Octave 3.8, jcobi/4

Percentage Accurate: 15.9% → 84.2%

Time: 4.9s

Alternatives: 8

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: 15.9% 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: 84.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;i \leq 5.9 \cdot 10^{+128}:\\ \;\;\;\;\frac{\left(\beta + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta\right)}}{\mathsf{fma}\left(2, i, \beta\right) - -1} \cdot \frac{\frac{\mathsf{fma}\left(\beta + i, i, \beta \cdot \alpha\right)}{\mathsf{fma}\left(2, i, \beta\right)}}{\mathsf{fma}\left(2, i, \beta\right) - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.125 \cdot i}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - 1}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= i 5.9e+128)
   (*
    (/ (* (+ beta i) (/ i (fma 2.0 i beta))) (- (fma 2.0 i beta) -1.0))
    (/
     (/ (fma (+ beta i) i (* beta alpha)) (fma 2.0 i beta))
     (- (fma 2.0 i beta) 1.0)))
   (/ (* 0.125 i) (- (fma 2.0 i (+ beta alpha)) 1.0))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 5.9e+128) {
		tmp = (((beta + i) * (i / fma(2.0, i, beta))) / (fma(2.0, i, beta) - -1.0)) * ((fma((beta + i), i, (beta * alpha)) / fma(2.0, i, beta)) / (fma(2.0, i, beta) - 1.0));
	} else {
		tmp = (0.125 * i) / (fma(2.0, i, (beta + alpha)) - 1.0);
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (i <= 5.9e+128)
		tmp = Float64(Float64(Float64(Float64(beta + i) * Float64(i / fma(2.0, i, beta))) / Float64(fma(2.0, i, beta) - -1.0)) * Float64(Float64(fma(Float64(beta + i), i, Float64(beta * alpha)) / fma(2.0, i, beta)) / Float64(fma(2.0, i, beta) - 1.0)));
	else
		tmp = Float64(Float64(0.125 * i) / Float64(fma(2.0, i, Float64(beta + alpha)) - 1.0));
	end
	return tmp
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[i, 5.9e+128], 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[(N[(N[(beta + i), $MachinePrecision] * i + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] / N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 * i + beta), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.125 * i), $MachinePrecision] / N[(N[(2.0 * i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 5.89999999999999987e128

   1. Initial program 15.9%

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

   2. 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 15.9%

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

   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 15.9%

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

   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 15.9%

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

   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 15.9%

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

   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 15.9%

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

   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 15.9%

       \[\leadsto \frac{\frac{\left(i \cdot \left(\beta + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\beta + i\right)\right)}{\left(\beta + 2 \cdot i\right) \cdot \left(\beta + 2 \cdot i\right)}}{\left(\beta + 2 \cdot i\right) \cdot \left(\color{blue}{\beta} + 2 \cdot i\right) - 1} \]
   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 42.8%

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

   if 5.89999999999999987e128 < i

   1. Initial program 15.9%

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

   2. Taylor expanded in alpha around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 8.8

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

   4. Applied rewrites 8.8%

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

   6. Applied rewrites 8.8%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

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

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. +-commutative N/A

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

      15. lift-*.f64 N/A

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

   8. Applied rewrites 7.5%

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

   9. Taylor expanded in i around inf

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

       1. lower-*.f64 70.2

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

   11. Applied rewrites 70.2%

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

Alternative 2: 84.0% accurate, 0.6× speedup

Math

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

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (* t_0 t_0))
        (t_2 (* i (+ (+ alpha beta) i))))
   (if (<= (/ (/ (* t_2 (+ (* beta alpha) t_2)) t_1) (- t_1 1.0)) INFINITY)
     (*
      (/
       (fma (+ beta i) i (* beta alpha))
       (* (fma 2.0 i beta) (fma 2.0 i beta)))
      (/ (* (+ beta i) i) (fma (fma 2.0 i beta) (fma 2.0 i beta) -1.0)))
     (/ (fma -0.125 beta (fma 0.0625 i (* 0.125 beta))) i))))

C

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

Julia

alpha, beta, i = sort([alpha, beta, i])
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))
	tmp = 0.0
	if (Float64(Float64(Float64(t_2 * Float64(Float64(beta * alpha) + t_2)) / t_1) / Float64(t_1 - 1.0)) <= Inf)
		tmp = Float64(Float64(fma(Float64(beta + i), i, Float64(beta * alpha)) / Float64(fma(2.0, i, beta) * fma(2.0, i, beta))) * Float64(Float64(Float64(beta + i) * i) / fma(fma(2.0, i, beta), fma(2.0, i, beta), -1.0)));
	else
		tmp = Float64(fma(-0.125, beta, fma(0.0625, i, Float64(0.125 * beta))) / i);
	end
	return tmp
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(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]}, If[LessEqual[N[(N[(N[(t$95$2 * N[(N[(beta * alpha), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 - 1.0), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(N[(beta + i), $MachinePrecision] * i + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 * i + beta), $MachinePrecision] * N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(beta + i), $MachinePrecision] * i), $MachinePrecision] / N[(N[(2.0 * i + beta), $MachinePrecision] * N[(2.0 * i + beta), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.125 * beta + N[(0.0625 * i + N[(0.125 * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 15.9%

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

   2. 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 15.9%

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

   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 15.9%

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

   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 15.9%

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

   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 15.9%

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

   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 15.9%

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

   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 15.9%

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

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

   21. Applied rewrites 37.9%

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

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

   1. Initial program 15.9%

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

   2. 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 77.3

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

   4. Applied rewrites 77.3%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]
   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 77.3

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

      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 77.3%

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

   7. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 77.3

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

   9. Applied rewrites 77.3%

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

   10. Taylor expanded in alpha around 0

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

       1. lower-fma.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-*.f64 77.3

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

   12. Applied rewrites 77.3%

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

Alternative 3: 82.8% accurate, 0.6× speedup

Math

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

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (* t_0 t_0))
        (t_2 (* i (+ (+ alpha beta) i))))
   (if (<= (/ (/ (* t_2 (+ (* beta alpha) t_2)) t_1) (- t_1 1.0)) INFINITY)
     (*
      (/ (+ beta i) (fma (fma 2.0 i beta) (fma 2.0 i beta) -1.0))
      (*
       (/ i (* (fma 2.0 i beta) (fma 2.0 i beta)))
       (fma (+ beta i) i (* beta alpha))))
     (/ (fma -0.125 beta (fma 0.0625 i (* 0.125 beta))) i))))

C

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

Julia

alpha, beta, i = sort([alpha, beta, i])
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))
	tmp = 0.0
	if (Float64(Float64(Float64(t_2 * Float64(Float64(beta * alpha) + t_2)) / t_1) / Float64(t_1 - 1.0)) <= Inf)
		tmp = Float64(Float64(Float64(beta + i) / fma(fma(2.0, i, beta), fma(2.0, i, beta), -1.0)) * Float64(Float64(i / Float64(fma(2.0, i, beta) * fma(2.0, i, beta))) * fma(Float64(beta + i), i, Float64(beta * alpha))));
	else
		tmp = Float64(fma(-0.125, beta, fma(0.0625, i, Float64(0.125 * beta))) / i);
	end
	return tmp
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(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]}, If[LessEqual[N[(N[(N[(t$95$2 * N[(N[(beta * alpha), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 - 1.0), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(beta + i), $MachinePrecision] / N[(N[(2.0 * i + beta), $MachinePrecision] * N[(2.0 * i + beta), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(i / N[(N[(2.0 * i + beta), $MachinePrecision] * N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(beta + i), $MachinePrecision] * i + N[(beta * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.125 * beta + N[(0.0625 * i + N[(0.125 * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 15.9%

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

   2. 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 15.9%

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

   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 15.9%

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

   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 15.9%

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

   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 15.9%

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

   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 15.9%

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

   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 15.9%

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

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

   21. Applied rewrites 17.4%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-/.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. times-frac N/A

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

   23. Applied rewrites 37.8%

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

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

   1. Initial program 15.9%

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

   2. 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 77.3

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

   4. Applied rewrites 77.3%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]
   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 77.3

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

      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 77.3%

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

   7. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 77.3

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

   9. Applied rewrites 77.3%

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

   10. Taylor expanded in alpha around 0

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

       1. lower-fma.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-*.f64 77.3

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

   12. Applied rewrites 77.3%

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

Alternative 4: 77.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.7 \cdot 10^{+158}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i \cdot \left(\alpha + i\right)}{\beta}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - 1}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.7e+158)
   0.0625
   (/ (/ (* i (+ alpha i)) beta) (- (fma 2.0 i (+ beta alpha)) 1.0))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.7e+158) {
		tmp = 0.0625;
	} else {
		tmp = ((i * (alpha + i)) / beta) / (fma(2.0, i, (beta + alpha)) - 1.0);
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.7e+158)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(i * Float64(alpha + i)) / beta) / Float64(fma(2.0, i, Float64(beta + alpha)) - 1.0));
	end
	return tmp
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 1.7e+158], 0.0625, N[(N[(N[(i * N[(alpha + i), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision] / N[(N[(2.0 * i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.7e158

   1. Initial program 15.9%

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

   2. Taylor expanded in i around inf

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

   4. Applied rewrites 70.7%

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

   if 1.7e158 < beta

   1. Initial program 15.9%

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

   2. Taylor expanded in alpha around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 8.8

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

   4. Applied rewrites 8.8%

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

   6. Applied rewrites 8.8%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

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

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. +-commutative N/A

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

      15. lift-*.f64 N/A

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

   8. Applied rewrites 7.5%

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

   9. Taylor expanded in beta around inf

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-+.f64 23.4

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

   11. Applied rewrites 23.4%

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

Alternative 5: 77.3% accurate, 4.4× speedup

Math

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

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (/ (fma -0.125 beta (fma 0.0625 i (* 0.125 beta))) i))

C

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

Julia

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

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := N[(N[(-0.125 * beta + N[(0.0625 * i + N[(0.125 * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]

Derivation

1. Initial program 15.9%

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

2. 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 77.3

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

4. Applied rewrites 77.3%

   \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]
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 77.3

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

   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 77.3%

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

7. Taylor expanded in i around 0

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

   1. lower-/.f64 N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-+.f64 77.3

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

9. Applied rewrites 77.3%

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

10. Taylor expanded in alpha around 0

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

    1. lower-fma.f64 N/A

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

    2. lower-fma.f64 N/A

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

    3. lower-*.f64 77.3

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

12. Applied rewrites 77.3%

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

Alternative 6: 74.0% accurate, 3.6× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+232}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125, \alpha, \mathsf{fma}\left(0.0625, i, 0.125 \cdot \alpha\right)\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{i}, -0.125, 0.125 \cdot \frac{\beta}{i}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 5e+232)
   (/ (fma -0.125 alpha (fma 0.0625 i (* 0.125 alpha))) i)
   (fma (/ beta i) -0.125 (* 0.125 (/ beta i)))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 5e+232) {
		tmp = fma(-0.125, alpha, fma(0.0625, i, (0.125 * alpha))) / i;
	} else {
		tmp = fma((beta / i), -0.125, (0.125 * (beta / i)));
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 5e+232)
		tmp = Float64(fma(-0.125, alpha, fma(0.0625, i, Float64(0.125 * alpha))) / i);
	else
		tmp = fma(Float64(beta / i), -0.125, Float64(0.125 * Float64(beta / i)));
	end
	return tmp
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 5e+232], N[(N[(-0.125 * alpha + N[(0.0625 * i + N[(0.125 * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], N[(N[(beta / i), $MachinePrecision] * -0.125 + N[(0.125 * N[(beta / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.99999999999999987e232

   1. Initial program 15.9%

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

   2. 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 77.3

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

   4. Applied rewrites 77.3%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]
   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 77.3

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

      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 77.3%

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

   7. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 77.3

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

   9. Applied rewrites 77.3%

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

   10. Taylor expanded in beta around 0

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

       1. lower-fma.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-*.f64 71.4

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

   12. Applied rewrites 71.4%

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

   if 4.99999999999999987e232 < beta

   1. Initial program 15.9%

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

   2. 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 77.3

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

   4. Applied rewrites 77.3%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]
   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 77.3

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

      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 77.3%

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

   7. Taylor expanded in alpha around 0

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

   9. Applied rewrites 77.3%

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

   10. Taylor expanded in alpha around 0

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

   12. Applied rewrites 77.3%

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

   13. Taylor expanded in beta around inf

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 10.1

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

   15. Applied rewrites 10.1%

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

Alternative 7: 73.9% accurate, 3.9× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+232}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{i}, -0.125, 0.125 \cdot \frac{\beta}{i}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 5e+232) 0.0625 (fma (/ beta i) -0.125 (* 0.125 (/ beta i)))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 5e+232) {
		tmp = 0.0625;
	} else {
		tmp = fma((beta / i), -0.125, (0.125 * (beta / i)));
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 5e+232)
		tmp = 0.0625;
	else
		tmp = fma(Float64(beta / i), -0.125, Float64(0.125 * Float64(beta / i)));
	end
	return tmp
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 5e+232], 0.0625, N[(N[(beta / i), $MachinePrecision] * -0.125 + N[(0.125 * N[(beta / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.99999999999999987e232

   1. Initial program 15.9%

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

   2. Taylor expanded in i around inf

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

   4. Applied rewrites 70.7%

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

   if 4.99999999999999987e232 < beta

   1. Initial program 15.9%

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

   2. 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 77.3

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

   4. Applied rewrites 77.3%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]
   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 77.3

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

      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 77.3%

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

   7. Taylor expanded in alpha around 0

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

   9. Applied rewrites 77.3%

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

   10. Taylor expanded in alpha around 0

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

   12. Applied rewrites 77.3%

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

   13. Taylor expanded in beta around inf

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 10.1

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

   15. Applied rewrites 10.1%

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

Alternative 8: 70.7% accurate, 75.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ 0.0625 \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i) :precision binary64 0.0625)

C

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

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
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

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

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	return 0.0625

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	return 0.0625
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp = code(alpha, beta, i)
	tmp = 0.0625;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := 0.0625

Derivation

1. Initial program 15.9%

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

2. Taylor expanded in i around inf

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

4. Applied rewrites 70.7%

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

Reproduce

herbie shell --seed 2025159 
(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)))