Octave 3.8, jcobi/4

Percentage Accurate: 16.1% → 83.6%

Time: 4.6s

Alternatives: 6

Speedup: 75.4×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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: 16.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 83.6% accurate, 0.5× speedup

Math

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

FPCore

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

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 = t_1 - 1.0;
	double t_3 = i * ((alpha + beta) + i);
	double t_4 = (beta + alpha) + i;
	double t_5 = fma(2.0, i, (beta + alpha));
	double tmp;
	if ((((t_3 * ((beta * alpha) + t_3)) / t_1) / t_2) <= ((double) INFINITY)) {
		tmp = (i * (t_4 * (fma(t_4, i, (beta * alpha)) / (t_5 * t_5)))) / t_2;
	} else {
		tmp = (0.0625 + (0.0625 * (fma(2.0, alpha, (2.0 * beta)) / i))) - (0.125 * ((alpha + beta) / i));
	}
	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(t_1 - 1.0)
	t_3 = Float64(i * Float64(Float64(alpha + beta) + i))
	t_4 = Float64(Float64(beta + alpha) + i)
	t_5 = fma(2.0, i, Float64(beta + alpha))
	tmp = 0.0
	if (Float64(Float64(Float64(t_3 * Float64(Float64(beta * alpha) + t_3)) / t_1) / t_2) <= Inf)
		tmp = Float64(Float64(i * Float64(t_4 * Float64(fma(t_4, i, Float64(beta * alpha)) / Float64(t_5 * t_5)))) / t_2);
	else
		tmp = Float64(Float64(0.0625 + Float64(0.0625 * Float64(fma(2.0, alpha, Float64(2.0 * beta)) / i))) - Float64(0.125 * Float64(Float64(alpha + beta) / i)));
	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[(t$95$1 - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(beta + alpha), $MachinePrecision] + i), $MachinePrecision]}, Block[{t$95$5 = N[(2.0 * i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$3 * N[(N[(beta * alpha), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision], Infinity], N[(N[(i * N[(t$95$4 * N[(N[(t$95$4 * i + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] / N[(t$95$5 * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(0.0625 + N[(0.0625 * N[(N[(2.0 * alpha + N[(2.0 * beta), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 * N[(N[(alpha + beta), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64))) < +inf.0

   1. Initial program 16.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 37.3%

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

   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 16.1%

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

   2. 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 77.1

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

   4. Applied rewrites 77.1%

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

Alternative 2: 80.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(i * Float64(Float64(alpha + beta) + i))
	t_2 = Float64(t_0 * t_0)
	t_3 = Float64(t_2 - 1.0)
	t_4 = fma(2.0, i, Float64(beta + alpha))
	tmp = 0.0
	if (Float64(Float64(Float64(t_1 * Float64(Float64(beta * alpha) + t_1)) / t_2) / t_3) <= Inf)
		tmp = Float64(Float64(i * Float64(Float64(Float64(beta + alpha) + i) * Float64(Float64(i * Float64(beta + i)) / Float64(t_4 * t_4)))) / t_3);
	else
		tmp = Float64(Float64(0.0625 + Float64(0.0625 * Float64(fma(2.0, alpha, Float64(2.0 * beta)) / i))) - Float64(0.125 * Float64(Float64(alpha + beta) / i)));
	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[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - 1.0), $MachinePrecision]}, Block[{t$95$4 = N[(2.0 * i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$1 * N[(N[(beta * alpha), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] / t$95$3), $MachinePrecision], Infinity], N[(N[(i * N[(N[(N[(beta + alpha), $MachinePrecision] + i), $MachinePrecision] * N[(N[(i * N[(beta + i), $MachinePrecision]), $MachinePrecision] / N[(t$95$4 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], N[(N[(0.0625 + N[(0.0625 * N[(N[(2.0 * alpha + N[(2.0 * beta), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 * N[(N[(alpha + beta), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64))) < +inf.0

   1. Initial program 16.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 37.3%

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

   4. Taylor expanded in alpha around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 36.0

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

   6. Applied rewrites 36.0%

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

   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 16.1%

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

   2. 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 77.1

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

   4. Applied rewrites 77.1%

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

Alternative 3: 78.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, i, \beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 2.9 \cdot 10^{+130}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(0.0625, i, \left(\beta + \alpha\right) \cdot -0.125\right) + 0.125 \cdot \beta\right) + 0.125 \cdot \alpha}{i}\\ \mathbf{else}:\\ \;\;\;\;i \cdot \frac{\frac{-1 \cdot \mathsf{fma}\left(-1, \alpha, -1 \cdot i\right)}{t\_0 - -1}}{t\_0 - 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma 2.0 i (+ beta alpha))))
   (if (<= beta 2.9e+130)
     (/
      (+
       (+ (fma 0.0625 i (* (+ beta alpha) -0.125)) (* 0.125 beta))
       (* 0.125 alpha))
      i)
     (*
      i
      (/ (/ (* -1.0 (fma -1.0 alpha (* -1.0 i))) (- t_0 -1.0)) (- t_0 1.0))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = fma(2.0, i, (beta + alpha));
	double tmp;
	if (beta <= 2.9e+130) {
		tmp = ((fma(0.0625, i, ((beta + alpha) * -0.125)) + (0.125 * beta)) + (0.125 * alpha)) / i;
	} else {
		tmp = i * (((-1.0 * fma(-1.0, alpha, (-1.0 * i))) / (t_0 - -1.0)) / (t_0 - 1.0));
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = fma(2.0, i, Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 2.9e+130)
		tmp = Float64(Float64(Float64(fma(0.0625, i, Float64(Float64(beta + alpha) * -0.125)) + Float64(0.125 * beta)) + Float64(0.125 * alpha)) / i);
	else
		tmp = Float64(i * Float64(Float64(Float64(-1.0 * fma(-1.0, alpha, Float64(-1.0 * i))) / Float64(t_0 - -1.0)) / Float64(t_0 - 1.0)));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(2.0 * i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2.9e+130], N[(N[(N[(N[(0.0625 * i + N[(N[(beta + alpha), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] + N[(0.125 * beta), $MachinePrecision]), $MachinePrecision] + N[(0.125 * alpha), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], N[(i * N[(N[(N[(-1.0 * N[(-1.0 * alpha + N[(-1.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - -1.0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.8999999999999999e130

   1. Initial program 16.1%

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

   2. 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 77.1

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

   4. Applied rewrites 77.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. add-to-fraction N/A

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

      6. lower-/.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-lft-out N/A

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

      10. lift-+.f64 N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 77.1

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 77.1

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

   6. Applied rewrites 77.1%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. distribute-neg-frac N/A

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

      13. lift-/.f64 N/A

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

      14. div-add-rev N/A

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

      15. lower-/.f64 N/A

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

   8. Applied rewrites 77.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-+r+ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. distribute-rgt-in N/A

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

      7. associate-+r+ N/A

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

      8. *-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 74.6

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

   10. Applied rewrites 74.6%

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

   if 2.8999999999999999e130 < beta

   1. Initial program 16.1%

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

      1. 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. lift-*.f64 N/A

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

   3. Applied rewrites 23.4%

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

   4. Taylor expanded in alpha around inf

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

   6. Applied rewrites 9.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-fma.f64 N/A

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

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

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   8. Applied rewrites 18.4%

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

   9. Taylor expanded in beta around -inf

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

       1. lower-*.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-*.f64 23.8

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

   11. Applied rewrites 23.8%

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

Alternative 4: 74.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.1 \cdot 10^{+203}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125, \beta, \mathsf{fma}\left(0.0625, i, 0.125 \cdot \beta\right)\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;i \cdot \frac{\frac{\alpha + i}{\beta}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 6.1e+203)
   (/ (fma -0.125 beta (fma 0.0625 i (* 0.125 beta))) i)
   (* i (/ (/ (+ alpha i) beta) (- (fma 2.0 i (+ beta alpha)) 1.0)))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 6.1e+203) {
		tmp = fma(-0.125, beta, fma(0.0625, i, (0.125 * beta))) / i;
	} else {
		tmp = i * (((alpha + i) / beta) / (fma(2.0, i, (beta + alpha)) - 1.0));
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 6.1e+203)
		tmp = Float64(fma(-0.125, beta, fma(0.0625, i, Float64(0.125 * beta))) / i);
	else
		tmp = Float64(i * Float64(Float64(Float64(alpha + i) / beta) / Float64(fma(2.0, i, Float64(beta + alpha)) - 1.0)));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 6.1e+203], N[(N[(-0.125 * beta + N[(0.0625 * i + N[(0.125 * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], N[(i * N[(N[(N[(alpha + i), $MachinePrecision] / beta), $MachinePrecision] / N[(N[(2.0 * i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.10000000000000014e203

   1. Initial program 16.1%

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

   2. 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 77.1

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

   4. Applied rewrites 77.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. add-to-fraction N/A

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

      6. lower-/.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-lft-out N/A

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

      10. lift-+.f64 N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 77.1

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 77.1

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

   6. Applied rewrites 77.1%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. distribute-neg-frac N/A

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

      13. lift-/.f64 N/A

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

      14. div-add-rev N/A

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

      15. lower-/.f64 N/A

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

   8. Applied rewrites 77.1%

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

   9. Taylor expanded in alpha around 0

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

   11. Applied rewrites 73.4%

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

   12. Taylor expanded in alpha around 0

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

   14. Applied rewrites 74.2%

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

   if 6.10000000000000014e203 < beta

   1. Initial program 16.1%

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

      1. 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. lift-*.f64 N/A

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

   3. Applied rewrites 23.4%

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

   4. Taylor expanded in alpha around inf

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

   6. Applied rewrites 9.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-fma.f64 N/A

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

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

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   8. Applied rewrites 18.4%

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

   9. Taylor expanded in beta around inf

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

       1. lower-/.f64 N/A

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

       2. lower-+.f64 13.2

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

   11. Applied rewrites 13.2%

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

Alternative 5: 74.2% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-0.125, \beta, \mathsf{fma}\left(0.0625, i, 0.125 \cdot \beta\right)\right)}{i} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (/ (fma -0.125 beta (fma 0.0625 i (* 0.125 beta))) i))

C

double code(double alpha, double beta, double i) {
	return fma(-0.125, beta, fma(0.0625, i, (0.125 * beta))) / i;
}

Julia

function code(alpha, beta, i)
	return Float64(fma(-0.125, beta, fma(0.0625, i, Float64(0.125 * beta))) / i)
end

Wolfram

code[alpha_, beta_, i_] := N[(N[(-0.125 * beta + N[(0.0625 * i + N[(0.125 * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]

Derivation

1. Initial program 16.1%

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

2. 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 77.1

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

4. Applied rewrites 77.1%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*r/ N/A

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

   5. add-to-fraction N/A

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

   6. lower-/.f64 N/A

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

   7. lift-fma.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. distribute-lft-out N/A

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

   10. lift-+.f64 N/A

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

   11. associate-*r* N/A

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

   12. metadata-eval N/A

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

   13. lower-fma.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 77.1

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lift-+.f64 77.1

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

6. Applied rewrites 77.1%

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

   1. lift--.f64 N/A

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

   2. sub-flip N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*r/ N/A

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

   8. +-commutative N/A

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

   9. lift-+.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. distribute-neg-frac N/A

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

   13. lift-/.f64 N/A

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

   14. div-add-rev N/A

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

   15. lower-/.f64 N/A

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

8. Applied rewrites 77.1%

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

9. Taylor expanded in alpha around 0

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

11. Applied rewrites 73.4%

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

12. Taylor expanded in alpha around 0

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

14. Applied rewrites 74.2%

    \[\leadsto \frac{\mathsf{fma}\left(-0.125, \beta, \mathsf{fma}\left(0.0625, i, 0.125 \cdot \beta\right)\right)}{i} \]
15. Add Preprocessing

Alternative 6: 70.8% accurate, 75.4× speedup

Math

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

FPCore

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

C

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

Fortran

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 16.1%

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

2. Taylor expanded in i around inf

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

4. Applied rewrites 70.8%

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

Reproduce

herbie shell --seed 2025156 
(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)))