 0.0625, -0.125 \cdot t\_4\right)}}\\ \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (fmin alpha beta) (fmax alpha beta)))
        (t_1 (* i (+ t_0 i)))
        (t_2 (+ t_0 (* 2.0 i)))
        (t_3 (* t_2 t_2))
        (t_4 (+ (fmax alpha beta) (fmin alpha beta)))
        (t_5 (fma 2.0 i t_4)))
   (if (<=
        (/
         (/ (* t_1 (+ (* (fmax alpha beta) (fmin alpha beta)) t_1)) t_3)
         (- t_3 1.0))
        INFINITY)
     (*
      (/
       (* i (+ (fmax alpha beta) i))
       (- (pow (+ (fmax alpha beta) (* 2.0 i)) 2.0) 1.0))
      (/ (* (+ t_4 i) i) (* t_5 t_5)))
     (/ 1.0 (/ i (fma (fma t_4 2.0 i) 0.0625 (* -0.125 t_4)))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = fmin(alpha, beta) + fmax(alpha, beta);
	double t_1 = i * (t_0 + i);
	double t_2 = t_0 + (2.0 * i);
	double t_3 = t_2 * t_2;
	double t_4 = fmax(alpha, beta) + fmin(alpha, beta);
	double t_5 = fma(2.0, i, t_4);
	double tmp;
	if ((((t_1 * ((fmax(alpha, beta) * fmin(alpha, beta)) + t_1)) / t_3) / (t_3 - 1.0)) <= ((double) INFINITY)) {
		tmp = ((i * (fmax(alpha, beta) + i)) / (pow((fmax(alpha, beta) + (2.0 * i)), 2.0) - 1.0)) * (((t_4 + i) * i) / (t_5 * t_5));
	} else {
		tmp = 1.0 / (i / fma(fma(t_4, 2.0, i), 0.0625, (-0.125 * t_4)));
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(fmin(alpha, beta) + fmax(alpha, beta))
	t_1 = Float64(i * Float64(t_0 + i))
	t_2 = Float64(t_0 + Float64(2.0 * i))
	t_3 = Float64(t_2 * t_2)
	t_4 = Float64(fmax(alpha, beta) + fmin(alpha, beta))
	t_5 = fma(2.0, i, t_4)
	tmp = 0.0
	if (Float64(Float64(Float64(t_1 * Float64(Float64(fmax(alpha, beta) * fmin(alpha, beta)) + t_1)) / t_3) / Float64(t_3 - 1.0)) <= Inf)
		tmp = Float64(Float64(Float64(i * Float64(fmax(alpha, beta) + i)) / Float64((Float64(fmax(alpha, beta) + Float64(2.0 * i)) ^ 2.0) - 1.0)) * Float64(Float64(Float64(t_4 + i) * i) / Float64(t_5 * t_5)));
	else
		tmp = Float64(1.0 / Float64(i / fma(fma(t_4, 2.0, i), 0.0625, Float64(-0.125 * t_4))));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[Min[alpha, beta], $MachinePrecision] + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(i * N[(t$95$0 + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[Max[alpha, beta], $MachinePrecision] + N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(2.0 * i + t$95$4), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$1 * N[(N[(N[Max[alpha, beta], $MachinePrecision] * N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] / N[(t$95$3 - 1.0), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(i * N[(N[Max[alpha, beta], $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[(N[Max[alpha, beta], $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$4 + i), $MachinePrecision] * i), $MachinePrecision] / N[(t$95$5 * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(i / N[(N[(t$95$4 * 2.0 + i), $MachinePrecision] * 0.0625 + N[(-0.125 * t$95$4), $MachinePrecision]), $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))) < +inf.0

   1. Initial program 16.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

   3. Applied rewrites 37.8%

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

   4. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 34.6

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

   6. Applied rewrites 34.6%

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

   if +inf.0 < (/.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 16.6%

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

   2. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i}} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \color{blue}{\frac{1}{8} \cdot \frac{\alpha + \beta}{i}} \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \color{blue}{\frac{1}{8}} \cdot \frac{\alpha + \beta}{i} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \color{blue}{\frac{\alpha + \beta}{i}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{\color{blue}{i}} \]

      9. lower-+.f64 77.0

         \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   4. Applied rewrites 77.0%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]

   5. Taylor expanded in i around 0

      \[\leadsto \frac{\left(\frac{1}{16} \cdot i + \frac{1}{16} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right)\right) - \frac{1}{8} \cdot \left(\alpha + \beta\right)}{\color{blue}{i}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{16} \cdot i + \frac{1}{16} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right)\right) - \frac{1}{8} \cdot \left(\alpha + \beta\right)}{i} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{16} \cdot i + \frac{1}{16} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right)\right) - \frac{1}{8} \cdot \left(\alpha + \beta\right)}{i} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{16}, i, \frac{1}{16} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right)\right) - \frac{1}{8} \cdot \left(\alpha + \beta\right)}{i} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{16}, i, \frac{1}{16} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right)\right) - \frac{1}{8} \cdot \left(\alpha + \beta\right)}{i} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{16}, i, \frac{1}{16} \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) - \frac{1}{8} \cdot \left(\alpha + \beta\right)}{i} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{16}, i, \frac{1}{16} \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) - \frac{1}{8} \cdot \left(\alpha + \beta\right)}{i} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{16}, i, \frac{1}{16} \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) - \frac{1}{8} \cdot \left(\alpha + \beta\right)}{i} \]

      8. lower-+.f64 77.0

         \[\leadsto \frac{\mathsf{fma}\left(0.0625, i, 0.0625 \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) - 0.125 \cdot \left(\alpha + \beta\right)}{i} \]

   7. Applied rewrites 77.0%

      \[\leadsto \frac{\mathsf{fma}\left(0.0625, i, 0.0625 \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) - 0.125 \cdot \left(\alpha + \beta\right)}{\color{blue}{i}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{16}, i, \frac{1}{16} \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) - \frac{1}{8} \cdot \left(\alpha + \beta\right)}{i} \]

      2. div-flip N/A

         \[\leadsto \frac{1}{\frac{i}{\color{blue}{\mathsf{fma}\left(\frac{1}{16}, i, \frac{1}{16} \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) - \frac{1}{8} \cdot \left(\alpha + \beta\right)}}} \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \frac{1}{\frac{i}{\color{blue}{\mathsf{fma}\left(\frac{1}{16}, i, \frac{1}{16} \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) - \frac{1}{8} \cdot \left(\alpha + \beta\right)}}} \]

      4. lower-unsound-/.f64 77.0

         \[\leadsto \frac{1}{\frac{i}{\mathsf{fma}\left(0.0625, i, 0.0625 \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) - \color{blue}{0.125 \cdot \left(\alpha + \beta\right)}}} \]

      5. lift--.f64 N/A

         \[\leadsto \frac{1}{\frac{i}{\mathsf{fma}\left(\frac{1}{16}, i, \frac{1}{16} \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) - \frac{1}{8} \cdot \color{blue}{\left(\alpha + \beta\right)}}} \]

      6. sub-flip N/A

         \[\leadsto \frac{1}{\frac{i}{\mathsf{fma}\left(\frac{1}{16}, i, \frac{1}{16} \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) + \left(\mathsf{neg}\left(\frac{1}{8} \cdot \left(\alpha + \beta\right)\right)\right)}} \]

      7. lift-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{i}{\left(\frac{1}{16} \cdot i + \frac{1}{16} \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) + \left(\mathsf{neg}\left(\frac{1}{8} \cdot \left(\alpha + \beta\right)\right)\right)}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{i}{\left(\frac{1}{16} \cdot i + \frac{1}{16} \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) + \left(\mathsf{neg}\left(\frac{1}{8} \cdot \left(\alpha + \beta\right)\right)\right)}} \]

      9. distribute-lft-out N/A

         \[\leadsto \frac{1}{\frac{i}{\frac{1}{16} \cdot \left(i + \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) + \left(\mathsf{neg}\left(\frac{1}{8} \cdot \left(\alpha + \beta\right)\right)\right)}} \]

      10. *-commutative N/A

          \[\leadsto \frac{1}{\frac{i}{\left(i + \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) \cdot \frac{1}{16} + \left(\mathsf{neg}\left(\frac{1}{8} \cdot \left(\alpha + \beta\right)\right)\right)}} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{i}{\mathsf{fma}\left(i + \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right), \frac{1}{16}, \mathsf{neg}\left(\frac{1}{8} \cdot \left(\alpha + \beta\right)\right)\right)}} \]

   9. Applied rewrites 77.0%

      \[\leadsto \frac{1}{\frac{i}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\beta + \alpha, 2, i\right), 0.0625, -0.125 \cdot \left(\beta + \alpha\right)\right)}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 79.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{max}\left(\alpha, \beta\right) \leq 3.8 \cdot 10^{+133}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i \cdot \left(\mathsf{min}\left(\alpha, \beta\right) + i\right)}{\mathsf{max}\left(\alpha, \beta\right)}}{\mathsf{fma}\left(2, i, \mathsf{max}\left(\alpha, \beta\right) + \mathsf{min}\left(\alpha, \beta\right)\right) - 1}\\ \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= (fmax alpha beta) 3.8e+133)
   0.0625
   (/
    (/ (* i (+ (fmin alpha beta) i)) (fmax alpha beta))
    (- (fma 2.0 i (+ (fmax alpha beta) (fmin alpha beta))) 1.0))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (fmax(alpha, beta) <= 3.8e+133) {
		tmp = 0.0625;
	} else {
		tmp = ((i * (fmin(alpha, beta) + i)) / fmax(alpha, beta)) / (fma(2.0, i, (fmax(alpha, beta) + fmin(alpha, beta))) - 1.0);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (fmax(alpha, beta) <= 3.8e+133)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(i * Float64(fmin(alpha, beta) + i)) / fmax(alpha, beta)) / Float64(fma(2.0, i, Float64(fmax(alpha, beta) + fmin(alpha, beta))) - 1.0));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[N[Max[alpha, beta], $MachinePrecision], 3.8e+133], 0.0625, N[(N[(N[(i * N[(N[Min[alpha, beta], $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision] / N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 * i + N[(N[Max[alpha, beta], $MachinePrecision] + N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.8000000000000002e133

   1. Initial program 16.6%

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

   2. Taylor expanded in i around inf

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

   4. Applied rewrites 70.4%

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

   if 3.8000000000000002e133 < beta

   1. Initial program 16.6%

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

   2. Taylor expanded in alpha around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 13.5

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

   4. Applied rewrites 13.5%

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

   5. Taylor expanded in beta around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 10.0

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

   7. Applied rewrites 10.0%

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

   9. Applied rewrites 17.2%

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

   10. Taylor expanded in beta around inf

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-+.f64 12.6

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

   12. Applied rewrites 12.6%

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

Alternative 6: 77.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\alpha, \beta\right) + \mathsf{max}\left(\alpha, \beta\right)\\ t_1 := t\_0 + 2 \cdot i\\ t_2 := t\_1 \cdot t\_1\\ t_3 := i \cdot \left(t\_0 + i\right)\\ \mathbf{if}\;\frac{\frac{t\_3 \cdot \left(\mathsf{max}\left(\alpha, \beta\right) \cdot \mathsf{min}\left(\alpha, \beta\right) + t\_3\right)}{t\_2}}{t\_2 - 1} \leq 0.004:\\ \;\;\;\;\frac{i \cdot i}{\mathsf{max}\left(\alpha, \beta\right) \cdot \mathsf{max}\left(\alpha, \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.0625, i, 0.125 \cdot \mathsf{max}\left(\alpha, \beta\right)\right) - 0.125 \cdot t\_0}{i}\\ \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (fmin alpha beta) (fmax alpha beta)))
        (t_1 (+ t_0 (* 2.0 i)))
        (t_2 (* t_1 t_1))
        (t_3 (* i (+ t_0 i))))
   (if (<=
        (/
         (/ (* t_3 (+ (* (fmax alpha beta) (fmin alpha beta)) t_3)) t_2)
         (- t_2 1.0))
        0.004)
     (/ (* i i) (* (fmax alpha beta) (fmax alpha beta)))
     (/ (- (fma 0.0625 i (* 0.125 (fmax alpha beta))) (* 0.125 t_0)) i))))

C

double code(double alpha, double beta, double i) {
	double t_0 = fmin(alpha, beta) + fmax(alpha, beta);
	double t_1 = t_0 + (2.0 * i);
	double t_2 = t_1 * t_1;
	double t_3 = i * (t_0 + i);
	double tmp;
	if ((((t_3 * ((fmax(alpha, beta) * fmin(alpha, beta)) + t_3)) / t_2) / (t_2 - 1.0)) <= 0.004) {
		tmp = (i * i) / (fmax(alpha, beta) * fmax(alpha, beta));
	} else {
		tmp = (fma(0.0625, i, (0.125 * fmax(alpha, beta))) - (0.125 * t_0)) / i;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(fmin(alpha, beta) + fmax(alpha, beta))
	t_1 = Float64(t_0 + Float64(2.0 * i))
	t_2 = Float64(t_1 * t_1)
	t_3 = Float64(i * Float64(t_0 + i))
	tmp = 0.0
	if (Float64(Float64(Float64(t_3 * Float64(Float64(fmax(alpha, beta) * fmin(alpha, beta)) + t_3)) / t_2) / Float64(t_2 - 1.0)) <= 0.004)
		tmp = Float64(Float64(i * i) / Float64(fmax(alpha, beta) * fmax(alpha, beta)));
	else
		tmp = Float64(Float64(fma(0.0625, i, Float64(0.125 * fmax(alpha, beta))) - Float64(0.125 * t_0)) / i);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[Min[alpha, beta], $MachinePrecision] + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(i * N[(t$95$0 + i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$3 * N[(N[(N[Max[alpha, beta], $MachinePrecision] * N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] / N[(t$95$2 - 1.0), $MachinePrecision]), $MachinePrecision], 0.004], N[(N[(i * i), $MachinePrecision] / N[(N[Max[alpha, beta], $MachinePrecision] * N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.0625 * i + N[(0.125 * N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 * t$95$0), $MachinePrecision]), $MachinePrecision] / i), $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 16.6%

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

   2. Taylor expanded in alpha around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 13.5

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

   4. Applied rewrites 13.5%

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

   5. Taylor expanded in beta around inf

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

      1. lower-pow.f64 4.7

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

   7. Applied rewrites 4.7%

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

   9. Applied rewrites 4.7%

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

   10. Taylor expanded in beta around 0

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

   12. Applied rewrites 9.0%

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

   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 16.6%

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

   2. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i}} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \color{blue}{\frac{1}{8} \cdot \frac{\alpha + \beta}{i}} \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \color{blue}{\frac{1}{8}} \cdot \frac{\alpha + \beta}{i} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \color{blue}{\frac{\alpha + \beta}{i}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{\color{blue}{i}} \]

      9. lower-+.f64 77.0

         \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   4. Applied rewrites 77.0%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]

   5. Taylor expanded in i around 0

      \[\leadsto \frac{\left(\frac{1}{16} \cdot i + \frac{1}{16} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right)\right) - \frac{1}{8} \cdot \left(\alpha + \beta\right)}{\color{blue}{i}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{16} \cdot i + \frac{1}{16} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right)\right) - \frac{1}{8} \cdot \left(\alpha + \beta\right)}{i} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{16} \cdot i + \frac{1}{16} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right)\right) - \frac{1}{8} \cdot \left(\alpha + \beta\right)}{i} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{16}, i, \frac{1}{16} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right)\right) - \frac{1}{8} \cdot \left(\alpha + \beta\right)}{i} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{16}, i, \frac{1}{16} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right)\right) - \frac{1}{8} \cdot \left(\alpha + \beta\right)}{i} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{16}, i, \frac{1}{16} \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) - \frac{1}{8} \cdot \left(\alpha + \beta\right)}{i} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{16}, i, \frac{1}{16} \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) - \frac{1}{8} \cdot \left(\alpha + \beta\right)}{i} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{16}, i, \frac{1}{16} \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) - \frac{1}{8} \cdot \left(\alpha + \beta\right)}{i} \]

      8. lower-+.f64 77.0

         \[\leadsto \frac{\mathsf{fma}\left(0.0625, i, 0.0625 \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) - 0.125 \cdot \left(\alpha + \beta\right)}{i} \]

   7. Applied rewrites 77.0%

      \[\leadsto \frac{\mathsf{fma}\left(0.0625, i, 0.0625 \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) - 0.125 \cdot \left(\alpha + \beta\right)}{\color{blue}{i}} \]

   8. Taylor expanded in alpha around 0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{16}, i, \frac{1}{8} \cdot \beta\right) - \frac{1}{8} \cdot \left(\alpha + \beta\right)}{i} \]
   9. Step-by-step derivation

      1. lower-*.f64 73.0

         \[\leadsto \frac{\mathsf{fma}\left(0.0625, i, 0.125 \cdot \beta\right) - 0.125 \cdot \left(\alpha + \beta\right)}{i} \]

   10. Applied rewrites 73.0%

       \[\leadsto \frac{\mathsf{fma}\left(0.0625, i, 0.125 \cdot \beta\right) - 0.125 \cdot \left(\alpha + \beta\right)}{i} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 74.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\alpha, \beta\right) + \mathsf{max}\left(\alpha, \beta\right)\\ \mathbf{if}\;\mathsf{max}\left(\alpha, \beta\right) \leq 1.5 \cdot 10^{+243}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125, t\_0, 0.125 \cdot t\_0\right)}{i}\\ \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (fmin alpha beta) (fmax alpha beta))))
   (if (<= (fmax alpha beta) 1.5e+243)
     0.0625
     (/ (fma -0.125 t_0 (* 0.125 t_0)) i))))

C

double code(double alpha, double beta, double i) {
	double t_0 = fmin(alpha, beta) + fmax(alpha, beta);
	double tmp;
	if (fmax(alpha, beta) <= 1.5e+243) {
		tmp = 0.0625;
	} else {
		tmp = fma(-0.125, t_0, (0.125 * t_0)) / i;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(fmin(alpha, beta) + fmax(alpha, beta))
	tmp = 0.0
	if (fmax(alpha, beta) <= 1.5e+243)
		tmp = 0.0625;
	else
		tmp = Float64(fma(-0.125, t_0, Float64(0.125 * t_0)) / i);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[Min[alpha, beta], $MachinePrecision] + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Max[alpha, beta], $MachinePrecision], 1.5e+243], 0.0625, N[(N[(-0.125 * t$95$0 + N[(0.125 * t$95$0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.49999999999999992e243

   1. Initial program 16.6%

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

   2. Taylor expanded in i around inf

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

   4. Applied rewrites 70.4%

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

   if 1.49999999999999992e243 < beta

   1. Initial program 16.6%

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

   2. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i}} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \color{blue}{\frac{1}{8} \cdot \frac{\alpha + \beta}{i}} \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \color{blue}{\frac{1}{8}} \cdot \frac{\alpha + \beta}{i} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \color{blue}{\frac{\alpha + \beta}{i}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{\color{blue}{i}} \]

      9. lower-+.f64 77.0

         \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   4. Applied rewrites 77.0%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]

   5. Taylor expanded in i around 0

      \[\leadsto \frac{\left(\frac{1}{16} \cdot i + \frac{1}{16} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right)\right) - \frac{1}{8} \cdot \left(\alpha + \beta\right)}{\color{blue}{i}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{16} \cdot i + \frac{1}{16} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right)\right) - \frac{1}{8} \cdot \left(\alpha + \beta\right)}{i} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{16} \cdot i + \frac{1}{16} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right)\right) - \frac{1}{8} \cdot \left(\alpha + \beta\right)}{i} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{16}, i, \frac{1}{16} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right)\right) - \frac{1}{8} \cdot \left(\alpha + \beta\right)}{i} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{16}, i, \frac{1}{16} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right)\right) - \frac{1}{8} \cdot \left(\alpha + \beta\right)}{i} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{16}, i, \frac{1}{16} \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) - \frac{1}{8} \cdot \left(\alpha + \beta\right)}{i} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{16}, i, \frac{1}{16} \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) - \frac{1}{8} \cdot \left(\alpha + \beta\right)}{i} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{16}, i, \frac{1}{16} \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) - \frac{1}{8} \cdot \left(\alpha + \beta\right)}{i} \]

      8. lower-+.f64 77.0

         \[\leadsto \frac{\mathsf{fma}\left(0.0625, i, 0.0625 \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) - 0.125 \cdot \left(\alpha + \beta\right)}{i} \]

   7. Applied rewrites 77.0%

      \[\leadsto \frac{\mathsf{fma}\left(0.0625, i, 0.0625 \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) - 0.125 \cdot \left(\alpha + \beta\right)}{\color{blue}{i}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{16}, i, \frac{1}{16} \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) - \frac{1}{8} \cdot \left(\alpha + \beta\right)}{i} \]

      2. div-flip N/A

         \[\leadsto \frac{1}{\frac{i}{\color{blue}{\mathsf{fma}\left(\frac{1}{16}, i, \frac{1}{16} \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) - \frac{1}{8} \cdot \left(\alpha + \beta\right)}}} \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \frac{1}{\frac{i}{\color{blue}{\mathsf{fma}\left(\frac{1}{16}, i, \frac{1}{16} \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) - \frac{1}{8} \cdot \left(\alpha + \beta\right)}}} \]

      4. lower-unsound-/.f64 77.0

         \[\leadsto \frac{1}{\frac{i}{\mathsf{fma}\left(0.0625, i, 0.0625 \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) - \color{blue}{0.125 \cdot \left(\alpha + \beta\right)}}} \]

      5. lift--.f64 N/A

         \[\leadsto \frac{1}{\frac{i}{\mathsf{fma}\left(\frac{1}{16}, i, \frac{1}{16} \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) - \frac{1}{8} \cdot \color{blue}{\left(\alpha + \beta\right)}}} \]

      6. sub-flip N/A

         \[\leadsto \frac{1}{\frac{i}{\mathsf{fma}\left(\frac{1}{16}, i, \frac{1}{16} \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) + \left(\mathsf{neg}\left(\frac{1}{8} \cdot \left(\alpha + \beta\right)\right)\right)}} \]

      7. lift-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{i}{\left(\frac{1}{16} \cdot i + \frac{1}{16} \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) + \left(\mathsf{neg}\left(\frac{1}{8} \cdot \left(\alpha + \beta\right)\right)\right)}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{i}{\left(\frac{1}{16} \cdot i + \frac{1}{16} \cdot \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) + \left(\mathsf{neg}\left(\frac{1}{8} \cdot \left(\alpha + \beta\right)\right)\right)}} \]

      9. distribute-lft-out N/A

         \[\leadsto \frac{1}{\frac{i}{\frac{1}{16} \cdot \left(i + \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) + \left(\mathsf{neg}\left(\frac{1}{8} \cdot \left(\alpha + \beta\right)\right)\right)}} \]

      10. *-commutative N/A

          \[\leadsto \frac{1}{\frac{i}{\left(i + \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)\right) \cdot \frac{1}{16} + \left(\mathsf{neg}\left(\frac{1}{8} \cdot \left(\alpha + \beta\right)\right)\right)}} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{i}{\mathsf{fma}\left(i + \mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right), \frac{1}{16}, \mathsf{neg}\left(\frac{1}{8} \cdot \left(\alpha + \beta\right)\right)\right)}} \]

   9. Applied rewrites 77.0%

      \[\leadsto \frac{1}{\frac{i}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\beta + \alpha, 2, i\right), 0.0625, -0.125 \cdot \left(\beta + \alpha\right)\right)}}} \]

   10. Taylor expanded in i around 0

       \[\leadsto \frac{\frac{-1}{8} \cdot \left(\alpha + \beta\right) + \frac{1}{8} \cdot \left(\alpha + \beta\right)}{i} \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{\frac{-1}{8} \cdot \left(\alpha + \beta\right) + \frac{1}{8} \cdot \left(\alpha + \beta\right)}{i} \]

       2. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \alpha + \beta, \frac{1}{8} \cdot \left(\alpha + \beta\right)\right)}{i} \]

       3. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \alpha + \beta, \frac{1}{8} \cdot \left(\alpha + \beta\right)\right)}{i} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \alpha + \beta, \frac{1}{8} \cdot \left(\alpha + \beta\right)\right)}{i} \]

       5. lower-+.f64 10.1

          \[\leadsto \frac{\mathsf{fma}\left(-0.125, \alpha + \beta, 0.125 \cdot \left(\alpha + \beta\right)\right)}{i} \]

   12. Applied rewrites 10.1%

       \[\leadsto \frac{\mathsf{fma}\left(-0.125, \alpha + \beta, 0.125 \cdot \left(\alpha + \beta\right)\right)}{i} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 73.7% accurate, 3.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{max}\left(\alpha, \beta\right) \leq 1.5 \cdot 10^{+243}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot i}{\mathsf{max}\left(\alpha, \beta\right) \cdot \mathsf{max}\left(\alpha, \beta\right)}\\ \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= (fmax alpha beta) 1.5e+243)
   0.0625
   (/ (* i i) (* (fmax alpha beta) (fmax alpha beta)))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (fmax(alpha, beta) <= 1.5e+243) {
		tmp = 0.0625;
	} else {
		tmp = (i * i) / (fmax(alpha, beta) * fmax(alpha, beta));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(alpha, beta, i)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (fmax(alpha, beta) <= 1.5d+243) then
        tmp = 0.0625d0
    else
        tmp = (i * i) / (fmax(alpha, beta) * fmax(alpha, beta))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (fmax(alpha, beta) <= 1.5e+243) {
		tmp = 0.0625;
	} else {
		tmp = (i * i) / (fmax(alpha, beta) * fmax(alpha, beta));
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if fmax(alpha, beta) <= 1.5e+243:
		tmp = 0.0625
	else:
		tmp = (i * i) / (fmax(alpha, beta) * fmax(alpha, beta))
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (fmax(alpha, beta) <= 1.5e+243)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i * i) / Float64(fmax(alpha, beta) * fmax(alpha, beta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (max(alpha, beta) <= 1.5e+243)
		tmp = 0.0625;
	else
		tmp = (i * i) / (max(alpha, beta) * max(alpha, beta));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[N[Max[alpha, beta], $MachinePrecision], 1.5e+243], 0.0625, N[(N[(i * i), $MachinePrecision] / N[(N[Max[alpha, beta], $MachinePrecision] * N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.49999999999999992e243

   1. Initial program 16.6%

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

   2. Taylor expanded in i around inf

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

   4. Applied rewrites 70.4%

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

   if 1.49999999999999992e243 < beta

   1. Initial program 16.6%

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

   2. Taylor expanded in alpha around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 13.5

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

   4. Applied rewrites 13.5%

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

   5. Taylor expanded in beta around inf

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

      1. lower-pow.f64 4.7

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

   7. Applied rewrites 4.7%

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

   9. Applied rewrites 4.7%

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

   10. Taylor expanded in beta around 0

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

   12. Applied rewrites 9.0%

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

Alternative 9: 70.4% accurate, 75.4× speedup

Math

\[0.0625 \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(alpha, beta, i)
use fmin_fmax_functions
    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 16.6%

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

2. Taylor expanded in i around inf

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

4. Applied rewrites 70.4%

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

Reproduce

herbie shell --seed 2025173 
(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)))