Octave 3.8, jcobi/4

Percentage Accurate: 15.8% → 84.1%

Time: 5.4s

Alternatives: 9

Speedup: 115.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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: 15.8% 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: 84.1% accurate, 0.5× speedup

Math

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

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

C

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

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = Float64(i * Float64(Float64(alpha + beta) + i))
	t_1 = Float64(Float64(alpha + beta) + Float64(i + i))
	t_2 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_3 = Float64(t_2 * t_2)
	t_4 = Float64(t_3 - 1.0)
	t_5 = Float64(0.125 * Float64(beta / i))
	tmp = 0.0
	if (Float64(Float64(Float64(t_0 * Float64(Float64(beta * alpha) + t_0)) / t_3) / t_4) <= Inf)
		tmp = Float64(Float64(Float64(t_0 / t_1) * Float64(fma(beta, alpha, t_0) / t_1)) / t_4);
	else
		tmp = Float64(Float64(0.0625 + t_5) - t_5);
	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[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] + N[(i + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 - 1.0), $MachinePrecision]}, Block[{t$95$5 = N[(0.125 * N[(beta / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$0 * N[(N[(beta * alpha), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] / t$95$4), $MachinePrecision], Infinity], N[(N[(N[(t$95$0 / t$95$1), $MachinePrecision] * N[(N[(beta * alpha + t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision], N[(N[(0.0625 + t$95$5), $MachinePrecision] - t$95$5), $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 47.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. lift-*.f64 N/A

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

   3. Applied rewrites 99.6%

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

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

   1. Initial program 0.0%

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

   2. 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. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 76.2

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

   4. Applied rewrites 76.2%

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

   5. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-+.f64 76.2

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

   7. Applied rewrites 76.2%

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

   8. Taylor expanded in alpha around 0

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

   10. Applied rewrites 76.2%

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

   11. Taylor expanded in alpha around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 76.2

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

   13. Applied rewrites 76.2%

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

Alternative 2: 80.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := t\_0 \cdot t\_0\\ t_2 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ \mathbf{if}\;\frac{\frac{t\_2 \cdot \left(\beta \cdot \alpha + t\_2\right)}{t\_1}}{t\_1 - 1} \leq 0.0005:\\ \;\;\;\;\frac{i \cdot \left(\alpha + i\right)}{\left(i \cdot i\right) \cdot \left(4 + \frac{\mathsf{fma}\left(4, \beta, \frac{\beta \cdot \beta}{i}\right)}{i}\right) - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.0625, i, 0.125 \cdot \beta\right)}{i} - 0.125 \cdot \frac{\beta}{i}\\ \end{array} \end{array} \]

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)) 0.0005)
     (/
      (* i (+ alpha i))
      (- (* (* i i) (+ 4.0 (/ (fma 4.0 beta (/ (* beta beta) i)) i))) 1.0))
     (- (/ (fma 0.0625 i (* 0.125 beta)) 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)) <= 0.0005) {
		tmp = (i * (alpha + i)) / (((i * i) * (4.0 + (fma(4.0, beta, ((beta * beta) / i)) / i))) - 1.0);
	} else {
		tmp = (fma(0.0625, i, (0.125 * beta)) / 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)) <= 0.0005)
		tmp = Float64(Float64(i * Float64(alpha + i)) / Float64(Float64(Float64(i * i) * Float64(4.0 + Float64(fma(4.0, beta, Float64(Float64(beta * beta) / i)) / i))) - 1.0));
	else
		tmp = Float64(Float64(fma(0.0625, i, Float64(0.125 * beta)) / i) - 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_] := 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], 0.0005], N[(N[(i * N[(alpha + i), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(i * i), $MachinePrecision] * N[(4.0 + N[(N[(4.0 * beta + N[(N[(beta * beta), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.0625 * i + N[(0.125 * beta), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] - N[(0.125 * N[(beta / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64))) < 5.0000000000000001e-4

   1. Initial program 98.7%

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

   2. Taylor expanded in beta around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 91.0

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

   4. Applied rewrites 91.0%

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

   5. Taylor expanded in alpha around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lift-*.f64 91.0

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

   7. Applied rewrites 91.0%

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

   8. Taylor expanded in i around -inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 90.7

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

   10. Applied rewrites 90.7%

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

   11. Taylor expanded in i around inf

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

       1. lower-/.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. pow2 N/A

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

       4. lift-/.f64 N/A

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

       5. lift-*.f64 90.7

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

   13. Applied rewrites 90.7%

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

   if 5.0000000000000001e-4 < (/.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 13.2%

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

   2. 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. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 79.8

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

   4. Applied rewrites 79.8%

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

   5. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-+.f64 79.9

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

   7. Applied rewrites 79.9%

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

   8. Taylor expanded in alpha around 0

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

   10. Applied rewrites 79.8%

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

   11. Taylor expanded in alpha around 0

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

       1. lower-*.f64 79.9

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

   13. Applied rewrites 79.9%

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

Alternative 3: 80.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := t\_0 \cdot t\_0\\ t_2 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ t_3 := \beta + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{t\_2 \cdot \left(\beta \cdot \alpha + t\_2\right)}{t\_1}}{t\_1 - 1} \leq 0.0005:\\ \;\;\;\;\frac{i \cdot i}{t\_3 \cdot t\_3 - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.0625, i, 0.125 \cdot \beta\right)}{i} - 0.125 \cdot \frac{\beta}{i}\\ \end{array} \end{array} \]

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)))
        (t_3 (+ beta (* 2.0 i))))
   (if (<= (/ (/ (* t_2 (+ (* beta alpha) t_2)) t_1) (- t_1 1.0)) 0.0005)
     (/ (* i i) (- (* t_3 t_3) 1.0))
     (- (/ (fma 0.0625 i (* 0.125 beta)) 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 t_3 = beta + (2.0 * i);
	double tmp;
	if ((((t_2 * ((beta * alpha) + t_2)) / t_1) / (t_1 - 1.0)) <= 0.0005) {
		tmp = (i * i) / ((t_3 * t_3) - 1.0);
	} else {
		tmp = (fma(0.0625, i, (0.125 * beta)) / 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))
	t_3 = Float64(beta + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(t_2 * Float64(Float64(beta * alpha) + t_2)) / t_1) / Float64(t_1 - 1.0)) <= 0.0005)
		tmp = Float64(Float64(i * i) / Float64(Float64(t_3 * t_3) - 1.0));
	else
		tmp = Float64(Float64(fma(0.0625, i, Float64(0.125 * beta)) / i) - 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_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(beta + N[(2.0 * 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], 0.0005], N[(N[(i * i), $MachinePrecision] / N[(N[(t$95$3 * t$95$3), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.0625 * i + N[(0.125 * beta), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] - N[(0.125 * N[(beta / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64))) < 5.0000000000000001e-4

   1. Initial program 98.7%

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

   2. Taylor expanded in beta around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 91.0

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

   4. Applied rewrites 91.0%

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

   5. Taylor expanded in alpha around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lift-*.f64 91.0

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

   7. Applied rewrites 91.0%

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

   8. Taylor expanded in alpha around 0

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

   10. Applied rewrites 86.5%

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

   if 5.0000000000000001e-4 < (/.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 13.2%

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

   2. 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. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 79.8

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

   4. Applied rewrites 79.8%

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

   5. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-+.f64 79.9

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

   7. Applied rewrites 79.9%

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

   8. Taylor expanded in alpha around 0

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

   10. Applied rewrites 79.8%

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

   11. Taylor expanded in alpha around 0

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

       1. lower-*.f64 79.9

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

   13. Applied rewrites 79.9%

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

Alternative 4: 80.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := t\_0 \cdot t\_0\\ t_2 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ t_3 := \beta + 2 \cdot i\\ t_4 := 0.125 \cdot \frac{\beta}{i}\\ \mathbf{if}\;\frac{\frac{t\_2 \cdot \left(\beta \cdot \alpha + t\_2\right)}{t\_1}}{t\_1 - 1} \leq 0.0005:\\ \;\;\;\;\frac{i \cdot i}{t\_3 \cdot t\_3 - 1}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + t\_4\right) - t\_4\\ \end{array} \end{array} \]

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)))
        (t_3 (+ beta (* 2.0 i)))
        (t_4 (* 0.125 (/ beta i))))
   (if (<= (/ (/ (* t_2 (+ (* beta alpha) t_2)) t_1) (- t_1 1.0)) 0.0005)
     (/ (* i i) (- (* t_3 t_3) 1.0))
     (- (+ 0.0625 t_4) t_4))))

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 t_3 = beta + (2.0 * i);
	double t_4 = 0.125 * (beta / i);
	double tmp;
	if ((((t_2 * ((beta * alpha) + t_2)) / t_1) / (t_1 - 1.0)) <= 0.0005) {
		tmp = (i * i) / ((t_3 * t_3) - 1.0);
	} else {
		tmp = (0.0625 + t_4) - t_4;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    t_1 = t_0 * t_0
    t_2 = i * ((alpha + beta) + i)
    t_3 = beta + (2.0d0 * i)
    t_4 = 0.125d0 * (beta / i)
    if ((((t_2 * ((beta * alpha) + t_2)) / t_1) / (t_1 - 1.0d0)) <= 0.0005d0) then
        tmp = (i * i) / ((t_3 * t_3) - 1.0d0)
    else
        tmp = (0.0625d0 + t_4) - t_4
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = t_0 * t_0;
	double t_2 = i * ((alpha + beta) + i);
	double t_3 = beta + (2.0 * i);
	double t_4 = 0.125 * (beta / i);
	double tmp;
	if ((((t_2 * ((beta * alpha) + t_2)) / t_1) / (t_1 - 1.0)) <= 0.0005) {
		tmp = (i * i) / ((t_3 * t_3) - 1.0);
	} else {
		tmp = (0.0625 + t_4) - t_4;
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	t_1 = t_0 * t_0
	t_2 = i * ((alpha + beta) + i)
	t_3 = beta + (2.0 * i)
	t_4 = 0.125 * (beta / i)
	tmp = 0
	if (((t_2 * ((beta * alpha) + t_2)) / t_1) / (t_1 - 1.0)) <= 0.0005:
		tmp = (i * i) / ((t_3 * t_3) - 1.0)
	else:
		tmp = (0.0625 + t_4) - t_4
	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))
	t_3 = Float64(beta + Float64(2.0 * i))
	t_4 = Float64(0.125 * Float64(beta / i))
	tmp = 0.0
	if (Float64(Float64(Float64(t_2 * Float64(Float64(beta * alpha) + t_2)) / t_1) / Float64(t_1 - 1.0)) <= 0.0005)
		tmp = Float64(Float64(i * i) / Float64(Float64(t_3 * t_3) - 1.0));
	else
		tmp = Float64(Float64(0.0625 + t_4) - t_4);
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	t_1 = t_0 * t_0;
	t_2 = i * ((alpha + beta) + i);
	t_3 = beta + (2.0 * i);
	t_4 = 0.125 * (beta / i);
	tmp = 0.0;
	if ((((t_2 * ((beta * alpha) + t_2)) / t_1) / (t_1 - 1.0)) <= 0.0005)
		tmp = (i * i) / ((t_3 * t_3) - 1.0);
	else
		tmp = (0.0625 + t_4) - t_4;
	end
	tmp_2 = 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]}, Block[{t$95$3 = N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(0.125 * N[(beta / 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], 0.0005], N[(N[(i * i), $MachinePrecision] / N[(N[(t$95$3 * t$95$3), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.0625 + t$95$4), $MachinePrecision] - t$95$4), $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))) < 5.0000000000000001e-4

   1. Initial program 98.7%

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

   2. Taylor expanded in beta around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 91.0

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

   4. Applied rewrites 91.0%

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

   5. Taylor expanded in alpha around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lift-*.f64 91.0

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

   7. Applied rewrites 91.0%

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

   8. Taylor expanded in alpha around 0

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

   10. Applied rewrites 86.5%

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

   if 5.0000000000000001e-4 < (/.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 13.2%

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

   2. 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. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 79.8

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

   4. Applied rewrites 79.8%

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

   5. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-+.f64 79.9

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

   7. Applied rewrites 79.9%

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

   8. Taylor expanded in alpha around 0

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

   10. Applied rewrites 79.8%

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

   11. Taylor expanded in alpha around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 79.9

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

   13. Applied rewrites 79.9%

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

Alternative 5: 80.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_2 := t\_1 \cdot t\_1\\ t_3 := 0.125 \cdot \frac{\beta}{i}\\ \mathbf{if}\;\frac{\frac{t\_0 \cdot \left(\beta \cdot \alpha + t\_0\right)}{t\_2}}{t\_2 - 1} \leq 0.0005:\\ \;\;\;\;\frac{i \cdot i}{\mathsf{fma}\left(4, \beta \cdot i, \beta \cdot \beta\right) - 1}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + t\_3\right) - t\_3\\ \end{array} \end{array} \]

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 (* i (+ (+ alpha beta) i)))
        (t_1 (+ (+ alpha beta) (* 2.0 i)))
        (t_2 (* t_1 t_1))
        (t_3 (* 0.125 (/ beta i))))
   (if (<= (/ (/ (* t_0 (+ (* beta alpha) t_0)) t_2) (- t_2 1.0)) 0.0005)
     (/ (* i i) (- (fma 4.0 (* beta i) (* beta beta)) 1.0))
     (- (+ 0.0625 t_3) t_3))))

C

assert(alpha < beta && beta < i);
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;
	double t_3 = 0.125 * (beta / i);
	double tmp;
	if ((((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0)) <= 0.0005) {
		tmp = (i * i) / (fma(4.0, (beta * i), (beta * beta)) - 1.0);
	} else {
		tmp = (0.0625 + t_3) - t_3;
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
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)
	t_3 = Float64(0.125 * Float64(beta / i))
	tmp = 0.0
	if (Float64(Float64(Float64(t_0 * Float64(Float64(beta * alpha) + t_0)) / t_2) / Float64(t_2 - 1.0)) <= 0.0005)
		tmp = Float64(Float64(i * i) / Float64(fma(4.0, Float64(beta * i), Float64(beta * beta)) - 1.0));
	else
		tmp = Float64(Float64(0.0625 + t_3) - t_3);
	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[(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]}, Block[{t$95$3 = N[(0.125 * N[(beta / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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], 0.0005], N[(N[(i * i), $MachinePrecision] / N[(N[(4.0 * N[(beta * i), $MachinePrecision] + N[(beta * beta), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.0625 + t$95$3), $MachinePrecision] - t$95$3), $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))) < 5.0000000000000001e-4

   1. Initial program 98.7%

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

   2. Taylor expanded in beta around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 91.0

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

   4. Applied rewrites 91.0%

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

   5. Taylor expanded in alpha around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lift-*.f64 91.0

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

   7. Applied rewrites 91.0%

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

   8. Taylor expanded in alpha around 0

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

   10. Applied rewrites 86.5%

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

   11. Taylor expanded in i around 0

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

       1. lower-fma.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 86.5

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

   13. Applied rewrites 86.5%

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

   if 5.0000000000000001e-4 < (/.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 13.2%

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

   2. 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. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 79.8

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

   4. Applied rewrites 79.8%

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

   5. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-+.f64 79.9

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

   7. Applied rewrites 79.9%

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

   8. Taylor expanded in alpha around 0

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

   10. Applied rewrites 79.8%

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

   11. Taylor expanded in alpha around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 79.9

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

   13. Applied rewrites 79.9%

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

Alternative 6: 74.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.5 \cdot 10^{+228}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{else}:\\ \;\;\;\;0.125 \cdot \frac{\beta}{i} - 0.125 \cdot \frac{\alpha + \beta}{i}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 7.5e+228)
   (+ 0.0625 (/ 0.015625 (* i i)))
   (- (* 0.125 (/ beta i)) (* 0.125 (/ (+ alpha beta) i)))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 7.5e+228) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else {
		tmp = (0.125 * (beta / i)) - (0.125 * ((alpha + beta) / i));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (beta <= 7.5d+228) then
        tmp = 0.0625d0 + (0.015625d0 / (i * i))
    else
        tmp = (0.125d0 * (beta / i)) - (0.125d0 * ((alpha + beta) / i))
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 7.5e+228) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else {
		tmp = (0.125 * (beta / i)) - (0.125 * ((alpha + beta) / i));
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 7.5e+228:
		tmp = 0.0625 + (0.015625 / (i * i))
	else:
		tmp = (0.125 * (beta / i)) - (0.125 * ((alpha + beta) / i))
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 7.5e+228)
		tmp = Float64(0.0625 + Float64(0.015625 / Float64(i * i)));
	else
		tmp = Float64(Float64(0.125 * Float64(beta / i)) - Float64(0.125 * Float64(Float64(alpha + beta) / i)));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 7.5e+228)
		tmp = 0.0625 + (0.015625 / (i * i));
	else
		tmp = (0.125 * (beta / i)) - (0.125 * ((alpha + beta) / i));
	end
	tmp_2 = 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, 7.5e+228], N[(0.0625 + N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.125 * N[(beta / i), $MachinePrecision]), $MachinePrecision] - N[(0.125 * N[(N[(alpha + beta), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.49999999999999994e228

   1. Initial program 18.8%

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

   2. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 15.0%

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

   5. Taylor expanded in i around inf

      \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{{\alpha}^{2}}{{i}^{2}}\right) - \color{blue}{\frac{1}{256} \cdot \frac{4 \cdot \left({\alpha}^{2} - 1\right) + \left(4 \cdot {\alpha}^{2} + 16 \cdot {\alpha}^{2}\right)}{{i}^{2}}} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{{\alpha}^{2}}{{i}^{2}}\right) - \frac{1}{256} \cdot \color{blue}{\frac{4 \cdot \left({\alpha}^{2} - 1\right) + \left(4 \cdot {\alpha}^{2} + 16 \cdot {\alpha}^{2}\right)}{{i}^{2}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{{\alpha}^{2}}{{i}^{2}}\right) - \frac{1}{256} \cdot \frac{\color{blue}{4 \cdot \left({\alpha}^{2} - 1\right) + \left(4 \cdot {\alpha}^{2} + 16 \cdot {\alpha}^{2}\right)}}{{i}^{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{{\alpha}^{2}}{{i}^{2}}\right) - \frac{1}{256} \cdot \frac{4 \cdot \left({\alpha}^{2} - 1\right) + \color{blue}{\left(4 \cdot {\alpha}^{2} + 16 \cdot {\alpha}^{2}\right)}}{{i}^{2}} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{{\alpha}^{2}}{{i}^{2}}\right) - \frac{1}{256} \cdot \frac{4 \cdot \left({\alpha}^{2} - 1\right) + \left(4 \cdot {\alpha}^{2} + \color{blue}{16 \cdot {\alpha}^{2}}\right)}{{i}^{2}} \]

      5. unpow2 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\alpha \cdot \alpha}{{i}^{2}}\right) - \frac{1}{256} \cdot \frac{4 \cdot \left({\alpha}^{2} - 1\right) + \left(4 \cdot {\alpha}^{2} + \color{blue}{16} \cdot {\alpha}^{2}\right)}{{i}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\alpha \cdot \alpha}{{i}^{2}}\right) - \frac{1}{256} \cdot \frac{4 \cdot \left({\alpha}^{2} - 1\right) + \left(4 \cdot {\alpha}^{2} + \color{blue}{16} \cdot {\alpha}^{2}\right)}{{i}^{2}} \]

      7. pow2 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\alpha \cdot \alpha}{i \cdot i}\right) - \frac{1}{256} \cdot \frac{4 \cdot \left({\alpha}^{2} - 1\right) + \left(4 \cdot {\alpha}^{2} + 16 \cdot \color{blue}{{\alpha}^{2}}\right)}{{i}^{2}} \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\alpha \cdot \alpha}{i \cdot i}\right) - \frac{1}{256} \cdot \frac{4 \cdot \left({\alpha}^{2} - 1\right) + \left(4 \cdot {\alpha}^{2} + 16 \cdot \color{blue}{{\alpha}^{2}}\right)}{{i}^{2}} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\alpha \cdot \alpha}{i \cdot i}\right) - \frac{1}{256} \cdot \frac{4 \cdot \left({\alpha}^{2} - 1\right) + \left(4 \cdot {\alpha}^{2} + 16 \cdot {\alpha}^{2}\right)}{\color{blue}{{i}^{2}}} \]

   7. Applied rewrites 81.8%

      \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{\alpha \cdot \alpha}{i \cdot i}\right) - \color{blue}{0.00390625 \cdot \frac{\mathsf{fma}\left(4, \alpha \cdot \alpha - 1, \mathsf{fma}\left(4, \alpha \cdot \alpha, 16 \cdot \left(\alpha \cdot \alpha\right)\right)\right)}{i \cdot i}} \]

   8. Taylor expanded in alpha around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 82.2

         \[\leadsto 0.0625 + 0.015625 \cdot \frac{1}{i \cdot i} \]

   10. Applied rewrites 82.2%

       \[\leadsto 0.0625 + 0.015625 \cdot \color{blue}{\frac{1}{i \cdot i}} \]

   11. Taylor expanded in i around 0

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

       1. lower-/.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 82.2

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

   13. Applied rewrites 82.2%

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

   if 7.49999999999999994e228 < beta

   1. Initial program 0.0%

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

   2. 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. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 47.0

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

   4. Applied rewrites 47.0%

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

   5. Taylor expanded in beta around inf

      \[\leadsto \frac{1}{8} \cdot \frac{\beta}{i} - \color{blue}{\frac{1}{8}} \cdot \frac{\alpha + \beta}{i} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{8} \cdot \frac{\beta}{i} - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      2. lower-/.f64 35.4

         \[\leadsto 0.125 \cdot \frac{\beta}{i} - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   7. Applied rewrites 35.4%

      \[\leadsto 0.125 \cdot \frac{\beta}{i} - \color{blue}{0.125} \cdot \frac{\alpha + \beta}{i} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 74.4% accurate, 4.1× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.9 \cdot 10^{+229}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot i}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.9e+229)
   (+ 0.0625 (/ 0.015625 (* i i)))
   (/ (* i i) (* beta beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.9e+229) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else {
		tmp = (i * i) / (beta * beta);
	}
	return tmp;
}

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
    real(8) :: tmp
    if (beta <= 1.9d+229) then
        tmp = 0.0625d0 + (0.015625d0 / (i * i))
    else
        tmp = (i * i) / (beta * beta)
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.9e+229) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else {
		tmp = (i * i) / (beta * beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.9e+229:
		tmp = 0.0625 + (0.015625 / (i * i))
	else:
		tmp = (i * i) / (beta * beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.9e+229)
		tmp = Float64(0.0625 + Float64(0.015625 / Float64(i * i)));
	else
		tmp = Float64(Float64(i * i) / Float64(beta * beta));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1.9e+229)
		tmp = 0.0625 + (0.015625 / (i * i));
	else
		tmp = (i * i) / (beta * beta);
	end
	tmp_2 = 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, 1.9e+229], N[(0.0625 + N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i * i), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.90000000000000009e229

   1. Initial program 18.8%

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

   2. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 15.0%

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

   5. Taylor expanded in i around inf

      \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{{\alpha}^{2}}{{i}^{2}}\right) - \color{blue}{\frac{1}{256} \cdot \frac{4 \cdot \left({\alpha}^{2} - 1\right) + \left(4 \cdot {\alpha}^{2} + 16 \cdot {\alpha}^{2}\right)}{{i}^{2}}} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{{\alpha}^{2}}{{i}^{2}}\right) - \frac{1}{256} \cdot \color{blue}{\frac{4 \cdot \left({\alpha}^{2} - 1\right) + \left(4 \cdot {\alpha}^{2} + 16 \cdot {\alpha}^{2}\right)}{{i}^{2}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{{\alpha}^{2}}{{i}^{2}}\right) - \frac{1}{256} \cdot \frac{\color{blue}{4 \cdot \left({\alpha}^{2} - 1\right) + \left(4 \cdot {\alpha}^{2} + 16 \cdot {\alpha}^{2}\right)}}{{i}^{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{{\alpha}^{2}}{{i}^{2}}\right) - \frac{1}{256} \cdot \frac{4 \cdot \left({\alpha}^{2} - 1\right) + \color{blue}{\left(4 \cdot {\alpha}^{2} + 16 \cdot {\alpha}^{2}\right)}}{{i}^{2}} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{{\alpha}^{2}}{{i}^{2}}\right) - \frac{1}{256} \cdot \frac{4 \cdot \left({\alpha}^{2} - 1\right) + \left(4 \cdot {\alpha}^{2} + \color{blue}{16 \cdot {\alpha}^{2}}\right)}{{i}^{2}} \]

      5. unpow2 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\alpha \cdot \alpha}{{i}^{2}}\right) - \frac{1}{256} \cdot \frac{4 \cdot \left({\alpha}^{2} - 1\right) + \left(4 \cdot {\alpha}^{2} + \color{blue}{16} \cdot {\alpha}^{2}\right)}{{i}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\alpha \cdot \alpha}{{i}^{2}}\right) - \frac{1}{256} \cdot \frac{4 \cdot \left({\alpha}^{2} - 1\right) + \left(4 \cdot {\alpha}^{2} + \color{blue}{16} \cdot {\alpha}^{2}\right)}{{i}^{2}} \]

      7. pow2 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\alpha \cdot \alpha}{i \cdot i}\right) - \frac{1}{256} \cdot \frac{4 \cdot \left({\alpha}^{2} - 1\right) + \left(4 \cdot {\alpha}^{2} + 16 \cdot \color{blue}{{\alpha}^{2}}\right)}{{i}^{2}} \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\alpha \cdot \alpha}{i \cdot i}\right) - \frac{1}{256} \cdot \frac{4 \cdot \left({\alpha}^{2} - 1\right) + \left(4 \cdot {\alpha}^{2} + 16 \cdot \color{blue}{{\alpha}^{2}}\right)}{{i}^{2}} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\alpha \cdot \alpha}{i \cdot i}\right) - \frac{1}{256} \cdot \frac{4 \cdot \left({\alpha}^{2} - 1\right) + \left(4 \cdot {\alpha}^{2} + 16 \cdot {\alpha}^{2}\right)}{\color{blue}{{i}^{2}}} \]

   7. Applied rewrites 81.8%

      \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{\alpha \cdot \alpha}{i \cdot i}\right) - \color{blue}{0.00390625 \cdot \frac{\mathsf{fma}\left(4, \alpha \cdot \alpha - 1, \mathsf{fma}\left(4, \alpha \cdot \alpha, 16 \cdot \left(\alpha \cdot \alpha\right)\right)\right)}{i \cdot i}} \]

   8. Taylor expanded in alpha around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 82.1

         \[\leadsto 0.0625 + 0.015625 \cdot \frac{1}{i \cdot i} \]

   10. Applied rewrites 82.1%

       \[\leadsto 0.0625 + 0.015625 \cdot \color{blue}{\frac{1}{i \cdot i}} \]

   11. Taylor expanded in i around 0

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

       1. lower-/.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 82.1

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

   13. Applied rewrites 82.1%

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

   if 1.90000000000000009e229 < beta

   1. Initial program 0.0%

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

   2. Taylor expanded in beta around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 33.2

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

   4. Applied rewrites 33.2%

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

   5. Taylor expanded in alpha around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lift-*.f64 33.2

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

   7. Applied rewrites 33.2%

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

   8. Taylor expanded in alpha around 0

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

   10. Applied rewrites 33.4%

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

   11. Taylor expanded in beta around inf

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

       1. pow2 N/A

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

       2. lift-*.f64 33.4

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

   13. Applied rewrites 33.4%

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

Alternative 8: 71.6% accurate, 5.8× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ 0.0625 + \frac{0.015625}{i \cdot i} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i) :precision binary64 (+ 0.0625 (/ 0.015625 (* i i))))

C

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

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 + (0.015625d0 / (i * i))
end function

Java

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

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	return 0.0625 + (0.015625 / (i * i))

Julia

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

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp = code(alpha, beta, i)
	tmp = 0.0625 + (0.015625 / (i * i));
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := N[(0.0625 + N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 15.8%

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

2. Taylor expanded in beta around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-+.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. lower-*.f64 N/A

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

4. Applied rewrites 12.9%

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

5. Taylor expanded in i around inf

   \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{{\alpha}^{2}}{{i}^{2}}\right) - \color{blue}{\frac{1}{256} \cdot \frac{4 \cdot \left({\alpha}^{2} - 1\right) + \left(4 \cdot {\alpha}^{2} + 16 \cdot {\alpha}^{2}\right)}{{i}^{2}}} \]
6. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{{\alpha}^{2}}{{i}^{2}}\right) - \frac{1}{256} \cdot \color{blue}{\frac{4 \cdot \left({\alpha}^{2} - 1\right) + \left(4 \cdot {\alpha}^{2} + 16 \cdot {\alpha}^{2}\right)}{{i}^{2}}} \]

   2. lower-+.f64 N/A

      \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{{\alpha}^{2}}{{i}^{2}}\right) - \frac{1}{256} \cdot \frac{\color{blue}{4 \cdot \left({\alpha}^{2} - 1\right) + \left(4 \cdot {\alpha}^{2} + 16 \cdot {\alpha}^{2}\right)}}{{i}^{2}} \]

   3. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{{\alpha}^{2}}{{i}^{2}}\right) - \frac{1}{256} \cdot \frac{4 \cdot \left({\alpha}^{2} - 1\right) + \color{blue}{\left(4 \cdot {\alpha}^{2} + 16 \cdot {\alpha}^{2}\right)}}{{i}^{2}} \]

   4. lower-/.f64 N/A

      \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{{\alpha}^{2}}{{i}^{2}}\right) - \frac{1}{256} \cdot \frac{4 \cdot \left({\alpha}^{2} - 1\right) + \left(4 \cdot {\alpha}^{2} + \color{blue}{16 \cdot {\alpha}^{2}}\right)}{{i}^{2}} \]

   5. unpow2 N/A

      \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\alpha \cdot \alpha}{{i}^{2}}\right) - \frac{1}{256} \cdot \frac{4 \cdot \left({\alpha}^{2} - 1\right) + \left(4 \cdot {\alpha}^{2} + \color{blue}{16} \cdot {\alpha}^{2}\right)}{{i}^{2}} \]

   6. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\alpha \cdot \alpha}{{i}^{2}}\right) - \frac{1}{256} \cdot \frac{4 \cdot \left({\alpha}^{2} - 1\right) + \left(4 \cdot {\alpha}^{2} + \color{blue}{16} \cdot {\alpha}^{2}\right)}{{i}^{2}} \]

   7. pow2 N/A

      \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\alpha \cdot \alpha}{i \cdot i}\right) - \frac{1}{256} \cdot \frac{4 \cdot \left({\alpha}^{2} - 1\right) + \left(4 \cdot {\alpha}^{2} + 16 \cdot \color{blue}{{\alpha}^{2}}\right)}{{i}^{2}} \]

   8. lift-*.f64 N/A

      \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\alpha \cdot \alpha}{i \cdot i}\right) - \frac{1}{256} \cdot \frac{4 \cdot \left({\alpha}^{2} - 1\right) + \left(4 \cdot {\alpha}^{2} + 16 \cdot \color{blue}{{\alpha}^{2}}\right)}{{i}^{2}} \]

   9. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\alpha \cdot \alpha}{i \cdot i}\right) - \frac{1}{256} \cdot \frac{4 \cdot \left({\alpha}^{2} - 1\right) + \left(4 \cdot {\alpha}^{2} + 16 \cdot {\alpha}^{2}\right)}{\color{blue}{{i}^{2}}} \]

7. Applied rewrites 71.0%

   \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{\alpha \cdot \alpha}{i \cdot i}\right) - \color{blue}{0.00390625 \cdot \frac{\mathsf{fma}\left(4, \alpha \cdot \alpha - 1, \mathsf{fma}\left(4, \alpha \cdot \alpha, 16 \cdot \left(\alpha \cdot \alpha\right)\right)\right)}{i \cdot i}} \]

8. Taylor expanded in alpha around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 71.6

      \[\leadsto 0.0625 + 0.015625 \cdot \frac{1}{i \cdot i} \]

10. Applied rewrites 71.6%

    \[\leadsto 0.0625 + 0.015625 \cdot \color{blue}{\frac{1}{i \cdot i}} \]

11. Taylor expanded in i around 0

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

    1. lower-/.f64 N/A

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

    2. pow2 N/A

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

    3. lift-*.f64 71.6

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

13. Applied rewrites 71.6%

    \[\leadsto 0.0625 + \frac{0.015625}{i \cdot i} \]
14. Add Preprocessing

Alternative 9: 71.3% accurate, 115.0× 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 15.8%

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

2. Taylor expanded in i around inf

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

4. Applied rewrites 71.3%

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

Reproduce

herbie shell --seed 2025101 
(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)))