Octave 3.8, jcobi/1

Percentage Accurate: 74.4% → 99.7%

Time: 8.1s

Alternatives: 14

Speedup: 0.6×

Specification

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

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = (((beta - alpha) / ((alpha + beta) + 2.0d0)) + 1.0d0) / 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = (((beta - alpha) / ((alpha + beta) + 2.0d0)) + 1.0d0) / 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + \left(\alpha + \beta\right)\\ t_1 := \frac{\beta}{t\_0}\\ t_2 := \frac{\alpha}{t\_0}\\ \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\frac{1 + \beta}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_1 \cdot t\_1 - t\_2 \cdot t\_2}{t\_1 + t\_2} + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ alpha beta))) (t_1 (/ beta t_0)) (t_2 (/ alpha t_0)))
   (if (<= (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0) 5e-10)
     (/ (+ 1.0 beta) alpha)
     (/ (+ (/ (- (* t_1 t_1) (* t_2 t_2)) (+ t_1 t_2)) 1.0) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double t_1 = beta / t_0;
	double t_2 = alpha / t_0;
	double tmp;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 5e-10) {
		tmp = (1.0 + beta) / alpha;
	} else {
		tmp = ((((t_1 * t_1) - (t_2 * t_2)) / (t_1 + t_2)) + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 2.0d0 + (alpha + beta)
    t_1 = beta / t_0
    t_2 = alpha / t_0
    if (((((beta - alpha) / ((alpha + beta) + 2.0d0)) + 1.0d0) / 2.0d0) <= 5d-10) then
        tmp = (1.0d0 + beta) / alpha
    else
        tmp = ((((t_1 * t_1) - (t_2 * t_2)) / (t_1 + t_2)) + 1.0d0) / 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 t_1 = beta / t_0;
	double t_2 = alpha / t_0;
	double tmp;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 5e-10) {
		tmp = (1.0 + beta) / alpha;
	} else {
		tmp = ((((t_1 * t_1) - (t_2 * t_2)) / (t_1 + t_2)) + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = 2.0 + (alpha + beta)
	t_1 = beta / t_0
	t_2 = alpha / t_0
	tmp = 0
	if ((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 5e-10:
		tmp = (1.0 + beta) / alpha
	else:
		tmp = ((((t_1 * t_1) - (t_2 * t_2)) / (t_1 + t_2)) + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(alpha + beta))
	t_1 = Float64(beta / t_0)
	t_2 = Float64(alpha / t_0)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) + 1.0) / 2.0) <= 5e-10)
		tmp = Float64(Float64(1.0 + beta) / alpha);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(t_1 * t_1) - Float64(t_2 * t_2)) / Float64(t_1 + t_2)) + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = 2.0 + (alpha + beta);
	t_1 = beta / t_0;
	t_2 = alpha / t_0;
	tmp = 0.0;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 5e-10)
		tmp = (1.0 + beta) / alpha;
	else
		tmp = ((((t_1 * t_1) - (t_2 * t_2)) / (t_1 + t_2)) + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(beta / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(alpha / t$95$0), $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], 5e-10], N[(N[(1.0 + beta), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 + t$95$2), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 6.9%

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

   3. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-+.f64 99.5

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

   5. Applied rewrites 99.5%

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

   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 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. flip-- N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 100.0%

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

Alternative 2: 91.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-10}:\\ \;\;\;\;{\alpha}^{-1}\\ \mathbf{elif}\;t\_0 \leq 0.8:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\alpha, 0.125, -0.25\right), \alpha, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0)))
   (if (<= t_0 5e-10)
     (pow alpha -1.0)
     (if (<= t_0 0.8)
       (fma (fma alpha 0.125 -0.25) alpha 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 <= 5e-10) {
		tmp = pow(alpha, -1.0);
	} else if (t_0 <= 0.8) {
		tmp = fma(fma(alpha, 0.125, -0.25), alpha, 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 <= 5e-10)
		tmp = alpha ^ -1.0;
	elseif (t_0 <= 0.8)
		tmp = fma(fma(alpha, 0.125, -0.25), alpha, 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, 5e-10], N[Power[alpha, -1.0], $MachinePrecision], If[LessEqual[t$95$0, 0.8], N[(N[(alpha * 0.125 + -0.25), $MachinePrecision] * alpha + 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)) < 5.00000000000000031e-10

   1. Initial program 6.9%

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

   3. Taylor expanded in beta around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

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

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

      8. distribute-lft-neg-in N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. metadata-eval 6.6

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

   5. Applied rewrites 6.6%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 80.7%

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

   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)) < 0.80000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in beta around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

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

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

      8. distribute-lft-neg-in N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. metadata-eval 97.8

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

   5. Applied rewrites 97.8%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 96.7%

      \[\leadsto \mathsf{fma}\left(0.125 \cdot \alpha - 0.25, \color{blue}{\alpha}, 0.5\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 96.7%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\alpha, 0.125, -0.25\right), \alpha, 0.5\right) \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in beta around inf

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

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

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

      2. metadata-eval N/A

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

      3. div-add N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. associate-*r/ N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. associate-/l* N/A

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

      11. metadata-eval N/A

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

      12. associate-*l* N/A

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

      13. metadata-eval N/A

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

      14. *-lft-identity N/A

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

      15. +-commutative N/A

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

      16. lower--.f64 N/A

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

      17. +-commutative N/A

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

      18. div-add-rev N/A

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

      19. lower-/.f64 N/A

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

      20. lower-+.f64 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in alpha around 0

      \[\leadsto 1 - \frac{1}{\beta} \]
   7. Step-by-step derivation

   8. Applied rewrites 98.4%

      \[\leadsto 1 - \frac{1}{\beta} \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 5 \cdot 10^{-10}:\\ \;\;\;\;{\alpha}^{-1}\\ \mathbf{elif}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 0.8:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\alpha, 0.125, -0.25\right), \alpha, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-10}:\\ \;\;\;\;{\alpha}^{-1}\\ \mathbf{elif}\;t\_0 \leq 0.8:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\alpha, 0.125, -0.25\right), \alpha, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0)))
   (if (<= t_0 5e-10)
     (pow alpha -1.0)
     (if (<= t_0 0.8)
       (fma (fma alpha 0.125 -0.25) alpha 0.5)
       (- 1.0 (/ alpha 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 <= 5e-10) {
		tmp = pow(alpha, -1.0);
	} else if (t_0 <= 0.8) {
		tmp = fma(fma(alpha, 0.125, -0.25), alpha, 0.5);
	} else {
		tmp = 1.0 - (alpha / 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 <= 5e-10)
		tmp = alpha ^ -1.0;
	elseif (t_0 <= 0.8)
		tmp = fma(fma(alpha, 0.125, -0.25), alpha, 0.5);
	else
		tmp = Float64(1.0 - Float64(alpha / 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, 5e-10], N[Power[alpha, -1.0], $MachinePrecision], If[LessEqual[t$95$0, 0.8], N[(N[(alpha * 0.125 + -0.25), $MachinePrecision] * alpha + 0.5), $MachinePrecision], N[(1.0 - N[(alpha / 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)) < 5.00000000000000031e-10

   1. Initial program 6.9%

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

   3. Taylor expanded in beta around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

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

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

      8. distribute-lft-neg-in N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. metadata-eval 6.6

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

   5. Applied rewrites 6.6%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 80.7%

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

   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)) < 0.80000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in beta around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

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

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

      8. distribute-lft-neg-in N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. metadata-eval 97.8

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

   5. Applied rewrites 97.8%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 96.7%

      \[\leadsto \mathsf{fma}\left(0.125 \cdot \alpha - 0.25, \color{blue}{\alpha}, 0.5\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 96.7%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\alpha, 0.125, -0.25\right), \alpha, 0.5\right) \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in beta around inf

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

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

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

      2. metadata-eval N/A

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

      3. div-add N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. associate-*r/ N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. associate-/l* N/A

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

      11. metadata-eval N/A

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

      12. associate-*l* N/A

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

      13. metadata-eval N/A

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

      14. *-lft-identity N/A

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

      15. +-commutative N/A

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

      16. lower--.f64 N/A

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

      17. +-commutative N/A

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

      18. div-add-rev N/A

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

      19. lower-/.f64 N/A

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

      20. lower-+.f64 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 98.2%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 5 \cdot 10^{-10}:\\ \;\;\;\;{\alpha}^{-1}\\ \mathbf{elif}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 0.8:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\alpha, 0.125, -0.25\right), \alpha, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\alpha}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-10}:\\ \;\;\;\;{\alpha}^{-1}\\ \mathbf{elif}\;t\_0 \leq 0.8:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\alpha, 0.125, -0.25\right), \alpha, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0)))
   (if (<= t_0 5e-10)
     (pow alpha -1.0)
     (if (<= t_0 0.8) (fma (fma alpha 0.125 -0.25) alpha 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 = pow(alpha, -1.0);
	} else if (t_0 <= 0.8) {
		tmp = fma(fma(alpha, 0.125, -0.25), alpha, 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 = alpha ^ -1.0;
	elseif (t_0 <= 0.8)
		tmp = fma(fma(alpha, 0.125, -0.25), alpha, 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, 5e-10], N[Power[alpha, -1.0], $MachinePrecision], If[LessEqual[t$95$0, 0.8], N[(N[(alpha * 0.125 + -0.25), $MachinePrecision] * alpha + 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)) < 5.00000000000000031e-10

   1. Initial program 6.9%

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

   3. Taylor expanded in beta around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

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

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

      8. distribute-lft-neg-in N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. metadata-eval 6.6

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

   5. Applied rewrites 6.6%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 80.7%

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

   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)) < 0.80000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in beta around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

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

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

      8. distribute-lft-neg-in N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. metadata-eval 97.8

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

   5. Applied rewrites 97.8%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 96.7%

      \[\leadsto \mathsf{fma}\left(0.125 \cdot \alpha - 0.25, \color{blue}{\alpha}, 0.5\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 96.7%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\alpha, 0.125, -0.25\right), \alpha, 0.5\right) \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in beta around inf

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

   5. Applied rewrites 97.8%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 5 \cdot 10^{-10}:\\ \;\;\;\;{\alpha}^{-1}\\ \mathbf{elif}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 0.8:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\alpha, 0.125, -0.25\right), \alpha, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-10}:\\ \;\;\;\;{\alpha}^{-1}\\ \mathbf{elif}\;t\_0 \leq 0.8:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \alpha, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0)))
   (if (<= t_0 5e-10)
     (pow alpha -1.0)
     (if (<= t_0 0.8) (fma -0.25 alpha 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 = pow(alpha, -1.0);
	} else if (t_0 <= 0.8) {
		tmp = fma(-0.25, alpha, 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 = alpha ^ -1.0;
	elseif (t_0 <= 0.8)
		tmp = fma(-0.25, alpha, 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, 5e-10], N[Power[alpha, -1.0], $MachinePrecision], If[LessEqual[t$95$0, 0.8], N[(-0.25 * alpha + 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)) < 5.00000000000000031e-10

   1. Initial program 6.9%

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

   3. Taylor expanded in beta around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

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

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

      8. distribute-lft-neg-in N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. metadata-eval 6.6

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

   5. Applied rewrites 6.6%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 80.7%

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

   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)) < 0.80000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in beta around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

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

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

      8. distribute-lft-neg-in N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. metadata-eval 97.8

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

   5. Applied rewrites 97.8%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 96.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in beta around inf

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

   5. Applied rewrites 97.8%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 5 \cdot 10^{-10}:\\ \;\;\;\;{\alpha}^{-1}\\ \mathbf{elif}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 0.8:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \alpha, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\frac{1 + \beta}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 0.8:\\ \;\;\;\;\mathsf{fma}\left(\frac{\alpha}{2 + \alpha}, -0.5, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta - \left(1 + \alpha\right)}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0)))
   (if (<= t_0 5e-10)
     (/ (+ 1.0 beta) alpha)
     (if (<= t_0 0.8)
       (fma (/ alpha (+ 2.0 alpha)) -0.5 0.5)
       (/ (- beta (+ 1.0 alpha)) 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 <= 5e-10) {
		tmp = (1.0 + beta) / alpha;
	} else if (t_0 <= 0.8) {
		tmp = fma((alpha / (2.0 + alpha)), -0.5, 0.5);
	} else {
		tmp = (beta - (1.0 + alpha)) / 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 <= 5e-10)
		tmp = Float64(Float64(1.0 + beta) / alpha);
	elseif (t_0 <= 0.8)
		tmp = fma(Float64(alpha / Float64(2.0 + alpha)), -0.5, 0.5);
	else
		tmp = Float64(Float64(beta - Float64(1.0 + alpha)) / 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, 5e-10], N[(N[(1.0 + beta), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.8], N[(N[(alpha / N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision] * -0.5 + 0.5), $MachinePrecision], N[(N[(beta - N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision] / beta), $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)) < 5.00000000000000031e-10

   1. Initial program 6.9%

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

   3. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-+.f64 99.5

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

   5. Applied rewrites 99.5%

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

   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)) < 0.80000000000000004

   1. Initial program 100.0%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing
   3. 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. lower-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in beta around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 97.8

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

   7. Applied rewrites 97.8%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in beta around inf

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

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

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

      2. metadata-eval N/A

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

      3. div-add N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. associate-*r/ N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. associate-/l* N/A

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

      11. metadata-eval N/A

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

      12. associate-*l* N/A

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

      13. metadata-eval N/A

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

      14. *-lft-identity N/A

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

      15. +-commutative N/A

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

      16. lower--.f64 N/A

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

      17. +-commutative N/A

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

      18. div-add-rev N/A

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

      19. lower-/.f64 N/A

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

      20. lower-+.f64 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in beta around 0

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

   8. Applied rewrites 98.7%

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

Alternative 7: 97.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\frac{1 + \beta}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 0.8:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \alpha, 0.125\right), \alpha, -0.25\right), \alpha, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta - \left(1 + \alpha\right)}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0)))
   (if (<= t_0 5e-10)
     (/ (+ 1.0 beta) alpha)
     (if (<= t_0 0.8)
       (fma (fma (fma -0.0625 alpha 0.125) alpha -0.25) alpha 0.5)
       (/ (- beta (+ 1.0 alpha)) 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 <= 5e-10) {
		tmp = (1.0 + beta) / alpha;
	} else if (t_0 <= 0.8) {
		tmp = fma(fma(fma(-0.0625, alpha, 0.125), alpha, -0.25), alpha, 0.5);
	} else {
		tmp = (beta - (1.0 + alpha)) / 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 <= 5e-10)
		tmp = Float64(Float64(1.0 + beta) / alpha);
	elseif (t_0 <= 0.8)
		tmp = fma(fma(fma(-0.0625, alpha, 0.125), alpha, -0.25), alpha, 0.5);
	else
		tmp = Float64(Float64(beta - Float64(1.0 + alpha)) / 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, 5e-10], N[(N[(1.0 + beta), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.8], N[(N[(N[(-0.0625 * alpha + 0.125), $MachinePrecision] * alpha + -0.25), $MachinePrecision] * alpha + 0.5), $MachinePrecision], N[(N[(beta - N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision] / beta), $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)) < 5.00000000000000031e-10

   1. Initial program 6.9%

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

   3. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-+.f64 99.5

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

   5. Applied rewrites 99.5%

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

   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)) < 0.80000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in beta around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

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

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

      8. distribute-lft-neg-in N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. metadata-eval 97.8

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

   5. Applied rewrites 97.8%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 95.5%

      \[\leadsto 0.5 \]

   9. Taylor expanded in alpha around 0

      \[\leadsto \frac{1}{2} + \color{blue}{\alpha \cdot \left(\alpha \cdot \left(\frac{1}{8} + \frac{-1}{16} \cdot \alpha\right) - \frac{1}{4}\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 96.9%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \alpha, 0.125\right), \alpha, -0.25\right), \color{blue}{\alpha}, 0.5\right) \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in beta around inf

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

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

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

      2. metadata-eval N/A

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

      3. div-add N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. associate-*r/ N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. associate-/l* N/A

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

      11. metadata-eval N/A

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

      12. associate-*l* N/A

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

      13. metadata-eval N/A

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

      14. *-lft-identity N/A

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

      15. +-commutative N/A

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

      16. lower--.f64 N/A

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

      17. +-commutative N/A

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

      18. div-add-rev N/A

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

      19. lower-/.f64 N/A

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

      20. lower-+.f64 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in beta around 0

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

   8. Applied rewrites 98.7%

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

Alternative 8: 97.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\frac{1 + \beta}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 0.8:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\alpha, 0.125, -0.25\right), \alpha, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta - \left(1 + \alpha\right)}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0)))
   (if (<= t_0 5e-10)
     (/ (+ 1.0 beta) alpha)
     (if (<= t_0 0.8)
       (fma (fma alpha 0.125 -0.25) alpha 0.5)
       (/ (- beta (+ 1.0 alpha)) 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 <= 5e-10) {
		tmp = (1.0 + beta) / alpha;
	} else if (t_0 <= 0.8) {
		tmp = fma(fma(alpha, 0.125, -0.25), alpha, 0.5);
	} else {
		tmp = (beta - (1.0 + alpha)) / 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 <= 5e-10)
		tmp = Float64(Float64(1.0 + beta) / alpha);
	elseif (t_0 <= 0.8)
		tmp = fma(fma(alpha, 0.125, -0.25), alpha, 0.5);
	else
		tmp = Float64(Float64(beta - Float64(1.0 + alpha)) / 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, 5e-10], N[(N[(1.0 + beta), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.8], N[(N[(alpha * 0.125 + -0.25), $MachinePrecision] * alpha + 0.5), $MachinePrecision], N[(N[(beta - N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision] / beta), $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)) < 5.00000000000000031e-10

   1. Initial program 6.9%

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

   3. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-+.f64 99.5

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

   5. Applied rewrites 99.5%

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

   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)) < 0.80000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in beta around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

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

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

      8. distribute-lft-neg-in N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. metadata-eval 97.8

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

   5. Applied rewrites 97.8%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 96.7%

      \[\leadsto \mathsf{fma}\left(0.125 \cdot \alpha - 0.25, \color{blue}{\alpha}, 0.5\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 96.7%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\alpha, 0.125, -0.25\right), \alpha, 0.5\right) \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in beta around inf

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

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

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

      2. metadata-eval N/A

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

      3. div-add N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. associate-*r/ N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. associate-/l* N/A

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

      11. metadata-eval N/A

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

      12. associate-*l* N/A

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

      13. metadata-eval N/A

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

      14. *-lft-identity N/A

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

      15. +-commutative N/A

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

      16. lower--.f64 N/A

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

      17. +-commutative N/A

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

      18. div-add-rev N/A

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

      19. lower-/.f64 N/A

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

      20. lower-+.f64 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in beta around 0

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

   8. Applied rewrites 98.7%

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

Alternative 9: 97.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\frac{1 + \beta}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 0.8:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\alpha, 0.125, -0.25\right), \alpha, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0)))
   (if (<= t_0 5e-10)
     (/ (+ 1.0 beta) alpha)
     (if (<= t_0 0.8)
       (fma (fma alpha 0.125 -0.25) alpha 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 <= 5e-10) {
		tmp = (1.0 + beta) / alpha;
	} else if (t_0 <= 0.8) {
		tmp = fma(fma(alpha, 0.125, -0.25), alpha, 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 <= 5e-10)
		tmp = Float64(Float64(1.0 + beta) / alpha);
	elseif (t_0 <= 0.8)
		tmp = fma(fma(alpha, 0.125, -0.25), alpha, 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, 5e-10], N[(N[(1.0 + beta), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.8], N[(N[(alpha * 0.125 + -0.25), $MachinePrecision] * alpha + 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)) < 5.00000000000000031e-10

   1. Initial program 6.9%

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

   3. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-+.f64 99.5

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

   5. Applied rewrites 99.5%

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

   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)) < 0.80000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in beta around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

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

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

      8. distribute-lft-neg-in N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. metadata-eval 97.8

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

   5. Applied rewrites 97.8%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 96.7%

      \[\leadsto \mathsf{fma}\left(0.125 \cdot \alpha - 0.25, \color{blue}{\alpha}, 0.5\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 96.7%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\alpha, 0.125, -0.25\right), \alpha, 0.5\right) \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in beta around inf

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

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

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

      2. metadata-eval N/A

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

      3. div-add N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. associate-*r/ N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. associate-/l* N/A

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

      11. metadata-eval N/A

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

      12. associate-*l* N/A

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

      13. metadata-eval N/A

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

      14. *-lft-identity N/A

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

      15. +-commutative N/A

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

      16. lower--.f64 N/A

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

      17. +-commutative N/A

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

      18. div-add-rev N/A

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

      19. lower-/.f64 N/A

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

      20. lower-+.f64 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in alpha around 0

      \[\leadsto 1 - \frac{1}{\beta} \]
   7. Step-by-step derivation

   8. Applied rewrites 98.4%

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

Alternative 10: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\frac{1 + \beta}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta - \alpha}{\mathsf{fma}\left(\alpha + \beta, 2, 4\right)} + 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0) 5e-10)
   (/ (+ 1.0 beta) alpha)
   (+ (/ (- beta alpha) (fma (+ alpha beta) 2.0 4.0)) 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 = (1.0 + beta) / alpha;
	} else {
		tmp = ((beta - alpha) / fma((alpha + beta), 2.0, 4.0)) + 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(Float64(1.0 + beta) / alpha);
	else
		tmp = Float64(Float64(Float64(beta - alpha) / fma(Float64(alpha + beta), 2.0, 4.0)) + 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[(N[(1.0 + beta), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] * 2.0 + 4.0), $MachinePrecision]), $MachinePrecision] + 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 6.9%

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

   3. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-+.f64 99.5

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

   5. Applied rewrites 99.5%

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

   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 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. flip-- N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. frac-add N/A

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

   6. Applied rewrites 100.0%

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

Alternative 11: 97.9% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0) 5e-10)
   (/ (+ 1.0 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) <= 5e-10) {
		tmp = (1.0 + 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) <= 5e-10)
		tmp = Float64(Float64(1.0 + 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], 5e-10], N[(N[(1.0 + beta), $MachinePrecision] / alpha), $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)) < 5.00000000000000031e-10

   1. Initial program 6.9%

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

   3. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-+.f64 99.5

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

   5. Applied rewrites 99.5%

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

   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 100.0%

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

   3. Taylor expanded in alpha around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

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

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

      9. distribute-lft-neg-in N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. metadata-eval 98.2

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

   5. Applied rewrites 98.2%

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

Alternative 12: 70.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (((((beta - alpha) / ((alpha + beta) + 2.0d0)) + 1.0d0) / 2.0d0) <= 0.8d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 65.2%

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

   3. Taylor expanded in beta around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

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

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

      8. distribute-lft-neg-in N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. metadata-eval 63.7

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

   5. Applied rewrites 63.7%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 61.8%

      \[\leadsto 0.5 \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in beta around inf

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

   5. Applied rewrites 97.8%

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

Alternative 13: 71.4% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 69.2%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. flip-- N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 69.2%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. frac-add N/A

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

   6. Applied rewrites 69.2%

      \[\leadsto \color{blue}{\frac{\beta - \alpha}{\mathsf{fma}\left(\alpha + \beta, 2, 4\right)} + 0.5} \]

   7. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 66.9

         \[\leadsto \frac{\beta}{\color{blue}{\mathsf{fma}\left(2, \beta, 4\right)}} + 0.5 \]

   9. Applied rewrites 66.9%

      \[\leadsto \color{blue}{\frac{\beta}{\mathsf{fma}\left(2, \beta, 4\right)}} + 0.5 \]

   10. Taylor expanded in beta around 0

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

   12. Applied rewrites 65.8%

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

   if 2 < beta

   1. Initial program 85.4%

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

   3. Taylor expanded in beta around inf

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

   5. Applied rewrites 83.8%

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

Alternative 14: 36.6% accurate, 35.0× speedup

Math

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

FPCore

(FPCore (alpha beta) :precision binary64 1.0)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 1.0d0
end function

Java

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

Python

def code(alpha, beta):
	return 1.0

Julia

function code(alpha, beta)
	return 1.0
end

MATLAB

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

Wolfram

code[alpha_, beta_] := 1.0

Derivation

1. Initial program 74.2%

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

3. Taylor expanded in beta around inf

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

5. Applied rewrites 35.4%

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

Reproduce

herbie shell --seed 2024360 
(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))