Octave 3.8, jcobi/3

Percentage Accurate: 94.1% → 99.8%

Time: 6.5s

Alternatives: 21

Speedup: 1.7×

Specification

\[\alpha > -1 \land \beta > -1\]

Math

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 1.0))))
   (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) t_0) t_0) (+ t_0 1.0))))

C

double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + (2.0 * 1.0);
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 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)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    t_0 = (alpha + beta) + (2.0d0 * 1.0d0)
    code = (((((alpha + beta) + (beta * alpha)) + 1.0d0) / t_0) / t_0) / (t_0 + 1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.1% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 1.0))))
   (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) t_0) t_0) (+ t_0 1.0))))

C

double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + (2.0 * 1.0);
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 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)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    t_0 = (alpha + beta) + (2.0d0 * 1.0d0)
    code = (((((alpha + beta) + (beta * alpha)) + 1.0d0) / t_0) / t_0) / (t_0 + 1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := -2 - \mathsf{max}\left(\alpha, \beta\right)\\ t_1 := \left(\mathsf{min}\left(\alpha, \beta\right) + \mathsf{max}\left(\alpha, \beta\right)\right) + 2 \cdot 1\\ t_2 := t\_0 - \mathsf{min}\left(\alpha, \beta\right)\\ t_3 := \mathsf{min}\left(\alpha, \beta\right) - t\_0\\ \mathbf{if}\;\mathsf{max}\left(\alpha, \beta\right) \leq 8 \cdot 10^{+15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{min}\left(\alpha, \beta\right) - -1, \mathsf{max}\left(\alpha, \beta\right), \mathsf{min}\left(\alpha, \beta\right)\right), t\_3, t\_3 \cdot 1\right) \cdot \frac{-1}{t\_2}}{\left(t\_2 \cdot t\_2\right) \cdot \left(\left(\mathsf{max}\left(\alpha, \beta\right) + \mathsf{min}\left(\alpha, \beta\right)\right) - -3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 - \frac{1}{\mathsf{max}\left(\alpha, \beta\right)}\right) + \frac{\mathsf{min}\left(\alpha, \beta\right)}{t\_3} \cdot \mathsf{max}\left(\alpha, \beta\right)}{t\_1}}{t\_1 + 1}\\ \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (- -2.0 (fmax alpha beta)))
        (t_1 (+ (+ (fmin alpha beta) (fmax alpha beta)) (* 2.0 1.0)))
        (t_2 (- t_0 (fmin alpha beta)))
        (t_3 (- (fmin alpha beta) t_0)))
   (if (<= (fmax alpha beta) 8e+15)
     (/
      (*
       (fma
        (fma (- (fmin alpha beta) -1.0) (fmax alpha beta) (fmin alpha beta))
        t_3
        (* t_3 1.0))
       (/ -1.0 t_2))
      (* (* t_2 t_2) (- (+ (fmax alpha beta) (fmin alpha beta)) -3.0)))
     (/
      (/
       (+
        (- 1.0 (/ 1.0 (fmax alpha beta)))
        (* (/ (fmin alpha beta) t_3) (fmax alpha beta)))
       t_1)
      (+ t_1 1.0)))))

C

double code(double alpha, double beta) {
	double t_0 = -2.0 - fmax(alpha, beta);
	double t_1 = (fmin(alpha, beta) + fmax(alpha, beta)) + (2.0 * 1.0);
	double t_2 = t_0 - fmin(alpha, beta);
	double t_3 = fmin(alpha, beta) - t_0;
	double tmp;
	if (fmax(alpha, beta) <= 8e+15) {
		tmp = (fma(fma((fmin(alpha, beta) - -1.0), fmax(alpha, beta), fmin(alpha, beta)), t_3, (t_3 * 1.0)) * (-1.0 / t_2)) / ((t_2 * t_2) * ((fmax(alpha, beta) + fmin(alpha, beta)) - -3.0));
	} else {
		tmp = (((1.0 - (1.0 / fmax(alpha, beta))) + ((fmin(alpha, beta) / t_3) * fmax(alpha, beta))) / t_1) / (t_1 + 1.0);
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(-2.0 - fmax(alpha, beta))
	t_1 = Float64(Float64(fmin(alpha, beta) + fmax(alpha, beta)) + Float64(2.0 * 1.0))
	t_2 = Float64(t_0 - fmin(alpha, beta))
	t_3 = Float64(fmin(alpha, beta) - t_0)
	tmp = 0.0
	if (fmax(alpha, beta) <= 8e+15)
		tmp = Float64(Float64(fma(fma(Float64(fmin(alpha, beta) - -1.0), fmax(alpha, beta), fmin(alpha, beta)), t_3, Float64(t_3 * 1.0)) * Float64(-1.0 / t_2)) / Float64(Float64(t_2 * t_2) * Float64(Float64(fmax(alpha, beta) + fmin(alpha, beta)) - -3.0)));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - Float64(1.0 / fmax(alpha, beta))) + Float64(Float64(fmin(alpha, beta) / t_3) * fmax(alpha, beta))) / t_1) / Float64(t_1 + 1.0));
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(-2.0 - N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Min[alpha, beta], $MachinePrecision] + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision] + N[(2.0 * 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 - N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Min[alpha, beta], $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[N[Max[alpha, beta], $MachinePrecision], 8e+15], N[(N[(N[(N[(N[(N[Min[alpha, beta], $MachinePrecision] - -1.0), $MachinePrecision] * N[Max[alpha, beta], $MachinePrecision] + N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision] * t$95$3 + N[(t$95$3 * 1.0), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$2 * t$95$2), $MachinePrecision] * N[(N[(N[Max[alpha, beta], $MachinePrecision] + N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision] - -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 - N[(1.0 / N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Min[alpha, beta], $MachinePrecision] / t$95$3), $MachinePrecision] * N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 8e15

   1. Initial program 94.1%

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

   3. Applied rewrites 60.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\alpha - -1, \beta, \alpha\right), \alpha - \left(-2 - \beta\right), \left(\alpha - \left(-2 - \beta\right)\right) \cdot 1\right) \cdot \frac{-1}{\left(-2 - \beta\right) - \alpha}}{\left(\left(\left(-2 - \beta\right) - \alpha\right) \cdot \left(\left(-2 - \beta\right) - \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) - -3\right)}} \]

   if 8e15 < beta

   1. Initial program 94.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. div-add N/A

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

      6. sum-to-mult N/A

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

      7. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 94.1%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. div-add N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lift--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. lower--.f64 99.8%

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

   5. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sum-to-mult-rev N/A

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

      5. lift-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

      8. lower-+.f64 N/A

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in beta around inf

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 41.0%

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

   10. Applied rewrites 41.0%

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

Alternative 2: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \left(\mathsf{min}\left(\alpha, \beta\right) + \mathsf{max}\left(\alpha, \beta\right)\right) + 2 \cdot 1\\ t_1 := -2 - \mathsf{max}\left(\alpha, \beta\right)\\ t_2 := \mathsf{min}\left(\alpha, \beta\right) - t\_1\\ \mathbf{if}\;\mathsf{max}\left(\alpha, \beta\right) \leq 5000000:\\ \;\;\;\;\frac{-1}{-3 - \left(\mathsf{max}\left(\alpha, \beta\right) + \mathsf{min}\left(\alpha, \beta\right)\right)} \cdot \frac{-1 - \mathsf{fma}\left(\mathsf{min}\left(\alpha, \beta\right) - -1, \mathsf{max}\left(\alpha, \beta\right), \mathsf{min}\left(\alpha, \beta\right)\right)}{\left(t\_1 - \mathsf{min}\left(\alpha, \beta\right)\right) \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 - \frac{1}{\mathsf{max}\left(\alpha, \beta\right)}\right) + \frac{\mathsf{min}\left(\alpha, \beta\right)}{t\_2} \cdot \mathsf{max}\left(\alpha, \beta\right)}{t\_0}}{t\_0 + 1}\\ \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ (fmin alpha beta) (fmax alpha beta)) (* 2.0 1.0)))
        (t_1 (- -2.0 (fmax alpha beta)))
        (t_2 (- (fmin alpha beta) t_1)))
   (if (<= (fmax alpha beta) 5000000.0)
     (*
      (/ -1.0 (- -3.0 (+ (fmax alpha beta) (fmin alpha beta))))
      (/
       (-
        -1.0
        (fma (- (fmin alpha beta) -1.0) (fmax alpha beta) (fmin alpha beta)))
       (* (- t_1 (fmin alpha beta)) t_2)))
     (/
      (/
       (+
        (- 1.0 (/ 1.0 (fmax alpha beta)))
        (* (/ (fmin alpha beta) t_2) (fmax alpha beta)))
       t_0)
      (+ t_0 1.0)))))

C

double code(double alpha, double beta) {
	double t_0 = (fmin(alpha, beta) + fmax(alpha, beta)) + (2.0 * 1.0);
	double t_1 = -2.0 - fmax(alpha, beta);
	double t_2 = fmin(alpha, beta) - t_1;
	double tmp;
	if (fmax(alpha, beta) <= 5000000.0) {
		tmp = (-1.0 / (-3.0 - (fmax(alpha, beta) + fmin(alpha, beta)))) * ((-1.0 - fma((fmin(alpha, beta) - -1.0), fmax(alpha, beta), fmin(alpha, beta))) / ((t_1 - fmin(alpha, beta)) * t_2));
	} else {
		tmp = (((1.0 - (1.0 / fmax(alpha, beta))) + ((fmin(alpha, beta) / t_2) * fmax(alpha, beta))) / t_0) / (t_0 + 1.0);
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(fmin(alpha, beta) + fmax(alpha, beta)) + Float64(2.0 * 1.0))
	t_1 = Float64(-2.0 - fmax(alpha, beta))
	t_2 = Float64(fmin(alpha, beta) - t_1)
	tmp = 0.0
	if (fmax(alpha, beta) <= 5000000.0)
		tmp = Float64(Float64(-1.0 / Float64(-3.0 - Float64(fmax(alpha, beta) + fmin(alpha, beta)))) * Float64(Float64(-1.0 - fma(Float64(fmin(alpha, beta) - -1.0), fmax(alpha, beta), fmin(alpha, beta))) / Float64(Float64(t_1 - fmin(alpha, beta)) * t_2)));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - Float64(1.0 / fmax(alpha, beta))) + Float64(Float64(fmin(alpha, beta) / t_2) * fmax(alpha, beta))) / t_0) / Float64(t_0 + 1.0));
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(N[Min[alpha, beta], $MachinePrecision] + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision] + N[(2.0 * 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-2.0 - N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Min[alpha, beta], $MachinePrecision] - t$95$1), $MachinePrecision]}, If[LessEqual[N[Max[alpha, beta], $MachinePrecision], 5000000.0], N[(N[(-1.0 / N[(-3.0 - N[(N[Max[alpha, beta], $MachinePrecision] + N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(-1.0 - N[(N[(N[Min[alpha, beta], $MachinePrecision] - -1.0), $MachinePrecision] * N[Max[alpha, beta], $MachinePrecision] + N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$1 - N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 - N[(1.0 / N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Min[alpha, beta], $MachinePrecision] / t$95$2), $MachinePrecision] * N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5e6

   1. Initial program 94.1%

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

   3. Applied rewrites 92.6%

      \[\leadsto \color{blue}{\frac{-1}{-3 - \left(\beta + \alpha\right)} \cdot \frac{-1 - \mathsf{fma}\left(\alpha - -1, \beta, \alpha\right)}{\left(\left(-2 - \beta\right) - \alpha\right) \cdot \left(\alpha - \left(-2 - \beta\right)\right)}} \]

   if 5e6 < beta

   1. Initial program 94.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. div-add N/A

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

      6. sum-to-mult N/A

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

      7. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 94.1%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. div-add N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lift--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. lower--.f64 99.8%

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

   5. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sum-to-mult-rev N/A

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

      5. lift-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

      8. lower-+.f64 N/A

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in beta around inf

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 41.0%

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

   10. Applied rewrites 41.0%

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

Alternative 3: 99.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.1%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. associate-+l+ N/A

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

   5. div-add N/A

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

   6. sum-to-mult N/A

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

   7. lower-unsound-*.f64 N/A

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

3. Applied rewrites 94.1%

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

   1. lift-/.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. div-add N/A

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

   4. associate-/l* N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. frac-2neg N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 N/A

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

   10. lift--.f64 N/A

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

   11. sub-negate-rev N/A

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

   12. lower--.f64 99.8%

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

5. Applied rewrites 99.8%

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

Alternative 4: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.1%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. associate-+l+ N/A

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

   5. div-add N/A

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

   6. sum-to-mult N/A

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

   7. lower-unsound-*.f64 N/A

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

3. Applied rewrites 94.1%

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

   1. lift-/.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. div-add N/A

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

   4. associate-/l* N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. frac-2neg N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 N/A

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

   10. lift--.f64 N/A

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

   11. sub-negate-rev N/A

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

   12. lower--.f64 99.8%

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

5. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. sum-to-mult-rev N/A

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

   5. lift-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. associate-+r+ N/A

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

   8. lower-+.f64 N/A

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

7. Applied rewrites 99.8%

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

Alternative 5: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\alpha, \beta\right) - \left(-2 - \mathsf{max}\left(\alpha, \beta\right)\right)\\ t_1 := \left(\mathsf{min}\left(\alpha, \beta\right) + \mathsf{max}\left(\alpha, \beta\right)\right) + 2 \cdot 1\\ \mathbf{if}\;\mathsf{max}\left(\alpha, \beta\right) \leq 2 \cdot 10^{+144}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(\mathsf{min}\left(\alpha, \beta\right) - -1, \mathsf{max}\left(\alpha, \beta\right), \mathsf{min}\left(\alpha, \beta\right)\right) - -1}{t\_0}}{\left(\mathsf{max}\left(\alpha, \beta\right) + \mathsf{min}\left(\alpha, \beta\right)\right) - -3}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 \cdot \left(-1 \cdot \mathsf{min}\left(\alpha, \beta\right) - 1\right)}{t\_1}}{t\_1 + 1}\\ \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (- (fmin alpha beta) (- -2.0 (fmax alpha beta))))
        (t_1 (+ (+ (fmin alpha beta) (fmax alpha beta)) (* 2.0 1.0))))
   (if (<= (fmax alpha beta) 2e+144)
     (/
      (/
       (/
        (-
         (fma (- (fmin alpha beta) -1.0) (fmax alpha beta) (fmin alpha beta))
         -1.0)
        t_0)
       (- (+ (fmax alpha beta) (fmin alpha beta)) -3.0))
      t_0)
     (/ (/ (* -1.0 (- (* -1.0 (fmin alpha beta)) 1.0)) t_1) (+ t_1 1.0)))))

C

double code(double alpha, double beta) {
	double t_0 = fmin(alpha, beta) - (-2.0 - fmax(alpha, beta));
	double t_1 = (fmin(alpha, beta) + fmax(alpha, beta)) + (2.0 * 1.0);
	double tmp;
	if (fmax(alpha, beta) <= 2e+144) {
		tmp = (((fma((fmin(alpha, beta) - -1.0), fmax(alpha, beta), fmin(alpha, beta)) - -1.0) / t_0) / ((fmax(alpha, beta) + fmin(alpha, beta)) - -3.0)) / t_0;
	} else {
		tmp = ((-1.0 * ((-1.0 * fmin(alpha, beta)) - 1.0)) / t_1) / (t_1 + 1.0);
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(fmin(alpha, beta) - Float64(-2.0 - fmax(alpha, beta)))
	t_1 = Float64(Float64(fmin(alpha, beta) + fmax(alpha, beta)) + Float64(2.0 * 1.0))
	tmp = 0.0
	if (fmax(alpha, beta) <= 2e+144)
		tmp = Float64(Float64(Float64(Float64(fma(Float64(fmin(alpha, beta) - -1.0), fmax(alpha, beta), fmin(alpha, beta)) - -1.0) / t_0) / Float64(Float64(fmax(alpha, beta) + fmin(alpha, beta)) - -3.0)) / t_0);
	else
		tmp = Float64(Float64(Float64(-1.0 * Float64(Float64(-1.0 * fmin(alpha, beta)) - 1.0)) / t_1) / Float64(t_1 + 1.0));
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[Min[alpha, beta], $MachinePrecision] - N[(-2.0 - N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Min[alpha, beta], $MachinePrecision] + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision] + N[(2.0 * 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Max[alpha, beta], $MachinePrecision], 2e+144], N[(N[(N[(N[(N[(N[(N[Min[alpha, beta], $MachinePrecision] - -1.0), $MachinePrecision] * N[Max[alpha, beta], $MachinePrecision] + N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(N[Max[alpha, beta], $MachinePrecision] + N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision] - -3.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(-1.0 * N[(N[(-1.0 * N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.00000000000000005e144

   1. Initial program 94.1%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 94.1%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\mathsf{fma}\left(\alpha - -1, \beta, \alpha\right) - -1}{\alpha - \left(-2 - \beta\right)}}{\left(\beta + \alpha\right) - -3}}{\alpha - \left(-2 - \beta\right)}} \]

   if 2.00000000000000005e144 < beta

   1. Initial program 94.1%

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

   2. Taylor expanded in beta around -inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 37.6%

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

   4. Applied rewrites 37.6%

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

Alternative 6: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \left(-2 - \mathsf{max}\left(\alpha, \beta\right)\right) - \mathsf{min}\left(\alpha, \beta\right)\\ t_1 := \left(\mathsf{min}\left(\alpha, \beta\right) + \mathsf{max}\left(\alpha, \beta\right)\right) + 2 \cdot 1\\ \mathbf{if}\;\mathsf{max}\left(\alpha, \beta\right) \leq 4 \cdot 10^{+141}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{min}\left(\alpha, \beta\right) - -1, \mathsf{max}\left(\alpha, \beta\right), \mathsf{min}\left(\alpha, \beta\right)\right) - -1}{\left(\mathsf{max}\left(\alpha, \beta\right) + \mathsf{min}\left(\alpha, \beta\right)\right) - -3}}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 \cdot \left(-1 \cdot \mathsf{min}\left(\alpha, \beta\right) - 1\right)}{t\_1}}{t\_1 + 1}\\ \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (- (- -2.0 (fmax alpha beta)) (fmin alpha beta)))
        (t_1 (+ (+ (fmin alpha beta) (fmax alpha beta)) (* 2.0 1.0))))
   (if (<= (fmax alpha beta) 4e+141)
     (/
      (/
       (-
        (fma (- (fmin alpha beta) -1.0) (fmax alpha beta) (fmin alpha beta))
        -1.0)
       (- (+ (fmax alpha beta) (fmin alpha beta)) -3.0))
      (* t_0 t_0))
     (/ (/ (* -1.0 (- (* -1.0 (fmin alpha beta)) 1.0)) t_1) (+ t_1 1.0)))))

C

double code(double alpha, double beta) {
	double t_0 = (-2.0 - fmax(alpha, beta)) - fmin(alpha, beta);
	double t_1 = (fmin(alpha, beta) + fmax(alpha, beta)) + (2.0 * 1.0);
	double tmp;
	if (fmax(alpha, beta) <= 4e+141) {
		tmp = ((fma((fmin(alpha, beta) - -1.0), fmax(alpha, beta), fmin(alpha, beta)) - -1.0) / ((fmax(alpha, beta) + fmin(alpha, beta)) - -3.0)) / (t_0 * t_0);
	} else {
		tmp = ((-1.0 * ((-1.0 * fmin(alpha, beta)) - 1.0)) / t_1) / (t_1 + 1.0);
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(-2.0 - fmax(alpha, beta)) - fmin(alpha, beta))
	t_1 = Float64(Float64(fmin(alpha, beta) + fmax(alpha, beta)) + Float64(2.0 * 1.0))
	tmp = 0.0
	if (fmax(alpha, beta) <= 4e+141)
		tmp = Float64(Float64(Float64(fma(Float64(fmin(alpha, beta) - -1.0), fmax(alpha, beta), fmin(alpha, beta)) - -1.0) / Float64(Float64(fmax(alpha, beta) + fmin(alpha, beta)) - -3.0)) / Float64(t_0 * t_0));
	else
		tmp = Float64(Float64(Float64(-1.0 * Float64(Float64(-1.0 * fmin(alpha, beta)) - 1.0)) / t_1) / Float64(t_1 + 1.0));
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(-2.0 - N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision] - N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Min[alpha, beta], $MachinePrecision] + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision] + N[(2.0 * 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Max[alpha, beta], $MachinePrecision], 4e+141], N[(N[(N[(N[(N[(N[Min[alpha, beta], $MachinePrecision] - -1.0), $MachinePrecision] * N[Max[alpha, beta], $MachinePrecision] + N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] / N[(N[(N[Max[alpha, beta], $MachinePrecision] + N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision] - -3.0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-1.0 * N[(N[(-1.0 * N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.00000000000000007e141

   1. Initial program 94.1%

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

   3. Applied rewrites 92.7%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\alpha - -1, \beta, \alpha\right) - -1}{\left(\beta + \alpha\right) - -3}}{\left(\left(-2 - \beta\right) - \alpha\right) \cdot \left(\left(-2 - \beta\right) - \alpha\right)}} \]

   if 4.00000000000000007e141 < beta

   1. Initial program 94.1%

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

   2. Taylor expanded in beta around -inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 37.6%

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

   4. Applied rewrites 37.6%

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

Alternative 7: 99.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \left(-2 - \mathsf{max}\left(\alpha, \beta\right)\right) - \mathsf{min}\left(\alpha, \beta\right)\\ t_1 := \left(\mathsf{min}\left(\alpha, \beta\right) + \mathsf{max}\left(\alpha, \beta\right)\right) + 2 \cdot 1\\ \mathbf{if}\;\mathsf{max}\left(\alpha, \beta\right) \leq 4 \cdot 10^{+141}:\\ \;\;\;\;\frac{\frac{\left(\mathsf{max}\left(\alpha, \beta\right) - -1\right) \cdot \left(\mathsf{min}\left(\alpha, \beta\right) - -1\right)}{t\_0 \cdot t\_0}}{\mathsf{min}\left(\alpha, \beta\right) - \left(-3 - \mathsf{max}\left(\alpha, \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 \cdot \left(-1 \cdot \mathsf{min}\left(\alpha, \beta\right) - 1\right)}{t\_1}}{t\_1 + 1}\\ \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (- (- -2.0 (fmax alpha beta)) (fmin alpha beta)))
        (t_1 (+ (+ (fmin alpha beta) (fmax alpha beta)) (* 2.0 1.0))))
   (if (<= (fmax alpha beta) 4e+141)
     (/
      (/ (* (- (fmax alpha beta) -1.0) (- (fmin alpha beta) -1.0)) (* t_0 t_0))
      (- (fmin alpha beta) (- -3.0 (fmax alpha beta))))
     (/ (/ (* -1.0 (- (* -1.0 (fmin alpha beta)) 1.0)) t_1) (+ t_1 1.0)))))

C

double code(double alpha, double beta) {
	double t_0 = (-2.0 - fmax(alpha, beta)) - fmin(alpha, beta);
	double t_1 = (fmin(alpha, beta) + fmax(alpha, beta)) + (2.0 * 1.0);
	double tmp;
	if (fmax(alpha, beta) <= 4e+141) {
		tmp = (((fmax(alpha, beta) - -1.0) * (fmin(alpha, beta) - -1.0)) / (t_0 * t_0)) / (fmin(alpha, beta) - (-3.0 - fmax(alpha, beta)));
	} else {
		tmp = ((-1.0 * ((-1.0 * fmin(alpha, beta)) - 1.0)) / t_1) / (t_1 + 1.0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((-2.0d0) - fmax(alpha, beta)) - fmin(alpha, beta)
    t_1 = (fmin(alpha, beta) + fmax(alpha, beta)) + (2.0d0 * 1.0d0)
    if (fmax(alpha, beta) <= 4d+141) then
        tmp = (((fmax(alpha, beta) - (-1.0d0)) * (fmin(alpha, beta) - (-1.0d0))) / (t_0 * t_0)) / (fmin(alpha, beta) - ((-3.0d0) - fmax(alpha, beta)))
    else
        tmp = (((-1.0d0) * (((-1.0d0) * fmin(alpha, beta)) - 1.0d0)) / t_1) / (t_1 + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (-2.0 - fmax(alpha, beta)) - fmin(alpha, beta);
	double t_1 = (fmin(alpha, beta) + fmax(alpha, beta)) + (2.0 * 1.0);
	double tmp;
	if (fmax(alpha, beta) <= 4e+141) {
		tmp = (((fmax(alpha, beta) - -1.0) * (fmin(alpha, beta) - -1.0)) / (t_0 * t_0)) / (fmin(alpha, beta) - (-3.0 - fmax(alpha, beta)));
	} else {
		tmp = ((-1.0 * ((-1.0 * fmin(alpha, beta)) - 1.0)) / t_1) / (t_1 + 1.0);
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (-2.0 - fmax(alpha, beta)) - fmin(alpha, beta)
	t_1 = (fmin(alpha, beta) + fmax(alpha, beta)) + (2.0 * 1.0)
	tmp = 0
	if fmax(alpha, beta) <= 4e+141:
		tmp = (((fmax(alpha, beta) - -1.0) * (fmin(alpha, beta) - -1.0)) / (t_0 * t_0)) / (fmin(alpha, beta) - (-3.0 - fmax(alpha, beta)))
	else:
		tmp = ((-1.0 * ((-1.0 * fmin(alpha, beta)) - 1.0)) / t_1) / (t_1 + 1.0)
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(-2.0 - fmax(alpha, beta)) - fmin(alpha, beta))
	t_1 = Float64(Float64(fmin(alpha, beta) + fmax(alpha, beta)) + Float64(2.0 * 1.0))
	tmp = 0.0
	if (fmax(alpha, beta) <= 4e+141)
		tmp = Float64(Float64(Float64(Float64(fmax(alpha, beta) - -1.0) * Float64(fmin(alpha, beta) - -1.0)) / Float64(t_0 * t_0)) / Float64(fmin(alpha, beta) - Float64(-3.0 - fmax(alpha, beta))));
	else
		tmp = Float64(Float64(Float64(-1.0 * Float64(Float64(-1.0 * fmin(alpha, beta)) - 1.0)) / t_1) / Float64(t_1 + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (-2.0 - max(alpha, beta)) - min(alpha, beta);
	t_1 = (min(alpha, beta) + max(alpha, beta)) + (2.0 * 1.0);
	tmp = 0.0;
	if (max(alpha, beta) <= 4e+141)
		tmp = (((max(alpha, beta) - -1.0) * (min(alpha, beta) - -1.0)) / (t_0 * t_0)) / (min(alpha, beta) - (-3.0 - max(alpha, beta)));
	else
		tmp = ((-1.0 * ((-1.0 * min(alpha, beta)) - 1.0)) / t_1) / (t_1 + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(-2.0 - N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision] - N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Min[alpha, beta], $MachinePrecision] + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision] + N[(2.0 * 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Max[alpha, beta], $MachinePrecision], 4e+141], N[(N[(N[(N[(N[Max[alpha, beta], $MachinePrecision] - -1.0), $MachinePrecision] * N[(N[Min[alpha, beta], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[Min[alpha, beta], $MachinePrecision] - N[(-3.0 - N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-1.0 * N[(N[(-1.0 * N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.00000000000000007e141

   1. Initial program 94.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. div-add N/A

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

      6. sum-to-mult N/A

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

      7. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 94.1%

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

   5. Applied rewrites 92.6%

      \[\leadsto \color{blue}{\frac{\frac{\left(\beta - -1\right) \cdot \left(\alpha - -1\right)}{\left(\left(-2 - \beta\right) - \alpha\right) \cdot \left(\left(-2 - \beta\right) - \alpha\right)}}{\alpha - \left(-3 - \beta\right)}} \]

   if 4.00000000000000007e141 < beta

   1. Initial program 94.1%

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

   2. Taylor expanded in beta around -inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 37.6%

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

   4. Applied rewrites 37.6%

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

Alternative 8: 99.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \left(-2 - \mathsf{max}\left(\alpha, \beta\right)\right) - \mathsf{min}\left(\alpha, \beta\right)\\ t_1 := \mathsf{min}\left(\alpha, \beta\right) - \left(-3 - \mathsf{max}\left(\alpha, \beta\right)\right)\\ t_2 := \mathsf{min}\left(\alpha, \beta\right) - -1\\ \mathbf{if}\;\mathsf{max}\left(\alpha, \beta\right) \leq 4 \cdot 10^{+141}:\\ \;\;\;\;\frac{\frac{\left(\mathsf{max}\left(\alpha, \beta\right) - -1\right) \cdot t\_2}{t\_0 \cdot t\_0}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_2}{\mathsf{max}\left(\alpha, \beta\right)}}{t\_1}\\ \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (- (- -2.0 (fmax alpha beta)) (fmin alpha beta)))
        (t_1 (- (fmin alpha beta) (- -3.0 (fmax alpha beta))))
        (t_2 (- (fmin alpha beta) -1.0)))
   (if (<= (fmax alpha beta) 4e+141)
     (/ (/ (* (- (fmax alpha beta) -1.0) t_2) (* t_0 t_0)) t_1)
     (/ (/ t_2 (fmax alpha beta)) t_1))))

C

double code(double alpha, double beta) {
	double t_0 = (-2.0 - fmax(alpha, beta)) - fmin(alpha, beta);
	double t_1 = fmin(alpha, beta) - (-3.0 - fmax(alpha, beta));
	double t_2 = fmin(alpha, beta) - -1.0;
	double tmp;
	if (fmax(alpha, beta) <= 4e+141) {
		tmp = (((fmax(alpha, beta) - -1.0) * t_2) / (t_0 * t_0)) / t_1;
	} else {
		tmp = (t_2 / fmax(alpha, beta)) / t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = ((-2.0d0) - fmax(alpha, beta)) - fmin(alpha, beta)
    t_1 = fmin(alpha, beta) - ((-3.0d0) - fmax(alpha, beta))
    t_2 = fmin(alpha, beta) - (-1.0d0)
    if (fmax(alpha, beta) <= 4d+141) then
        tmp = (((fmax(alpha, beta) - (-1.0d0)) * t_2) / (t_0 * t_0)) / t_1
    else
        tmp = (t_2 / fmax(alpha, beta)) / t_1
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (-2.0 - fmax(alpha, beta)) - fmin(alpha, beta);
	double t_1 = fmin(alpha, beta) - (-3.0 - fmax(alpha, beta));
	double t_2 = fmin(alpha, beta) - -1.0;
	double tmp;
	if (fmax(alpha, beta) <= 4e+141) {
		tmp = (((fmax(alpha, beta) - -1.0) * t_2) / (t_0 * t_0)) / t_1;
	} else {
		tmp = (t_2 / fmax(alpha, beta)) / t_1;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (-2.0 - fmax(alpha, beta)) - fmin(alpha, beta)
	t_1 = fmin(alpha, beta) - (-3.0 - fmax(alpha, beta))
	t_2 = fmin(alpha, beta) - -1.0
	tmp = 0
	if fmax(alpha, beta) <= 4e+141:
		tmp = (((fmax(alpha, beta) - -1.0) * t_2) / (t_0 * t_0)) / t_1
	else:
		tmp = (t_2 / fmax(alpha, beta)) / t_1
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(-2.0 - fmax(alpha, beta)) - fmin(alpha, beta))
	t_1 = Float64(fmin(alpha, beta) - Float64(-3.0 - fmax(alpha, beta)))
	t_2 = Float64(fmin(alpha, beta) - -1.0)
	tmp = 0.0
	if (fmax(alpha, beta) <= 4e+141)
		tmp = Float64(Float64(Float64(Float64(fmax(alpha, beta) - -1.0) * t_2) / Float64(t_0 * t_0)) / t_1);
	else
		tmp = Float64(Float64(t_2 / fmax(alpha, beta)) / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (-2.0 - max(alpha, beta)) - min(alpha, beta);
	t_1 = min(alpha, beta) - (-3.0 - max(alpha, beta));
	t_2 = min(alpha, beta) - -1.0;
	tmp = 0.0;
	if (max(alpha, beta) <= 4e+141)
		tmp = (((max(alpha, beta) - -1.0) * t_2) / (t_0 * t_0)) / t_1;
	else
		tmp = (t_2 / max(alpha, beta)) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(-2.0 - N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision] - N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Min[alpha, beta], $MachinePrecision] - N[(-3.0 - N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Min[alpha, beta], $MachinePrecision] - -1.0), $MachinePrecision]}, If[LessEqual[N[Max[alpha, beta], $MachinePrecision], 4e+141], N[(N[(N[(N[(N[Max[alpha, beta], $MachinePrecision] - -1.0), $MachinePrecision] * t$95$2), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(t$95$2 / N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.00000000000000007e141

   1. Initial program 94.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. div-add N/A

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

      6. sum-to-mult N/A

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

      7. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 94.1%

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

   5. Applied rewrites 92.6%

      \[\leadsto \color{blue}{\frac{\frac{\left(\beta - -1\right) \cdot \left(\alpha - -1\right)}{\left(\left(-2 - \beta\right) - \alpha\right) \cdot \left(\left(-2 - \beta\right) - \alpha\right)}}{\alpha - \left(-3 - \beta\right)}} \]

   if 4.00000000000000007e141 < beta

   1. Initial program 94.1%

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

   2. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 29.6%

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

   4. Applied rewrites 29.6%

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

      1. metadata-eval 29.6%

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

      2. metadata-eval 29.6%

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

   6. Applied rewrites 29.6%

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

Alternative 9: 99.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\alpha, \beta\right) - -1\\ t_1 := \mathsf{min}\left(\alpha, \beta\right) - \left(-2 - \mathsf{max}\left(\alpha, \beta\right)\right)\\ \mathbf{if}\;\mathsf{max}\left(\alpha, \beta\right) \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\frac{\left(\mathsf{max}\left(\alpha, \beta\right) - -1\right) \cdot t\_0}{t\_1 \cdot \left(\left(\left(\mathsf{max}\left(\alpha, \beta\right) + \mathsf{min}\left(\alpha, \beta\right)\right) - -3\right) \cdot t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0}{\mathsf{max}\left(\alpha, \beta\right)}}{\mathsf{min}\left(\alpha, \beta\right) - \left(-3 - \mathsf{max}\left(\alpha, \beta\right)\right)}\\ \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (- (fmin alpha beta) -1.0))
        (t_1 (- (fmin alpha beta) (- -2.0 (fmax alpha beta)))))
   (if (<= (fmax alpha beta) 5e+16)
     (/
      (* (- (fmax alpha beta) -1.0) t_0)
      (* t_1 (* (- (+ (fmax alpha beta) (fmin alpha beta)) -3.0) t_1)))
     (/
      (/ t_0 (fmax alpha beta))
      (- (fmin alpha beta) (- -3.0 (fmax alpha beta)))))))

C

double code(double alpha, double beta) {
	double t_0 = fmin(alpha, beta) - -1.0;
	double t_1 = fmin(alpha, beta) - (-2.0 - fmax(alpha, beta));
	double tmp;
	if (fmax(alpha, beta) <= 5e+16) {
		tmp = ((fmax(alpha, beta) - -1.0) * t_0) / (t_1 * (((fmax(alpha, beta) + fmin(alpha, beta)) - -3.0) * t_1));
	} else {
		tmp = (t_0 / fmax(alpha, beta)) / (fmin(alpha, beta) - (-3.0 - fmax(alpha, beta)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = fmin(alpha, beta) - (-1.0d0)
    t_1 = fmin(alpha, beta) - ((-2.0d0) - fmax(alpha, beta))
    if (fmax(alpha, beta) <= 5d+16) then
        tmp = ((fmax(alpha, beta) - (-1.0d0)) * t_0) / (t_1 * (((fmax(alpha, beta) + fmin(alpha, beta)) - (-3.0d0)) * t_1))
    else
        tmp = (t_0 / fmax(alpha, beta)) / (fmin(alpha, beta) - ((-3.0d0) - fmax(alpha, beta)))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = fmin(alpha, beta) - -1.0;
	double t_1 = fmin(alpha, beta) - (-2.0 - fmax(alpha, beta));
	double tmp;
	if (fmax(alpha, beta) <= 5e+16) {
		tmp = ((fmax(alpha, beta) - -1.0) * t_0) / (t_1 * (((fmax(alpha, beta) + fmin(alpha, beta)) - -3.0) * t_1));
	} else {
		tmp = (t_0 / fmax(alpha, beta)) / (fmin(alpha, beta) - (-3.0 - fmax(alpha, beta)));
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = fmin(alpha, beta) - -1.0
	t_1 = fmin(alpha, beta) - (-2.0 - fmax(alpha, beta))
	tmp = 0
	if fmax(alpha, beta) <= 5e+16:
		tmp = ((fmax(alpha, beta) - -1.0) * t_0) / (t_1 * (((fmax(alpha, beta) + fmin(alpha, beta)) - -3.0) * t_1))
	else:
		tmp = (t_0 / fmax(alpha, beta)) / (fmin(alpha, beta) - (-3.0 - fmax(alpha, beta)))
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(fmin(alpha, beta) - -1.0)
	t_1 = Float64(fmin(alpha, beta) - Float64(-2.0 - fmax(alpha, beta)))
	tmp = 0.0
	if (fmax(alpha, beta) <= 5e+16)
		tmp = Float64(Float64(Float64(fmax(alpha, beta) - -1.0) * t_0) / Float64(t_1 * Float64(Float64(Float64(fmax(alpha, beta) + fmin(alpha, beta)) - -3.0) * t_1)));
	else
		tmp = Float64(Float64(t_0 / fmax(alpha, beta)) / Float64(fmin(alpha, beta) - Float64(-3.0 - fmax(alpha, beta))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = min(alpha, beta) - -1.0;
	t_1 = min(alpha, beta) - (-2.0 - max(alpha, beta));
	tmp = 0.0;
	if (max(alpha, beta) <= 5e+16)
		tmp = ((max(alpha, beta) - -1.0) * t_0) / (t_1 * (((max(alpha, beta) + min(alpha, beta)) - -3.0) * t_1));
	else
		tmp = (t_0 / max(alpha, beta)) / (min(alpha, beta) - (-3.0 - max(alpha, beta)));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[Min[alpha, beta], $MachinePrecision] - -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Min[alpha, beta], $MachinePrecision] - N[(-2.0 - N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Max[alpha, beta], $MachinePrecision], 5e+16], N[(N[(N[(N[Max[alpha, beta], $MachinePrecision] - -1.0), $MachinePrecision] * t$95$0), $MachinePrecision] / N[(t$95$1 * N[(N[(N[(N[Max[alpha, beta], $MachinePrecision] + N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision] - -3.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision] / N[(N[Min[alpha, beta], $MachinePrecision] - N[(-3.0 - N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5e16

   1. Initial program 94.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

   3. Applied rewrites 84.0%

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

      1. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\alpha - -1, \beta, \alpha\right) - -1}}{\left(\alpha - \left(-2 - \beta\right)\right) \cdot \left(\left(\left(\beta + \alpha\right) - -3\right) \cdot \left(\alpha - \left(-2 - \beta\right)\right)\right)} \]

      2. lift-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\alpha - -1\right) \cdot \beta + \alpha\right)} - -1}{\left(\alpha - \left(-2 - \beta\right)\right) \cdot \left(\left(\left(\beta + \alpha\right) - -3\right) \cdot \left(\alpha - \left(-2 - \beta\right)\right)\right)} \]

      3. associate--l+ N/A

         \[\leadsto \frac{\color{blue}{\left(\alpha - -1\right) \cdot \beta + \left(\alpha - -1\right)}}{\left(\alpha - \left(-2 - \beta\right)\right) \cdot \left(\left(\left(\beta + \alpha\right) - -3\right) \cdot \left(\alpha - \left(-2 - \beta\right)\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\beta \cdot \left(\alpha - -1\right)} + \left(\alpha - -1\right)}{\left(\alpha - \left(-2 - \beta\right)\right) \cdot \left(\left(\left(\beta + \alpha\right) - -3\right) \cdot \left(\alpha - \left(-2 - \beta\right)\right)\right)} \]

      5. lift--.f64 N/A

         \[\leadsto \frac{\beta \cdot \color{blue}{\left(\alpha - -1\right)} + \left(\alpha - -1\right)}{\left(\alpha - \left(-2 - \beta\right)\right) \cdot \left(\left(\left(\beta + \alpha\right) - -3\right) \cdot \left(\alpha - \left(-2 - \beta\right)\right)\right)} \]

      6. sub-flip N/A

         \[\leadsto \frac{\beta \cdot \color{blue}{\left(\alpha + \left(\mathsf{neg}\left(-1\right)\right)\right)} + \left(\alpha - -1\right)}{\left(\alpha - \left(-2 - \beta\right)\right) \cdot \left(\left(\left(\beta + \alpha\right) - -3\right) \cdot \left(\alpha - \left(-2 - \beta\right)\right)\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\beta \cdot \left(\alpha + \color{blue}{1}\right) + \left(\alpha - -1\right)}{\left(\alpha - \left(-2 - \beta\right)\right) \cdot \left(\left(\left(\beta + \alpha\right) - -3\right) \cdot \left(\alpha - \left(-2 - \beta\right)\right)\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{\beta \cdot \color{blue}{\left(1 + \alpha\right)} + \left(\alpha - -1\right)}{\left(\alpha - \left(-2 - \beta\right)\right) \cdot \left(\left(\left(\beta + \alpha\right) - -3\right) \cdot \left(\alpha - \left(-2 - \beta\right)\right)\right)} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\beta \cdot \color{blue}{\left(1 + \alpha\right)} + \left(\alpha - -1\right)}{\left(\alpha - \left(-2 - \beta\right)\right) \cdot \left(\left(\left(\beta + \alpha\right) - -3\right) \cdot \left(\alpha - \left(-2 - \beta\right)\right)\right)} \]

      10. sub-flip N/A

          \[\leadsto \frac{\beta \cdot \left(1 + \alpha\right) + \color{blue}{\left(\alpha + \left(\mathsf{neg}\left(-1\right)\right)\right)}}{\left(\alpha - \left(-2 - \beta\right)\right) \cdot \left(\left(\left(\beta + \alpha\right) - -3\right) \cdot \left(\alpha - \left(-2 - \beta\right)\right)\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\beta \cdot \left(1 + \alpha\right) + \left(\alpha + \color{blue}{1}\right)}{\left(\alpha - \left(-2 - \beta\right)\right) \cdot \left(\left(\left(\beta + \alpha\right) - -3\right) \cdot \left(\alpha - \left(-2 - \beta\right)\right)\right)} \]

      12. +-commutative N/A

          \[\leadsto \frac{\beta \cdot \left(1 + \alpha\right) + \color{blue}{\left(1 + \alpha\right)}}{\left(\alpha - \left(-2 - \beta\right)\right) \cdot \left(\left(\left(\beta + \alpha\right) - -3\right) \cdot \left(\alpha - \left(-2 - \beta\right)\right)\right)} \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{\beta \cdot \left(1 + \alpha\right) + \color{blue}{\left(1 + \alpha\right)}}{\left(\alpha - \left(-2 - \beta\right)\right) \cdot \left(\left(\left(\beta + \alpha\right) - -3\right) \cdot \left(\alpha - \left(-2 - \beta\right)\right)\right)} \]

      14. distribute-lft1-in N/A

          \[\leadsto \frac{\color{blue}{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}}{\left(\alpha - \left(-2 - \beta\right)\right) \cdot \left(\left(\left(\beta + \alpha\right) - -3\right) \cdot \left(\alpha - \left(-2 - \beta\right)\right)\right)} \]

      15. +-commutative N/A

          \[\leadsto \frac{\color{blue}{\left(1 + \beta\right)} \cdot \left(1 + \alpha\right)}{\left(\alpha - \left(-2 - \beta\right)\right) \cdot \left(\left(\left(\beta + \alpha\right) - -3\right) \cdot \left(\alpha - \left(-2 - \beta\right)\right)\right)} \]

      16. lift-+.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(1 + \beta\right)} \cdot \left(1 + \alpha\right)}{\left(\alpha - \left(-2 - \beta\right)\right) \cdot \left(\left(\left(\beta + \alpha\right) - -3\right) \cdot \left(\alpha - \left(-2 - \beta\right)\right)\right)} \]

      17. lower-*.f64 84.0%

          \[\leadsto \frac{\color{blue}{\left(1 + \beta\right) \cdot \left(1 + \alpha\right)}}{\left(\alpha - \left(-2 - \beta\right)\right) \cdot \left(\left(\left(\beta + \alpha\right) - -3\right) \cdot \left(\alpha - \left(-2 - \beta\right)\right)\right)} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(1 + \beta\right)} \cdot \left(1 + \alpha\right)}{\left(\alpha - \left(-2 - \beta\right)\right) \cdot \left(\left(\left(\beta + \alpha\right) - -3\right) \cdot \left(\alpha - \left(-2 - \beta\right)\right)\right)} \]

      19. +-commutative N/A

          \[\leadsto \frac{\color{blue}{\left(\beta + 1\right)} \cdot \left(1 + \alpha\right)}{\left(\alpha - \left(-2 - \beta\right)\right) \cdot \left(\left(\left(\beta + \alpha\right) - -3\right) \cdot \left(\alpha - \left(-2 - \beta\right)\right)\right)} \]

      20. add-flip N/A

          \[\leadsto \frac{\color{blue}{\left(\beta - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(1 + \alpha\right)}{\left(\alpha - \left(-2 - \beta\right)\right) \cdot \left(\left(\left(\beta + \alpha\right) - -3\right) \cdot \left(\alpha - \left(-2 - \beta\right)\right)\right)} \]

      21. metadata-eval N/A

          \[\leadsto \frac{\left(\beta - \color{blue}{-1}\right) \cdot \left(1 + \alpha\right)}{\left(\alpha - \left(-2 - \beta\right)\right) \cdot \left(\left(\left(\beta + \alpha\right) - -3\right) \cdot \left(\alpha - \left(-2 - \beta\right)\right)\right)} \]

      22. lower--.f64 84.0%

          \[\leadsto \frac{\color{blue}{\left(\beta - -1\right)} \cdot \left(1 + \alpha\right)}{\left(\alpha - \left(-2 - \beta\right)\right) \cdot \left(\left(\left(\beta + \alpha\right) - -3\right) \cdot \left(\alpha - \left(-2 - \beta\right)\right)\right)} \]

      23. lift-+.f64 N/A

          \[\leadsto \frac{\left(\beta - -1\right) \cdot \color{blue}{\left(1 + \alpha\right)}}{\left(\alpha - \left(-2 - \beta\right)\right) \cdot \left(\left(\left(\beta + \alpha\right) - -3\right) \cdot \left(\alpha - \left(-2 - \beta\right)\right)\right)} \]

      24. +-commutative N/A

          \[\leadsto \frac{\left(\beta - -1\right) \cdot \color{blue}{\left(\alpha + 1\right)}}{\left(\alpha - \left(-2 - \beta\right)\right) \cdot \left(\left(\left(\beta + \alpha\right) - -3\right) \cdot \left(\alpha - \left(-2 - \beta\right)\right)\right)} \]

      25. metadata-eval N/A

          \[\leadsto \frac{\left(\beta - -1\right) \cdot \left(\alpha + \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)}\right)}{\left(\alpha - \left(-2 - \beta\right)\right) \cdot \left(\left(\left(\beta + \alpha\right) - -3\right) \cdot \left(\alpha - \left(-2 - \beta\right)\right)\right)} \]

      26. sub-flip N/A

          \[\leadsto \frac{\left(\beta - -1\right) \cdot \color{blue}{\left(\alpha - -1\right)}}{\left(\alpha - \left(-2 - \beta\right)\right) \cdot \left(\left(\left(\beta + \alpha\right) - -3\right) \cdot \left(\alpha - \left(-2 - \beta\right)\right)\right)} \]

      27. lift--.f64 84.0%

          \[\leadsto \frac{\left(\beta - -1\right) \cdot \color{blue}{\left(\alpha - -1\right)}}{\left(\alpha - \left(-2 - \beta\right)\right) \cdot \left(\left(\left(\beta + \alpha\right) - -3\right) \cdot \left(\alpha - \left(-2 - \beta\right)\right)\right)} \]

   5. Applied rewrites 84.0%

      \[\leadsto \frac{\color{blue}{\left(\beta - -1\right) \cdot \left(\alpha - -1\right)}}{\left(\alpha - \left(-2 - \beta\right)\right) \cdot \left(\left(\left(\beta + \alpha\right) - -3\right) \cdot \left(\alpha - \left(-2 - \beta\right)\right)\right)} \]

   if 5e16 < beta

   1. Initial program 94.1%

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

   2. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 29.6%

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

   4. Applied rewrites 29.6%

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

      1. metadata-eval 29.6%

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

      2. metadata-eval 29.6%

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

   6. Applied rewrites 29.6%

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

Alternative 10: 99.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := -2 - \mathsf{max}\left(\alpha, \beta\right)\\ t_1 := \mathsf{min}\left(\alpha, \beta\right) - -1\\ \mathbf{if}\;\mathsf{max}\left(\alpha, \beta\right) \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\frac{\left(\mathsf{max}\left(\alpha, \beta\right) - -1\right) \cdot t\_1}{\left(t\_0 - \mathsf{min}\left(\alpha, \beta\right)\right) \cdot \left(\left(\left(-3 - \mathsf{min}\left(\alpha, \beta\right)\right) - \mathsf{max}\left(\alpha, \beta\right)\right) \cdot \left(\mathsf{min}\left(\alpha, \beta\right) - t\_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_1}{\mathsf{max}\left(\alpha, \beta\right)}}{\mathsf{min}\left(\alpha, \beta\right) - \left(-3 - \mathsf{max}\left(\alpha, \beta\right)\right)}\\ \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (- -2.0 (fmax alpha beta))) (t_1 (- (fmin alpha beta) -1.0)))
   (if (<= (fmax alpha beta) 5e+16)
     (/
      (* (- (fmax alpha beta) -1.0) t_1)
      (*
       (- t_0 (fmin alpha beta))
       (*
        (- (- -3.0 (fmin alpha beta)) (fmax alpha beta))
        (- (fmin alpha beta) t_0))))
     (/
      (/ t_1 (fmax alpha beta))
      (- (fmin alpha beta) (- -3.0 (fmax alpha beta)))))))

C

double code(double alpha, double beta) {
	double t_0 = -2.0 - fmax(alpha, beta);
	double t_1 = fmin(alpha, beta) - -1.0;
	double tmp;
	if (fmax(alpha, beta) <= 5e+16) {
		tmp = ((fmax(alpha, beta) - -1.0) * t_1) / ((t_0 - fmin(alpha, beta)) * (((-3.0 - fmin(alpha, beta)) - fmax(alpha, beta)) * (fmin(alpha, beta) - t_0)));
	} else {
		tmp = (t_1 / fmax(alpha, beta)) / (fmin(alpha, beta) - (-3.0 - fmax(alpha, beta)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (-2.0d0) - fmax(alpha, beta)
    t_1 = fmin(alpha, beta) - (-1.0d0)
    if (fmax(alpha, beta) <= 5d+16) then
        tmp = ((fmax(alpha, beta) - (-1.0d0)) * t_1) / ((t_0 - fmin(alpha, beta)) * ((((-3.0d0) - fmin(alpha, beta)) - fmax(alpha, beta)) * (fmin(alpha, beta) - t_0)))
    else
        tmp = (t_1 / fmax(alpha, beta)) / (fmin(alpha, beta) - ((-3.0d0) - fmax(alpha, beta)))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = -2.0 - fmax(alpha, beta);
	double t_1 = fmin(alpha, beta) - -1.0;
	double tmp;
	if (fmax(alpha, beta) <= 5e+16) {
		tmp = ((fmax(alpha, beta) - -1.0) * t_1) / ((t_0 - fmin(alpha, beta)) * (((-3.0 - fmin(alpha, beta)) - fmax(alpha, beta)) * (fmin(alpha, beta) - t_0)));
	} else {
		tmp = (t_1 / fmax(alpha, beta)) / (fmin(alpha, beta) - (-3.0 - fmax(alpha, beta)));
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = -2.0 - fmax(alpha, beta)
	t_1 = fmin(alpha, beta) - -1.0
	tmp = 0
	if fmax(alpha, beta) <= 5e+16:
		tmp = ((fmax(alpha, beta) - -1.0) * t_1) / ((t_0 - fmin(alpha, beta)) * (((-3.0 - fmin(alpha, beta)) - fmax(alpha, beta)) * (fmin(alpha, beta) - t_0)))
	else:
		tmp = (t_1 / fmax(alpha, beta)) / (fmin(alpha, beta) - (-3.0 - fmax(alpha, beta)))
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(-2.0 - fmax(alpha, beta))
	t_1 = Float64(fmin(alpha, beta) - -1.0)
	tmp = 0.0
	if (fmax(alpha, beta) <= 5e+16)
		tmp = Float64(Float64(Float64(fmax(alpha, beta) - -1.0) * t_1) / Float64(Float64(t_0 - fmin(alpha, beta)) * Float64(Float64(Float64(-3.0 - fmin(alpha, beta)) - fmax(alpha, beta)) * Float64(fmin(alpha, beta) - t_0))));
	else
		tmp = Float64(Float64(t_1 / fmax(alpha, beta)) / Float64(fmin(alpha, beta) - Float64(-3.0 - fmax(alpha, beta))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = -2.0 - max(alpha, beta);
	t_1 = min(alpha, beta) - -1.0;
	tmp = 0.0;
	if (max(alpha, beta) <= 5e+16)
		tmp = ((max(alpha, beta) - -1.0) * t_1) / ((t_0 - min(alpha, beta)) * (((-3.0 - min(alpha, beta)) - max(alpha, beta)) * (min(alpha, beta) - t_0)));
	else
		tmp = (t_1 / max(alpha, beta)) / (min(alpha, beta) - (-3.0 - max(alpha, beta)));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(-2.0 - N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Min[alpha, beta], $MachinePrecision] - -1.0), $MachinePrecision]}, If[LessEqual[N[Max[alpha, beta], $MachinePrecision], 5e+16], N[(N[(N[(N[Max[alpha, beta], $MachinePrecision] - -1.0), $MachinePrecision] * t$95$1), $MachinePrecision] / N[(N[(t$95$0 - N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(-3.0 - N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision] - N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision] * N[(N[Min[alpha, beta], $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision] / N[(N[Min[alpha, beta], $MachinePrecision] - N[(-3.0 - N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5e16

   1. Initial program 94.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. div-add N/A

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

      6. sum-to-mult N/A

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

      7. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 94.1%

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

   5. Applied rewrites 84.0%

      \[\leadsto \color{blue}{\frac{\left(\beta - -1\right) \cdot \left(\alpha - -1\right)}{\left(\left(-2 - \beta\right) - \alpha\right) \cdot \left(\left(\left(-3 - \alpha\right) - \beta\right) \cdot \left(\alpha - \left(-2 - \beta\right)\right)\right)}} \]

   if 5e16 < beta

   1. Initial program 94.1%

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

   2. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 29.6%

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

   4. Applied rewrites 29.6%

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

      1. metadata-eval 29.6%

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

      2. metadata-eval 29.6%

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

   6. Applied rewrites 29.6%

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

Alternative 11: 98.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{max}\left(\alpha, \beta\right) \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\frac{1 + \mathsf{max}\left(\alpha, \beta\right)}{\mathsf{fma}\left(\mathsf{max}\left(\alpha, \beta\right), 4, \mathsf{fma}\left(\mathsf{max}\left(\alpha, \beta\right), \mathsf{max}\left(\alpha, \beta\right), 4\right)\right) \cdot \left(3 + \mathsf{max}\left(\alpha, \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{min}\left(\alpha, \beta\right) - -1}{\mathsf{max}\left(\alpha, \beta\right)}}{\mathsf{min}\left(\alpha, \beta\right) - \left(-3 - \mathsf{max}\left(\alpha, \beta\right)\right)}\\ \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (fmax alpha beta) 5e+16)
   (/
    (+ 1.0 (fmax alpha beta))
    (*
     (fma (fmax alpha beta) 4.0 (fma (fmax alpha beta) (fmax alpha beta) 4.0))
     (+ 3.0 (fmax alpha beta))))
   (/
    (/ (- (fmin alpha beta) -1.0) (fmax alpha beta))
    (- (fmin alpha beta) (- -3.0 (fmax alpha beta))))))

C

double code(double alpha, double beta) {
	double tmp;
	if (fmax(alpha, beta) <= 5e+16) {
		tmp = (1.0 + fmax(alpha, beta)) / (fma(fmax(alpha, beta), 4.0, fma(fmax(alpha, beta), fmax(alpha, beta), 4.0)) * (3.0 + fmax(alpha, beta)));
	} else {
		tmp = ((fmin(alpha, beta) - -1.0) / fmax(alpha, beta)) / (fmin(alpha, beta) - (-3.0 - fmax(alpha, beta)));
	}
	return tmp;
}

Julia

function code(alpha, beta)
	tmp = 0.0
	if (fmax(alpha, beta) <= 5e+16)
		tmp = Float64(Float64(1.0 + fmax(alpha, beta)) / Float64(fma(fmax(alpha, beta), 4.0, fma(fmax(alpha, beta), fmax(alpha, beta), 4.0)) * Float64(3.0 + fmax(alpha, beta))));
	else
		tmp = Float64(Float64(Float64(fmin(alpha, beta) - -1.0) / fmax(alpha, beta)) / Float64(fmin(alpha, beta) - Float64(-3.0 - fmax(alpha, beta))));
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[Max[alpha, beta], $MachinePrecision], 5e+16], N[(N[(1.0 + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Max[alpha, beta], $MachinePrecision] * 4.0 + N[(N[Max[alpha, beta], $MachinePrecision] * N[Max[alpha, beta], $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision] * N[(3.0 + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Min[alpha, beta], $MachinePrecision] - -1.0), $MachinePrecision] / N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision] / N[(N[Min[alpha, beta], $MachinePrecision] - N[(-3.0 - N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5e16

   1. Initial program 94.1%

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

   2. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{1 + \beta}{\color{blue}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{1 + \beta}{\color{blue}{{\left(2 + \beta\right)}^{2}} \cdot \left(3 + \beta\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \color{blue}{\left(3 + \beta\right)}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(\color{blue}{3} + \beta\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)} \]

      6. lower-+.f64 66.2%

         \[\leadsto \frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \color{blue}{\beta}\right)} \]

   4. Applied rewrites 66.2%

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

      1. lift-pow.f64 N/A

         \[\leadsto \frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(\color{blue}{3} + \beta\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)} \]

      3. sum-square-pow N/A

         \[\leadsto \frac{1 + \beta}{\left(\left({2}^{2} + 2 \cdot \left(2 \cdot \beta\right)\right) + {\beta}^{2}\right) \cdot \left(\color{blue}{3} + \beta\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{1 + \beta}{\left(\left({2}^{2} + 2 \cdot \left(2 \cdot \beta\right)\right) + {\beta}^{2}\right) \cdot \left(\color{blue}{3} + \beta\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{1 + \beta}{\left(\left(2 \cdot \left(2 \cdot \beta\right) + {2}^{2}\right) + {\beta}^{2}\right) \cdot \left(3 + \beta\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{1 + \beta}{\left(\left(\left(2 \cdot 2\right) \cdot \beta + {2}^{2}\right) + {\beta}^{2}\right) \cdot \left(3 + \beta\right)} \]

      7. unpow2 N/A

         \[\leadsto \frac{1 + \beta}{\left(\left({2}^{2} \cdot \beta + {2}^{2}\right) + {\beta}^{2}\right) \cdot \left(3 + \beta\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{1 + \beta}{\left(\mathsf{fma}\left({2}^{2}, \beta, {2}^{2}\right) + {\beta}^{2}\right) \cdot \left(3 + \beta\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{1 + \beta}{\left(\mathsf{fma}\left(4, \beta, {2}^{2}\right) + {\beta}^{2}\right) \cdot \left(3 + \beta\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{1 + \beta}{\left(\mathsf{fma}\left(4, \beta, 4\right) + {\beta}^{2}\right) \cdot \left(3 + \beta\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{1 + \beta}{\left(\mathsf{fma}\left(4, \beta, 4\right) + \beta \cdot \beta\right) \cdot \left(3 + \beta\right)} \]

      12. lower-*.f64 66.2%

          \[\leadsto \frac{1 + \beta}{\left(\mathsf{fma}\left(4, \beta, 4\right) + \beta \cdot \beta\right) \cdot \left(3 + \beta\right)} \]

   6. Applied rewrites 66.2%

      \[\leadsto \frac{1 + \beta}{\left(\mathsf{fma}\left(4, \beta, 4\right) + \beta \cdot \beta\right) \cdot \left(\color{blue}{3} + \beta\right)} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{1 + \beta}{\left(\mathsf{fma}\left(4, \beta, 4\right) + \beta \cdot \beta\right) \cdot \left(\color{blue}{3} + \beta\right)} \]

      2. lift-fma.f64 N/A

         \[\leadsto \frac{1 + \beta}{\left(\left(4 \cdot \beta + 4\right) + \beta \cdot \beta\right) \cdot \left(3 + \beta\right)} \]

      3. associate-+l+ N/A

         \[\leadsto \frac{1 + \beta}{\left(4 \cdot \beta + \left(4 + \beta \cdot \beta\right)\right) \cdot \left(\color{blue}{3} + \beta\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{1 + \beta}{\left(\beta \cdot 4 + \left(4 + \beta \cdot \beta\right)\right) \cdot \left(3 + \beta\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{1 + \beta}{\mathsf{fma}\left(\beta, 4, 4 + \beta \cdot \beta\right) \cdot \left(\color{blue}{3} + \beta\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{1 + \beta}{\mathsf{fma}\left(\beta, 4, \beta \cdot \beta + 4\right) \cdot \left(3 + \beta\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1 + \beta}{\mathsf{fma}\left(\beta, 4, \beta \cdot \beta + 4\right) \cdot \left(3 + \beta\right)} \]

      8. lower-fma.f64 66.2%

         \[\leadsto \frac{1 + \beta}{\mathsf{fma}\left(\beta, 4, \mathsf{fma}\left(\beta, \beta, 4\right)\right) \cdot \left(3 + \beta\right)} \]

   8. Applied rewrites 66.2%

      \[\leadsto \frac{1 + \beta}{\mathsf{fma}\left(\beta, 4, \mathsf{fma}\left(\beta, \beta, 4\right)\right) \cdot \left(\color{blue}{3} + \beta\right)} \]

   if 5e16 < beta

   1. Initial program 94.1%

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

   2. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 29.6%

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

   4. Applied rewrites 29.6%

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

      1. metadata-eval 29.6%

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

      2. metadata-eval 29.6%

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

   6. Applied rewrites 29.6%

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

Alternative 12: 98.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{max}\left(\alpha, \beta\right) \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\frac{1 + \mathsf{max}\left(\alpha, \beta\right)}{12 + \mathsf{max}\left(\alpha, \beta\right) \cdot \left(16 + \mathsf{max}\left(\alpha, \beta\right) \cdot \left(7 + \mathsf{max}\left(\alpha, \beta\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{min}\left(\alpha, \beta\right) - -1}{\mathsf{max}\left(\alpha, \beta\right)}}{\mathsf{min}\left(\alpha, \beta\right) - \left(-3 - \mathsf{max}\left(\alpha, \beta\right)\right)}\\ \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (fmax alpha beta) 5e+16)
   (/
    (+ 1.0 (fmax alpha beta))
    (+
     12.0
     (*
      (fmax alpha beta)
      (+ 16.0 (* (fmax alpha beta) (+ 7.0 (fmax alpha beta)))))))
   (/
    (/ (- (fmin alpha beta) -1.0) (fmax alpha beta))
    (- (fmin alpha beta) (- -3.0 (fmax alpha beta))))))

C

double code(double alpha, double beta) {
	double tmp;
	if (fmax(alpha, beta) <= 5e+16) {
		tmp = (1.0 + fmax(alpha, beta)) / (12.0 + (fmax(alpha, beta) * (16.0 + (fmax(alpha, beta) * (7.0 + fmax(alpha, beta))))));
	} else {
		tmp = ((fmin(alpha, beta) - -1.0) / fmax(alpha, beta)) / (fmin(alpha, beta) - (-3.0 - fmax(alpha, beta)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (fmax(alpha, beta) <= 5d+16) then
        tmp = (1.0d0 + fmax(alpha, beta)) / (12.0d0 + (fmax(alpha, beta) * (16.0d0 + (fmax(alpha, beta) * (7.0d0 + fmax(alpha, beta))))))
    else
        tmp = ((fmin(alpha, beta) - (-1.0d0)) / fmax(alpha, beta)) / (fmin(alpha, beta) - ((-3.0d0) - fmax(alpha, beta)))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (fmax(alpha, beta) <= 5e+16) {
		tmp = (1.0 + fmax(alpha, beta)) / (12.0 + (fmax(alpha, beta) * (16.0 + (fmax(alpha, beta) * (7.0 + fmax(alpha, beta))))));
	} else {
		tmp = ((fmin(alpha, beta) - -1.0) / fmax(alpha, beta)) / (fmin(alpha, beta) - (-3.0 - fmax(alpha, beta)));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if fmax(alpha, beta) <= 5e+16:
		tmp = (1.0 + fmax(alpha, beta)) / (12.0 + (fmax(alpha, beta) * (16.0 + (fmax(alpha, beta) * (7.0 + fmax(alpha, beta))))))
	else:
		tmp = ((fmin(alpha, beta) - -1.0) / fmax(alpha, beta)) / (fmin(alpha, beta) - (-3.0 - fmax(alpha, beta)))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (fmax(alpha, beta) <= 5e+16)
		tmp = Float64(Float64(1.0 + fmax(alpha, beta)) / Float64(12.0 + Float64(fmax(alpha, beta) * Float64(16.0 + Float64(fmax(alpha, beta) * Float64(7.0 + fmax(alpha, beta)))))));
	else
		tmp = Float64(Float64(Float64(fmin(alpha, beta) - -1.0) / fmax(alpha, beta)) / Float64(fmin(alpha, beta) - Float64(-3.0 - fmax(alpha, beta))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (max(alpha, beta) <= 5e+16)
		tmp = (1.0 + max(alpha, beta)) / (12.0 + (max(alpha, beta) * (16.0 + (max(alpha, beta) * (7.0 + max(alpha, beta))))));
	else
		tmp = ((min(alpha, beta) - -1.0) / max(alpha, beta)) / (min(alpha, beta) - (-3.0 - max(alpha, beta)));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[Max[alpha, beta], $MachinePrecision], 5e+16], N[(N[(1.0 + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision] / N[(12.0 + N[(N[Max[alpha, beta], $MachinePrecision] * N[(16.0 + N[(N[Max[alpha, beta], $MachinePrecision] * N[(7.0 + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Min[alpha, beta], $MachinePrecision] - -1.0), $MachinePrecision] / N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision] / N[(N[Min[alpha, beta], $MachinePrecision] - N[(-3.0 - N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5e16

   1. Initial program 94.1%

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

   2. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{1 + \beta}{\color{blue}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{1 + \beta}{\color{blue}{{\left(2 + \beta\right)}^{2}} \cdot \left(3 + \beta\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \color{blue}{\left(3 + \beta\right)}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(\color{blue}{3} + \beta\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)} \]

      6. lower-+.f64 66.2%

         \[\leadsto \frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \color{blue}{\beta}\right)} \]

   4. Applied rewrites 66.2%

      \[\leadsto \color{blue}{\frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}} \]

   5. Taylor expanded in beta around 0

      \[\leadsto \frac{1 + \beta}{12 + \color{blue}{\beta \cdot \left(16 + \beta \cdot \left(7 + \beta\right)\right)}} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1 + \beta}{12 + \beta \cdot \color{blue}{\left(16 + \beta \cdot \left(7 + \beta\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1 + \beta}{12 + \beta \cdot \left(16 + \color{blue}{\beta \cdot \left(7 + \beta\right)}\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{1 + \beta}{12 + \beta \cdot \left(16 + \beta \cdot \color{blue}{\left(7 + \beta\right)}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1 + \beta}{12 + \beta \cdot \left(16 + \beta \cdot \left(7 + \color{blue}{\beta}\right)\right)} \]

      5. lower-+.f64 66.2%

         \[\leadsto \frac{1 + \beta}{12 + \beta \cdot \left(16 + \beta \cdot \left(7 + \beta\right)\right)} \]

   7. Applied rewrites 66.2%

      \[\leadsto \frac{1 + \beta}{12 + \color{blue}{\beta \cdot \left(16 + \beta \cdot \left(7 + \beta\right)\right)}} \]

   if 5e16 < beta

   1. Initial program 94.1%

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

   2. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 29.6%

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

   4. Applied rewrites 29.6%

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

      1. metadata-eval 29.6%

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

      2. metadata-eval 29.6%

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

   6. Applied rewrites 29.6%

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

Alternative 13: 98.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\alpha, \beta\right) - -2\\ \mathbf{if}\;\mathsf{max}\left(\alpha, \beta\right) \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{\mathsf{max}\left(\alpha, \beta\right) - -1}{\mathsf{max}\left(\alpha, \beta\right) - -3}}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{min}\left(\alpha, \beta\right) - -1}{\mathsf{max}\left(\alpha, \beta\right)}}{\mathsf{min}\left(\alpha, \beta\right) - \left(-3 - \mathsf{max}\left(\alpha, \beta\right)\right)}\\ \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (- (fmax alpha beta) -2.0)))
   (if (<= (fmax alpha beta) 5e+16)
     (/ (/ (- (fmax alpha beta) -1.0) (- (fmax alpha beta) -3.0)) (* t_0 t_0))
     (/
      (/ (- (fmin alpha beta) -1.0) (fmax alpha beta))
      (- (fmin alpha beta) (- -3.0 (fmax alpha beta)))))))

C

double code(double alpha, double beta) {
	double t_0 = fmax(alpha, beta) - -2.0;
	double tmp;
	if (fmax(alpha, beta) <= 5e+16) {
		tmp = ((fmax(alpha, beta) - -1.0) / (fmax(alpha, beta) - -3.0)) / (t_0 * t_0);
	} else {
		tmp = ((fmin(alpha, beta) - -1.0) / fmax(alpha, beta)) / (fmin(alpha, beta) - (-3.0 - fmax(alpha, beta)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = fmax(alpha, beta) - (-2.0d0)
    if (fmax(alpha, beta) <= 5d+16) then
        tmp = ((fmax(alpha, beta) - (-1.0d0)) / (fmax(alpha, beta) - (-3.0d0))) / (t_0 * t_0)
    else
        tmp = ((fmin(alpha, beta) - (-1.0d0)) / fmax(alpha, beta)) / (fmin(alpha, beta) - ((-3.0d0) - fmax(alpha, beta)))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = fmax(alpha, beta) - -2.0;
	double tmp;
	if (fmax(alpha, beta) <= 5e+16) {
		tmp = ((fmax(alpha, beta) - -1.0) / (fmax(alpha, beta) - -3.0)) / (t_0 * t_0);
	} else {
		tmp = ((fmin(alpha, beta) - -1.0) / fmax(alpha, beta)) / (fmin(alpha, beta) - (-3.0 - fmax(alpha, beta)));
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = fmax(alpha, beta) - -2.0
	tmp = 0
	if fmax(alpha, beta) <= 5e+16:
		tmp = ((fmax(alpha, beta) - -1.0) / (fmax(alpha, beta) - -3.0)) / (t_0 * t_0)
	else:
		tmp = ((fmin(alpha, beta) - -1.0) / fmax(alpha, beta)) / (fmin(alpha, beta) - (-3.0 - fmax(alpha, beta)))
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(fmax(alpha, beta) - -2.0)
	tmp = 0.0
	if (fmax(alpha, beta) <= 5e+16)
		tmp = Float64(Float64(Float64(fmax(alpha, beta) - -1.0) / Float64(fmax(alpha, beta) - -3.0)) / Float64(t_0 * t_0));
	else
		tmp = Float64(Float64(Float64(fmin(alpha, beta) - -1.0) / fmax(alpha, beta)) / Float64(fmin(alpha, beta) - Float64(-3.0 - fmax(alpha, beta))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = max(alpha, beta) - -2.0;
	tmp = 0.0;
	if (max(alpha, beta) <= 5e+16)
		tmp = ((max(alpha, beta) - -1.0) / (max(alpha, beta) - -3.0)) / (t_0 * t_0);
	else
		tmp = ((min(alpha, beta) - -1.0) / max(alpha, beta)) / (min(alpha, beta) - (-3.0 - max(alpha, beta)));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[Max[alpha, beta], $MachinePrecision] - -2.0), $MachinePrecision]}, If[LessEqual[N[Max[alpha, beta], $MachinePrecision], 5e+16], N[(N[(N[(N[Max[alpha, beta], $MachinePrecision] - -1.0), $MachinePrecision] / N[(N[Max[alpha, beta], $MachinePrecision] - -3.0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Min[alpha, beta], $MachinePrecision] - -1.0), $MachinePrecision] / N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision] / N[(N[Min[alpha, beta], $MachinePrecision] - N[(-3.0 - N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5e16

   1. Initial program 94.1%

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

   2. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{1 + \beta}{\color{blue}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{1 + \beta}{\color{blue}{{\left(2 + \beta\right)}^{2}} \cdot \left(3 + \beta\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \color{blue}{\left(3 + \beta\right)}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(\color{blue}{3} + \beta\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)} \]

      6. lower-+.f64 66.2%

         \[\leadsto \frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \color{blue}{\beta}\right)} \]

   4. Applied rewrites 66.2%

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

      1. lift-/.f64 N/A

         \[\leadsto \frac{1 + \beta}{\color{blue}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \color{blue}{\left(3 + \beta\right)}} \]

      3. *-commutative N/A

         \[\leadsto \frac{1 + \beta}{\left(3 + \beta\right) \cdot \color{blue}{{\left(2 + \beta\right)}^{2}}} \]

      4. associate-/r* N/A

         \[\leadsto \frac{\frac{1 + \beta}{3 + \beta}}{\color{blue}{{\left(2 + \beta\right)}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1 + \beta}{3 + \beta}}{\color{blue}{{\left(2 + \beta\right)}^{2}}} \]

      6. lower-/.f64 69.3%

         \[\leadsto \frac{\frac{1 + \beta}{3 + \beta}}{{\color{blue}{\left(2 + \beta\right)}}^{2}} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\frac{1 + \beta}{3 + \beta}}{{\left(\color{blue}{2} + \beta\right)}^{2}} \]

      8. +-commutative N/A

         \[\leadsto \frac{\frac{\beta + 1}{3 + \beta}}{{\left(\color{blue}{2} + \beta\right)}^{2}} \]

      9. add-flip N/A

         \[\leadsto \frac{\frac{\beta - \left(\mathsf{neg}\left(1\right)\right)}{3 + \beta}}{{\left(\color{blue}{2} + \beta\right)}^{2}} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{\beta - -1}{3 + \beta}}{{\left(2 + \beta\right)}^{2}} \]

      11. lower--.f64 69.3%

          \[\leadsto \frac{\frac{\beta - -1}{3 + \beta}}{{\left(\color{blue}{2} + \beta\right)}^{2}} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{\beta - -1}{3 + \beta}}{{\left(2 + \color{blue}{\beta}\right)}^{2}} \]

      13. +-commutative N/A

          \[\leadsto \frac{\frac{\beta - -1}{\beta + 3}}{{\left(2 + \color{blue}{\beta}\right)}^{2}} \]

      14. add-flip N/A

          \[\leadsto \frac{\frac{\beta - -1}{\beta - \left(\mathsf{neg}\left(3\right)\right)}}{{\left(2 + \color{blue}{\beta}\right)}^{2}} \]

      15. metadata-eval N/A

          \[\leadsto \frac{\frac{\beta - -1}{\beta - -3}}{{\left(2 + \beta\right)}^{2}} \]

      16. lower--.f64 69.3%

          \[\leadsto \frac{\frac{\beta - -1}{\beta - -3}}{{\left(2 + \color{blue}{\beta}\right)}^{2}} \]

      17. lift-pow.f64 N/A

          \[\leadsto \frac{\frac{\beta - -1}{\beta - -3}}{{\left(2 + \beta\right)}^{\color{blue}{2}}} \]

      18. unpow2 N/A

          \[\leadsto \frac{\frac{\beta - -1}{\beta - -3}}{\left(2 + \beta\right) \cdot \color{blue}{\left(2 + \beta\right)}} \]

      19. lower-*.f64 69.3%

          \[\leadsto \frac{\frac{\beta - -1}{\beta - -3}}{\left(2 + \beta\right) \cdot \color{blue}{\left(2 + \beta\right)}} \]

      20. lift-+.f64 N/A

          \[\leadsto \frac{\frac{\beta - -1}{\beta - -3}}{\left(2 + \beta\right) \cdot \left(\color{blue}{2} + \beta\right)} \]

      21. +-commutative N/A

          \[\leadsto \frac{\frac{\beta - -1}{\beta - -3}}{\left(\beta + 2\right) \cdot \left(\color{blue}{2} + \beta\right)} \]

      22. add-flip-rev N/A

          \[\leadsto \frac{\frac{\beta - -1}{\beta - -3}}{\left(\beta - \left(\mathsf{neg}\left(2\right)\right)\right) \cdot \left(\color{blue}{2} + \beta\right)} \]

      23. metadata-eval N/A

          \[\leadsto \frac{\frac{\beta - -1}{\beta - -3}}{\left(\beta - -2\right) \cdot \left(2 + \beta\right)} \]

      24. lower--.f64 69.3%

          \[\leadsto \frac{\frac{\beta - -1}{\beta - -3}}{\left(\beta - -2\right) \cdot \left(\color{blue}{2} + \beta\right)} \]

      25. lift-+.f64 N/A

          \[\leadsto \frac{\frac{\beta - -1}{\beta - -3}}{\left(\beta - -2\right) \cdot \left(2 + \color{blue}{\beta}\right)} \]

      26. +-commutative N/A

          \[\leadsto \frac{\frac{\beta - -1}{\beta - -3}}{\left(\beta - -2\right) \cdot \left(\beta + \color{blue}{2}\right)} \]

      27. add-flip-rev N/A

          \[\leadsto \frac{\frac{\beta - -1}{\beta - -3}}{\left(\beta - -2\right) \cdot \left(\beta - \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)} \]

      28. metadata-eval N/A

          \[\leadsto \frac{\frac{\beta - -1}{\beta - -3}}{\left(\beta - -2\right) \cdot \left(\beta - -2\right)} \]

      29. lower--.f64 69.3%

          \[\leadsto \frac{\frac{\beta - -1}{\beta - -3}}{\left(\beta - -2\right) \cdot \left(\beta - \color{blue}{-2}\right)} \]

   6. Applied rewrites 69.3%

      \[\leadsto \frac{\frac{\beta - -1}{\beta - -3}}{\color{blue}{\left(\beta - -2\right) \cdot \left(\beta - -2\right)}} \]

   if 5e16 < beta

   1. Initial program 94.1%

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

   2. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 29.6%

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

   4. Applied rewrites 29.6%

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

      1. metadata-eval 29.6%

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

      2. metadata-eval 29.6%

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

   6. Applied rewrites 29.6%

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

Alternative 14: 98.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\alpha, \beta\right) - -2\\ \mathbf{if}\;\mathsf{max}\left(\alpha, \beta\right) \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\frac{\mathsf{max}\left(\alpha, \beta\right) - -1}{\left(\mathsf{max}\left(\alpha, \beta\right) - -3\right) \cdot \left(t\_0 \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{min}\left(\alpha, \beta\right) - -1}{\mathsf{max}\left(\alpha, \beta\right)}}{\mathsf{min}\left(\alpha, \beta\right) - \left(-3 - \mathsf{max}\left(\alpha, \beta\right)\right)}\\ \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (- (fmax alpha beta) -2.0)))
   (if (<= (fmax alpha beta) 5e+16)
     (/ (- (fmax alpha beta) -1.0) (* (- (fmax alpha beta) -3.0) (* t_0 t_0)))
     (/
      (/ (- (fmin alpha beta) -1.0) (fmax alpha beta))
      (- (fmin alpha beta) (- -3.0 (fmax alpha beta)))))))

C

double code(double alpha, double beta) {
	double t_0 = fmax(alpha, beta) - -2.0;
	double tmp;
	if (fmax(alpha, beta) <= 5e+16) {
		tmp = (fmax(alpha, beta) - -1.0) / ((fmax(alpha, beta) - -3.0) * (t_0 * t_0));
	} else {
		tmp = ((fmin(alpha, beta) - -1.0) / fmax(alpha, beta)) / (fmin(alpha, beta) - (-3.0 - fmax(alpha, beta)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = fmax(alpha, beta) - (-2.0d0)
    if (fmax(alpha, beta) <= 5d+16) then
        tmp = (fmax(alpha, beta) - (-1.0d0)) / ((fmax(alpha, beta) - (-3.0d0)) * (t_0 * t_0))
    else
        tmp = ((fmin(alpha, beta) - (-1.0d0)) / fmax(alpha, beta)) / (fmin(alpha, beta) - ((-3.0d0) - fmax(alpha, beta)))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = fmax(alpha, beta) - -2.0;
	double tmp;
	if (fmax(alpha, beta) <= 5e+16) {
		tmp = (fmax(alpha, beta) - -1.0) / ((fmax(alpha, beta) - -3.0) * (t_0 * t_0));
	} else {
		tmp = ((fmin(alpha, beta) - -1.0) / fmax(alpha, beta)) / (fmin(alpha, beta) - (-3.0 - fmax(alpha, beta)));
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = fmax(alpha, beta) - -2.0
	tmp = 0
	if fmax(alpha, beta) <= 5e+16:
		tmp = (fmax(alpha, beta) - -1.0) / ((fmax(alpha, beta) - -3.0) * (t_0 * t_0))
	else:
		tmp = ((fmin(alpha, beta) - -1.0) / fmax(alpha, beta)) / (fmin(alpha, beta) - (-3.0 - fmax(alpha, beta)))
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(fmax(alpha, beta) - -2.0)
	tmp = 0.0
	if (fmax(alpha, beta) <= 5e+16)
		tmp = Float64(Float64(fmax(alpha, beta) - -1.0) / Float64(Float64(fmax(alpha, beta) - -3.0) * Float64(t_0 * t_0)));
	else
		tmp = Float64(Float64(Float64(fmin(alpha, beta) - -1.0) / fmax(alpha, beta)) / Float64(fmin(alpha, beta) - Float64(-3.0 - fmax(alpha, beta))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = max(alpha, beta) - -2.0;
	tmp = 0.0;
	if (max(alpha, beta) <= 5e+16)
		tmp = (max(alpha, beta) - -1.0) / ((max(alpha, beta) - -3.0) * (t_0 * t_0));
	else
		tmp = ((min(alpha, beta) - -1.0) / max(alpha, beta)) / (min(alpha, beta) - (-3.0 - max(alpha, beta)));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[Max[alpha, beta], $MachinePrecision] - -2.0), $MachinePrecision]}, If[LessEqual[N[Max[alpha, beta], $MachinePrecision], 5e+16], N[(N[(N[Max[alpha, beta], $MachinePrecision] - -1.0), $MachinePrecision] / N[(N[(N[Max[alpha, beta], $MachinePrecision] - -3.0), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Min[alpha, beta], $MachinePrecision] - -1.0), $MachinePrecision] / N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision] / N[(N[Min[alpha, beta], $MachinePrecision] - N[(-3.0 - N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5e16

   1. Initial program 94.1%

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

   2. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{1 + \beta}{\color{blue}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{1 + \beta}{\color{blue}{{\left(2 + \beta\right)}^{2}} \cdot \left(3 + \beta\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \color{blue}{\left(3 + \beta\right)}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(\color{blue}{3} + \beta\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)} \]

      6. lower-+.f64 66.2%

         \[\leadsto \frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \color{blue}{\beta}\right)} \]

   4. Applied rewrites 66.2%

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

      1. lift-+.f64 N/A

         \[\leadsto \frac{1 + \beta}{\color{blue}{{\left(2 + \beta\right)}^{2}} \cdot \left(3 + \beta\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\beta + 1}{\color{blue}{{\left(2 + \beta\right)}^{2}} \cdot \left(3 + \beta\right)} \]

      3. add-flip N/A

         \[\leadsto \frac{\beta - \left(\mathsf{neg}\left(1\right)\right)}{\color{blue}{{\left(2 + \beta\right)}^{2}} \cdot \left(3 + \beta\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\beta - -1}{{\left(2 + \beta\right)}^{\color{blue}{2}} \cdot \left(3 + \beta\right)} \]

      5. lower--.f64 66.2%

         \[\leadsto \frac{\beta - -1}{\color{blue}{{\left(2 + \beta\right)}^{2}} \cdot \left(3 + \beta\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\beta - -1}{{\left(2 + \beta\right)}^{2} \cdot \color{blue}{\left(3 + \beta\right)}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\beta - -1}{\left(3 + \beta\right) \cdot \color{blue}{{\left(2 + \beta\right)}^{2}}} \]

      8. lower-*.f64 66.2%

         \[\leadsto \frac{\beta - -1}{\left(3 + \beta\right) \cdot \color{blue}{{\left(2 + \beta\right)}^{2}}} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\beta - -1}{\left(3 + \beta\right) \cdot {\color{blue}{\left(2 + \beta\right)}}^{2}} \]

      10. +-commutative N/A

          \[\leadsto \frac{\beta - -1}{\left(\beta + 3\right) \cdot {\color{blue}{\left(2 + \beta\right)}}^{2}} \]

      11. add-flip N/A

          \[\leadsto \frac{\beta - -1}{\left(\beta - \left(\mathsf{neg}\left(3\right)\right)\right) \cdot {\color{blue}{\left(2 + \beta\right)}}^{2}} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\beta - -1}{\left(\beta - -3\right) \cdot {\left(2 + \color{blue}{\beta}\right)}^{2}} \]

      13. lower--.f64 66.2%

          \[\leadsto \frac{\beta - -1}{\left(\beta - -3\right) \cdot {\color{blue}{\left(2 + \beta\right)}}^{2}} \]

      14. lift-pow.f64 N/A

          \[\leadsto \frac{\beta - -1}{\left(\beta - -3\right) \cdot {\left(2 + \beta\right)}^{\color{blue}{2}}} \]

      15. unpow2 N/A

          \[\leadsto \frac{\beta - -1}{\left(\beta - -3\right) \cdot \left(\left(2 + \beta\right) \cdot \color{blue}{\left(2 + \beta\right)}\right)} \]

      16. lower-*.f64 66.2%

          \[\leadsto \frac{\beta - -1}{\left(\beta - -3\right) \cdot \left(\left(2 + \beta\right) \cdot \color{blue}{\left(2 + \beta\right)}\right)} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{\beta - -1}{\left(\beta - -3\right) \cdot \left(\left(2 + \beta\right) \cdot \left(\color{blue}{2} + \beta\right)\right)} \]

      18. +-commutative N/A

          \[\leadsto \frac{\beta - -1}{\left(\beta - -3\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\color{blue}{2} + \beta\right)\right)} \]

      19. add-flip-rev N/A

          \[\leadsto \frac{\beta - -1}{\left(\beta - -3\right) \cdot \left(\left(\beta - \left(\mathsf{neg}\left(2\right)\right)\right) \cdot \left(\color{blue}{2} + \beta\right)\right)} \]

      20. metadata-eval N/A

          \[\leadsto \frac{\beta - -1}{\left(\beta - -3\right) \cdot \left(\left(\beta - -2\right) \cdot \left(2 + \beta\right)\right)} \]

      21. lower--.f64 66.2%

          \[\leadsto \frac{\beta - -1}{\left(\beta - -3\right) \cdot \left(\left(\beta - -2\right) \cdot \left(\color{blue}{2} + \beta\right)\right)} \]

      22. lift-+.f64 N/A

          \[\leadsto \frac{\beta - -1}{\left(\beta - -3\right) \cdot \left(\left(\beta - -2\right) \cdot \left(2 + \color{blue}{\beta}\right)\right)} \]

      23. +-commutative N/A

          \[\leadsto \frac{\beta - -1}{\left(\beta - -3\right) \cdot \left(\left(\beta - -2\right) \cdot \left(\beta + \color{blue}{2}\right)\right)} \]

      24. add-flip-rev N/A

          \[\leadsto \frac{\beta - -1}{\left(\beta - -3\right) \cdot \left(\left(\beta - -2\right) \cdot \left(\beta - \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)} \]

      25. metadata-eval N/A

          \[\leadsto \frac{\beta - -1}{\left(\beta - -3\right) \cdot \left(\left(\beta - -2\right) \cdot \left(\beta - -2\right)\right)} \]

      26. lower--.f64 66.2%

          \[\leadsto \frac{\beta - -1}{\left(\beta - -3\right) \cdot \left(\left(\beta - -2\right) \cdot \left(\beta - \color{blue}{-2}\right)\right)} \]

   6. Applied rewrites 66.2%

      \[\leadsto \frac{\beta - -1}{\color{blue}{\left(\beta - -3\right) \cdot \left(\left(\beta - -2\right) \cdot \left(\beta - -2\right)\right)}} \]

   if 5e16 < beta

   1. Initial program 94.1%

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

   2. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 29.6%

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

   4. Applied rewrites 29.6%

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

      1. metadata-eval 29.6%

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

      2. metadata-eval 29.6%

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

   6. Applied rewrites 29.6%

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

Alternative 15: 97.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{max}\left(\alpha, \beta\right) \leq 21500:\\ \;\;\;\;\frac{1 + \mathsf{max}\left(\alpha, \beta\right)}{12 + \mathsf{max}\left(\alpha, \beta\right) \cdot \left(16 + 7 \cdot \mathsf{max}\left(\alpha, \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{min}\left(\alpha, \beta\right) - -1}{\mathsf{max}\left(\alpha, \beta\right)}}{\mathsf{min}\left(\alpha, \beta\right) - \left(-3 - \mathsf{max}\left(\alpha, \beta\right)\right)}\\ \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (fmax alpha beta) 21500.0)
   (/
    (+ 1.0 (fmax alpha beta))
    (+ 12.0 (* (fmax alpha beta) (+ 16.0 (* 7.0 (fmax alpha beta))))))
   (/
    (/ (- (fmin alpha beta) -1.0) (fmax alpha beta))
    (- (fmin alpha beta) (- -3.0 (fmax alpha beta))))))

C

double code(double alpha, double beta) {
	double tmp;
	if (fmax(alpha, beta) <= 21500.0) {
		tmp = (1.0 + fmax(alpha, beta)) / (12.0 + (fmax(alpha, beta) * (16.0 + (7.0 * fmax(alpha, beta)))));
	} else {
		tmp = ((fmin(alpha, beta) - -1.0) / fmax(alpha, beta)) / (fmin(alpha, beta) - (-3.0 - fmax(alpha, beta)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (fmax(alpha, beta) <= 21500.0d0) then
        tmp = (1.0d0 + fmax(alpha, beta)) / (12.0d0 + (fmax(alpha, beta) * (16.0d0 + (7.0d0 * fmax(alpha, beta)))))
    else
        tmp = ((fmin(alpha, beta) - (-1.0d0)) / fmax(alpha, beta)) / (fmin(alpha, beta) - ((-3.0d0) - fmax(alpha, beta)))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (fmax(alpha, beta) <= 21500.0) {
		tmp = (1.0 + fmax(alpha, beta)) / (12.0 + (fmax(alpha, beta) * (16.0 + (7.0 * fmax(alpha, beta)))));
	} else {
		tmp = ((fmin(alpha, beta) - -1.0) / fmax(alpha, beta)) / (fmin(alpha, beta) - (-3.0 - fmax(alpha, beta)));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if fmax(alpha, beta) <= 21500.0:
		tmp = (1.0 + fmax(alpha, beta)) / (12.0 + (fmax(alpha, beta) * (16.0 + (7.0 * fmax(alpha, beta)))))
	else:
		tmp = ((fmin(alpha, beta) - -1.0) / fmax(alpha, beta)) / (fmin(alpha, beta) - (-3.0 - fmax(alpha, beta)))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (fmax(alpha, beta) <= 21500.0)
		tmp = Float64(Float64(1.0 + fmax(alpha, beta)) / Float64(12.0 + Float64(fmax(alpha, beta) * Float64(16.0 + Float64(7.0 * fmax(alpha, beta))))));
	else
		tmp = Float64(Float64(Float64(fmin(alpha, beta) - -1.0) / fmax(alpha, beta)) / Float64(fmin(alpha, beta) - Float64(-3.0 - fmax(alpha, beta))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (max(alpha, beta) <= 21500.0)
		tmp = (1.0 + max(alpha, beta)) / (12.0 + (max(alpha, beta) * (16.0 + (7.0 * max(alpha, beta)))));
	else
		tmp = ((min(alpha, beta) - -1.0) / max(alpha, beta)) / (min(alpha, beta) - (-3.0 - max(alpha, beta)));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[Max[alpha, beta], $MachinePrecision], 21500.0], N[(N[(1.0 + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision] / N[(12.0 + N[(N[Max[alpha, beta], $MachinePrecision] * N[(16.0 + N[(7.0 * N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Min[alpha, beta], $MachinePrecision] - -1.0), $MachinePrecision] / N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision] / N[(N[Min[alpha, beta], $MachinePrecision] - N[(-3.0 - N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 21500

   1. Initial program 94.1%

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

   2. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{1 + \beta}{\color{blue}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{1 + \beta}{\color{blue}{{\left(2 + \beta\right)}^{2}} \cdot \left(3 + \beta\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \color{blue}{\left(3 + \beta\right)}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(\color{blue}{3} + \beta\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)} \]

      6. lower-+.f64 66.2%

         \[\leadsto \frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \color{blue}{\beta}\right)} \]

   4. Applied rewrites 66.2%

      \[\leadsto \color{blue}{\frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}} \]

   5. Taylor expanded in beta around 0

      \[\leadsto \frac{1 + \beta}{12 + \color{blue}{\beta \cdot \left(16 + 7 \cdot \beta\right)}} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1 + \beta}{12 + \beta \cdot \color{blue}{\left(16 + 7 \cdot \beta\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1 + \beta}{12 + \beta \cdot \left(16 + \color{blue}{7 \cdot \beta}\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{1 + \beta}{12 + \beta \cdot \left(16 + 7 \cdot \color{blue}{\beta}\right)} \]

      4. lower-*.f64 58.5%

         \[\leadsto \frac{1 + \beta}{12 + \beta \cdot \left(16 + 7 \cdot \beta\right)} \]

   7. Applied rewrites 58.5%

      \[\leadsto \frac{1 + \beta}{12 + \color{blue}{\beta \cdot \left(16 + 7 \cdot \beta\right)}} \]

   if 21500 < beta

   1. Initial program 94.1%

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

   2. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 29.6%

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

   4. Applied rewrites 29.6%

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

      1. metadata-eval 29.6%

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

      2. metadata-eval 29.6%

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

   6. Applied rewrites 29.6%

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

Alternative 16: 97.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{max}\left(\alpha, \beta\right) \leq 1.7:\\ \;\;\;\;0.08333333333333333 + \mathsf{max}\left(\alpha, \beta\right) \cdot \left(\mathsf{max}\left(\alpha, \beta\right) \cdot \left(0.024691358024691357 \cdot \mathsf{max}\left(\alpha, \beta\right) - 0.011574074074074073\right) - 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{min}\left(\alpha, \beta\right) - -1}{\mathsf{max}\left(\alpha, \beta\right)}}{\mathsf{min}\left(\alpha, \beta\right) - \left(-3 - \mathsf{max}\left(\alpha, \beta\right)\right)}\\ \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (fmax alpha beta) 1.7)
   (+
    0.08333333333333333
    (*
     (fmax alpha beta)
     (-
      (*
       (fmax alpha beta)
       (- (* 0.024691358024691357 (fmax alpha beta)) 0.011574074074074073))
      0.027777777777777776)))
   (/
    (/ (- (fmin alpha beta) -1.0) (fmax alpha beta))
    (- (fmin alpha beta) (- -3.0 (fmax alpha beta))))))

C

double code(double alpha, double beta) {
	double tmp;
	if (fmax(alpha, beta) <= 1.7) {
		tmp = 0.08333333333333333 + (fmax(alpha, beta) * ((fmax(alpha, beta) * ((0.024691358024691357 * fmax(alpha, beta)) - 0.011574074074074073)) - 0.027777777777777776));
	} else {
		tmp = ((fmin(alpha, beta) - -1.0) / fmax(alpha, beta)) / (fmin(alpha, beta) - (-3.0 - fmax(alpha, beta)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (fmax(alpha, beta) <= 1.7d0) then
        tmp = 0.08333333333333333d0 + (fmax(alpha, beta) * ((fmax(alpha, beta) * ((0.024691358024691357d0 * fmax(alpha, beta)) - 0.011574074074074073d0)) - 0.027777777777777776d0))
    else
        tmp = ((fmin(alpha, beta) - (-1.0d0)) / fmax(alpha, beta)) / (fmin(alpha, beta) - ((-3.0d0) - fmax(alpha, beta)))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (fmax(alpha, beta) <= 1.7) {
		tmp = 0.08333333333333333 + (fmax(alpha, beta) * ((fmax(alpha, beta) * ((0.024691358024691357 * fmax(alpha, beta)) - 0.011574074074074073)) - 0.027777777777777776));
	} else {
		tmp = ((fmin(alpha, beta) - -1.0) / fmax(alpha, beta)) / (fmin(alpha, beta) - (-3.0 - fmax(alpha, beta)));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if fmax(alpha, beta) <= 1.7:
		tmp = 0.08333333333333333 + (fmax(alpha, beta) * ((fmax(alpha, beta) * ((0.024691358024691357 * fmax(alpha, beta)) - 0.011574074074074073)) - 0.027777777777777776))
	else:
		tmp = ((fmin(alpha, beta) - -1.0) / fmax(alpha, beta)) / (fmin(alpha, beta) - (-3.0 - fmax(alpha, beta)))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (fmax(alpha, beta) <= 1.7)
		tmp = Float64(0.08333333333333333 + Float64(fmax(alpha, beta) * Float64(Float64(fmax(alpha, beta) * Float64(Float64(0.024691358024691357 * fmax(alpha, beta)) - 0.011574074074074073)) - 0.027777777777777776)));
	else
		tmp = Float64(Float64(Float64(fmin(alpha, beta) - -1.0) / fmax(alpha, beta)) / Float64(fmin(alpha, beta) - Float64(-3.0 - fmax(alpha, beta))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (max(alpha, beta) <= 1.7)
		tmp = 0.08333333333333333 + (max(alpha, beta) * ((max(alpha, beta) * ((0.024691358024691357 * max(alpha, beta)) - 0.011574074074074073)) - 0.027777777777777776));
	else
		tmp = ((min(alpha, beta) - -1.0) / max(alpha, beta)) / (min(alpha, beta) - (-3.0 - max(alpha, beta)));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[Max[alpha, beta], $MachinePrecision], 1.7], N[(0.08333333333333333 + N[(N[Max[alpha, beta], $MachinePrecision] * N[(N[(N[Max[alpha, beta], $MachinePrecision] * N[(N[(0.024691358024691357 * N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision] - 0.011574074074074073), $MachinePrecision]), $MachinePrecision] - 0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Min[alpha, beta], $MachinePrecision] - -1.0), $MachinePrecision] / N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision] / N[(N[Min[alpha, beta], $MachinePrecision] - N[(-3.0 - N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.69999999999999996

   1. Initial program 94.1%

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

   2. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{1 + \beta}{\color{blue}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{1 + \beta}{\color{blue}{{\left(2 + \beta\right)}^{2}} \cdot \left(3 + \beta\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \color{blue}{\left(3 + \beta\right)}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(\color{blue}{3} + \beta\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)} \]

      6. lower-+.f64 66.2%

         \[\leadsto \frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \color{blue}{\beta}\right)} \]

   4. Applied rewrites 66.2%

      \[\leadsto \color{blue}{\frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}} \]

   5. Taylor expanded in beta around 0

      \[\leadsto \frac{1}{12} + \color{blue}{\beta \cdot \left(\beta \cdot \left(\frac{2}{81} \cdot \beta - \frac{5}{432}\right) - \frac{1}{36}\right)} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{12} + \beta \cdot \color{blue}{\left(\beta \cdot \left(\frac{2}{81} \cdot \beta - \frac{5}{432}\right) - \frac{1}{36}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{12} + \beta \cdot \left(\beta \cdot \left(\frac{2}{81} \cdot \beta - \frac{5}{432}\right) - \color{blue}{\frac{1}{36}}\right) \]

      3. lower--.f64 N/A

         \[\leadsto \frac{1}{12} + \beta \cdot \left(\beta \cdot \left(\frac{2}{81} \cdot \beta - \frac{5}{432}\right) - \frac{1}{36}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{12} + \beta \cdot \left(\beta \cdot \left(\frac{2}{81} \cdot \beta - \frac{5}{432}\right) - \frac{1}{36}\right) \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{12} + \beta \cdot \left(\beta \cdot \left(\frac{2}{81} \cdot \beta - \frac{5}{432}\right) - \frac{1}{36}\right) \]

      6. lower-*.f64 44.0%

         \[\leadsto 0.08333333333333333 + \beta \cdot \left(\beta \cdot \left(0.024691358024691357 \cdot \beta - 0.011574074074074073\right) - 0.027777777777777776\right) \]

   7. Applied rewrites 44.0%

      \[\leadsto 0.08333333333333333 + \color{blue}{\beta \cdot \left(\beta \cdot \left(0.024691358024691357 \cdot \beta - 0.011574074074074073\right) - 0.027777777777777776\right)} \]

   if 1.69999999999999996 < beta

   1. Initial program 94.1%

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

   2. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 29.6%

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

   4. Applied rewrites 29.6%

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

      1. metadata-eval 29.6%

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

      2. metadata-eval 29.6%

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

   6. Applied rewrites 29.6%

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

Alternative 17: 97.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{max}\left(\alpha, \beta\right) \leq 1.2:\\ \;\;\;\;0.08333333333333333 + \mathsf{max}\left(\alpha, \beta\right) \cdot \left(-0.011574074074074073 \cdot \mathsf{max}\left(\alpha, \beta\right) - 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{min}\left(\alpha, \beta\right) - -1}{\mathsf{max}\left(\alpha, \beta\right)}}{\mathsf{min}\left(\alpha, \beta\right) - \left(-3 - \mathsf{max}\left(\alpha, \beta\right)\right)}\\ \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (fmax alpha beta) 1.2)
   (+
    0.08333333333333333
    (*
     (fmax alpha beta)
     (- (* -0.011574074074074073 (fmax alpha beta)) 0.027777777777777776)))
   (/
    (/ (- (fmin alpha beta) -1.0) (fmax alpha beta))
    (- (fmin alpha beta) (- -3.0 (fmax alpha beta))))))

C

double code(double alpha, double beta) {
	double tmp;
	if (fmax(alpha, beta) <= 1.2) {
		tmp = 0.08333333333333333 + (fmax(alpha, beta) * ((-0.011574074074074073 * fmax(alpha, beta)) - 0.027777777777777776));
	} else {
		tmp = ((fmin(alpha, beta) - -1.0) / fmax(alpha, beta)) / (fmin(alpha, beta) - (-3.0 - fmax(alpha, beta)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (fmax(alpha, beta) <= 1.2d0) then
        tmp = 0.08333333333333333d0 + (fmax(alpha, beta) * (((-0.011574074074074073d0) * fmax(alpha, beta)) - 0.027777777777777776d0))
    else
        tmp = ((fmin(alpha, beta) - (-1.0d0)) / fmax(alpha, beta)) / (fmin(alpha, beta) - ((-3.0d0) - fmax(alpha, beta)))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (fmax(alpha, beta) <= 1.2) {
		tmp = 0.08333333333333333 + (fmax(alpha, beta) * ((-0.011574074074074073 * fmax(alpha, beta)) - 0.027777777777777776));
	} else {
		tmp = ((fmin(alpha, beta) - -1.0) / fmax(alpha, beta)) / (fmin(alpha, beta) - (-3.0 - fmax(alpha, beta)));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if fmax(alpha, beta) <= 1.2:
		tmp = 0.08333333333333333 + (fmax(alpha, beta) * ((-0.011574074074074073 * fmax(alpha, beta)) - 0.027777777777777776))
	else:
		tmp = ((fmin(alpha, beta) - -1.0) / fmax(alpha, beta)) / (fmin(alpha, beta) - (-3.0 - fmax(alpha, beta)))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (fmax(alpha, beta) <= 1.2)
		tmp = Float64(0.08333333333333333 + Float64(fmax(alpha, beta) * Float64(Float64(-0.011574074074074073 * fmax(alpha, beta)) - 0.027777777777777776)));
	else
		tmp = Float64(Float64(Float64(fmin(alpha, beta) - -1.0) / fmax(alpha, beta)) / Float64(fmin(alpha, beta) - Float64(-3.0 - fmax(alpha, beta))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (max(alpha, beta) <= 1.2)
		tmp = 0.08333333333333333 + (max(alpha, beta) * ((-0.011574074074074073 * max(alpha, beta)) - 0.027777777777777776));
	else
		tmp = ((min(alpha, beta) - -1.0) / max(alpha, beta)) / (min(alpha, beta) - (-3.0 - max(alpha, beta)));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[Max[alpha, beta], $MachinePrecision], 1.2], N[(0.08333333333333333 + N[(N[Max[alpha, beta], $MachinePrecision] * N[(N[(-0.011574074074074073 * N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision] - 0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Min[alpha, beta], $MachinePrecision] - -1.0), $MachinePrecision] / N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision] / N[(N[Min[alpha, beta], $MachinePrecision] - N[(-3.0 - N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.19999999999999996

   1. Initial program 94.1%

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

   2. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{1 + \beta}{\color{blue}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{1 + \beta}{\color{blue}{{\left(2 + \beta\right)}^{2}} \cdot \left(3 + \beta\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \color{blue}{\left(3 + \beta\right)}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(\color{blue}{3} + \beta\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)} \]

      6. lower-+.f64 66.2%

         \[\leadsto \frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \color{blue}{\beta}\right)} \]

   4. Applied rewrites 66.2%

      \[\leadsto \color{blue}{\frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}} \]

   5. Taylor expanded in beta around 0

      \[\leadsto \frac{1}{12} + \color{blue}{\beta \cdot \left(\frac{-5}{432} \cdot \beta - \frac{1}{36}\right)} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{12} + \beta \cdot \color{blue}{\left(\frac{-5}{432} \cdot \beta - \frac{1}{36}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{12} + \beta \cdot \left(\frac{-5}{432} \cdot \beta - \color{blue}{\frac{1}{36}}\right) \]

      3. lower--.f64 N/A

         \[\leadsto \frac{1}{12} + \beta \cdot \left(\frac{-5}{432} \cdot \beta - \frac{1}{36}\right) \]

      4. lower-*.f64 43.7%

         \[\leadsto 0.08333333333333333 + \beta \cdot \left(-0.011574074074074073 \cdot \beta - 0.027777777777777776\right) \]

   7. Applied rewrites 43.7%

      \[\leadsto 0.08333333333333333 + \color{blue}{\beta \cdot \left(-0.011574074074074073 \cdot \beta - 0.027777777777777776\right)} \]

   if 1.19999999999999996 < beta

   1. Initial program 94.1%

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

   2. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 29.6%

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

   4. Applied rewrites 29.6%

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

      1. metadata-eval 29.6%

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

      2. metadata-eval 29.6%

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

   6. Applied rewrites 29.6%

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

Alternative 18: 97.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{max}\left(\alpha, \beta\right) \leq 1.2:\\ \;\;\;\;0.08333333333333333 + \mathsf{max}\left(\alpha, \beta\right) \cdot \left(-0.011574074074074073 \cdot \mathsf{max}\left(\alpha, \beta\right) - 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \mathsf{min}\left(\alpha, \beta\right)}{\mathsf{max}\left(\alpha, \beta\right)}}{3 + \mathsf{max}\left(\alpha, \beta\right)}\\ \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (fmax alpha beta) 1.2)
   (+
    0.08333333333333333
    (*
     (fmax alpha beta)
     (- (* -0.011574074074074073 (fmax alpha beta)) 0.027777777777777776)))
   (/
    (/ (+ 1.0 (fmin alpha beta)) (fmax alpha beta))
    (+ 3.0 (fmax alpha beta)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (fmax(alpha, beta) <= 1.2) {
		tmp = 0.08333333333333333 + (fmax(alpha, beta) * ((-0.011574074074074073 * fmax(alpha, beta)) - 0.027777777777777776));
	} else {
		tmp = ((1.0 + fmin(alpha, beta)) / fmax(alpha, beta)) / (3.0 + fmax(alpha, beta));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (fmax(alpha, beta) <= 1.2d0) then
        tmp = 0.08333333333333333d0 + (fmax(alpha, beta) * (((-0.011574074074074073d0) * fmax(alpha, beta)) - 0.027777777777777776d0))
    else
        tmp = ((1.0d0 + fmin(alpha, beta)) / fmax(alpha, beta)) / (3.0d0 + fmax(alpha, beta))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (fmax(alpha, beta) <= 1.2) {
		tmp = 0.08333333333333333 + (fmax(alpha, beta) * ((-0.011574074074074073 * fmax(alpha, beta)) - 0.027777777777777776));
	} else {
		tmp = ((1.0 + fmin(alpha, beta)) / fmax(alpha, beta)) / (3.0 + fmax(alpha, beta));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if fmax(alpha, beta) <= 1.2:
		tmp = 0.08333333333333333 + (fmax(alpha, beta) * ((-0.011574074074074073 * fmax(alpha, beta)) - 0.027777777777777776))
	else:
		tmp = ((1.0 + fmin(alpha, beta)) / fmax(alpha, beta)) / (3.0 + fmax(alpha, beta))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (fmax(alpha, beta) <= 1.2)
		tmp = Float64(0.08333333333333333 + Float64(fmax(alpha, beta) * Float64(Float64(-0.011574074074074073 * fmax(alpha, beta)) - 0.027777777777777776)));
	else
		tmp = Float64(Float64(Float64(1.0 + fmin(alpha, beta)) / fmax(alpha, beta)) / Float64(3.0 + fmax(alpha, beta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (max(alpha, beta) <= 1.2)
		tmp = 0.08333333333333333 + (max(alpha, beta) * ((-0.011574074074074073 * max(alpha, beta)) - 0.027777777777777776));
	else
		tmp = ((1.0 + min(alpha, beta)) / max(alpha, beta)) / (3.0 + max(alpha, beta));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[Max[alpha, beta], $MachinePrecision], 1.2], N[(0.08333333333333333 + N[(N[Max[alpha, beta], $MachinePrecision] * N[(N[(-0.011574074074074073 * N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision] - 0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision] / N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.19999999999999996

   1. Initial program 94.1%

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

   2. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{1 + \beta}{\color{blue}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{1 + \beta}{\color{blue}{{\left(2 + \beta\right)}^{2}} \cdot \left(3 + \beta\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \color{blue}{\left(3 + \beta\right)}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(\color{blue}{3} + \beta\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)} \]

      6. lower-+.f64 66.2%

         \[\leadsto \frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \color{blue}{\beta}\right)} \]

   4. Applied rewrites 66.2%

      \[\leadsto \color{blue}{\frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}} \]

   5. Taylor expanded in beta around 0

      \[\leadsto \frac{1}{12} + \color{blue}{\beta \cdot \left(\frac{-5}{432} \cdot \beta - \frac{1}{36}\right)} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{12} + \beta \cdot \color{blue}{\left(\frac{-5}{432} \cdot \beta - \frac{1}{36}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{12} + \beta \cdot \left(\frac{-5}{432} \cdot \beta - \color{blue}{\frac{1}{36}}\right) \]

      3. lower--.f64 N/A

         \[\leadsto \frac{1}{12} + \beta \cdot \left(\frac{-5}{432} \cdot \beta - \frac{1}{36}\right) \]

      4. lower-*.f64 43.7%

         \[\leadsto 0.08333333333333333 + \beta \cdot \left(-0.011574074074074073 \cdot \beta - 0.027777777777777776\right) \]

   7. Applied rewrites 43.7%

      \[\leadsto 0.08333333333333333 + \color{blue}{\beta \cdot \left(-0.011574074074074073 \cdot \beta - 0.027777777777777776\right)} \]

   if 1.19999999999999996 < beta

   1. Initial program 94.1%

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

   2. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 29.6%

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

   4. Applied rewrites 29.6%

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

   5. Taylor expanded in alpha around 0

      \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{\color{blue}{3 + \beta}} \]
   6. Step-by-step derivation

      1. lower-+.f64 29.5%

         \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{3 + \color{blue}{\beta}} \]

   7. Applied rewrites 29.5%

      \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{\color{blue}{3 + \beta}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 46.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{max}\left(\alpha, \beta\right) \leq 1.35:\\ \;\;\;\;0.08333333333333333 + \mathsf{max}\left(\alpha, \beta\right) \cdot \left(-0.011574074074074073 \cdot \mathsf{max}\left(\alpha, \beta\right) - 0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \mathsf{min}\left(\alpha, \beta\right)}{\mathsf{max}\left(\alpha, \beta\right)}}{3 + \mathsf{min}\left(\alpha, \beta\right)}\\ \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (fmax alpha beta) 1.35)
   (+
    0.08333333333333333
    (*
     (fmax alpha beta)
     (- (* -0.011574074074074073 (fmax alpha beta)) 0.027777777777777776)))
   (/
    (/ (+ 1.0 (fmin alpha beta)) (fmax alpha beta))
    (+ 3.0 (fmin alpha beta)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (fmax(alpha, beta) <= 1.35) {
		tmp = 0.08333333333333333 + (fmax(alpha, beta) * ((-0.011574074074074073 * fmax(alpha, beta)) - 0.027777777777777776));
	} else {
		tmp = ((1.0 + fmin(alpha, beta)) / fmax(alpha, beta)) / (3.0 + fmin(alpha, beta));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (fmax(alpha, beta) <= 1.35d0) then
        tmp = 0.08333333333333333d0 + (fmax(alpha, beta) * (((-0.011574074074074073d0) * fmax(alpha, beta)) - 0.027777777777777776d0))
    else
        tmp = ((1.0d0 + fmin(alpha, beta)) / fmax(alpha, beta)) / (3.0d0 + fmin(alpha, beta))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (fmax(alpha, beta) <= 1.35) {
		tmp = 0.08333333333333333 + (fmax(alpha, beta) * ((-0.011574074074074073 * fmax(alpha, beta)) - 0.027777777777777776));
	} else {
		tmp = ((1.0 + fmin(alpha, beta)) / fmax(alpha, beta)) / (3.0 + fmin(alpha, beta));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if fmax(alpha, beta) <= 1.35:
		tmp = 0.08333333333333333 + (fmax(alpha, beta) * ((-0.011574074074074073 * fmax(alpha, beta)) - 0.027777777777777776))
	else:
		tmp = ((1.0 + fmin(alpha, beta)) / fmax(alpha, beta)) / (3.0 + fmin(alpha, beta))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (fmax(alpha, beta) <= 1.35)
		tmp = Float64(0.08333333333333333 + Float64(fmax(alpha, beta) * Float64(Float64(-0.011574074074074073 * fmax(alpha, beta)) - 0.027777777777777776)));
	else
		tmp = Float64(Float64(Float64(1.0 + fmin(alpha, beta)) / fmax(alpha, beta)) / Float64(3.0 + fmin(alpha, beta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (max(alpha, beta) <= 1.35)
		tmp = 0.08333333333333333 + (max(alpha, beta) * ((-0.011574074074074073 * max(alpha, beta)) - 0.027777777777777776));
	else
		tmp = ((1.0 + min(alpha, beta)) / max(alpha, beta)) / (3.0 + min(alpha, beta));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[Max[alpha, beta], $MachinePrecision], 1.35], N[(0.08333333333333333 + N[(N[Max[alpha, beta], $MachinePrecision] * N[(N[(-0.011574074074074073 * N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision] - 0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision] / N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[Min[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.3500000000000001

   1. Initial program 94.1%

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

   2. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{1 + \beta}{\color{blue}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{1 + \beta}{\color{blue}{{\left(2 + \beta\right)}^{2}} \cdot \left(3 + \beta\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \color{blue}{\left(3 + \beta\right)}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(\color{blue}{3} + \beta\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)} \]

      6. lower-+.f64 66.2%

         \[\leadsto \frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \color{blue}{\beta}\right)} \]

   4. Applied rewrites 66.2%

      \[\leadsto \color{blue}{\frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}} \]

   5. Taylor expanded in beta around 0

      \[\leadsto \frac{1}{12} + \color{blue}{\beta \cdot \left(\frac{-5}{432} \cdot \beta - \frac{1}{36}\right)} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{12} + \beta \cdot \color{blue}{\left(\frac{-5}{432} \cdot \beta - \frac{1}{36}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{12} + \beta \cdot \left(\frac{-5}{432} \cdot \beta - \color{blue}{\frac{1}{36}}\right) \]

      3. lower--.f64 N/A

         \[\leadsto \frac{1}{12} + \beta \cdot \left(\frac{-5}{432} \cdot \beta - \frac{1}{36}\right) \]

      4. lower-*.f64 43.7%

         \[\leadsto 0.08333333333333333 + \beta \cdot \left(-0.011574074074074073 \cdot \beta - 0.027777777777777776\right) \]

   7. Applied rewrites 43.7%

      \[\leadsto 0.08333333333333333 + \color{blue}{\beta \cdot \left(-0.011574074074074073 \cdot \beta - 0.027777777777777776\right)} \]

   if 1.3500000000000001 < beta

   1. Initial program 94.1%

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

   2. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 29.6%

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

   4. Applied rewrites 29.6%

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

   5. Taylor expanded in beta around 0

      \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{\color{blue}{3 + \alpha}} \]
   6. Step-by-step derivation

      1. lower-+.f64 4.2%

         \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{3 + \color{blue}{\alpha}} \]

   7. Applied rewrites 4.2%

      \[\leadsto \frac{\frac{1 + \alpha}{\beta}}{\color{blue}{3 + \alpha}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 44.5% accurate, 2.7× speedup

Math

\[\frac{1 + \mathsf{max}\left(\alpha, \beta\right)}{12 + 16 \cdot \mathsf{max}\left(\alpha, \beta\right)} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (/ (+ 1.0 (fmax alpha beta)) (+ 12.0 (* 16.0 (fmax alpha beta)))))

C

double code(double alpha, double beta) {
	return (1.0 + fmax(alpha, beta)) / (12.0 + (16.0 * fmax(alpha, beta)));
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = (1.0d0 + fmax(alpha, beta)) / (12.0d0 + (16.0d0 * fmax(alpha, beta)))
end function

Java

public static double code(double alpha, double beta) {
	return (1.0 + fmax(alpha, beta)) / (12.0 + (16.0 * fmax(alpha, beta)));
}

Python

def code(alpha, beta):
	return (1.0 + fmax(alpha, beta)) / (12.0 + (16.0 * fmax(alpha, beta)))

Julia

function code(alpha, beta)
	return Float64(Float64(1.0 + fmax(alpha, beta)) / Float64(12.0 + Float64(16.0 * fmax(alpha, beta))))
end

MATLAB

function tmp = code(alpha, beta)
	tmp = (1.0 + max(alpha, beta)) / (12.0 + (16.0 * max(alpha, beta)));
end

Wolfram

code[alpha_, beta_] := N[(N[(1.0 + N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision] / N[(12.0 + N[(16.0 * N[Max[alpha, beta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.1%

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

2. Taylor expanded in alpha around 0

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

   1. lower-/.f64 N/A

      \[\leadsto \frac{1 + \beta}{\color{blue}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}} \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{1 + \beta}{\color{blue}{{\left(2 + \beta\right)}^{2}} \cdot \left(3 + \beta\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \color{blue}{\left(3 + \beta\right)}} \]

   4. lower-pow.f64 N/A

      \[\leadsto \frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(\color{blue}{3} + \beta\right)} \]

   5. lower-+.f64 N/A

      \[\leadsto \frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)} \]

   6. lower-+.f64 66.2%

      \[\leadsto \frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \color{blue}{\beta}\right)} \]

4. Applied rewrites 66.2%

   \[\leadsto \color{blue}{\frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}} \]

5. Taylor expanded in beta around 0

   \[\leadsto \frac{1 + \beta}{12 + \color{blue}{16 \cdot \beta}} \]
6. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \frac{1 + \beta}{12 + 16 \cdot \color{blue}{\beta}} \]

   2. lower-*.f64 44.5%

      \[\leadsto \frac{1 + \beta}{12 + 16 \cdot \beta} \]

7. Applied rewrites 44.5%

   \[\leadsto \frac{1 + \beta}{12 + \color{blue}{16 \cdot \beta}} \]
8. Add Preprocessing

Alternative 21: 44.0% accurate, 50.2× speedup

Math

\[0.08333333333333333 \]

FPCore

(FPCore (alpha beta) :precision binary64 0.08333333333333333)

C

double code(double alpha, double beta) {
	return 0.08333333333333333;
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.08333333333333333d0
end function

Java

public static double code(double alpha, double beta) {
	return 0.08333333333333333;
}

Python

def code(alpha, beta):
	return 0.08333333333333333

Julia

function code(alpha, beta)
	return 0.08333333333333333
end

MATLAB

function tmp = code(alpha, beta)
	tmp = 0.08333333333333333;
end

Wolfram

code[alpha_, beta_] := 0.08333333333333333

Derivation

1. Initial program 94.1%

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

2. Taylor expanded in alpha around 0

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

   1. lower-/.f64 N/A

      \[\leadsto \frac{1 + \beta}{\color{blue}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}} \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{1 + \beta}{\color{blue}{{\left(2 + \beta\right)}^{2}} \cdot \left(3 + \beta\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \color{blue}{\left(3 + \beta\right)}} \]

   4. lower-pow.f64 N/A

      \[\leadsto \frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(\color{blue}{3} + \beta\right)} \]

   5. lower-+.f64 N/A

      \[\leadsto \frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)} \]

   6. lower-+.f64 66.2%

      \[\leadsto \frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \color{blue}{\beta}\right)} \]

4. Applied rewrites 66.2%

   \[\leadsto \color{blue}{\frac{1 + \beta}{{\left(2 + \beta\right)}^{2} \cdot \left(3 + \beta\right)}} \]

5. Taylor expanded in beta around 0

   \[\leadsto \frac{1}{12} \]
6. Step-by-step derivation

7. Applied rewrites 44.0%

   \[\leadsto 0.08333333333333333 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025179 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :precision binary64
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ (+ alpha beta) (* 2.0 1.0)) 1.0)))