Octave 3.8, jcobi/4

Percentage Accurate: 16.5% → 99.7%

Time: 6.5s

Alternatives: 11

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: 16.5% 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: 99.7% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 16.5%

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

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

   3. lift-*.f64 N/A

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

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

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

3. Applied rewrites 37.7%

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

   1. lift-/.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. div-add N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. times-frac N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. pow2 N/A

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

5. Applied rewrites 93.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. associate-*r/ N/A

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

   6. lower-/.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 99.7

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-+.f64 99.7

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lift-+.f64 99.7

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

7. Applied rewrites 99.7%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-+.f64 N/A

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

   7. associate-/l* N/A

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-+.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lift-+.f64 N/A

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

   15. lower-/.f64 99.7

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lift-+.f64 99.7

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

9. Applied rewrites 99.7%

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

Alternative 2: 85.2% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (*
  (/ (* (+ beta i) (/ i (fma 2.0 i beta))) (- (fma 2.0 i beta) -1.0))
  (/
   (fma i (/ (+ beta i) (fma 2.0 i beta)) (* alpha (/ beta (fma 2.0 i beta))))
   (- (fma 2.0 i beta) 1.0))))

C

double code(double alpha, double beta, double i) {
	return (((beta + i) * (i / fma(2.0, i, beta))) / (fma(2.0, i, beta) - -1.0)) * (fma(i, ((beta + i) / fma(2.0, i, beta)), (alpha * (beta / fma(2.0, i, beta)))) / (fma(2.0, i, beta) - 1.0));
}

Julia

function code(alpha, beta, i)
	return Float64(Float64(Float64(Float64(beta + i) * Float64(i / fma(2.0, i, beta))) / Float64(fma(2.0, i, beta) - -1.0)) * Float64(fma(i, Float64(Float64(beta + i) / fma(2.0, i, beta)), Float64(alpha * Float64(beta / fma(2.0, i, beta)))) / Float64(fma(2.0, i, beta) - 1.0)))
end

Wolfram

code[alpha_, beta_, i_] := N[(N[(N[(N[(beta + i), $MachinePrecision] * N[(i / N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 * i + beta), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(i * N[(N[(beta + i), $MachinePrecision] / N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision] + N[(alpha * N[(beta / N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 * i + beta), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 16.5%

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

2. Taylor expanded in alpha around 0

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

4. Applied rewrites 15.7%

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

5. Taylor expanded in alpha around 0

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

7. Applied rewrites 16.7%

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

8. Taylor expanded in alpha around 0

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

10. Applied rewrites 16.8%

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

11. Taylor expanded in alpha around 0

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

13. Applied rewrites 16.9%

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

14. Taylor expanded in alpha around 0

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

16. Applied rewrites 15.6%

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

17. Taylor expanded in alpha around 0

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

19. Applied rewrites 15.2%

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

    1. lift-/.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. times-frac N/A

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

21. Applied rewrites 36.6%

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

    1. lift-/.f64 N/A

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

    2. lift-fma.f64 N/A

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

    3. div-add N/A

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

    4. *-commutative N/A

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

    5. associate-/l* N/A

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

    6. lower-fma.f64 N/A

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

    7. lower-/.f64 N/A

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

    8. lift-*.f64 N/A

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

    9. *-commutative N/A

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

    10. associate-/l* N/A

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

    11. lower-*.f64 N/A

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

    12. lower-/.f64 85.2

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

23. Applied rewrites 85.2%

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

Alternative 3: 78.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, i, \beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 1.7 \cdot 10^{+172}:\\ \;\;\;\;\frac{\left(\beta + \alpha\right) + i}{t\_0 - 1} \cdot \frac{i \cdot 0.25}{t\_0 - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\beta + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta\right)}}{\mathsf{fma}\left(2, i, \beta\right) - -1} \cdot \frac{-1 \cdot \mathsf{fma}\left(-1, \alpha, -1 \cdot i\right)}{\mathsf{fma}\left(2, i, \beta\right) - 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma 2.0 i (+ beta alpha))))
   (if (<= beta 1.7e+172)
     (* (/ (+ (+ beta alpha) i) (- t_0 1.0)) (/ (* i 0.25) (- t_0 -1.0)))
     (*
      (/ (* (+ beta i) (/ i (fma 2.0 i beta))) (- (fma 2.0 i beta) -1.0))
      (/ (* -1.0 (fma -1.0 alpha (* -1.0 i))) (- (fma 2.0 i beta) 1.0))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = fma(2.0, i, (beta + alpha));
	double tmp;
	if (beta <= 1.7e+172) {
		tmp = (((beta + alpha) + i) / (t_0 - 1.0)) * ((i * 0.25) / (t_0 - -1.0));
	} else {
		tmp = (((beta + i) * (i / fma(2.0, i, beta))) / (fma(2.0, i, beta) - -1.0)) * ((-1.0 * fma(-1.0, alpha, (-1.0 * i))) / (fma(2.0, i, beta) - 1.0));
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = fma(2.0, i, Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 1.7e+172)
		tmp = Float64(Float64(Float64(Float64(beta + alpha) + i) / Float64(t_0 - 1.0)) * Float64(Float64(i * 0.25) / Float64(t_0 - -1.0)));
	else
		tmp = Float64(Float64(Float64(Float64(beta + i) * Float64(i / fma(2.0, i, beta))) / Float64(fma(2.0, i, beta) - -1.0)) * Float64(Float64(-1.0 * fma(-1.0, alpha, Float64(-1.0 * i))) / Float64(fma(2.0, i, beta) - 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, 1.7e+172], N[(N[(N[(N[(beta + alpha), $MachinePrecision] + i), $MachinePrecision] / N[(t$95$0 - 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(i * 0.25), $MachinePrecision] / N[(t$95$0 - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(beta + i), $MachinePrecision] * N[(i / N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 * i + beta), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(-1.0 * N[(-1.0 * alpha + N[(-1.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 * i + beta), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.6999999999999999e172

   1. Initial program 16.5%

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

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

      3. lift-*.f64 N/A

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

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

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

   3. Applied rewrites 37.7%

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

   4. Taylor expanded in i around inf

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

   6. Applied rewrites 71.6%

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

   if 1.6999999999999999e172 < beta

   1. Initial program 16.5%

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

   2. Taylor expanded in alpha around 0

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

   4. Applied rewrites 15.7%

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

   5. Taylor expanded in alpha around 0

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

   7. Applied rewrites 16.7%

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

   8. Taylor expanded in alpha around 0

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

   10. Applied rewrites 16.8%

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

   11. Taylor expanded in alpha around 0

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

   13. Applied rewrites 16.9%

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

   14. Taylor expanded in alpha around 0

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

   16. Applied rewrites 15.6%

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

   17. Taylor expanded in alpha around 0

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

   19. Applied rewrites 15.2%

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

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. times-frac N/A

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

   21. Applied rewrites 36.6%

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

   22. Taylor expanded in beta around -inf

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

       1. lower-*.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-*.f64 27.9

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

   24. Applied rewrites 27.9%

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

Alternative 4: 78.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, i, \beta + \alpha\right)\\ t_1 := t\_0 - -1\\ t_2 := \frac{\left(\beta + \alpha\right) + i}{t\_0 - 1}\\ \mathbf{if}\;\beta \leq 1.7 \cdot 10^{+172}:\\ \;\;\;\;t\_2 \cdot \frac{i \cdot 0.25}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \frac{i \cdot \frac{\alpha + i}{\beta}}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma 2.0 i (+ beta alpha)))
        (t_1 (- t_0 -1.0))
        (t_2 (/ (+ (+ beta alpha) i) (- t_0 1.0))))
   (if (<= beta 1.7e+172)
     (* t_2 (/ (* i 0.25) t_1))
     (* t_2 (/ (* i (/ (+ alpha i) beta)) t_1)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = fma(2.0, i, (beta + alpha));
	double t_1 = t_0 - -1.0;
	double t_2 = ((beta + alpha) + i) / (t_0 - 1.0);
	double tmp;
	if (beta <= 1.7e+172) {
		tmp = t_2 * ((i * 0.25) / t_1);
	} else {
		tmp = t_2 * ((i * ((alpha + i) / beta)) / t_1);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = fma(2.0, i, Float64(beta + alpha))
	t_1 = Float64(t_0 - -1.0)
	t_2 = Float64(Float64(Float64(beta + alpha) + i) / Float64(t_0 - 1.0))
	tmp = 0.0
	if (beta <= 1.7e+172)
		tmp = Float64(t_2 * Float64(Float64(i * 0.25) / t_1));
	else
		tmp = Float64(t_2 * Float64(Float64(i * Float64(Float64(alpha + i) / beta)) / t_1));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(2.0 * i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - -1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(beta + alpha), $MachinePrecision] + i), $MachinePrecision] / N[(t$95$0 - 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.7e+172], N[(t$95$2 * N[(N[(i * 0.25), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(N[(i * N[(N[(alpha + i), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.6999999999999999e172

   1. Initial program 16.5%

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

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

      3. lift-*.f64 N/A

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

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

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

   3. Applied rewrites 37.7%

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

   4. Taylor expanded in i around inf

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

   6. Applied rewrites 71.6%

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

   if 1.6999999999999999e172 < beta

   1. Initial program 16.5%

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

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

      3. lift-*.f64 N/A

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

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

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

   3. Applied rewrites 37.7%

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

   4. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 16.5

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

   6. Applied rewrites 16.5%

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

Alternative 5: 78.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, i, \beta + \alpha\right)\\ t_1 := t\_0 - -1\\ t_2 := t\_0 - 1\\ \mathbf{if}\;\beta \leq 2.55 \cdot 10^{+172}:\\ \;\;\;\;\frac{\left(\beta + \alpha\right) + i}{t\_2} \cdot \frac{i \cdot 0.25}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i \cdot \left(\alpha + i\right)}{t\_1}}{t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma 2.0 i (+ beta alpha))) (t_1 (- t_0 -1.0)) (t_2 (- t_0 1.0)))
   (if (<= beta 2.55e+172)
     (* (/ (+ (+ beta alpha) i) t_2) (/ (* i 0.25) t_1))
     (/ (/ (* i (+ alpha i)) t_1) t_2))))

C

double code(double alpha, double beta, double i) {
	double t_0 = fma(2.0, i, (beta + alpha));
	double t_1 = t_0 - -1.0;
	double t_2 = t_0 - 1.0;
	double tmp;
	if (beta <= 2.55e+172) {
		tmp = (((beta + alpha) + i) / t_2) * ((i * 0.25) / t_1);
	} else {
		tmp = ((i * (alpha + i)) / t_1) / t_2;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = fma(2.0, i, Float64(beta + alpha))
	t_1 = Float64(t_0 - -1.0)
	t_2 = Float64(t_0 - 1.0)
	tmp = 0.0
	if (beta <= 2.55e+172)
		tmp = Float64(Float64(Float64(Float64(beta + alpha) + i) / t_2) * Float64(Float64(i * 0.25) / t_1));
	else
		tmp = Float64(Float64(Float64(i * Float64(alpha + i)) / t_1) / t_2);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(2.0 * i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - -1.0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 - 1.0), $MachinePrecision]}, If[LessEqual[beta, 2.55e+172], N[(N[(N[(N[(beta + alpha), $MachinePrecision] + i), $MachinePrecision] / t$95$2), $MachinePrecision] * N[(N[(i * 0.25), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(i * N[(alpha + i), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.55e172

   1. Initial program 16.5%

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

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

      3. lift-*.f64 N/A

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

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

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

   3. Applied rewrites 37.7%

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

   4. Taylor expanded in i around inf

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

   6. Applied rewrites 71.6%

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

   if 2.55e172 < beta

   1. Initial program 16.5%

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

   2. Taylor expanded in beta around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 13.3

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

   4. Applied rewrites 13.3%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

   6. Applied rewrites 17.3%

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

Alternative 6: 78.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, i, \beta + \alpha\right)\\ \mathbf{if}\;\alpha \leq 8.6 \cdot 10^{-13}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i \cdot \left(\alpha + i\right)}{t\_0 - -1}}{t\_0 - 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma 2.0 i (+ beta alpha))))
   (if (<= alpha 8.6e-13)
     (- (+ 0.0625 (* 0.125 (/ beta i))) (* 0.125 (/ (+ alpha beta) i)))
     (/ (/ (* i (+ alpha 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 (alpha <= 8.6e-13) {
		tmp = (0.0625 + (0.125 * (beta / i))) - (0.125 * ((alpha + beta) / i));
	} else {
		tmp = ((i * (alpha + 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 (alpha <= 8.6e-13)
		tmp = Float64(Float64(0.0625 + Float64(0.125 * Float64(beta / i))) - Float64(0.125 * Float64(Float64(alpha + beta) / i)));
	else
		tmp = Float64(Float64(Float64(i * Float64(alpha + 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[alpha, 8.6e-13], N[(N[(0.0625 + N[(0.125 * N[(beta / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 * N[(N[(alpha + beta), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(i * N[(alpha + i), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - -1.0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if alpha < 8.5999999999999997e-13

   1. Initial program 16.5%

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

   2. 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. Taylor expanded in alpha around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 73.2

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

   7. Applied rewrites 73.2%

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

   if 8.5999999999999997e-13 < alpha

   1. Initial program 16.5%

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

   2. Taylor expanded in beta around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 13.3

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

   4. Applied rewrites 13.3%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

   6. Applied rewrites 17.3%

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

Alternative 7: 77.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := t\_0 \cdot t\_0\\ t_2 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ t_3 := \frac{\beta + \alpha}{i}\\ \mathbf{if}\;\frac{\frac{t\_2 \cdot \left(\beta \cdot \alpha + t\_2\right)}{t\_1}}{t\_1 - 1} \leq 2 \cdot 10^{-42}:\\ \;\;\;\;i \cdot \frac{\alpha + i}{{\beta}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_3, -0.125, \mathsf{fma}\left(t\_3, 0.125, 0.0625\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (* t_0 t_0))
        (t_2 (* i (+ (+ alpha beta) i)))
        (t_3 (/ (+ beta alpha) i)))
   (if (<= (/ (/ (* t_2 (+ (* beta alpha) t_2)) t_1) (- t_1 1.0)) 2e-42)
     (* i (/ (+ alpha i) (pow beta 2.0)))
     (fma t_3 -0.125 (fma t_3 0.125 0.0625)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = t_0 * t_0;
	double t_2 = i * ((alpha + beta) + i);
	double t_3 = (beta + alpha) / i;
	double tmp;
	if ((((t_2 * ((beta * alpha) + t_2)) / t_1) / (t_1 - 1.0)) <= 2e-42) {
		tmp = i * ((alpha + i) / pow(beta, 2.0));
	} else {
		tmp = fma(t_3, -0.125, fma(t_3, 0.125, 0.0625));
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(i * Float64(Float64(alpha + beta) + i))
	t_3 = Float64(Float64(beta + alpha) / i)
	tmp = 0.0
	if (Float64(Float64(Float64(t_2 * Float64(Float64(beta * alpha) + t_2)) / t_1) / Float64(t_1 - 1.0)) <= 2e-42)
		tmp = Float64(i * Float64(Float64(alpha + i) / (beta ^ 2.0)));
	else
		tmp = fma(t_3, -0.125, fma(t_3, 0.125, 0.0625));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(beta + alpha), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$2 * N[(N[(beta * alpha), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 - 1.0), $MachinePrecision]), $MachinePrecision], 2e-42], N[(i * N[(N[(alpha + i), $MachinePrecision] / N[Power[beta, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 * -0.125 + N[(t$95$3 * 0.125 + 0.0625), $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))) < 2.00000000000000008e-42

   1. Initial program 16.5%

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

   2. Taylor expanded in alpha around 0

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

   4. Applied rewrites 15.7%

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

   5. Taylor expanded in alpha around 0

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

   7. Applied rewrites 16.7%

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

   8. Taylor expanded in alpha around 0

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

   10. Applied rewrites 16.8%

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

   11. Taylor expanded in alpha around 0

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

   13. Applied rewrites 16.9%

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

   14. Taylor expanded in alpha around 0

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

   16. Applied rewrites 15.6%

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

   17. Taylor expanded in alpha around 0

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

   19. Applied rewrites 15.2%

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

       1. lift-/.f64 N/A

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

       2. mult-flip N/A

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

   21. Applied rewrites 33.2%

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

   22. Taylor expanded in alpha around inf

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

   24. Applied rewrites 10.2%

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

   25. Taylor expanded in beta around inf

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

       1. lower-/.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-pow.f64 9.7

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

   27. Applied rewrites 9.7%

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

   if 2.00000000000000008e-42 < (/.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 16.5%

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

   2. 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

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. metadata-eval 77.1

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

   6. Applied rewrites 77.2%

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

Alternative 8: 74.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := t\_0 \cdot t\_0\\ t_2 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ t_3 := \frac{\beta + \alpha}{i}\\ \mathbf{if}\;\frac{\frac{t\_2 \cdot \left(\beta \cdot \alpha + t\_2\right)}{t\_1}}{t\_1 - 1} \leq 2 \cdot 10^{-42}:\\ \;\;\;\;\frac{i \cdot \left(\alpha + i\right)}{{\beta}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_3, -0.125, \mathsf{fma}\left(t\_3, 0.125, 0.0625\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (* t_0 t_0))
        (t_2 (* i (+ (+ alpha beta) i)))
        (t_3 (/ (+ beta alpha) i)))
   (if (<= (/ (/ (* t_2 (+ (* beta alpha) t_2)) t_1) (- t_1 1.0)) 2e-42)
     (/ (* i (+ alpha i)) (pow beta 2.0))
     (fma t_3 -0.125 (fma t_3 0.125 0.0625)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = t_0 * t_0;
	double t_2 = i * ((alpha + beta) + i);
	double t_3 = (beta + alpha) / i;
	double tmp;
	if ((((t_2 * ((beta * alpha) + t_2)) / t_1) / (t_1 - 1.0)) <= 2e-42) {
		tmp = (i * (alpha + i)) / pow(beta, 2.0);
	} else {
		tmp = fma(t_3, -0.125, fma(t_3, 0.125, 0.0625));
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(i * Float64(Float64(alpha + beta) + i))
	t_3 = Float64(Float64(beta + alpha) / i)
	tmp = 0.0
	if (Float64(Float64(Float64(t_2 * Float64(Float64(beta * alpha) + t_2)) / t_1) / Float64(t_1 - 1.0)) <= 2e-42)
		tmp = Float64(Float64(i * Float64(alpha + i)) / (beta ^ 2.0));
	else
		tmp = fma(t_3, -0.125, fma(t_3, 0.125, 0.0625));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(beta + alpha), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$2 * N[(N[(beta * alpha), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 - 1.0), $MachinePrecision]), $MachinePrecision], 2e-42], N[(N[(i * N[(alpha + i), $MachinePrecision]), $MachinePrecision] / N[Power[beta, 2.0], $MachinePrecision]), $MachinePrecision], N[(t$95$3 * -0.125 + N[(t$95$3 * 0.125 + 0.0625), $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))) < 2.00000000000000008e-42

   1. Initial program 16.5%

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

   2. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-pow.f64 9.2

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

   4. Applied rewrites 9.2%

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

   if 2.00000000000000008e-42 < (/.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 16.5%

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

   2. 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

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. metadata-eval 77.1

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

   6. Applied rewrites 77.2%

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

Alternative 9: 74.5% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta + \alpha}{i}\\ \mathsf{fma}\left(t\_0, -0.125, \mathsf{fma}\left(t\_0, 0.125, 0.0625\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (/ (+ beta alpha) i))) (fma t_0 -0.125 (fma t_0 0.125 0.0625))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (beta + alpha) / i;
	return fma(t_0, -0.125, fma(t_0, 0.125, 0.0625));
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(beta + alpha) / i)
	return fma(t_0, -0.125, fma(t_0, 0.125, 0.0625))
end

Wolfram

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

Derivation

1. Initial program 16.5%

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

2. 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

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

   3. fp-cancel-sub-sign-inv N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-+.f64 N/A

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

   10. metadata-eval 77.1

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

6. Applied rewrites 77.2%

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

Alternative 10: 70.6% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 16.5%

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

2. 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

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

   3. +-commutative N/A

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

   4. associate--l+ N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*r/ N/A

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

   8. lift-fma.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. distribute-lft-out N/A

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

   11. lift-+.f64 N/A

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

   12. associate-*r* N/A

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

   13. metadata-eval N/A

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

   14. *-commutative N/A

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

   15. associate-*l/ N/A

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

   16. mult-flip N/A

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

   17. associate-*l* N/A

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

   18. lower-fma.f64 N/A

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

6. Applied rewrites 70.2%

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

7. Taylor expanded in alpha around 0

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

9. Applied rewrites 69.6%

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

10. Taylor expanded in alpha around 0

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

12. Applied rewrites 70.4%

    \[\leadsto \mathsf{fma}\left(\beta, \frac{1}{i} \cdot 0.125, 0.0625 - \frac{\beta}{i} \cdot 0.125\right) \]
13. Step-by-step derivation

    1. lift-fma.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. associate-*r* N/A

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

    4. lift-/.f64 N/A

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

    5. mult-flip N/A

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

    6. lift-/.f64 N/A

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

    7. lift-*.f64 N/A

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

    8. lift--.f64 N/A

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

    9. associate-+r- N/A

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

    10. lower--.f64 N/A

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

14. Applied rewrites 74.5%

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

Alternative 11: 61.0% 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 16.5%

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

2. Taylor expanded in i around inf

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

4. Applied rewrites 70.6%

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

Reproduce

herbie shell --seed 2025155 
(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)))