Octave 3.8, jcobi/1

Percentage Accurate: 74.7% → 99.3%

Time: 4.6s

Alternatives: 11

Speedup: 0.5×

Specification

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

Math

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

FPCore

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

C

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

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
    code = (((beta - alpha) / ((alpha + beta) + 2.0d0)) + 1.0d0) / 2.0d0
end function

Java

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

Python

def code(alpha, beta):
	return (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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
    code = (((beta - alpha) / ((alpha + beta) + 2.0d0)) + 1.0d0) / 2.0d0
end function

Java

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

Python

def code(alpha, beta):
	return (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 0:\\ \;\;\;\;0.5 \cdot \frac{2 + 2 \cdot \beta}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\beta \cdot \left(1 + \frac{\alpha}{\beta}\right) + 2} + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0) 0.0)
   (* 0.5 (/ (+ 2.0 (* 2.0 beta)) alpha))
   (/ (+ (/ (- beta alpha) (+ (* beta (+ 1.0 (/ alpha beta))) 2.0)) 1.0) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 0.0) {
		tmp = 0.5 * ((2.0 + (2.0 * beta)) / alpha);
	} else {
		tmp = (((beta - alpha) / ((beta * (1.0 + (alpha / beta))) + 2.0)) + 1.0) / 2.0;
	}
	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) :: tmp
    if (((((beta - alpha) / ((alpha + beta) + 2.0d0)) + 1.0d0) / 2.0d0) <= 0.0d0) then
        tmp = 0.5d0 * ((2.0d0 + (2.0d0 * beta)) / alpha)
    else
        tmp = (((beta - alpha) / ((beta * (1.0d0 + (alpha / beta))) + 2.0d0)) + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 0.0) {
		tmp = 0.5 * ((2.0 + (2.0 * beta)) / alpha);
	} else {
		tmp = (((beta - alpha) / ((beta * (1.0 + (alpha / beta))) + 2.0)) + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if ((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 0.0:
		tmp = 0.5 * ((2.0 + (2.0 * beta)) / alpha)
	else:
		tmp = (((beta - alpha) / ((beta * (1.0 + (alpha / beta))) + 2.0)) + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) + 1.0) / 2.0) <= 0.0)
		tmp = Float64(0.5 * Float64(Float64(2.0 + Float64(2.0 * beta)) / alpha));
	else
		tmp = Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(beta * Float64(1.0 + Float64(alpha / beta))) + 2.0)) + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 0.0)
		tmp = 0.5 * ((2.0 + (2.0 * beta)) / alpha);
	else
		tmp = (((beta - alpha) / ((beta * (1.0 + (alpha / beta))) + 2.0)) + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], 0.0], N[(0.5 * N[(N[(2.0 + N[(2.0 * beta), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta * N[(1.0 + N[(alpha / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.0

   1. Initial program 74.7%

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

   2. Taylor expanded in alpha around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{2 + 2 \cdot \beta}{\alpha}} \]

   4. Applied rewrites 29.4%

      \[\leadsto \color{blue}{0.5 \cdot \frac{2 + 2 \cdot \beta}{\alpha}} \]

   if 0.0 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64))

   1. Initial program 74.7%

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

   2. Taylor expanded in beta around inf

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

   4. Applied rewrites 74.4%

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

Alternative 2: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;0.5 \cdot \frac{2 + 2 \cdot \beta}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0)))
   (if (<= t_0 0.0) (* 0.5 (/ (+ 2.0 (* 2.0 beta)) alpha)) t_0)))

C

double code(double alpha, double beta) {
	double t_0 = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 0.5 * ((2.0 + (2.0 * beta)) / alpha);
	} else {
		tmp = t_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) :: tmp
    t_0 = (((beta - alpha) / ((alpha + beta) + 2.0d0)) + 1.0d0) / 2.0d0
    if (t_0 <= 0.0d0) then
        tmp = 0.5d0 * ((2.0d0 + (2.0d0 * beta)) / alpha)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 0.5 * ((2.0 + (2.0 * beta)) / alpha);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0
	tmp = 0
	if t_0 <= 0.0:
		tmp = 0.5 * ((2.0 + (2.0 * beta)) / alpha)
	else:
		tmp = t_0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) + 1.0) / 2.0)
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(0.5 * Float64(Float64(2.0 + Float64(2.0 * beta)) / alpha));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = 0.5 * ((2.0 + (2.0 * beta)) / alpha);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(0.5 * N[(N[(2.0 + N[(2.0 * beta), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.0

   1. Initial program 74.7%

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

   2. Taylor expanded in alpha around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{2 + 2 \cdot \beta}{\alpha}} \]

   4. Applied rewrites 29.4%

      \[\leadsto \color{blue}{0.5 \cdot \frac{2 + 2 \cdot \beta}{\alpha}} \]

   if 0.0 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64))

   1. Initial program 74.7%

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

Alternative 3: 98.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 0.02:\\ \;\;\;\;0.5 \cdot \frac{2 + 2 \cdot \beta}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{2 + \beta} + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0) 0.02)
   (* 0.5 (/ (+ 2.0 (* 2.0 beta)) alpha))
   (/ (+ (/ (- beta alpha) (+ 2.0 beta)) 1.0) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 0.02) {
		tmp = 0.5 * ((2.0 + (2.0 * beta)) / alpha);
	} else {
		tmp = (((beta - alpha) / (2.0 + beta)) + 1.0) / 2.0;
	}
	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) :: tmp
    if (((((beta - alpha) / ((alpha + beta) + 2.0d0)) + 1.0d0) / 2.0d0) <= 0.02d0) then
        tmp = 0.5d0 * ((2.0d0 + (2.0d0 * beta)) / alpha)
    else
        tmp = (((beta - alpha) / (2.0d0 + beta)) + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 0.02) {
		tmp = 0.5 * ((2.0 + (2.0 * beta)) / alpha);
	} else {
		tmp = (((beta - alpha) / (2.0 + beta)) + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if ((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 0.02:
		tmp = 0.5 * ((2.0 + (2.0 * beta)) / alpha)
	else:
		tmp = (((beta - alpha) / (2.0 + beta)) + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) + 1.0) / 2.0) <= 0.02)
		tmp = Float64(0.5 * Float64(Float64(2.0 + Float64(2.0 * beta)) / alpha));
	else
		tmp = Float64(Float64(Float64(Float64(beta - alpha) / Float64(2.0 + beta)) + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 0.02)
		tmp = 0.5 * ((2.0 + (2.0 * beta)) / alpha);
	else
		tmp = (((beta - alpha) / (2.0 + beta)) + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], 0.02], N[(0.5 * N[(N[(2.0 + N[(2.0 * beta), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.0200000000000000004

   1. Initial program 74.7%

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

   2. Taylor expanded in alpha around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{2 + 2 \cdot \beta}{\alpha}} \]

   4. Applied rewrites 29.4%

      \[\leadsto \color{blue}{0.5 \cdot \frac{2 + 2 \cdot \beta}{\alpha}} \]

   if 0.0200000000000000004 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64))

   1. Initial program 74.7%

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

   2. Taylor expanded in alpha around 0

      \[\leadsto \frac{\frac{\beta - \alpha}{\color{blue}{2 + \beta}} + 1}{2} \]

   4. Applied rewrites 71.9%

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

Alternative 4: 98.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 0.02:\\ \;\;\;\;0.5 \cdot \frac{2 + 2 \cdot \beta}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(1 + \frac{\beta}{2 + \beta}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0) 0.02)
   (* 0.5 (/ (+ 2.0 (* 2.0 beta)) alpha))
   (* 0.5 (+ 1.0 (/ beta (+ 2.0 beta))))))

C

double code(double alpha, double beta) {
	double tmp;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 0.02) {
		tmp = 0.5 * ((2.0 + (2.0 * beta)) / alpha);
	} else {
		tmp = 0.5 * (1.0 + (beta / (2.0 + 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 (((((beta - alpha) / ((alpha + beta) + 2.0d0)) + 1.0d0) / 2.0d0) <= 0.02d0) then
        tmp = 0.5d0 * ((2.0d0 + (2.0d0 * beta)) / alpha)
    else
        tmp = 0.5d0 * (1.0d0 + (beta / (2.0d0 + beta)))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 0.02) {
		tmp = 0.5 * ((2.0 + (2.0 * beta)) / alpha);
	} else {
		tmp = 0.5 * (1.0 + (beta / (2.0 + beta)));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if ((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 0.02:
		tmp = 0.5 * ((2.0 + (2.0 * beta)) / alpha)
	else:
		tmp = 0.5 * (1.0 + (beta / (2.0 + beta)))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) + 1.0) / 2.0) <= 0.02)
		tmp = Float64(0.5 * Float64(Float64(2.0 + Float64(2.0 * beta)) / alpha));
	else
		tmp = Float64(0.5 * Float64(1.0 + Float64(beta / Float64(2.0 + beta))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 0.02)
		tmp = 0.5 * ((2.0 + (2.0 * beta)) / alpha);
	else
		tmp = 0.5 * (1.0 + (beta / (2.0 + beta)));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], 0.02], N[(0.5 * N[(N[(2.0 + N[(2.0 * beta), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(1.0 + N[(beta / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.0200000000000000004

   1. Initial program 74.7%

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

   2. Taylor expanded in alpha around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{2 + 2 \cdot \beta}{\alpha}} \]

   4. Applied rewrites 29.4%

      \[\leadsto \color{blue}{0.5 \cdot \frac{2 + 2 \cdot \beta}{\alpha}} \]

   if 0.0200000000000000004 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64))

   1. Initial program 74.7%

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

   2. Taylor expanded in alpha around 0

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

   4. Applied rewrites 72.6%

      \[\leadsto \color{blue}{0.5 \cdot \left(1 + \frac{\beta}{2 + \beta}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 73.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8.8 \cdot 10^{-24}:\\ \;\;\;\;0.5 \cdot \left(1 - \frac{\alpha}{2 + \alpha}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(1 + \frac{\beta}{2 + \beta}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 8.8e-24)
   (* 0.5 (- 1.0 (/ alpha (+ 2.0 alpha))))
   (* 0.5 (+ 1.0 (/ beta (+ 2.0 beta))))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.8e-24) {
		tmp = 0.5 * (1.0 - (alpha / (2.0 + alpha)));
	} else {
		tmp = 0.5 * (1.0 + (beta / (2.0 + 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 (beta <= 8.8d-24) then
        tmp = 0.5d0 * (1.0d0 - (alpha / (2.0d0 + alpha)))
    else
        tmp = 0.5d0 * (1.0d0 + (beta / (2.0d0 + beta)))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 8.8e-24) {
		tmp = 0.5 * (1.0 - (alpha / (2.0 + alpha)));
	} else {
		tmp = 0.5 * (1.0 + (beta / (2.0 + beta)));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 8.8e-24:
		tmp = 0.5 * (1.0 - (alpha / (2.0 + alpha)))
	else:
		tmp = 0.5 * (1.0 + (beta / (2.0 + beta)))
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 8.8e-24)
		tmp = Float64(0.5 * Float64(1.0 - Float64(alpha / Float64(2.0 + alpha))));
	else
		tmp = Float64(0.5 * Float64(1.0 + Float64(beta / Float64(2.0 + beta))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 8.8e-24)
		tmp = 0.5 * (1.0 - (alpha / (2.0 + alpha)));
	else
		tmp = 0.5 * (1.0 + (beta / (2.0 + beta)));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 8.8e-24], N[(0.5 * N[(1.0 - N[(alpha / N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(1.0 + N[(beta / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 8.80000000000000006e-24

   1. Initial program 74.7%

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

   2. Taylor expanded in beta around 0

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

   4. Applied rewrites 49.5%

      \[\leadsto \color{blue}{0.5 \cdot \left(1 - \frac{\alpha}{2 + \alpha}\right)} \]

   if 8.80000000000000006e-24 < beta

   1. Initial program 74.7%

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

   2. Taylor expanded in alpha around 0

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

   4. Applied rewrites 72.6%

      \[\leadsto \color{blue}{0.5 \cdot \left(1 + \frac{\beta}{2 + \beta}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 73.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 560000:\\ \;\;\;\;0.5 \cdot \left(1 - \frac{\alpha}{2 + \alpha}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 560000.0)
   (* 0.5 (- 1.0 (/ alpha (+ 2.0 alpha))))
   (- 1.0 (/ 1.0 beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 560000.0) {
		tmp = 0.5 * (1.0 - (alpha / (2.0 + alpha)));
	} else {
		tmp = 1.0 - (1.0 / 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 (beta <= 560000.0d0) then
        tmp = 0.5d0 * (1.0d0 - (alpha / (2.0d0 + alpha)))
    else
        tmp = 1.0d0 - (1.0d0 / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 560000.0) {
		tmp = 0.5 * (1.0 - (alpha / (2.0 + alpha)));
	} else {
		tmp = 1.0 - (1.0 / beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 560000.0:
		tmp = 0.5 * (1.0 - (alpha / (2.0 + alpha)))
	else:
		tmp = 1.0 - (1.0 / beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 560000.0)
		tmp = Float64(0.5 * Float64(1.0 - Float64(alpha / Float64(2.0 + alpha))));
	else
		tmp = Float64(1.0 - Float64(1.0 / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 560000.0)
		tmp = 0.5 * (1.0 - (alpha / (2.0 + alpha)));
	else
		tmp = 1.0 - (1.0 / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 560000.0], N[(0.5 * N[(1.0 - N[(alpha / N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(1.0 / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.6e5

   1. Initial program 74.7%

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

   2. Taylor expanded in beta around 0

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

   4. Applied rewrites 49.5%

      \[\leadsto \color{blue}{0.5 \cdot \left(1 - \frac{\alpha}{2 + \alpha}\right)} \]

   if 5.6e5 < beta

   1. Initial program 74.7%

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

   2. Taylor expanded in beta around inf

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

   4. Applied rewrites 30.0%

      \[\leadsto \color{blue}{1 + -0.5 \cdot \frac{2 + 2 \cdot \alpha}{\beta}} \]

   5. Taylor expanded in alpha around 0

      \[\leadsto 1 - \color{blue}{\frac{1}{\beta}} \]

   7. Applied rewrites 30.3%

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

Alternative 7: 72.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;0.5 + \beta \cdot \left(0.25 + -0.125 \cdot \beta\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.0)
   (+ 0.5 (* beta (+ 0.25 (* -0.125 beta))))
   (- 1.0 (/ 1.0 beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = 0.5 + (beta * (0.25 + (-0.125 * beta)));
	} else {
		tmp = 1.0 - (1.0 / 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 (beta <= 2.0d0) then
        tmp = 0.5d0 + (beta * (0.25d0 + ((-0.125d0) * beta)))
    else
        tmp = 1.0d0 - (1.0d0 / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = 0.5 + (beta * (0.25 + (-0.125 * beta)));
	} else {
		tmp = 1.0 - (1.0 / beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.0:
		tmp = 0.5 + (beta * (0.25 + (-0.125 * beta)))
	else:
		tmp = 1.0 - (1.0 / beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.0)
		tmp = Float64(0.5 + Float64(beta * Float64(0.25 + Float64(-0.125 * beta))));
	else
		tmp = Float64(1.0 - Float64(1.0 / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.0)
		tmp = 0.5 + (beta * (0.25 + (-0.125 * beta)));
	else
		tmp = 1.0 - (1.0 / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.0], N[(0.5 + N[(beta * N[(0.25 + N[(-0.125 * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(1.0 / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 74.7%

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

   2. Taylor expanded in alpha around 0

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

   4. Applied rewrites 72.6%

      \[\leadsto \color{blue}{0.5 \cdot \left(1 + \frac{\beta}{2 + \beta}\right)} \]

   5. Taylor expanded in beta around 0

      \[\leadsto \frac{1}{2} + \color{blue}{\beta \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot \beta\right)} \]

   7. Applied rewrites 44.1%

      \[\leadsto 0.5 + \color{blue}{\beta \cdot \left(0.25 + -0.125 \cdot \beta\right)} \]

   if 2 < beta

   1. Initial program 74.7%

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

   2. Taylor expanded in beta around inf

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

   4. Applied rewrites 30.0%

      \[\leadsto \color{blue}{1 + -0.5 \cdot \frac{2 + 2 \cdot \alpha}{\beta}} \]

   5. Taylor expanded in alpha around 0

      \[\leadsto 1 - \color{blue}{\frac{1}{\beta}} \]

   7. Applied rewrites 30.3%

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

Alternative 8: 72.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;0.5 + 0.25 \cdot \beta\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.0) (+ 0.5 (* 0.25 beta)) (- 1.0 (/ 1.0 beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = 0.5 + (0.25 * beta);
	} else {
		tmp = 1.0 - (1.0 / 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 (beta <= 2.0d0) then
        tmp = 0.5d0 + (0.25d0 * beta)
    else
        tmp = 1.0d0 - (1.0d0 / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = 0.5 + (0.25 * beta);
	} else {
		tmp = 1.0 - (1.0 / beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.0:
		tmp = 0.5 + (0.25 * beta)
	else:
		tmp = 1.0 - (1.0 / beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.0)
		tmp = Float64(0.5 + Float64(0.25 * beta));
	else
		tmp = Float64(1.0 - Float64(1.0 / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.0)
		tmp = 0.5 + (0.25 * beta);
	else
		tmp = 1.0 - (1.0 / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.0], N[(0.5 + N[(0.25 * beta), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(1.0 / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 74.7%

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

   2. Taylor expanded in alpha around 0

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

   4. Applied rewrites 72.6%

      \[\leadsto \color{blue}{0.5 \cdot \left(1 + \frac{\beta}{2 + \beta}\right)} \]

   5. Taylor expanded in beta around 0

      \[\leadsto \frac{1}{2} + \color{blue}{\frac{1}{4} \cdot \beta} \]

   7. Applied rewrites 45.3%

      \[\leadsto 0.5 + \color{blue}{0.25 \cdot \beta} \]

   if 2 < beta

   1. Initial program 74.7%

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

   2. Taylor expanded in beta around inf

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

   4. Applied rewrites 30.0%

      \[\leadsto \color{blue}{1 + -0.5 \cdot \frac{2 + 2 \cdot \alpha}{\beta}} \]

   5. Taylor expanded in alpha around 0

      \[\leadsto 1 - \color{blue}{\frac{1}{\beta}} \]

   7. Applied rewrites 30.3%

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

Alternative 9: 71.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;0.5 + 0.25 \cdot \beta\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.0) (+ 0.5 (* 0.25 beta)) 1.0))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = 0.5 + (0.25 * beta);
	} else {
		tmp = 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) :: tmp
    if (beta <= 2.0d0) then
        tmp = 0.5d0 + (0.25d0 * beta)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = 0.5 + (0.25 * beta);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.0:
		tmp = 0.5 + (0.25 * beta)
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.0)
		tmp = Float64(0.5 + Float64(0.25 * beta));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.0)
		tmp = 0.5 + (0.25 * beta);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.0], N[(0.5 + N[(0.25 * beta), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 74.7%

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

   2. Taylor expanded in alpha around 0

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

   4. Applied rewrites 72.6%

      \[\leadsto \color{blue}{0.5 \cdot \left(1 + \frac{\beta}{2 + \beta}\right)} \]

   5. Taylor expanded in beta around 0

      \[\leadsto \frac{1}{2} + \color{blue}{\frac{1}{4} \cdot \beta} \]

   7. Applied rewrites 45.3%

      \[\leadsto 0.5 + \color{blue}{0.25 \cdot \beta} \]

   if 2 < beta

   1. Initial program 74.7%

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

   2. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{1} \]

   4. Applied rewrites 37.5%

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

Alternative 10: 71.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 0.8:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0) 0.8)
   0.5
   1.0))

C

double code(double alpha, double beta) {
	double tmp;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 0.8) {
		tmp = 0.5;
	} else {
		tmp = 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) :: tmp
    if (((((beta - alpha) / ((alpha + beta) + 2.0d0)) + 1.0d0) / 2.0d0) <= 0.8d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 0.8) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if ((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 0.8:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) + 1.0) / 2.0) <= 0.8)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 0.8)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], 0.8], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.80000000000000004

   1. Initial program 74.7%

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

   2. Taylor expanded in alpha around 0

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

   4. Applied rewrites 72.6%

      \[\leadsto \color{blue}{0.5 \cdot \left(1 + \frac{\beta}{2 + \beta}\right)} \]

   5. Taylor expanded in beta around 0

      \[\leadsto \frac{1}{2} \]

   7. Applied rewrites 48.9%

      \[\leadsto 0.5 \]

   if 0.80000000000000004 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64))

   1. Initial program 74.7%

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

   2. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{1} \]

   4. Applied rewrites 37.5%

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

Alternative 11: 37.5% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 1.0)

C

double code(double alpha, double beta) {
	return 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
    code = 1.0d0
end function

Java

public static double code(double alpha, double beta) {
	return 1.0;
}

Python

def code(alpha, beta):
	return 1.0

Julia

function code(alpha, beta)
	return 1.0
end

MATLAB

function tmp = code(alpha, beta)
	tmp = 1.0;
end

Wolfram

code[alpha_, beta_] := 1.0

Derivation

1. Initial program 74.7%

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

2. Taylor expanded in beta around inf

   \[\leadsto \color{blue}{1} \]

4. Applied rewrites 37.5%

   \[\leadsto \color{blue}{1} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025159 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))