Octave 3.8, jcobi/1

Percentage Accurate: 74.8% → 99.8%

Time: 3.9s

Alternatives: 14

Speedup: 1.7×

Specification

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

Math

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

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: 74.8% accurate, 1.0× speedup

Math

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

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: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \alpha - \left(-2 - \beta\right)\\ \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 10^{-16}:\\ \;\;\;\;-0.5 \cdot \frac{-1 \cdot \beta - \left(2 + \beta\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{\beta}{t\_0} + 1\right) \cdot t\_0 - \alpha}{t\_0}}{2}\\ \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (- alpha (- -2.0 beta))))
   (if (<= (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0) 1e-16)
     (* -0.5 (/ (- (* -1.0 beta) (+ 2.0 beta)) alpha))
     (/ (/ (- (* (+ (/ beta t_0) 1.0) t_0) alpha) t_0) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = alpha - (-2.0 - beta);
	double tmp;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 1e-16) {
		tmp = -0.5 * (((-1.0 * beta) - (2.0 + beta)) / alpha);
	} else {
		tmp = (((((beta / t_0) + 1.0) * t_0) - alpha) / t_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) :: t_0
    real(8) :: tmp
    t_0 = alpha - ((-2.0d0) - beta)
    if (((((beta - alpha) / ((alpha + beta) + 2.0d0)) + 1.0d0) / 2.0d0) <= 1d-16) then
        tmp = (-0.5d0) * ((((-1.0d0) * beta) - (2.0d0 + beta)) / alpha)
    else
        tmp = (((((beta / t_0) + 1.0d0) * t_0) - alpha) / t_0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = alpha - (-2.0 - beta);
	double tmp;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 1e-16) {
		tmp = -0.5 * (((-1.0 * beta) - (2.0 + beta)) / alpha);
	} else {
		tmp = (((((beta / t_0) + 1.0) * t_0) - alpha) / t_0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = alpha - (-2.0 - beta)
	tmp = 0
	if ((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 1e-16:
		tmp = -0.5 * (((-1.0 * beta) - (2.0 + beta)) / alpha)
	else:
		tmp = (((((beta / t_0) + 1.0) * t_0) - alpha) / t_0) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(alpha - Float64(-2.0 - beta))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) + 1.0) / 2.0) <= 1e-16)
		tmp = Float64(-0.5 * Float64(Float64(Float64(-1.0 * beta) - Float64(2.0 + beta)) / alpha));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(beta / t_0) + 1.0) * t_0) - alpha) / t_0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = alpha - (-2.0 - beta);
	tmp = 0.0;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 1e-16)
		tmp = -0.5 * (((-1.0 * beta) - (2.0 + beta)) / alpha);
	else
		tmp = (((((beta / t_0) + 1.0) * t_0) - alpha) / t_0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(alpha - N[(-2.0 - beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], 1e-16], N[(-0.5 * N[(N[(N[(-1.0 * beta), $MachinePrecision] - N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(beta / t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision] - alpha), $MachinePrecision] / t$95$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)) < 9.9999999999999998e-17

   1. Initial program 74.8%

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

   2. Taylor expanded in alpha around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 29.3%

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

   4. Applied rewrites 29.3%

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

   if 9.9999999999999998e-17 < (/.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 74.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. div-sub N/A

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

      6. associate-+r- N/A

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

      7. sub-to-fraction N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 75.2%

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

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left(-2 - \alpha\right) - \beta\\ \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 10^{-16}:\\ \;\;\;\;-0.5 \cdot \frac{-1 \cdot \beta - \left(2 + \beta\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - \left(\beta - \alpha\right)}{t\_0 \cdot 2}\\ \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (- (- -2.0 alpha) beta)))
   (if (<= (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0) 1e-16)
     (* -0.5 (/ (- (* -1.0 beta) (+ 2.0 beta)) alpha))
     (/ (- t_0 (- beta alpha)) (* t_0 2.0)))))

C

double code(double alpha, double beta) {
	double t_0 = (-2.0 - alpha) - beta;
	double tmp;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 1e-16) {
		tmp = -0.5 * (((-1.0 * beta) - (2.0 + beta)) / alpha);
	} else {
		tmp = (t_0 - (beta - alpha)) / (t_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) :: t_0
    real(8) :: tmp
    t_0 = ((-2.0d0) - alpha) - beta
    if (((((beta - alpha) / ((alpha + beta) + 2.0d0)) + 1.0d0) / 2.0d0) <= 1d-16) then
        tmp = (-0.5d0) * ((((-1.0d0) * beta) - (2.0d0 + beta)) / alpha)
    else
        tmp = (t_0 - (beta - alpha)) / (t_0 * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (-2.0 - alpha) - beta;
	double tmp;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 1e-16) {
		tmp = -0.5 * (((-1.0 * beta) - (2.0 + beta)) / alpha);
	} else {
		tmp = (t_0 - (beta - alpha)) / (t_0 * 2.0);
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (-2.0 - alpha) - beta
	tmp = 0
	if ((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 1e-16:
		tmp = -0.5 * (((-1.0 * beta) - (2.0 + beta)) / alpha)
	else:
		tmp = (t_0 - (beta - alpha)) / (t_0 * 2.0)
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(-2.0 - alpha) - beta)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) + 1.0) / 2.0) <= 1e-16)
		tmp = Float64(-0.5 * Float64(Float64(Float64(-1.0 * beta) - Float64(2.0 + beta)) / alpha));
	else
		tmp = Float64(Float64(t_0 - Float64(beta - alpha)) / Float64(t_0 * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (-2.0 - alpha) - beta;
	tmp = 0.0;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 1e-16)
		tmp = -0.5 * (((-1.0 * beta) - (2.0 + beta)) / alpha);
	else
		tmp = (t_0 - (beta - alpha)) / (t_0 * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(-2.0 - alpha), $MachinePrecision] - beta), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], 1e-16], N[(-0.5 * N[(N[(N[(-1.0 * beta), $MachinePrecision] - N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * 2.0), $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)) < 9.9999999999999998e-17

   1. Initial program 74.8%

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

   2. Taylor expanded in alpha around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 29.3%

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

   4. Applied rewrites 29.3%

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

   if 9.9999999999999998e-17 < (/.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 74.8%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. frac-2neg N/A

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

      4. remove-double-neg N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. frac-2neg N/A

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

      9. add-to-fraction N/A

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

      10. remove-double-neg N/A

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

      11. associate-/l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 75.2%

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

Alternative 3: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 5 \cdot 10^{-10}:\\ \;\;\;\;-0.5 \cdot \frac{-1 \cdot \beta - \left(2 + \beta\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;0.5 - \left(\alpha - \beta\right) \cdot \left(\frac{-1}{\left(-2 - \alpha\right) - \beta} \cdot 0.5\right)\\ \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0) 5e-10)
   (* -0.5 (/ (- (* -1.0 beta) (+ 2.0 beta)) alpha))
   (- 0.5 (* (- alpha beta) (* (/ -1.0 (- (- -2.0 alpha) beta)) 0.5)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 5e-10) {
		tmp = -0.5 * (((-1.0 * beta) - (2.0 + beta)) / alpha);
	} else {
		tmp = 0.5 - ((alpha - beta) * ((-1.0 / ((-2.0 - alpha) - beta)) * 0.5));
	}
	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) <= 5d-10) then
        tmp = (-0.5d0) * ((((-1.0d0) * beta) - (2.0d0 + beta)) / alpha)
    else
        tmp = 0.5d0 - ((alpha - beta) * (((-1.0d0) / (((-2.0d0) - alpha) - beta)) * 0.5d0))
    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) <= 5e-10) {
		tmp = -0.5 * (((-1.0 * beta) - (2.0 + beta)) / alpha);
	} else {
		tmp = 0.5 - ((alpha - beta) * ((-1.0 / ((-2.0 - alpha) - beta)) * 0.5));
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if ((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 5e-10:
		tmp = -0.5 * (((-1.0 * beta) - (2.0 + beta)) / alpha)
	else:
		tmp = 0.5 - ((alpha - beta) * ((-1.0 / ((-2.0 - alpha) - beta)) * 0.5))
	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) <= 5e-10)
		tmp = Float64(-0.5 * Float64(Float64(Float64(-1.0 * beta) - Float64(2.0 + beta)) / alpha));
	else
		tmp = Float64(0.5 - Float64(Float64(alpha - beta) * Float64(Float64(-1.0 / Float64(Float64(-2.0 - alpha) - beta)) * 0.5)));
	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) <= 5e-10)
		tmp = -0.5 * (((-1.0 * beta) - (2.0 + beta)) / alpha);
	else
		tmp = 0.5 - ((alpha - beta) * ((-1.0 / ((-2.0 - alpha) - beta)) * 0.5));
	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], 5e-10], N[(-0.5 * N[(N[(N[(-1.0 * beta), $MachinePrecision] - N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision], N[(0.5 - N[(N[(alpha - beta), $MachinePrecision] * N[(N[(-1.0 / N[(N[(-2.0 - alpha), $MachinePrecision] - beta), $MachinePrecision]), $MachinePrecision] * 0.5), $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)) < 5.00000000000000031e-10

   1. Initial program 74.8%

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

   2. Taylor expanded in alpha around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 29.3%

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

   4. Applied rewrites 29.3%

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

   if 5.00000000000000031e-10 < (/.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 74.8%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. associate-/l* N/A

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

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

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

      9. lower--.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower--.f64 N/A

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

      15. mult-flip N/A

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

      16. lower-*.f64 N/A

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

   3. Applied rewrites 74.8%

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

Alternative 4: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 5 \cdot 10^{-10}:\\ \;\;\;\;-0.5 \cdot \frac{-1 \cdot \beta - \left(2 + \beta\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{\alpha - \left(-2 - \beta\right)}, 0.5, 0.5\right)\\ \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0) 5e-10)
   (* -0.5 (/ (- (* -1.0 beta) (+ 2.0 beta)) alpha))
   (fma (/ (- beta alpha) (- alpha (- -2.0 beta))) 0.5 0.5)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 5e-10) {
		tmp = -0.5 * (((-1.0 * beta) - (2.0 + beta)) / alpha);
	} else {
		tmp = fma(((beta - alpha) / (alpha - (-2.0 - beta))), 0.5, 0.5);
	}
	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) <= 5e-10)
		tmp = Float64(-0.5 * Float64(Float64(Float64(-1.0 * beta) - Float64(2.0 + beta)) / alpha));
	else
		tmp = fma(Float64(Float64(beta - alpha) / Float64(alpha - Float64(-2.0 - beta))), 0.5, 0.5);
	end
	return 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], 5e-10], N[(-0.5 * N[(N[(N[(-1.0 * beta), $MachinePrecision] - N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision], N[(N[(N[(beta - alpha), $MachinePrecision] / N[(alpha - N[(-2.0 - beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $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)) < 5.00000000000000031e-10

   1. Initial program 74.8%

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

   2. Taylor expanded in alpha around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 29.3%

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

   4. Applied rewrites 29.3%

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

   if 5.00000000000000031e-10 < (/.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 74.8%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. mult-flip N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. add-flip N/A

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

      9. associate-+l- N/A

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

      10. lower--.f64 N/A

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

      11. sub-flip-reverse N/A

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

      12. distribute-neg-out N/A

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

      13. add-flip N/A

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

      14. sub-negate N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval 74.8%

          \[\leadsto \mathsf{fma}\left(\frac{\beta - \alpha}{\alpha - \left(-2 - \beta\right)}, 0.5, \color{blue}{0.5}\right) \]

   3. Applied rewrites 74.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\beta - \alpha}{\alpha - \left(-2 - \beta\right)}, 0.5, 0.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 98.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 0.002:\\ \;\;\;\;-0.5 \cdot \frac{-1 \cdot \beta - \left(2 + \beta\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{\beta - -2}, 0.5, 0.5\right)\\ \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0) 0.002)
   (* -0.5 (/ (- (* -1.0 beta) (+ 2.0 beta)) alpha))
   (fma (/ beta (- beta -2.0)) 0.5 0.5)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 0.002) {
		tmp = -0.5 * (((-1.0 * beta) - (2.0 + beta)) / alpha);
	} else {
		tmp = fma((beta / (beta - -2.0)), 0.5, 0.5);
	}
	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) <= 0.002)
		tmp = Float64(-0.5 * Float64(Float64(Float64(-1.0 * beta) - Float64(2.0 + beta)) / alpha));
	else
		tmp = fma(Float64(beta / Float64(beta - -2.0)), 0.5, 0.5);
	end
	return 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], 0.002], N[(-0.5 * N[(N[(N[(-1.0 * beta), $MachinePrecision] - N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision], N[(N[(beta / N[(beta - -2.0), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $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)) < 2e-3

   1. Initial program 74.8%

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

   2. Taylor expanded in alpha around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 29.3%

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

   4. Applied rewrites 29.3%

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

   if 2e-3 < (/.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 74.8%

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

   2. Taylor expanded in alpha around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 72.8%

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

   4. Applied rewrites 72.8%

      \[\leadsto \color{blue}{0.5 \cdot \left(1 + \frac{\beta}{2 + \beta}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 72.8%

         \[\leadsto \mathsf{fma}\left(\frac{\beta}{2 + \beta}, \color{blue}{0.5}, 0.5\right) \]

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. add-flip-rev N/A

         \[\leadsto \mathsf{fma}\left(\frac{\beta}{\beta - \left(\mathsf{neg}\left(2\right)\right)}, \frac{1}{2}, \frac{1}{2}\right) \]

      10. metadata-eval N/A

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

      11. lower--.f64 72.8%

          \[\leadsto \mathsf{fma}\left(\frac{\beta}{\beta - -2}, 0.5, 0.5\right) \]

   6. Applied rewrites 72.8%

      \[\leadsto \mathsf{fma}\left(\frac{\beta}{\beta - -2}, \color{blue}{0.5}, 0.5\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 98.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 0.002:\\ \;\;\;\;0.5 \cdot \frac{2 + 2 \cdot \beta}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{\beta - -2}, 0.5, 0.5\right)\\ \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0) 0.002)
   (* 0.5 (/ (+ 2.0 (* 2.0 beta)) alpha))
   (fma (/ beta (- beta -2.0)) 0.5 0.5)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 0.002) {
		tmp = 0.5 * ((2.0 + (2.0 * beta)) / alpha);
	} else {
		tmp = fma((beta / (beta - -2.0)), 0.5, 0.5);
	}
	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) <= 0.002)
		tmp = Float64(0.5 * Float64(Float64(2.0 + Float64(2.0 * beta)) / alpha));
	else
		tmp = fma(Float64(beta / Float64(beta - -2.0)), 0.5, 0.5);
	end
	return 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], 0.002], N[(0.5 * N[(N[(2.0 + N[(2.0 * beta), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision], N[(N[(beta / N[(beta - -2.0), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $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)) < 2e-3

   1. Initial program 74.8%

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

   2. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 29.3%

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

   4. Applied rewrites 29.3%

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

   if 2e-3 < (/.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 74.8%

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

   2. Taylor expanded in alpha around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 72.8%

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

   4. Applied rewrites 72.8%

      \[\leadsto \color{blue}{0.5 \cdot \left(1 + \frac{\beta}{2 + \beta}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 72.8%

         \[\leadsto \mathsf{fma}\left(\frac{\beta}{2 + \beta}, \color{blue}{0.5}, 0.5\right) \]

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. add-flip-rev N/A

         \[\leadsto \mathsf{fma}\left(\frac{\beta}{\beta - \left(\mathsf{neg}\left(2\right)\right)}, \frac{1}{2}, \frac{1}{2}\right) \]

      10. metadata-eval N/A

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

      11. lower--.f64 72.8%

          \[\leadsto \mathsf{fma}\left(\frac{\beta}{\beta - -2}, 0.5, 0.5\right) \]

   6. Applied rewrites 72.8%

      \[\leadsto \mathsf{fma}\left(\frac{\beta}{\beta - -2}, \color{blue}{0.5}, 0.5\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 93.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{\alpha - -2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{\beta - -2}, 0.5, 0.5\right)\\ \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.1e-6)
   (/ 1.0 (- alpha -2.0))
   (fma (/ beta (- beta -2.0)) 0.5 0.5)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.1e-6) {
		tmp = 1.0 / (alpha - -2.0);
	} else {
		tmp = fma((beta / (beta - -2.0)), 0.5, 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.1e-6)
		tmp = Float64(1.0 / Float64(alpha - -2.0));
	else
		tmp = fma(Float64(beta / Float64(beta - -2.0)), 0.5, 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 3.1e-6], N[(1.0 / N[(alpha - -2.0), $MachinePrecision]), $MachinePrecision], N[(N[(beta / N[(beta - -2.0), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.1e-6

   1. Initial program 74.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. div-sub N/A

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

      6. associate-+r- N/A

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

      7. sub-to-fraction N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 75.2%

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

   4. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1}{2 + \alpha}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-+.f64 69.7%

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

   6. Applied rewrites 69.7%

      \[\leadsto \color{blue}{\frac{1}{2 + \alpha}} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip-rev N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 69.7%

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

   8. Applied rewrites 69.7%

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

   if 3.1e-6 < beta

   1. Initial program 74.8%

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

   2. Taylor expanded in alpha around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 72.8%

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

   4. Applied rewrites 72.8%

      \[\leadsto \color{blue}{0.5 \cdot \left(1 + \frac{\beta}{2 + \beta}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 72.8%

         \[\leadsto \mathsf{fma}\left(\frac{\beta}{2 + \beta}, \color{blue}{0.5}, 0.5\right) \]

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. add-flip-rev N/A

         \[\leadsto \mathsf{fma}\left(\frac{\beta}{\beta - \left(\mathsf{neg}\left(2\right)\right)}, \frac{1}{2}, \frac{1}{2}\right) \]

      10. metadata-eval N/A

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

      11. lower--.f64 72.8%

          \[\leadsto \mathsf{fma}\left(\frac{\beta}{\beta - -2}, 0.5, 0.5\right) \]

   6. Applied rewrites 72.8%

      \[\leadsto \mathsf{fma}\left(\frac{\beta}{\beta - -2}, \color{blue}{0.5}, 0.5\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 92.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1760:\\ \;\;\;\;\frac{1}{\alpha - -2}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{\beta}\\ \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1760.0) (/ 1.0 (- alpha -2.0)) (- 1.0 (/ 1.0 beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1760.0) {
		tmp = 1.0 / (alpha - -2.0);
	} 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 <= 1760.0d0) then
        tmp = 1.0d0 / (alpha - (-2.0d0))
    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 <= 1760.0) {
		tmp = 1.0 / (alpha - -2.0);
	} else {
		tmp = 1.0 - (1.0 / beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 1760.0:
		tmp = 1.0 / (alpha - -2.0)
	else:
		tmp = 1.0 - (1.0 / beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1760.0)
		tmp = Float64(1.0 / Float64(alpha - -2.0));
	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 <= 1760.0)
		tmp = 1.0 / (alpha - -2.0);
	else
		tmp = 1.0 - (1.0 / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 1760.0], N[(1.0 / N[(alpha - -2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(1.0 / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1760

   1. Initial program 74.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. div-sub N/A

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

      6. associate-+r- N/A

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

      7. sub-to-fraction N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 75.2%

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

   4. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1}{2 + \alpha}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-+.f64 69.7%

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

   6. Applied rewrites 69.7%

      \[\leadsto \color{blue}{\frac{1}{2 + \alpha}} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip-rev N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 69.7%

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

   8. Applied rewrites 69.7%

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

   if 1760 < beta

   1. Initial program 74.8%

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

   2. Taylor expanded in alpha around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 72.8%

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

   4. Applied rewrites 72.8%

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

   5. Taylor expanded in beta around inf

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 29.9%

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

   7. Applied rewrites 29.9%

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

Alternative 9: 91.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\\ \mathbf{if}\;t\_0 \leq 0.002:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 0.6:\\ \;\;\;\;\mathsf{fma}\left(\alpha, -0.25, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{\beta}\\ \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0)))
   (if (<= t_0 0.002)
     (/ 1.0 alpha)
     (if (<= t_0 0.6) (fma alpha -0.25 0.5) (- 1.0 (/ 1.0 beta))))))

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 <= 0.002) {
		tmp = 1.0 / alpha;
	} else if (t_0 <= 0.6) {
		tmp = fma(alpha, -0.25, 0.5);
	} else {
		tmp = 1.0 - (1.0 / beta);
	}
	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 <= 0.002)
		tmp = Float64(1.0 / alpha);
	elseif (t_0 <= 0.6)
		tmp = fma(alpha, -0.25, 0.5);
	else
		tmp = Float64(1.0 - Float64(1.0 / beta));
	end
	return 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, 0.002], N[(1.0 / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.6], N[(alpha * -0.25 + 0.5), $MachinePrecision], N[(1.0 - N[(1.0 / beta), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 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)) < 2e-3

   1. Initial program 74.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. div-sub N/A

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

      6. associate-+r- N/A

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

      7. sub-to-fraction N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 75.2%

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

   4. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1}{2 + \alpha}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-+.f64 69.7%

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

   6. Applied rewrites 69.7%

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

   7. Taylor expanded in alpha around inf

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

      1. lower-/.f64 23.9%

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

   9. Applied rewrites 23.9%

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

   if 2e-3 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.599999999999999978

   1. Initial program 74.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. div-sub N/A

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

      6. associate-+r- N/A

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

      7. sub-to-fraction N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 75.2%

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

   4. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1}{2 + \alpha}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-+.f64 69.7%

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

   6. Applied rewrites 69.7%

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

   7. Taylor expanded in alpha around 0

      \[\leadsto \frac{1}{2} + \color{blue}{\frac{-1}{4} \cdot \alpha} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-*.f64 47.6%

         \[\leadsto 0.5 + -0.25 \cdot \alpha \]

   9. Applied rewrites 47.6%

      \[\leadsto 0.5 + \color{blue}{-0.25 \cdot \alpha} \]
   10. Step-by-step derivation

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

          \[\leadsto \frac{-1}{4} \cdot \alpha + \frac{1}{2} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \alpha + \frac{1}{2} \]

       4. *-commutative N/A

          \[\leadsto \alpha \cdot \frac{-1}{4} + \frac{1}{2} \]

       5. lower-fma.f64 47.6%

          \[\leadsto \mathsf{fma}\left(\alpha, -0.25, 0.5\right) \]

   11. Applied rewrites 47.6%

       \[\leadsto \mathsf{fma}\left(\alpha, -0.25, 0.5\right) \]

   if 0.599999999999999978 < (/.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 74.8%

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

   2. Taylor expanded in alpha around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 72.8%

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

   4. Applied rewrites 72.8%

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

   5. Taylor expanded in beta around inf

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 29.9%

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

   7. Applied rewrites 29.9%

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

Alternative 10: 91.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\\ \mathbf{if}\;t\_0 \leq 0.002:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 0.6:\\ \;\;\;\;\mathsf{fma}\left(\alpha, -0.25, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0)))
   (if (<= t_0 0.002)
     (/ 1.0 alpha)
     (if (<= t_0 0.6) (fma alpha -0.25 0.5) 1.0))))

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 <= 0.002) {
		tmp = 1.0 / alpha;
	} else if (t_0 <= 0.6) {
		tmp = fma(alpha, -0.25, 0.5);
	} else {
		tmp = 1.0;
	}
	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 <= 0.002)
		tmp = Float64(1.0 / alpha);
	elseif (t_0 <= 0.6)
		tmp = fma(alpha, -0.25, 0.5);
	else
		tmp = 1.0;
	end
	return 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, 0.002], N[(1.0 / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.6], N[(alpha * -0.25 + 0.5), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 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)) < 2e-3

   1. Initial program 74.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. div-sub N/A

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

      6. associate-+r- N/A

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

      7. sub-to-fraction N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 75.2%

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

   4. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1}{2 + \alpha}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-+.f64 69.7%

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

   6. Applied rewrites 69.7%

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

   7. Taylor expanded in alpha around inf

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

      1. lower-/.f64 23.9%

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

   9. Applied rewrites 23.9%

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

   if 2e-3 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.599999999999999978

   1. Initial program 74.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. div-sub N/A

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

      6. associate-+r- N/A

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

      7. sub-to-fraction N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 75.2%

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

   4. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1}{2 + \alpha}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-+.f64 69.7%

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

   6. Applied rewrites 69.7%

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

   7. Taylor expanded in alpha around 0

      \[\leadsto \frac{1}{2} + \color{blue}{\frac{-1}{4} \cdot \alpha} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-*.f64 47.6%

         \[\leadsto 0.5 + -0.25 \cdot \alpha \]

   9. Applied rewrites 47.6%

      \[\leadsto 0.5 + \color{blue}{-0.25 \cdot \alpha} \]
   10. Step-by-step derivation

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

          \[\leadsto \frac{-1}{4} \cdot \alpha + \frac{1}{2} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \alpha + \frac{1}{2} \]

       4. *-commutative N/A

          \[\leadsto \alpha \cdot \frac{-1}{4} + \frac{1}{2} \]

       5. lower-fma.f64 47.6%

          \[\leadsto \mathsf{fma}\left(\alpha, -0.25, 0.5\right) \]

   11. Applied rewrites 47.6%

       \[\leadsto \mathsf{fma}\left(\alpha, -0.25, 0.5\right) \]

   if 0.599999999999999978 < (/.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 74.8%

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

   2. Taylor expanded in beta around inf

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

   4. Applied rewrites 37.1%

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

Alternative 11: 91.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\\ \mathbf{if}\;t\_0 \leq 0.002:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 0.6:\\ \;\;\;\;\mathsf{fma}\left(0.25, \beta, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0)))
   (if (<= t_0 0.002)
     (/ 1.0 alpha)
     (if (<= t_0 0.6) (fma 0.25 beta 0.5) 1.0))))

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 <= 0.002) {
		tmp = 1.0 / alpha;
	} else if (t_0 <= 0.6) {
		tmp = fma(0.25, beta, 0.5);
	} else {
		tmp = 1.0;
	}
	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 <= 0.002)
		tmp = Float64(1.0 / alpha);
	elseif (t_0 <= 0.6)
		tmp = fma(0.25, beta, 0.5);
	else
		tmp = 1.0;
	end
	return 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, 0.002], N[(1.0 / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.6], N[(0.25 * beta + 0.5), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 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)) < 2e-3

   1. Initial program 74.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. div-sub N/A

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

      6. associate-+r- N/A

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

      7. sub-to-fraction N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 75.2%

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

   4. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1}{2 + \alpha}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-+.f64 69.7%

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

   6. Applied rewrites 69.7%

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

   7. Taylor expanded in alpha around inf

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

      1. lower-/.f64 23.9%

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

   9. Applied rewrites 23.9%

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

   if 2e-3 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.599999999999999978

   1. Initial program 74.8%

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

   2. Taylor expanded in alpha around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 72.8%

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

   4. Applied rewrites 72.8%

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

   5. Taylor expanded in beta around 0

      \[\leadsto \frac{1}{2} + \color{blue}{\frac{1}{4} \cdot \beta} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-*.f64 45.9%

         \[\leadsto 0.5 + 0.25 \cdot \beta \]

   7. Applied rewrites 45.9%

      \[\leadsto 0.5 + \color{blue}{0.25 \cdot \beta} \]
   8. Step-by-step derivation

   9. Applied rewrites 45.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.25, \beta, 0.5\right)} \]

   if 0.599999999999999978 < (/.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 74.8%

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

   2. Taylor expanded in beta around inf

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

   4. Applied rewrites 37.1%

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

Alternative 12: 91.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 0.6:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0)))
   (if (<= t_0 5e-10) (/ 1.0 alpha) (if (<= t_0 0.6) 0.5 1.0))))

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 <= 5e-10) {
		tmp = 1.0 / alpha;
	} else if (t_0 <= 0.6) {
		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) :: t_0
    real(8) :: tmp
    t_0 = (((beta - alpha) / ((alpha + beta) + 2.0d0)) + 1.0d0) / 2.0d0
    if (t_0 <= 5d-10) then
        tmp = 1.0d0 / alpha
    else if (t_0 <= 0.6d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0
    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 <= 5e-10) {
		tmp = 1.0 / alpha;
	} else if (t_0 <= 0.6) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0
	tmp = 0
	if t_0 <= 5e-10:
		tmp = 1.0 / alpha
	elif t_0 <= 0.6:
		tmp = 0.5
	else:
		tmp = 1.0
	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 <= 5e-10)
		tmp = Float64(1.0 / alpha);
	elseif (t_0 <= 0.6)
		tmp = 0.5;
	else
		tmp = 1.0;
	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 <= 5e-10)
		tmp = 1.0 / alpha;
	elseif (t_0 <= 0.6)
		tmp = 0.5;
	else
		tmp = 1.0;
	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, 5e-10], N[(1.0 / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.6], 0.5, 1.0]]]

Derivation

1. Split input into 3 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)) < 5.00000000000000031e-10

   1. Initial program 74.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. div-sub N/A

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

      6. associate-+r- N/A

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

      7. sub-to-fraction N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 75.2%

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

   4. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1}{2 + \alpha}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-+.f64 69.7%

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

   6. Applied rewrites 69.7%

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

   7. Taylor expanded in alpha around inf

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

      1. lower-/.f64 23.9%

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

   9. Applied rewrites 23.9%

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

   if 5.00000000000000031e-10 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.599999999999999978

   1. Initial program 74.8%

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

   2. Taylor expanded in alpha around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 72.8%

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

   4. Applied rewrites 72.8%

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

   5. Taylor expanded in beta around 0

      \[\leadsto \frac{1}{2} \]
   6. Step-by-step derivation

   7. Applied rewrites 49.3%

      \[\leadsto 0.5 \]

   if 0.599999999999999978 < (/.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 74.8%

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

   2. Taylor expanded in beta around inf

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

   4. Applied rewrites 37.1%

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

Alternative 13: 71.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 0.86:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0) 0.86)
   0.5
   1.0))

C

double code(double alpha, double beta) {
	double tmp;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 0.86) {
		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) <= 0.86d0) 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) <= 0.86) {
		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) <= 0.86:
		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) <= 0.86)
		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) <= 0.86)
		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], 0.86], 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)) < 0.859999999999999987

   1. Initial program 74.8%

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

   2. Taylor expanded in alpha around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 72.8%

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

   4. Applied rewrites 72.8%

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

   5. Taylor expanded in beta around 0

      \[\leadsto \frac{1}{2} \]
   6. Step-by-step derivation

   7. Applied rewrites 49.3%

      \[\leadsto 0.5 \]

   if 0.859999999999999987 < (/.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 74.8%

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

   2. Taylor expanded in beta around inf

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

   4. Applied rewrites 37.1%

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

Alternative 14: 37.1% accurate, 18.5× speedup

Math

\[1 \]

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 74.8%

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

2. Taylor expanded in beta around inf

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

4. Applied rewrites 37.1%

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

Reproduce

herbie shell --seed 2025183 
(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))